Topics in Remote Sensing of Soil Moisture Using L-Band Radar

Transcription

1 Topics in Remote Sensing of Soil Moisture Using L-Band Radar Dissertation Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Jeffrey D. Ouellette, B.S.E.C.E., M.S.E.C.E. Graduate Program in Electrical and Computer Engineering The Ohio State University 2015 Dissertation Committee: Joel T. Johnson, Advisor Robert J. Burkholder Christopher J. Baker

2 c Copyright by Jeffrey D. Ouellette 2015

3 Abstract Remote sensing of soil moisture has become a topic of increasing interest over the past several decades. Mapping of soil moisture content is a critical element in meteorological modeling, agricultural planning, disease spread monitoring, flood/landslide risk evaluation, studies of the Earth s water and carbon cycles, and many other environmental issues. The remote sensing of surface soil moisture conditions is necessary for fostering these large-scale studies in which deployment of in-situ soil moisture sampling networks (e.g. via dielectric probes) is not possible or economically feasible. While remote sensing soil moisture from radiometric measurements has a long history, the subject of inverting soil moisture from radar measurements (particularly in the presence of vegetation) is a difficult problem which remains the subject of active research. Due to the relatively high dielectric constant of water (e.g. when compared to soil), the normalized radar cross section of soil has been found to be particularly sensitive to soil moisture content. Synthetic Aperture Radar (SAR) is particularly attractive for soil moisture remote sensing, given SAR s capability to form images of the Earth s surface at high spatial resolutions when compared to passive approaches such as radiometry. L-band frequencies are of ii

4 particular interest due to their low susceptibility to atmospheric effects, their ability to penetrate through vegetation (and sense the underlying soil surface), and their high sensitivity to changes in soil moisture. This dissertation focuses on the inversion of soil moisture from radar returns using airborne and space borne instruments. The inversion techniques discussed herein specifically include change-detection-based and forward-modelbased methods. Chapter 1 provides the introduction to this dissertation, including motivations and a brief history of soil moisture remote sensing using radar. Chapter 2 describes the use of bistatic radar configurations (where the transmitter and receiver are not co-located) and discusses bistatic, polarimetric normalized radar cross section (NRCS) predictions from randomly rough surfaces using several approximate and numerically exact models. Chapter 3 describes the use of single- and two-layer rough surface scattering to explain airborne interferometric SAR (InSAR) decorrelations observed for bare soil surfaces. Chapter 4 presents a simulation study of the use of a compact polarimetric radar mode, its advantages for use in soil moisture remote sensing, and its performance in comparison to a traditional, fully-polarimetric system. Chapter 5 introduces a new time-series soil moisture retrieval method, tailored for use with the Soil Moisture Active/Passive (SMAP) radar system, and discusses results from this method using simulated and measured data. Chapter 6 explains recent advances in the detection of inland water bodies using the SMAP radar. The culmination of the research described herein aims to benefit current and future soil moisture remote sensing radar systems. iii

5 To my parents iv

6 Acknowledgements First and most importantly I would like to thank my advisor, Professor Joel Johnson, for his support. Without his expert guidance and infinite patience, the work presented here would not have been possible. I sincerely thank you, Professor Johnson, for giving me the opportunity to be a part of your research group at Ohio State University. I would like to also thank the other members of my dissertation committee: Professor Robert Burkholder and Professor Christopher Baker, for their suggestions. Their expert opinions have proven instrumental in improving the overall quality of this dissertation. I would also like to thank all of my friends and colleagues that I have met during graduate school. Their friendship and support has eased the burdens I have shouldered for the past several years. Finally, I extend my love and gratitude to my parents, whose unconditional love and support has been invaluable during my time in graduate school. v

7 Vita June 11, Born - Springfield, MA, US May B.S. Electrical and Computer Engineering Western New England College, Springfield, MA December M.S. Electrical and Computer Engineering The Ohio State University, Columbus, OH Publications Journal Publications S. B. Kim, J. D. Ouellette, J. J. van Zyl, and J. T. Johnson, Dual-copolarized approach to detect surface water extent using L-land radar, IEEE Transactions on Geoscience and Remote Sensing, in review J. D. Ouellette et al., A simulation study of compact polarimetry for radar retrieval of soil moisture, IEEE Transactions on Geoscience and Remote Sensing, vol. 52, no. 9, J. T. Johnson and J. D. Ouellette, A study of polarization features in bistatic scattering from rough surfaces, IEEE Transactions on Geoscience and Remote Sensing, vol. 52, no. 3, vi

8 A. Nekrasov, J. D. Ouellette, N. Majurec, and J. T. Johnson. A study of sea surface estimation from near nadiral cross-track-scanned backscatter data, Geoscience and Remote Sensing Letters, vol. 10, no. 6, Conference Publications J. D. Ouellette et al., A Study of Soil Moisture Estimation from Multi- Temporal L-band Radar Observations of Vegetated Surfaces, European Conference on SAR, proceedings. Abstract J. D. Ouellette, J. T. Johnson, N. Majurec, and A. Nekrasov, On the Estimation of Wind Vectors over the Sea Surface from Near-Nadir Radar Observations, URSI National Radio Science Meeting, proceedings J. D. Ouellette, J. T. Johnson and S. Hensley, Backscattered field correlations for rough surfaces with varying dielectric properties, International Geoscience and Remote Sensing Symposium (IGARSS), proceedings J. D. Ouellette et al., A simulation study of compact polarimetry for radar retrieval of soil moisture, International Geoscience and Remote Sensing Symposium (IGARSS), proceedings J. D. Ouellette and J. T. Johnson, Polarimetric bistatic scattering patterns of circularly polarized waves from ocean-like surfaces, URSI National Radio Science Meeting, proceedings J. D. Ouellette and J. T. Johnson, Studies of the bistatic scattering pattern of ocean-like surfaces, JPL workshop on GNSS Reflectometry vii

9 J. D. Ouellette and J. T. Johnson, Analytical models for predicting full bistatic scattering and polarimetry from three-dimensional exponentially correlated randomly rough surfaces, IEEE Antenna and Propagation Society Symposium, proceedings Fields of Study Major field: Electrical and Computer Engineering Studies in: Applied Electromagnetics Microwave Remote Sensing viii

10 Table of Contents Abstract Dedication Acknowledgements Vita ii iv v vi List of Figures xiii List of Tables xix 1 Introduction Motivation History/Background of Active Soil Moisture Remote Sensing Introduction to Scattering from Natural Scenes The Normalized Radar Cross Section Scattering Geometry Polarization Basis Analytical Methods for Rough Surface Scattering Overview of The Soil Moisture Active/Passive Mission Vegetation Water Content The Baseline SMAP Radar Soil Moisture Retrieval Algorithm 16 ix

11 1.7 Data Cubes Land Cover Classification Dielectric Model for Soil Dissertation Overview Properties of Bistatic Scattering from Rough Surfaces Motivation The Small Perturbation Method Physical Optics Second-Order Small Slope Approximation Validation of SSA2 Using Method of Moments Comparison of MoM, SSA2, and SSA3 Predictions with Measured Data for Backscattering Conclusions A Model-Based Study of Repeat-Pass InSAR Decorrelations Motivation Single-Layer SSA2 Model Two-Layer SPM1 Model Conclusions A Simulation Study of Compact Polarimetry for Soil Moisture Retrieval Motivation x

12 4.2 Introduction Review of Compact Polarimetry Reconstruction Techniques Retrieval Studies Monte Carlo Simulation of Fields Maximum-likelihood Retrieval Results Conclusions A New Time-Series Soil Moisture Inversion Algorithm for Vegetated Surfaces Using L-band Radar Backscatter Motivation History and Basic Formulation Analysis and Refinement Tools SMAPVEX12 Field Campaign GloSim SMAP Simulator Novel Modifications to the Alpha Approximation Method Selection of Bounds for Least-Squares Optimization A Simulation Study Including Additive Vegetation Effects Studies of Polarization Effects on the Alpha Method Further Simulation Studies of the Modified Alpha Method Application of the Modified Alpha Method to SMAP Radar Data xi

13 5.5 Conclusions Inland Water Body Detection using L-band Radar Motivation Algorithm History Algorithm Assessment Subtraction of Noise-Equivalent NRCS Adjustment for Calm, Open Water Conclusions Conclusions References xii

14 List of Figures 1.1 A visualization of volume, double-bounce, and surface scattering mechanisms in the context of radar remote sensing An illustration of the bistatic scattering geometry A visualization of a SMAP data cube NRCS values as a function of soil moisture, surface roughness, and vegetation biomass, according to the SMAP grass-type data cube Example plots of soil permittivity versus moisture content according to the Peplinski/Dobson/Ulaby dielectric model Full, bistatic scattering hemispheres for a randomly rough surface target, as predicted by the first-order Small Perturbation Method More examples of bistatic scattering hemispheres for a randomly rough surface target, as predicted by the first-order Small Perturbation Method Bistatic scattering versus scattered azimuth angle for rough surface targets of varying permittivity, as predicted by SPM xiii

15 2.4 Full, bistatic scattering hemispheres for a randomly rough surface target, as predicted by the second-order Small Slope Approximation Full, bistatic scattering hemispheres for VV polarization for several rough surface targets with varying roughness conditions, as predicted by the second-order Small Slope Approximation SSA2 and SPM1 VV NRCS predictions versus azimuth angle for several different permittivity conditions Comparison of SSA2 monostatic and bistatic NRCS predictions versus soil moisture conditions Full, bistatic scattering hemispheres for a randomly rough surface target, as predicted by the Method of Moments Comparison of MoM and SSA2 predictions for the full bistatic scatter hemisphere of a rough surface target Comparison of MoM, SSA2, and SSA3 predictions with backscatter measurements from soil scenes, collected during a Michiganbased field campaign A map of the main CanEx10 study region around Kenaston, Saskatchewan, Canada Example HH- and VV-polarized repeat-pass field correlation images collected by UAVSAR xiv

16 3.3 An illustration of the geometry considered for the single-layer case for repeat-pass field correlation studies Repeat-pass field correlations as predicted by SSA2 for a singlelayer case, plotted as a function of soil moisture An illustration of scattering from layered rough-surface media An illustration of the geometry considered for the single-layer case for repeat-pass field correlation studies Repeat-pass field correlations as predicted by SPM1 for a twolayer case, plotted as a function of soil moisture of the upper layer Repeat-pass field correlations as predicted by SPM1 for a twolayer case, plotted as a function of soil moisture of the upper layer Repeat-pass field correlations as predicted by SPM1 for a twolayer case, plotted as a function of soil moisture of the upper layer Attempted reconstruction of HH and VV NRCS values from compact-pol measurements Plot of the µ parameter for a slice of the grass-type data cube, compared with surface, volume, and double-bounce contributions predicted by the data cube model xv

17 4.3 Plots of RMS error in RMS height and soil moisture using a compact polarimetric radar mode with time-series estimation approach. Comparison with a comparable fully-polarimetric system is provided Plots of RMS error in RMS height and soil moisture using a compact polarimetric radar mode with snapshot estimation approach. Comparison with a comparable fully-polarimetric system is provided Comparison of CP and fully polarimetric (FP) RMS retrieval errors as a function of vegetation biomass Plots of HH- and VV-polarized alpha coefficient magnitudes versus surface soil moisture A map of the main SMAPVEX12 study region around Elm Creek, Manitoba, Canada Two scatter plots of retrieved versus measured soil moisture generated by using the single-polarization alpha method on SMAPVEX12 data Flow diagram for data-cube simulation of the alpha method for the purpose of considering additive vegetation contributions A plot of RMSE versus VWC and roughness for the original alpha method when applied to a grass-type data cube xvi

18 5.6 A plot of RMSE versus VWC and roughness for the alpha method, modified using the Freeman-Durden decomposition, applied to a grass-type data cube A plot of RMSE versus VWC and roughness for the alpha method, modified using the RVI parameter, applied to a grasstype data cube A scatter plot of retrieved versus measured soil moisture generated by using several polarization configurations of the alpha method on SMAPVEX12 data A soil moisture map for the modified alpha method when applied to the GloSim2014 SMAP simulator An RMSE map for the modified alpha method when applied to the GloSim2014 SMAP simulator A soil moisture truth map for generated by the GloSim2014 SMAP simulator A breakdown of RMSE versus land cover classification for the modified alpha method when applied to the GloSim2014 SMAP simulator A breakdown of RMSE versus both land cover classification and vegetation biomass for the modified alpha method when applied to the GloSim2014 SMAP simulator A soil moisture map for the modified alpha method when applied to measured SMAP radar data xvii

19 5.15 Comparisons between soil moisture estimates from the modified alpha method, applied to measured SMAP radar data, and ground-measured, upscaled soil moisture for the SoilSCAPE calibration/validation site Comparisons between soil moisture estimates from the modified alpha method, applied to measured SMAP radar data, and ground-measured, upscaled soil moisture for the TxSON calibration/validation site A flowchart for the original SMAP radar water body detection algorithm proposed by Kim et al Optical and radar imagery of the Assiniboine River in Manitoba, Canada Optical and radar imagery of Yellowstone Lake A flowchart for a modified version of the water body detection algorithm where the noise-equivalent NRCS is subtracted Zoomed-in radar imagery of Yellowstone Lake where the incidence angle is approximately 40 degrees A flowchart for the modified water body detection algorithm, including a threshold of HH-polarized backscatter xviii

20 List of Tables 1.1 Land Cover Types Considered for SMAP Radar Algorithms Radar Water Body Detection Algorithm Performance Before and After Thermal Noise Subtraction Radar Water Body Detection Algorithm Performance With and Without a Threshold on HH Backscatter False Alarm Rates for Water Detection Over Land, Versus Land Classification xix

21 Chapter 1 Introduction 1.1 Motivation Over the past several decades, interest in global soil moisture estimation by microwave remote sensing has increased steadily. Soil moisture content can provide significant insight into the Earth s climate cycles, thus fostering conditions for improved agricultural productivity (including more accurate crop yield forecasts), more accurate weather forecasting, better predictions of droughts, floods, and landslides, and more thorough investigation of the Earth s carbon cycle [1 4]. While ground-based soil moisture sensors can provide accurate measurements of surface soil moisture content, a satellite-based platform is required to derive global-scale soil moisture estimates. Knowledge of global-scale soil moisture climatology is particularly important for climate change monitoring and weather forecasting. Several satellite missions such as SMOS [5] and AMSR-E [6] have been motivated by the need for global-scale soil moisture 1

22 estimates, but typically provide infrequent revisit times and poor spatial resolution. Synthetic Aperture Radar (SAR) is capable of providing soil moisture estimates with higher spatial resolution, as demonstrated by the SMAP mission described in following sections. Given the numerous difficulties in estimating soil moisture under a vegetation canopy using radar, active microwave soil moisture remote sensing remains a subject of ongoing research. Some of the studies presented in this dissertation investigate the potential advantages of radar configurations previously unused for the purpose of soil moisture remote sensing. Firstly, the use of out-of-plane bistatic scattering geometries has seldom, if ever, been proposed for the purposes of remote sensing. Out-of-plane geometries exhibit unique polarization features that depend on surface conditions including soil moisture. Rough surface scattering models, including analytical and numerical methods, are used to investigate these polarization features and assess the utility and feasibility of using out-of-plane configurations. Secondly, the use of interferometric SAR (InSAR) is also explored, particularly for the case of a layered, bare surface in which the incident wave penetrates through the entire upper layer. In conventional soil moisture remote sensing, this penetration effect (particularly for very dry, sandy soil) may lead to inaccurate moisture predictions if not treated carefully. Recent measurement data from an airborne SAR platform has suggested that significant field decorrelations can result from changes in the soil moisture of the upper layer where multiple soil layers are present (even for a bare soil surface). This is 2

23 the case even for moderate separation between layers and a moderately moist upper layer. This suggests InSAR measurements could be used for remote sensing of changes in soil moisture in the case of multiple soil layers. Thirdly, a compact polarimetric radar mode is explored for the purpose of soil moisture remote sensing. In compact polarimetry, a single transmit polarization (usually circular or 45 degree linear) is utilized while horizontal and vertical polarizations are simultaneously received. Such a configuration allows for greater flexibility in radar system design when compared to a fullypolarimetric radar mode in which horizontally- and vertically-polarized pulses are usually interleaved in time. Moreover, circular polarization is known to be less significantly influenced by Faraday rotation when compared to a linear polarization basis [7 9]. A compact polarimetric radar mode is compared with a fully-polarimetric mode for the purposes of remote sensing of soil moisture. It is shown that in the presence of speckle and thermal noise, the error performance of soil moisture retrieval using a compact polarimetric mode is only slightly degraded when compared to a fully-polarimetric mode. Fourthly, a new time-series approach to soil moisture retrieval is explored using SMAP data. This new time series approach does not rely on the use of ancillary vegetation information as the SMAP baseline algorithm does, and does not make the use of a complicated forward model. This means the algorithm is easy to train and will not be influenced by errors in ancillary vegetation information. Results have shown that volumetric soil moisture estimates from this algorithm are generally within 8% RMSE, even for relatively dense 3

24 vegetation conditions. Lastly, in soil moisture remote sensing using radar, it is important that inland water bodies be flagged before soil moisture estimation is performed so that soil moisture estimates are not biased by radar returns from water surfaces. The detection of inland water bodies using L-band radar is discussed. Improvements have been made to the original SMAP water body detection algorithm. The old SMAP algorithm, which made use of the ratio between HH- and VV-polarized NRCS measurements, was susceptible to unacceptably high error in the noise-limited case where radar backscatter of the water s surface was close to the noise floor of the radar. 1.2 History/Background of Active Soil Moisture Remote Sensing Algorithms for deriving soil moisture estimates from radar backscatter measurements mainly fall into one of three categories: empirical, model-based, and change detection. An empirical method for deriving soil moisture content from radar measurements is based on a training data set to which curve fitting is applied. Empirically-derived curves are then used to invert properties of the soil surface and vegetation canopy from radar measurements. Model-based approaches try to recreate electromagnetic scattering physics through the use of models of the scattering physics of a vegetation-covered soil surface. Finally, change-detection approaches simply map temporal changes in radar backscat- 4

25 ter signatures to temporal changes in soil moisture (as opposed to providing an estimate of absolute soil moisture). Many studies have found that soil moisture content can be obtained through microwave radar measurements due to water s relatively high dielectric constant [10 13]. The observable radar measurement usually takes the form of the normalized radar cross-section. The normalized radar cross-section (NRCS) is a quantity which is convenient for describing a radar measurement of a distributed target. Since a distributed target is defined as a target which occupies a region larger than the antenna footprint, a simple radar cross-section (RCS) measurement is not very meaningful. The NRCS describes the RCS per unit area of such a target, and is a unitless quantity. In most remote sensing systems, the backscattered NRCS, or radar backscatter coefficient (σ 0 ), is the quantity from which properties of a distributed target are derived. 1.3 Introduction to Scattering from Natural Scenes The Normalized Radar Cross Section The normalized radar cross section (NRCS) is often a metric of interest when describing scattering from a distributed target. From a radar s perspective, a distributed target can be defined as a scene which occupies many consecutive range and azimuth cells. Distributed targets are many times the primary focus for imaging radars, including synthetic aperture radars. Distributed tar- 5

26 gets are also of interest when characterizing radar clutter, i.e. unwanted radar scattering returns from a background scene. Since the NRCS of a natural scene (soil, ocean, etc.) cannot usually be predicted deterministically, it is required that the scattered fields from such a scene be treated as a random variables [14 18]. Such a natural scene that can be represented as a single-valued continuous interface between the air and a dielectric medium is commonly referred to as a rough surface target. If the rough surface height is considered to be a stationary Gaussian random process, then the surface can be characterized completely by its covariance function, which is the autocorrelation of the surface height as a function of position. Many times the quantity of interest for a rough surface target is the ensembleaveraged NRCS. The so-called ensemble-average is the average over multiple realizations of a target described by a particular set of statistics. Several numerical and approximate methods exist for the purpose of computing the ensemble-averaged NRCS from a randomly rough surface. The Soil Moisture Active/Passive (SMAP) radar makes use of backscattered NRCS values (at L- band, 40 degrees incidence) for the purpose of soil moisture remote sensing. The NRCS of a soil surface covered by vegetation is often represented as the sum of three terms which respectively represent soil surface contributions, vegetation scattering (volume) contributions, and contributions due to interactions between the ground and the vegetation canopy. In other words, the total NRCS can be represented by: 6

27 σ t pq = σ s pq exp( τ) + σ v pq + σ sv pq (1.1) where σ t pq, σ s pq, σ v pq, and σ sv pq are the total backscattered signal for pq-polarization and its contributions by surface, volume, and surface-to-volume scattering mechanisms respectively. exp( τ) represents a modification to the surface contribution due to attenuation by two-way propagation through the vegetation canopy. Figure 1.1 provides an illustration for each of these scattering mechanisms [19, 20]. Note that other contributions to the total backscattered radar signature exist, such as higher-order multiple scattering between the ground and the vegetation canopy. These effects, however, are generally considered small enough to be negligible, particularly at lower microwave frequencies such as L-band. 7

28 Figure 1.1: A visualization of the three dominant scattering mechanisms generally considered in a model of radar backscattering from a vegetation canopy over a soil surface. These three contributors (from left to right) are volume scattering, double-bounce (surface-to-volume) scattering, and surface scattering [19, 20] Scattering Geometry Figure 1.2 illustrates the bistatic scattering geometry (where the transmitter and receiver are not necessarily co-located) and the conventions adopted in defining wave vectors and incidence/scattering angles [21]. An incident wave impinges on the rough surface target from polar angle θ i and azimuth angle φ i = 0. The scattered field is observed from polar angle θ s and azimuth angle φ s. φ s = 0 or 180 represents the plane of incidence. (θ s = θ i, φ s = 0 ) and (θ s = θ i, φ s = 180 ) respectively represent the specular and backscattering directions. 8

29 Figure 1.2: An illustration of the bistatic scattering geometry. Based on the geometry illustrated in 1.2 the incident and scattered wave vectors are: k i =ˆxk xi ẑk zi =k(ˆx sin θ i ẑ cos θ i ) (1.2) k s =ˆxk xs + ŷk ys + ẑk zs =k[sin θ s (ˆx cos φ s + ŷ sin φ s ) + ẑ cos θ s ] where ˆx, ŷ, and ẑ are unit vectors in the x, y, and z directions. The z components of the wave vectors in the dielectric region (below the rough surface interface) are: k 1zi =k ɛ r sin 2 θ i k 1zs =k ɛ r sin 2 θ s, (1.3) 9

30 where ɛ r is the relative permittivity of the target scene, and k ρi =k sin θ i k ρs =k sin θ s. (1.4) Note that θ i and θ s are defined to be between 0 and π/2, ergo both k ρi and k ρs are always positive. The next section defines conventions for describing the polarizations of incident and scattered waves Polarization Basis This subsection defines the polarization basis adopted for this dissertation. The analyses in following chapters almost exclusively consider plane wave illumination of an observed scene. A plane wave is defined as a wave whose phase is constant along any plane which is perpendicular to the direction the wave is traveling [22]. Consider an incident plane wave with electric field E i, given by E i = Ehĥ i + Ei vˆv (1.5) where ĥ and ˆv are unit vectors corresponding to horizontal and vertical polarizations respectively. The polarization unit vectors ĥ and ˆv correspond to ˆφ and ˆθ respectively in the standard spherical coordinate system described in [18]. The scattered field E s (also represented by a plane wave) from an observed scene can be expressed as: 10

31 E s = E s h E s v = eik0r r S E i h E i v = eik0r r S hh S hv S hv S vv E i h E i v (1.6) where S is the scattering matrix of the observed scene. Each entry S αβ of the scattering matrix is the scattering coefficient for an α-polarized scattered field produced by a β-polarized incident wave [18]. k 0 and r are the electromagnetic wavenumber and bistatic range, respectively. In Equation 1.6, the standard e iωt time convention is assumed and suppressed [18, 22]. The bistatic radar cross section (RCS) and scattering matrix of the observed scene are related through: σ αβ = 4π S αβ 2 (1.7) where σ αβ is the αβ-polarized RCS of the observed scene. The normalized radar cross section (NRCS, also known as the backscatter coefficient) is defined as: σ 0 αβ = σ αβ A (1.8) where A is the area of the target illuminated by the incident electromagnetic wave. For a random, distributed target, σ 0 αβ is a random function. The next section describes methods of estimating the ensemble average of σ 0 αβ for a rough surface target, which is often used to represent a soil surface. 11

32 1.3.4 Analytical Methods for Rough Surface Scattering While numerical methods for calculating electromagnetic scattering are often applied to the rough surface scattering problem, these methods are extremely computationally expensive, even for a single frequency. Since a rough surface is described as random, and since one is typically interested in calculating the ensemble-averaged scattered fields, one must apply a Monte Carlo technique when using a numerical method to calculate the scattered fields from a rough surface. In a Monte Carlo technique, multiple realizations of a random process are generated such that the statistics of that random process can be calculated [23]. For a surface with both long- and short-scale roughness features (such as an ocean surface), a large surface realization must be generated with fine sampling to accurately capture the statistics of the surface. These Monte Carlo simulations can quickly become expensive and time consuming, particularly for large and/or finely-sampled surface realizations. Analytical methods are approximate approaches to describe electromagnetic scattering from a rough surface, and are based on various assumptions of the observed scene. The allure of analytical methods is that they are much faster (by orders of magnitude) than running a Monte Carlo, full-wave simulation of a randomly rough, distributed target. One classic approach to analytically describing electromagnetic scattering from a rough surface is the Small Perturbation Method (SPM). SPM is applicable in the limit of small surface heights compared to the electromagnetic 12

33 wavelength. This implies that for fixed surface statistics, the SPM applies for low frequencies [24]. The SPM provides a perturbation series of field terms, which is truncated to the first-order term for the studies reported in this dissertation. In contrast to SPM, Physical Optics (PO) is a classic approach to describing rough surface scattering in the high-frequency limit [16, 17, 26, 27]. The PO approximation (otherwise known as the tangent plane approximation or the Kirchhoff approximation) is applicable for rough surfaces with large RMS heights, as long as the surface profile is smooth and thus can be represented locally as a tilted plane. As stated previously, SPM and PO are applicable in low and high frequency limits respectively. Some analytical methods attempt to bridge the gap between PO and SPM since the two methods do not share a common regime where both are applicable [28]. One such method is the so-called Small Slope Approximation (SSA). The SSA model provides a series of field terms which reduce to SPM and PO in the appropriate limits [29]. The applicable roughness regime for SSA is thus not as restrictive as SPM or PO, but is still restricted to small surface slopes. Surfaces with a small ratio between correlation length and RMS height (less than about three) should not be considered by SSA. This dissertation primarily makes use of the second-order SSA (SSA2), which uses the first two terms in the SSA field series (which is a perturbation series of quasi-slope), which yields three NRCS terms. Inclusion of the secondorder SSA term ensures compliance with SPM up to second order whilst main- 13

34 taining compliance with PO under the appropriate limit. SSA2 formulations do not have a simple closed form and must be evaluated numerically. 1.4 Overview of The Soil Moisture Active/Passive Mission Chapters 5 and 6 respectively describe a new soil moisture estimation technique and a method for detecting inland water surfaces, both specifically designed for use with Soil Moisture Active/Passive (SMAP) radar data. The SMAP mission provides an excellent opportunity to research soil moisture monitoring using instruments observing the Earth from a satellite platform. SMAP is a satellite equipped with an L-band SAR (1.26 GHz) and an L-band radiometer (1.41 GHz) which share a conically-scanned mesh reflector dish illuminated by a single feed horn. The SMAP radar provides incoherent NRCS values in HH, VV, and either HV or VH polarizations (one cross-polarized measurement, which can be selected by ground command) [30]. Due to downlink considerations, HV and VH are not reported simultaneously. Soil moisture products are made available through the use of active (radar), passive (radiometer), and combined active/passive approaches [1 4, 31]. The active and passive soil moisture products delivered by SMAP provide a compromise between spatial resolution and accuracy. Though radiometerderived soil moisture estimates tend to be more accurate, SAR imagery is ideal for estimating soil moisture in all-weather conditions with high spatial and 14

35 temporal resolution. The SMAP-related work described in this dissertation focuses on radar-only SMAP algorithms including techniques for soil moisture retrieval using the SMAP radar. The baseline soil moisture estimation algorithm for the SMAP radar relies on the use of forward models which forecast the ensemble-averaged NRCS of an observed scene based on several input parameters. The outputs of these forward models are organized into three-dimensional look-up tables known as data cubes [19]. The three dimensions of these data cubes correspond to soil surface roughness, soil permittivity, and a single vegetation biomass estimator called Vegetation Water Content (VWC). The soil permittivity can be directly mapped to soil moisture, given that the soil texture is known [32, 33]. A different data cube is generated for every land cover classification considered in the SMAP algorithm (bare soil, soybean, corn, deciduous trees, grass, etc.). More details on VWC, the SMAP baseline radar soil moisture retrieval algorithm, the data cube models, land cover classification, and soil dielectric mixing models are provided in the following sections. 1.5 Vegetation Water Content While the problem of inverting soil moisture from radar backscatter signatures of a bare surface (with little or no vegetation canopy) is well-documented in literature, the subject of soil moisture remote sensing using imaging radars over vegetated terrain remains an open problem. The first issue is how to 15

36 quantify the amount of vegetation present in the observed scene such that one may objectively observe the influence of the amount of vegetation biomass on radar backscatter. The SMAP parameter for quantifying the amount of vegetation biomass present in the observed scene is the so-called Vegetation Water Content (VWC). The VWC, as the name suggests, quantifies the amount of water suspended within a vegetation canopy. The VWC of a vegetation structure is measured by harvesting all of the crops/vegetation within a 1 m 2 patch of land, weighing the harvested vegetation, baking it in an oven until dry, and weighing it again [34 36]. The VWC is the difference between the wet and dry weights of the plants within the 1 m 2 area, and is typically expressed in units of kg/m 2 [37]. The VWC parameter is adopted in many of the studies described in following chapters. In general, a scene with VWC<0.25 kg/m 2 can be considered sparsely vegetated. Usually an accurate soil moisture estimation is not possible for very high vegetation biomass (VWC>5 kg/m 2 ) because the surface contribution to the backscattered signal is orders of magnitude smaller than the vegetation contribution. 1.6 The Baseline SMAP Radar Soil Moisture Retrieval Algorithm The current baseline radar soil moisture retrieval algorithm for SMAP is the time-series data cube approach [19, 20, 38, 39]. In this approach, a time 16

37 series of radar data in HH- and VV-polarizations is compared with the data cubes. The VWC of the observed scene is provided via an ancillary data source; VWC for SMAP is derived from the Moderate-Resolution Imaging Spectroradiometer (MODIS) Normalized-Difference Vegetation Index (NDVI) [40]. Since vegetation information is assumed to be provided, this reduces the number of unknowns when comparing measured NRCS data with the (simulated) data cubes. Roughness is assumed to be constant for the observed scene over the length of a single time-series of radar measurements. Therefore, for a time-series of N SMAP radar observations, there are 2N measurements (HH and VV channel NRCS data for each element of the time series) and N + 1 unknowns (N soil moisture values and 1 surface roughness value). The cost function to be minimized is therefore: C(s, n, ɛ r1, ɛ r2,..., ɛ rn ) = [σ 0 HH(t 1 ) σ 0 HH,dc(s, n, ɛ r1 )] 2 + [σ 0 V V (t 1 ) σ 0 V V,dc(s, n, ɛ r1 )] 2 + [σ 0 HH(t 2 ) σ 0 HH,dc(s, n, ɛ r2 )] 2 + [σ 0 V V (t 2 ) σ 0 V V,dc(s, n, ɛ r2 )] 2 (1.9) [σ 0 HH(t N ) σ 0 HH,dc(s, n, ɛ rn )] 2 + [σ 0 V V (t N ) σ 0 V V,dc(s, n, ɛ rn )] 2 where s is the RMS height of the soil surface, n is the number of discrete scatterers used to represent the vegetation canopy in the scattering model, ɛ ri is the relative permittivity of the soil surface (corresponding to soil moisture) at the i th element of the time series, σ 0 P P (t i) is the measured, P P -polarized 17

38 NRCS at time t i, and σp 0 P,dc (s, n, ɛ ri) represents the NRCS prediction from the corresponding data cube. When Equation 1.9 is minimized by comparing SMAP measurements with data cube predictions, the most likely roughness value and N soil moisture values are reported through the corresponding axes of the data cube. As the sliding window of the time series shifts (i.e. when new radar measurements are made), multiple soil moisture estimates will be made for the same day. However, only the most recent soil moisture estimate is retained, which corresponds to the N th time series element. The baseline time series length is N = 6. Given SMAP s revisit time of around three days (depending on the latitude of the observed scene), six time series elements spans about 18 days. 1.7 Data Cubes Visualizations of example data cubes is provided in Figure 1.3. This figure demonstrates that for each polarization, the backscattered NRCS is dependent on roughness, soil moisture (soil permittivity), and vegetation biomass. The dependence of backscatter on each of these parameters makes soil moisture inversion complicated, particularly in the presence of vegetation. A different set of data cubes (one cube for each polarization) is generated for several different land-cover types. For each land cover type, a different discrete scatterer model is used to represent scattering from the vegetation canopy [19]. 18

39 Figure 1.3: A visualization of three look-up tables generated for use in baseline soil moisture retrieval algorithm for the SMAP radar, referred to as a data cube [19]. The data cubes are generated independently for each polarization (HH, HV, and VV). ks, ɛ r, and V W C represent the axes of the data cube which correspond to surface roughness, soil permittivity, and vegetation water content, respectively. The line plots in Figure 1.4 show HH-, VV- and HV-polarized backscattered NRCS data cube predictions as functions of soil moisture, surface roughness, and vegetation biomass. In these plots, only one of these parameters is allowed to change, while the other two are held constant. Figure 1.4 illustrates that the polarimetric backscatter signature from a vegetation-covered soil surface must be treated as a multivariate function. Note that the HV-polarized NRCS only becomes significant for large vegetation conditions since the HV NRCS is chiefly generated by multiple scattering by a vegetation layer. 19

40 Figure 1.4: NRCS returns as functions of soil moisture (m v ), surface roughness (ks), and vegetation biomass (V W C), according to the SMAP grass-type data cube [20]. (a) HH- and VV-polarized NRCS values versus soil moisture, with ks and V W C held constant at 0.30 and 0.53 kg/m 2 respectively. (b) HH- and VV-polarized NRCS values versus surface roughness, with m v and V W C held constant at 0.13 and 0.53 kg/m 2 respectively. (c) HH-, VV- and HV-polarized NRCS values versus vegetation biomass, with m v and ks held constant at 0.13 and 0.30 respectively. 20

41 Some of the simulation studies in later chapters make specific use of the grass-type data cube used for SMAP. The grass-type data cube was produced by NASA JPL for L-band backscatter at 40 degrees incidence angle [20, 38]. The model uses the small perturbation method (SPM) to describe scattering from the rough soil surface, thereby placing a limit on the maximum surface RMS height which can be considered (about 1.5 cm at the 1.26 GHz frequency of interest: the approximate center frequency of the SMAP radar). The rough surface interface is described by an isotropic, exponential autocorrelation function. The ratio between the rough surface s correlation length and RMS height was set to 10 in all cases. RMS height values are often reported in terms of ks, where s is the surface RMS height and k = rad/m is the electromagnetic wavenumber at 1.26 GHz. Soil permittivity values are produced as a function of soil moisture using the Peplinski/Ulaby/Dobson empirical model [32]. The grass vegetation simulated consists of a single layer of cylindrical vegetation structures of specified sizes in a specified orientation distribution and with a fixed number density. Previous studies [18, 20, 41, 42] have shown that the scattering contributions of a dielectric cylinder above a perturbed surface are a function of many parameters, including orientation, diameter, length, permittivity, and height above the surface. In most cases, the geometry does not change significantly during the time-window of retrieval interest (e.g., a few weeks at maximum). Then only the cylinder dielectric constant varies as the vegetation water content of the medium is varied, which characterizes the scattering model. 21

42 The data cubes form a foundation for several of the simulation studies reported upon in later chapters. The data cubes have been trained using experimental data to more accurately represent the predicted radar backscatter signatures from scenes of a particular land classification. Since the data cubes have been trained using real SAR data concurrent with ground sampling, these simulation data sets provide a first step in retrieval algorithm development. Also, the organization of data cubes allows for error performance assessment as a function of roughness, vegetation biomass, and soil moisture content independently. The forward models which generate the data cubes are each a hybrid of a rough surface scattering model (which describes scattering from the randomly rough soil surface) and a discrete scatterer model (in which the discrete scatterers represent leaves, stems, branches, etc. within the vegetation canopy). Backscatter contributions due to soil surface scattering, vegetation scattering, and interactions between the ground and the vegetation canopy (referred to as double-bounce scattering) are all incoherently summed in the data cube models, in accordance with Equation 1.1. The shape, number density, size, etc. of discrete scatterers within the vegetation layer of the data cube model is dependent on the land cover classification which the model is representing. The next section describes the land cover classification used in SMAP and adopted in this dissertation. 22

43 1.8 Land Cover Classification As mentioned previously, data cubes are generated for each land cover classification separately. For SMAP, land cover classifications are determined via ancillary data sets for ground properties. The proposed land cover classifications are based on research and statistics provided by the International Geosphere-Biosphere Program (IGBP) and the United Nations Food and Agriculture Organization [43 45]. The 16 land cover classes which will be used by SMAP are provided in Table 1.1. Note that the croplands IGBP land cover classification has been partitioned into several different crop types. This is because radar scattering of different crop types behaves very differently depending on the physical structure of the vegetation canopy. Crop types are derived from agricultural land surveys. The land cover types listed in Table 1.1 are adopted in many of the studies in later chapters, since radar scattering physics differ significantly between different land cover types. 23

44 Table 1.1: Land Cover Types Considered for SMAP Radar Algorithms 1 Evergreen Needleleaf Forest 2 Evergreen Broadleaf Forest 3 Deciduous Needleleaf Forest 4 Deciduous Broadleaf Forest 5 Mixed Forests 6 Closed Shrublands 7 Open Shrublands 8 Woody Savannas 9 Savannas 10 Grasslands Croplands: Wheat, Rice, Corn, Soybean 15 Cropland / Natural Vegetation Mosaic 16 Barren / Sparsely Vegetated 1.9 Dielectric Model for Soil In general, the relative permittivity of soil can be approximated as a function of four parameters: moisture content, sand fraction, clay fraction, and electromagnetic frequency [32, 33]. The dielectric mixing model adopted in this dissertation is the Peplinski/Dobson/Ulaby model. Figure 1.5 shows the dependence of the real and imaginary parts of soil permittivity on moisture content for several different soil textures (sand and clay fractions) according to 24

45 the Peplinski/Ulaby/Dobson model whose formulation is provided in [32]. For modeling radar scattering, it is important to model both the real and imaginary parts of soil permittivity when phase information must be preserved. When phase information is not measured (which is the case for SMAP), only the magnitude of soil permittivity is important. For SMAP, soil sand fraction and clay fraction information is provided by a combination of datasets from the International Soil Reference and Information Centre (ISRIC), State Soil Geographic (STATSGO), the Australian Soil Resource Information System (ASRIS), the National Soil DataBase (NSDB), and the United Nations Food and Agriculture Organization (UN-FAO) [19, 30, 45 49]. The influence of soil texture on radar backscatter is explored in [50]. 25

46 Figure 1.5: Plots of real and imaginary parts of soil s relative permittivity (ɛ r ) versus moisture content according to the Peplinski/Dobson/Ulaby dielectric model [32]. For clay soil, the sand and clay fractions are 20% and 60% respectively. For loam, the sand and clay fractions are 40% and 20% respectively. For sandy loam, the sand and clay fractions are 60% and 10% respectively [51]. 26

47 1.10 Dissertation Overview Radar remote sensing of soil moisture is a diverse topic with many challenges. The inversion of soil moisture from a vegetated surface is a complex and multi-faceted problem. Each chapter of this dissertation discusses studies on a different topic within the field of soil moisture remote sensing using radar. Chapters 2 through 4 discuss largely simulation-based studies exploring concepts for future soil moisture remote sensing missions using L-band radar. Chapters 5 and 6 address issues directly related to soil moisture retrieval using the SMAP radar. Chapter 2 explores the problem of predicting bistatic scattering behaviors from randomly rough surfaces and its potential utility in existing and future soil moisture remote sensing systems. Predictions of full bistatic scattering hemispheres are shown for approximate and numerical rough surface scattering models. Results from these models, including bistatic polarization features outside the plane of incidence, are intercompared. It is found that the polarization features of the bistatic scattering pattern are highly dependent on surface parameters including soil moisture, particularly for out-of-plane scattering configurations. In Chapter 3, single- and two-layer rough surface scattering models are used to explain field decorrelations observed in repeat-pass airborne InSAR data. It is found that stratified rough surfaces can generate significant decorrelations when the soil moisture condition of the upper layer changes significantly 27

48 between repeated measurements. In Chapter 4, the use of compact polarimetry is proposed for soil moisture remote sensing. This polarization configuration is investigated through the use of a single, circular transmit polarization and two linear receive polarizations. Simulated soil moisture retrievals using such a system are compared with a comparable fully-polarimetric system in the presence of speckle noise (due to scintillations of the target scene) and system noise. Chapter 5 introduces a new change-detection-based algorithm for deriving soil moisture estimates from a time series of radar data. The algorithm is applicable for vegetated soil surfaces and has been shown to provide reasonable soil moisture estimates even in the presence of moderate to heavy vegetation biomass. This algorithm offers advantages over existing techniques in that it does not require the use of ancillary vegetation information, and it does not make use of a complicated forward model for inversion. The algorithm is applied to simulation data sets as well as airborne field campaign data and SMAP radar data. Chapter 6 describes a number of improvements upon the existing SMAP radar water body detection algorithm. These improvements ensure a lower detection error in the presence of the noise-limited case where the backscattered signal is close to or below the noise floor (which is usually the case for calm, open water). It is shown that false alarm rates over land (false water classification) is not significantly influenced by the modifications to the existing algorithm. The algorithm is assessed using NRCS data collected by an 28

49 airborne SAR over land and inland water bodies. Finally, Chapter 7 discusses the overall conclusions for this dissertation. Suggestions are made for possible future work in the interest of continuing the research topics described herein. 29

50 Chapter 2 Properties of Bistatic Scattering from Rough Surfaces 2.1 Motivation Classic radar configurations employ a monostatic configuration where the transmitter and receiver are co-located, but each of the scattering models mentioned in Chapter 1 are easily extensible to bistatic scattering configurations (where the transmitter and receiver are separated by an appreciable distance). The use of bistatic (and often passive) radar configurations is an emerging technology in the field of remote sensing which has seen increasing interest over the past decade. Passive, bistatic radar offers several advantages over traditional, monostatic radar configurations by using reflections from signalsof-opportunity [52]. Several studies have explored estimating soil moisture from bistatic radar configurations, including the use of reflected Global Navigation Satellite System (GNSS) signals [53 57]. Furthermore, since most scattering models including a vegetation layer over a soil surface include ground-to- 30

51 canopy interactions (see Equation 1.1), bistatic scattering from a soil surface is an important topic to explore for the purpose of electromagnetic modeling for remote sensing. Although most bistatic radar studies focus on the specular region due to received power considerations, the studies described here focus on bistatic geometries outside of the plane of incidence. Results from approximate and numerical models show unique polarization features outside of the plane of incidence which have received limited attention in literature [58 60]. These features depend on surface parameters, including soil moisture for land surfaces. The surfaces considered in these studies are zero-mean, stationary Gaussian random processes. As such, the covariance function of the surface completely describes the roughness statistics of the rough surface target in question The Small Perturbation Method While the SPM has been extensively studied for backscattering, relatively few studies have examined its applicability for bistatic configurations. There have been particularly few studies examining the SPM for configurations where the transmitter is not located in the plane of incidence [61 64]. The SPM provides a perturbation series of field terms, which is truncated to the firstorder term in this section. For a deterministic, periodic surface the scattered fields produced by an incident plane wave are of the form: 31

52 ζ αβ = h ki k s g αβ (k s, k i ) (2.1) according to the first-order SPM (SPM1) [16, 17, 25, 26]. In Equation 2.1, ζ is the scattered wave amplitude for a particular polarization represented by α and β. α is the polarization of the scattered wave (H or V ) propagating in the k s direction. β is the polarization of the incident wave (H or V ) propagating in the k i direction. h ki k s is the Fourier coefficient for the surface roughness corresponding to index k i k s. g αβ (k s, k i ) represents the SPM1 kernel function which is independent of surface roughness. This represents a Bragg scattering phenomenon; the scattered field in a particular direction is completely determined by the surface roughness for one particular Fourier coefficient. In the case of a random target, the contributions of Equation 2.1 from each surface Fourier coefficient are coherently summed, and the ensemble-average fields are computed by averaging scattered fields over multiple realizations of the target. The SPM1 kernel functions (which are multiplied by the surface Fourier coefficients in accordance with Equation 2.1) take the following form: g HH cos φ s cos θ s + ɛ r sin 2 θ s (2.2) g V H sin φ s ɛr sin 2 θ s ɛ r cos θ s + (2.3) ɛ r sin 2 θ s g HV sin φ s cos θ s + ɛ r sin 2 θ s (2.4) 32

53 g V V ɛ r sin θ s sin θ i cos φ s ɛr sin 2 θ s ɛr sin 2 θ i ɛ r cos θ s + ɛ r sin 2 θ s. (2.5) Note that these kernel functions do not have dependence on surface roughness parameters, but are multiplied by the Bragg Fourier coefficients (which are a consequence of surface roughness) in the SPM integral [25]. As a consequence of Equation 2.2, the SPM1 HH-polarized NRCS always vanishes in the cross-plane (the plane perpendicular to the plane of incidence). It is well-known that SPM1 VH- and HV-polarized scattering vanishes in the plane of incidence as a result of Equations 2.3 and 2.4 respectively. It can be shown through Equation 2.5 that the SPM1 VV-polarized NRCS also has a minimum region which is defined by a more complicated function in (k xs,k ys ) space [21]. Figures 2.1 and 2.2 show a sample of bistatic scattering hemispheres in each polarization for a rough surface target whose electromagnetic scattering physics is described by SPM1. These plots effectively show the ensembleaveraged SPM1 NRCS as a function of scattering angle for a particular incident wave which impinges on the target at a fixed angle θ i, measured from nadir. Each of these plots can be thought of as a top-down view of the full scattering hemisphere from a rough surface target, where the color scale represents the strength of the ensemble-averaged NRCS. The left half of the NRCS plots correspond to the backscattering region, while the right half of the NRCS plots correspond to the forward scattering region. Note that each polarization configuration exhibits a unique minimum region as a consequence of Equations 33

54 2.2 through 2.5. Note that for HH polarization, this minimum always occurs in the plane perpendicular to the plane of incidence. Cross-polarized NRCS values always vanish within the plane of incidence; this is well-known for SPM1. For VV polarization, the minimum is a more complicated function of scattering angle. The behavior of the SPM1 VV minimum has been found to vary as a function of target permittivity. 34

55 Figure 2.1: Full, bistatic scattering hemispheres (one plot for each polarization) for a randomly rough surface target, as predicted by SPM1. These plots effectively show the ensemble-averaged SPM1 NRCS as a function of scattering angle. For these plots, the incidence angle is θ i = 20, the permittivity of the dielectric medium is ɛ = 10 + i0.05, the surface RMS height is λ/20, and the correlation length is λ/2. Point-to-point correlations of the surface height are described by an isotropic, Gaussian correlation function. 35

56 Figure 2.2: Full, bistatic scattering hemispheres (one plot for each polarization) for a randomly rough surface target, as predicted by SPM1. These plots effectively show the ensemble-averaged SPM1 NRCS as a function of scattering angle. For these plots, the incidence angle is θ i = 40, the permittivity of the dielectric medium is ɛ = 10 + i0.05, the surface RMS height is λ/20, and the correlation length is λ/2. Point-to-point correlations of the surface height are described by an isotropic, Gaussian correlation function. 36

57 Figure 2.3 shows several line plots of ensemble-averaged NRCS predictions from SPM1 in HH and VV polarizations, as a function of azimuth angle (where the scattered polar angle is fixed). Note that for varying permittivity, the depth and location of the minimum in the VV pattern changes. Since permittivity and soil moisture are proportional [32], these results imply that tracking this minimum could be used for soil moisture remote sensing. Figure 2.3: Bistatic scattering versus scattered azimuth angle (with the scattered polar angle fixed at θ s = 40 ) in HH and V V polarizations for randomly rough surface targets of varying permittivity, as predicted by SPM1. For these plots, the incidence angle is θ i = 40, the surface RMS height is λ/20, and the correlation length is λ/2. Point-to-point correlations of the surface height are described by an isotropic, Gaussian correlation function. 37

58 The location of the minimum in the VV-polarized bistatic pattern for SPM1 has been derived analytically [21]. Consider the behavior of the VV-polarized SPM1 kernel function in Equation 2.5, which can be rewritten as: g V V = A cos φ s (2.6) where A = ɛ r sin θ s sin θ i (ɛr sin 2 θ s )(ɛ r sin 2 θ i ). (2.7) For a fixed scattered polar angle θ s, the amplitude of g V V is minimized where cos φ s = Re{A} (2.8) and at this minimum, g V V 2 Im{A} 2. (2.9) According to Equation 2.9, the SPM1 VV bistatic NRCS will vanish in the minimum region if A is real. A will be real if the surface permittivity ɛ r is real-valued or in the limit of very large amplitude of ɛ r. Otherwise, the VV NRCS does not vanish in the minimum region. If A is rewritten as: A = ɛ rk ρs k ρi k 1zi k 1zs (2.10) then the location of the VV bistatic minimum in (k xs, k ys ) space can be described with: 38

59 k 2 xs + k 2 ys = k xs k 1zi k 1zs 2 k ρi Re{ɛ rk 1zi k 1zs } (2.11) which depends on surface permittivity and incidence angle, but not surface roughness parameters Physical Optics While SPM is applicable in the low-frequency limit where the surface exhibits small surface slopes, Physical Optics (PO) applies in the high-frequency limit and likewise applies for small surface slopes. PO approximates the scattering matrix as: S(k s, k i ) P (k s, k i ) Q n e iq R dr (2.12) where R = ˆxx + ŷy + ẑh(x, y) (2.13) r = R ẑh(x, y) = ˆxx + ŷy (2.14) Q = k s k i (2.15) and Q n = ˆn (k s k i ). (2.16) ˆn is a unit vector normal to the surface profile, which is represented by h(x, y). P represents the PO kernel function which depends on electromagnetic frequency, incident and scattering angles, polarization, and surface permittivity. 39

60 In the perfectly-conducting limit, a minimum occurs in the PO bistatic pattern [21], in which both HH and VV-polarized NRCS values vanish on the contour (k xs k2 2k ρi ) 2 ( ) k + kys 2 2 = + k xsk zs k zi. (2.17) 2k ρi k ρi The minimum location described by Equation 2.17 is not consistent with SPM1 results. This discrepancy encourages the use of more complicated scattering models to track the behavior of this minimum. Furthermore, it is likely that the scattering physics within the null regions is dominated by higher-order scattering not captured by SPM1 or PO. It is thus prudent to move toward a higher-order scattering model with wider applicable regime of surface roughness Second-Order Small Slope Approximation Inclusion of the second-order SSA term ensures compliance with SPM up to second order whilst maintaining compliance with PO under the appropriate limit. It is expected that the second-order term in the SSA field series will be significant where bistatic minima occur since higher-order scattering effects may dominate in this region. The SSA2 compliance with both SPM2 and PO results is desirable since, as previously mentioned, the location of the bistatic null changes between the SPM and PO limits. The SSA has a form similar to that of physical optics, but augmented to include a series of terms having additional spatial functions in the integration 40

61 to achieve compliance with SPM in low frequency limit. The second-order SSA uses two terms to approximate the scattering amplitude, i.e.: S(k s, k i ) S 0 (k s, k i ) + S 1 (k s, k i ) (2.18) where S 0 (k s, k i ) = B Q n e iq R dr (2.19) and S 1 (k s, k i ) = i e iq R M 1 h(ξ)dξdr. (2.20) In Equations 2.19 and 2.20, B and M 1 represent the first-order SPM and second-order SSA kernel functions respectively. h(ξ) represents the Fourier transform of the surface profile h(x, y). The second-order SSA term S 1 (k s, k i ) makes the location of the VV-polarized bistatic minimum dependent on the surface roughness and correlation function [21]. Also note that since the M 1 kernel has a dependence on surface roughness, it is expected that the VV bistatic minimum predicted by SSA2 will depend on surface roughness conditions. This is in contrast to the minimum predicted by SPM1, which does not depend on surface roughness. Figure 2.4 shows an example of full bistatic scattering hemispheres predicted by SSA2. Note this is the same case as the one considered in Figure 2.1. Since this case is within the SPM (low-frequency) limit, it is expected that results from SSA2 will be similar to those shown in Figure

62 Figure 2.4: Full, bistatic scattering hemispheres (one plot for each polarization) for a randomly rough surface target, as predicted by the secondorder Small Slope Approximation. These plots effectively show the ensembleaveraged SSA2 NRCS as a function of scattering angle. For these plots, the incidence angle is θ i = 20, the permittivity of the dielectric medium is ɛ r = 10+i0.05, the surface RMS height is σ = λ/20, and the correlation length is L = λ/2. Point-to-point correlations of the surface height are described by an isotropic, Gaussian correlation function [21]. 42

63 Figure 2.5 shows VV-polarized SSA2 NRCS predictions for four different roughness cases. Note that the minimum in the VV-polarized bistatic pattern with changes in depth and location as the surface roughness varies, which is a consequence of the M 1 kernel function in Equation This confirms that the VV minimum varies as a function of surface parameters, as suggested by the differences in the minimum location according to SPM1 and PO theories. 43

64 Figure 2.5: Full, bistatic scattering hemispheres for VV polarization for several rough surface targets with varying roughness conditions, as predicted by the second-order Small Slope Approximation [65]. The RMS height of the surface considered is highlighted in the title of each plot. The incidence angle, dielectric constant, and correlation length are fixed for each case at θ i = 40, ɛ r = i0.16, and l = 0.29λ respectively. Point-to-point correlations of the rough surface height are described by a band-limited exponential autocorrelation function with a cut-off correlation length l s = 0.05λ [66]. 44

65 Figure 2.6 shows several line plots of ensemble-averaged NRCS predictions from SSA2 for VV polarization, as a function of azimuth angle (where the scattered polar angle is fixed). Note that for varying permittivity, the location of the minimum in the VV pattern changes according to both SSA2 and SPM1. While SSA2 and SPM1 predictions agree upon the locations of the VV-polarized bistatic minima, the depths of these minima are estimated to be much higher for SSA2. For SPM1, the minima are consistently predicted to be less than -50 db for each permittivity condition considered. The second-order term in the SSA2 field series results in a filling-in of these minimum regions. Figure 2.7 shows plots of ensemble-averaged NRCS predictions from SSA2 for monostatic and bistatic configurations where the scattering angle is fixed in elevation and azimuth. Note that for the bistatic configuration, the VVpolarized NRCS in the bistatic configuration is more sensitive to surface soil moisture conditions, though it exhibits lower returns when compared to the monostatic case. Figures 2.6 and 2.7 reaffirm findings from the SPM1 and suggest that out-of-plane bistatic radar configurations could be used in soil moisture remote sensing. 45

66 Figure 2.6: Bistatic scattering versus scattered azimuth angle (with the scattered polar angle fixed at θ s = 40 ) in VV polarization for randomly rough surface targets of varying permittivity, as predicted by SSA2 and SPM1. For these plots, the incidence angle is θ i = 40, the surface RMS height is λ/20, and the correlation length is λ/2. To more accurately represent a soil surface, point-to-point correlations of the surface height are described by an isotropic, band-limited exponential autocorrelation function with a cut-off correlation length l s = 0.05λ [66]. 46

67 Figure 2.7: SSA2-predicted monostatic and bistatic (scattering angles fixed at θ s = 60 and φ s = 44 ) NRCS values in HH and VV polarizations for a randomly rough target with varying soil moisture. For these plots, the incidence angle is θ i = 40, the surface RMS height is λ/20, and the correlation length is λ/2. To more accurately represent a soil surface, point-to-point correlations of the surface height are described by an isotropic, band-limited exponential autocorrelation function with a cut-off correlation length l s = 0.05λ [66]. Soil permittivity is calculated from volumetric soil moisture using the Peplinski/Ulaby/Dobson model with sand and clay fractions of 0.51 and 0.13 respectively [32]. 47

68 2.1.4 Validation of SSA2 Using Method of Moments Since higher-order scattering may be dominant in the minimum regions previously discussed, a numerical approach was used to verify the predictions by analytical models which were shown in previous subsections. The Method of Moments (MoM) was selected for use in verifying SPM and SSA results. The MoM integral equation formulation and solution are described in [67]. Because the bistatic minima described in previous sections occur over very small angular regions, it is required that the MoM implementation have a fine angular resolution. The angular resolution of such a surface scattering code is limited by the physical size of the finite aperture over which surface currents are computed. A surface size of 32 by 32 wavelengths was thus used in the MoM simulations to allow for an adequate angular resolution. The roughness conditions considered required a sampling rate of 16 points per wavelength, thereby generating a 512 by 512 grid of sample points. This results in a total of unknowns per surface realization, since four unknowns must be computed for every point on the surface profile. A total of 64 realizations was considered in order to perform a Monte Carlo simulation to calculate the ensemble averaged NRCS values [21]. Because of the large scale of the simulations required for the bistatic scattering problem, and the requirement for many Monte Carlo realizations, several methods were used to make MoM computations more efficient. An iterative solver has been implemented to solve the MoM matrix equation faster. A 48

69 canonical grid acceleration technique was implemented to accelerate matrixvector multiply floating-point operations [67]. Figure 2.8 shows bistatic NRCS plots similar to those shown in previous sections, but using the Method of Moments numerical model [21]. Since the case shown in Figure 2.8 is within the SPM limit, it is expected that these plots will be similar to those produced by SPM1 and SSA2 for the same case. The locations of bistatic minima in MoM results confirm predictions made by SPM1 and SSA2. Figure 2.9 shows a direct comparison between MoM and SSA2 predictions for a rough surface target in the SPM (low-frequency) limit. The plots show very good agreement between MoM and SSA2, except in the case where θ s is close to 90 degrees; MoM results are questionable in this region. These plots show that SSA2 can accurately predict the location and level of the bistatic minimum. Similar results have been produced for a number of other rough surface targets in [21] where SSA2 results are further verified. It is furthermore confirmed through use of the MoM in [21] that the bistatic minimum in VV polarization varies with roughness and permittivity in accordance with predictions by SSA2. These comparisons reaffirm the notion that out-of-plane bistatic radar configurations may have some utility in future studies of bistatic remote sensing of soil moisture. For large surface slopes, where SSA is not applicable, it is expected that large differences will arise between SSA and MoM NRCS predictions. 49

70 Figure 2.8: Full, bistatic scattering hemispheres (one plot for each polarization) for a randomly rough surface target, as predicted by Method of Moments. These plots effectively show the ensemble-averaged MoM NRCS as a function of scattering angle. For these plots, the incidence angle is θ i = 20, the permittivity of the dielectric medium is ɛ = 10 + i0.05, the surface RMS height is λ/20, and the correlation length is λ/2. Point-to-point correlations of the surface height are described by an isotropic, Gaussian correlation function [21]. 50

71 Figure 2.9: Comparison (difference, in db) between MoM and SSA2 results. For these plots, the incidence angle is θ i = 20, the permittivity of the dielectric medium is ɛ = 10 + i0.05, the surface RMS height is λ/20, and the correlation length is λ/2. Point-to-point correlations of the surface height are described by an isotropic, Gaussian correlation function [21]. This is a quantitative comparison between the results shown in Figures 2.4 and

72 2.1.5 Comparison of MoM, SSA2, and SSA3 Predictions with Measured Data for Backscattering Subsequent chapters of this dissertation are focused on radar backscatter rather than bistatic radar scattering. Given the tendency of SSA2 to overestimate VV-polarized NRCS values for backscattering from moderately rough surfaces, the third term in the SSA field series (described in [68]) is introduced here. Given the increased computational load with the introduction of the third-order term (which involves a four-dimensional integration over the third-order kernel function), no full bistatic scattering hemispheres have been generated using the third-order SSA (SSA3). Figure 2.10 shows a plot of MoM, SSA2, and SSA3 backscatter predictions compared with measured backscatter data collected from soils in the 1992 field campaign in Ypsilanti, Michigan [69]. The MoM results shown in this figure were provided by Shaowu Huang and Leung Tsang [70, 71]. These measured data cover a wide variety of soil moisture, texture, and roughness conditions. The field campaign data set includes measurements of soil permittivities, RMS heights, and correlation lengths which were used as inputs for the MoM, SSA2 and SSA3 models. An exponential correlation function was used to describe the simulated soil surface in both the MoM and SSA simulations. Note that MoM, SSA2, and SSA3 predictions for these data are similar, and agreement between the two models and measurement data is generally good. There is, however, a consistent overestimation of VV-polarized NRCS values in the case 52

73 of SSA2, which is partially corrected by SSA3. SSA3 exhibits slightly better agreement with MoM than SSA2. For MoM, the RMS difference between modeled and measured NRCS values is 1.64 db for HH polarization and 1.43 db for VV polarization. For SSA2, the RMS difference between modeled and measured NRCS values is 1.72 db for HH polarization and 2.42 db for VV polarization. For SSA3, the RMS difference between modeled and measured NRCS values is 1.47 db for HH polarization and 1.97 db for VV polarization. Figure 2.10: Comparison of MoM, SSA2, and SSA3 predictions with measured backscatter data taken during the 1992 field campaign in Ypsilanti, Michigan [69]. (a) Comparison of HH NRCS results; (b) comparison of VV NRCS results. 2.2 Conclusions Polarization features of bistatic scattering patterns from randomly rough surfaces have been explored through use of several analytical models, whose 53

74 predictions were compared with each other and with numerical (Method of Moments) results. First-order Small Perturbation Method (SPM) and Physical Optics (PO), applicable in the low- and high-frequency limits respectively, both predict polarization-dependent minima to occur for HH- and VVpolarized scattering. First-order SPM results have shown the depth and location of the VV-polarized minimum to be dependent on surface permittivity. The locations of these minima disagree between SPM and PO, which warranted further investigation. The second-order Small Slope Approximation (SSA2) was used to further explore the locations of minima within the bistatic scattering hemisphere. SSA2 reduces to the second-order SPM and PO in the appropriate limits, but can be applied to a larger range of surface roughness conditions than either of these methods. The wider range of applicability for SSA2 has proven useful in exploring the behavior of bistatic minima (particularly for VV polarization) as a function of roughness. SSA2 results confirm that the location of the minimum in the VV-polarizaed bistatic scattering pattern changes with increasing roughness. Given that higher-order scattered effects could dominate for bistatic configurations in which a minimum occurs, it was considered prudent to verify approximate methods such as SSA2 using a numerical model: Method of Moments (MoM). MoM results confirmed the depths and locations for bistatic minima predicted by SSA2. The sensitivity of the VV-polarized bistatic minimum on surface permit- 54

75 tivity and roughness suggests that the tracking of this minimum could have potential applications for remote sensing. The remote sensing of permittivity (e.g. soil moisture) is of particular interest in land applications. Soil moisture estimation using reflected GNSS signals is an emerging remote sensing technique which makes use of bistatic scattering configurations [53 55]. Furthermore, it might be possible to use the minima in the bistatic surface return in order to isolate and monitor vegetation effects. Despite the variety of potential uses for out-of-plane bistatic radar configurations, the use of the varying null for VV-polarization presents a number of difficulties. For example, it could be difficult to track the behavior of the minimum given the very small returns it produces and the narrow angular region over which it occurs. Furthermore, surfaces with large surface slopes (not considered in this study) may exhibit different scattering behaviors which are not easily captured by the scattering models considered here. Since balance of this dissertation focuses on the use of backscattered NRCS data for remote sensing, a comparison between MoM, SSA, and measured backscatter data was shown. These results showed good agreement between MoM, SSA2, and SSA3, and good agreement between both models and measured data. The third-order SSA term partially corrects for over-predictions in VV-polarized backscatter made by SSA2. 55

76 Chapter 3 A Model-Based Study of Repeat-Pass InSAR Decorrelations 3.1 Motivation When imaging terrain using L-band radar, it is usually assumed that radar pulses do not penetrate deeper than roughly 10 cm into a soil surface. As such, sub-surface effects are often ignored, particularly for airborne and spaceborne SAR platforms. During the 2010 Canadian Soil Moisture Campaign (CanEx10), however, airborne SAR measurements suggested that L-band radar observations can be significantly influenced by the presence of multiple soil layers in an observed scene. Electromagnetic field correlation plots using repeat-pass interferometric SAR (InSAR) have shown significant temporal decorrelations can occur on short time baselines (about one day), even for bare and sparsely vegetated soil surfaces. This study attempts to explain these decorrelation effects with a two-layer soil model where the uppermost layer experiences a change in soil moisture between repeated mea- 56

77 surements. The results shown here have strong implications for soil moisture change detection using L-band InSAR using the amplitudes of scattered field correlations. The correlation between electromagnetic field measurements by JPL s Uninhabited Aerial Vehicle SAR (UAVSAR) during CanEx10 was calculated using repeat-pass visits by the airborne SAR platform [36, 72, 73]. The pppolarized correlation coefficient ρ pp between repeated visits of an observed scene is given by: ρ pp := Ẽ pp,t1 Ẽ pp,t 2 Ẽpp,t 1 2 Ẽpp,t 2 2 (3.1) where Ẽpp,t i is the pp-polarized scattered field measured at time t i and denotes complex conjugation [74]. These correlation coefficients were calculated for relatively short time scales (about one day between repeat-pass observations). Decorrelations occurred even for bare and sparsely-vegetated surfaces at these time scales, which is not easily explained by vegetation effects (e.g. plant growth). Figure 3.1 provides a map of the CanEx10 study area, for which UAVSAR imagery was produced and ground sampling was conducted during the course of the field campaign. The main study region is highlighted by the purple rectangle and includes mostly agricultural fields surrounding the city of Kenaston, Saskatchewan. Figure 3.2 shows some example field correlation plots, measured by UAVSAR, over part of the CanEx10 study region. This figure includes two example images for both HH and VV polarizations. Most of the region being imaged is dominated by agricultural fields. For each radar 57

78 pixel within these images, the correlation coefficient between two scattered field measurements is calculated in accordance with Equation 3.1, after InSAR calibration has been performed [72]. Note that in each of the correlation images in Figure 3.2, the latency between repeated measurements is only about one day. For bare surfaces, it was expected that these field correlations would be close to unity, but this was often not the case in these results. Figure 3.1: A map of the main CanEx10 study region around Kenaston, Saskatchewan, Canada [72]. UAVSAR measurements and ground sampling (including soil moisture sampling) were conducted in this region during the field campaign. 58

79 Figure 3.2: Example HH- and VV-polarized repeat-pass field correlation images, calculated for UAVSAR radar measurements acquired during the CanEx soil moisture field campaign [72]. For these images, the latency between repeatpass acquisitions is about one day. These images include both vegetated and barren soil surfaces. This chapter attempts to explain and quantify the field decorrelations observed during the CanEx10 campaign. The simulation study described here uses rough surface scattering models to describe a bare soil surface in which the soil moisture conditions change between repeated visits of the observed scene. The first modeling approach discussed in this chapter implements the second-order Small Slope Approximation (SSA) to describe scattering from a 59

80 single soil surface, where the soil moisture is changed between repeated measurements. The second approach is a two-layer first-order Small Perturbation Method (SPM) in which the moisture of the upper soil layer is changed between repeat measurements, but the bottom layer remains constant. This study shows that significant field decorrelations can be expected when subsurface returns provide a significant contribution to the backscattered signal. 3.2 Single-Layer SSA2 Model In this formulation, a single rough surface is considered in which the surface soil moisture is change between repeated measurements. The rough surface interface is treated as a stationary, zero-mean Gaussian random process described by an isotropic, exponential spatial correlation function. A variety of approximate methods can be applied to compute backscatter from such a surface (Physical Optics, first-order Small Perturbation Method, first-order Small Slope Approximation, etc.). Most of these first-order models predict that the scattered field is proportional to a function of soil moisture (dielectric constant; ɛ) multiplied by a function of surface roughness. In other words, these models predict that the scattered field can be described by the following proportionality: Ẽ f(ɛ)g(σ, L) (3.2) where σ and L are the RMS height and correlation length describing the rough surface interface. Given Equation 3.2, the correlation between two scattered 60

81 fields measured at times 1 and 2 is then ρ := Ẽ t1 Ẽt 2 = f(ɛ t1)f (ɛ t2) Ẽt 1 2 Ẽt 2 2 f(ɛ t1 ) f(ɛ t2 ) (3.3) which always has unity amplitude. As demonstrated by Equation 3.3, in the single-layer case, first order models can predict a changing phase in the correlation, but never any amplitude changes. For a single layer, to predict a non-unity correlation one must use a higher-order scattering model. The Small Slope Approximation (SSA) model provides a series of field terms which reduces to SPM and PO in the appropriate limits [29]. See Subsections and for more details on the form of SSA and its applicable regime. Two of the terms in the SSA field series are considered in this section. Higher order theories such as second-order SPM or second-order SSA can take the following form: Ẽ f 1 (ɛ)g 1 (σ, L) + f 2 (ɛ)g 2 (ɛ, σ, L). (3.4) Therefore, the cross-correlation between two field measurements will not necessarily have unity amplitude. Second-order SSA predictions were examined for representative single-layer cases. Figure 3.3 illustrates the calculation of repeat-pass field correlations for the single-layer case. Scattered fields are calculated for time t 1 and t 2, where the soil moisture (permittivity) is allowed to change between t 1 and t 2. The 61

82 correlation coefficient ρ is calculated between Ẽs(t 1 ) and Ẽs(t 2 ). Figure 3.3: An illustration of the geometry considered for the single-layer case for repeat-pass field correlation studies. The correlation is measured between Ẽ s when measured at times t 1 and t 2. The soil moisture (and thus the permittivity) of the soil surface is fixed at time t 1, but allowed to change for time t 2. This allows for a study of the effects of a dry-down condition or rain event on the correlation. Figure 3.4 shows a plot of the correlation versus soil moisture at time t2 for a single layer, as predicted by SSA2. The case shown in this plot reflects the soil moisture and surface roughness conditions observed during the CanEx10 campaign. It is clear that under these conditions, the single-layer SSA2 model is unable to produce significant decorrelations between repeat-pass measurements. Small phase changes are detected, but these are questionable given the amplitude issues. It is prudent to move to a model using multiple layers to explain the decorrelations observed in UAVSAR/CanEx data. 62

83 Figure 3.4: Repeat-pass field correlations (plotted as 1- ρ in amplitude) as predicted by SSA2 for a single-layer case, plotted as a function of soil moisture at time t 2. The initial soil moisture condition of the surface at time t 1 is 0.2 cm 3 /cm 3. The soil moisture of the soil surface is converted to relative permittivity through the Peplinski/Dobson/Ulaby dielectric mixing model [32] with sand and clay fractions equal to 40% and 20% respectively. The RMS height of the surface is σ = λ and the correlation length is L = 3.948λ. The electromagnetic wavelength is λ = cm. The rough soil surface is described by a band-limited exponential autocorrelation function with a cut-off correlation length l s = 0.05λ. 63

84 3.3 Two-Layer SPM1 Model A first-order Small Perturbation Method model has been previously developed in [75] to describe scattering from an arbitrary number of layers, each separated by a randomly rough interface. As highlighted in previous sections, a first-order analytical model requires at least two stratified surfaces to produce field correlations of non-unity amplitude. Since SPM is used here, this model is restricted to the case of small surface roughness compared to wavelength. See Subsections and for more details on the form and limitations of SPM1. Figure 3.5 illustrates the geometry of the two-layer rough surface scattering problem. As with the single-layer analysis, in this formulation each rough surface interface is treated as a stationary Gaussian random process described by an isotropic, band-limited exponential spatial correlation function [66]. 64

85 Figure 3.5: An illustration of scattering from stratified, rough-surface media [75]. d represents the separation between layers, measured between their mean heights. ɛ i and µ i are respectively the electromagnetic permittivity and permeability of layer i. Each layer is separated by a randomly rough interface. According to the two-layer SPM1 model, the backscattered pp-polarized scattering coefficient has two contributions: S pp = g (1,0) pp (ɛ top, ɛ bot )W 1 (k x, k y ; σ top, L top ) + g (0,1) pp (ɛ top, ɛ bot )W 2 (k x, k y ; σ bot, L bot ) (3.5) where ɛ top and ɛ bot are the relative permittivities of the top and bottom layers respectively. Likewise, σ top and σ bot represent the RMS heights of the upper and lower interfaces and L top and L bot represent their correlation lengths. W i 65

86 is the 2-dimensional Fourier transform of the surface profile for the i th layer. g (i,j) pp is the two-layer SPM1 kernel function, where i is the order to which perturbations of the upper layer are considered and j is likewise the order to which perturbations of the bottom layer are considered. Being a first-order model, each of the terms in the two-layer SPM1 consider roughness effects for only one interface, while the remaining interface is approximated as perfectly flat (0 th order approximation). The first contribution to S pp in Equation 3.5 involves only the surface roughness of the upper layer, and the second term only involves the surface roughness of the lower layer. The correlation between fields having the form of Equation 3.5 can have non-unity amplitude. In this study, the surface roughness of the upper and lower layers are statistically independent, but for simplicity they are assumed to be identically distributed (their covariance functions, RMS heights, and correlation lengths are the same). Figure 3.6 illustrates the calculation of repeat-pass field correlations for the two-layer case. Scattered fields are calculated for time t 1 and t 2, where the soil moisture (permittivity) of the upper soil layer is allowed to change between t 1 and t 2. The correlation coefficient ρ is calculated between Ẽs(t 1 ) and Ẽs(t 2 ). Note that the soil moisture of the lower layer is not expected to change over short time scales and thus is fixed at 40% for the purposes of this study. 66

87 Figure 3.6: An illustration of the geometry considered for the two-layer case for repeat-pass field correlation studies. The correlation is measured between Ẽ s when measured at times t 1 and t 2. The soil moisture (and thus the permittivity) of the upper soil layer is fixed at time t 1, but allowed to change for time t 2. This allows for a study of the effects of a dry-down condition or rain event on the correlation. 67

88 Figures 3.7 through 3.9 show that significant field decorrelations can be expected for the two-layer case when the soil moisture of the upper layer changes between repeated radar measurements. The oscillatory nature of the amplitude and phase of field correlations with respect to changing soil moisture conditions is due to constructive and destructive interference between scattering contributions from the upper and lower soil layers. This suggests that InSAR measurements could be used for sensing changes in soil moisture in the case of a layered soil medium. The cases considered here are simplified by the fact the bottom layer s soil moisture content remains constant over time. Given the penetration of radar pulses into the lower soil layer, it seems that InSAR field correlation measurements could be used for sensing of deeper ( root-zone [19]) soil moisture conditions. 68

89 Figure 3.7: Repeat-pass field correlations (amplitude and phase) as predicted by SPM1 for a two-layer case, plotted as a function of soil moisture of the upper layer at time t 2. The initial soil moisture condition of upper layer at time t 1 is 0.2 cm 3 /cm 3. For this case, the RMS height of both surfaces is σ = λ and the correlation length for both surfaces is L = 3.948λ. The electromagnetic wavelength is λ = cm. The soil moisture condition of the bottom layer is 0.4 cm 3 /cm 3. The mean separation between layers is d = 10 cm. The rough soil interfaces are both described by a band-limited exponential autocorrelation function with a cut-off correlation length l s = 0.05λ. 69

90 Figure 3.8: Repeat-pass field correlations (amplitude and phase) as predicted by SPM1 for a two-layer case, plotted as a function of soil moisture of the upper layer at time t 2. The initial soil moisture condition of upper layer at time t 1 is 0.15 cm 3 /cm 3. For this case, the RMS height of both surfaces is σ = λ and the correlation length for both surfaces is L = 3.948λ. The electromagnetic wavelength is λ = cm. The soil moisture condition of the bottom layer is 0.4 cm 3 /cm 3. The mean separation between layers is d = 10 cm. The rough soil interfaces are both described by a band-limited exponential autocorrelation function with a cut-off correlation length l s = 0.05λ. 70

91 Figure 3.9: Repeat-pass field correlations (amplitude and phase) as predicted by SPM1 for a two-layer case, plotted as a function of soil moisture of the upper layer at time t 2. The initial soil moisture condition of upper layer at time t 1 is 0.15 cm 3 /cm 3. For this case, the RMS height of both surfaces is σ = λ and the correlation length for both surfaces is L = 3.948λ. The electromagnetic wavelength is λ = cm. The soil moisture condition of the bottom layer is 0.4 cm 3 /cm 3. The mean separation between layers is d = 15 cm. The rough soil interfaces are both described by a band-limited exponential autocorrelation function with a cut-off correlation length l s = 0.05λ. 71

92 3.4 Conclusions In this chapter, single- and two-layer models were used in an attempt to predict field decorrelations between repeat-pass measurements of a barren soil surface. While single-layer results failed to produce non-unity field correlations (even when using a second-order scattering model), two layer results often displayed significant decorrelations when the soil moisture of the upper layer changed between repeated measurements. The results shown here suggest that one can expect significant field decorrelations to occur for a bare surface during a dry-down condition or rain event, given that a lower soil layer is present which provides a significant contribution to the backscattered signal (i.e. the incident wave is able to penetrate through the upper soil layer). Such an event can produce these decorrelations given a revisit latency of just one day. This information could be used for remote sensing of soil moisture by means of a change-detection algorithm where the correlation coefficient is recalculated each time the observed scene is re-measured. 72

93 Chapter 4 A Simulation Study of Compact Polarimetry for Soil Moisture Retrieval 4.1 Motivation The analyses described in this chapter were originally performed to assess a compact polarimetric mode which was being considered for the SMAP radar. Ultimately, this radar mode was not adopted for SMAP because of the additional hardware requirements imposed by introducing a new radar mode. The SMAP radar employs an incoherent multi-polarization basis (HH, HV, and VV; not a fully polarimetric mode). Nevertheless, these analyses were deemed useful to the remote sensing community for consideration in the future design of radar systems for soil moisture remote sensing. In this chapter, a compact polarimetric radar mode is compared with a comparable fully-polarimetric system for the purpose of soil moisture remote sensing. Since a compact polarimetric mode only requires transmission of a single polarization, this allows for greater flexibility in system design. 73

94 4.2 Introduction Compact polarimetry is an alternative polarimetric radar mode which may allow for increased flexibility in radar system design when compared to a traditional fully-polarimetric radar mode using a linear polarization basis. Several past studies have highlighted the potential advantages of implementing a compact polarimetric radar imaging mode in active soil moisture remote sensing systems [76 78]. Traditional fully polarimetric (FP) modes typically utilize a linear polarization basis where the transmitter alternates between horizontally- and vertically-polarized pulses and the receiver simultaneously receives horizontal and vertical polarizations. For monostatic systems, four complex measured values (HH, VV, VH, and HV polarized fields) result; the equality of HV and VH returns for backscatter from reciprocal scenes then implies that only one of these quantities needs to be considered. In contrast, a compact polarimetric (CP) mode uses only one transmit polarization, typically implemented as either 45 degree linear or as circular polarization, and receives in two orthogonal polarizations (typically linear). At lower operating frequencies, the use of a circular transmit polarization is advantageous due to its lower susceptibility to Faraday rotation, though linear receive polarizations are still affected by this phenomenon. Two measured complex fields result, here called the RH and RV fields (assuming right circular polarization is transmitted and a linear receive polarization basis is used) [7 9]. So-called reconstruction techniques for compact polarimetry have pre- 74

95 viously been proposed in literature. In these reconstruction techniques, fully polarimetric information is approximately derived from compact polarimetric measurements. These strategies require some assumptions about the observed scene such as reflection symmetry and relationships between co- and cross-polarized normalized radar cross sections and the co-polarized correlation coefficient. A key question is the applicability of these assumptions to geophysical media. If such assumptions are applicable, so that FP information can be reconstructed to reasonable accuracy from CP measurements, then it should be expected that CP sensing performance should be similar to that of FP. Reference [78] presents a study of soil moisture retrieval performance with CP for bare surfaces. CP measurements from a bare surface in the form of RH and RV normalized radar cross-section (NRCS) values were simulated from measured FP data, and then used as HH and VV NRCS values in the Dubois empirical model [13] to invert soil moisture. The results showed reasonable performance in retrieving soil moisture for bare surfaces. Note that no field phase information was utilized in this process and that no attempt to reconstruct FP measurements (or even HH and VV NRCS values) was performed. The retrieval algorithm utilized in [13,78] is a snapshot approach: soil moisture information at a given time is retrieved directly from radar measurements at that time. Recent works (e.g. [38]) have shown the potential of time series methods for improving soil moisture retrieval performance. Examining CP performance in such algorithms is of interest, as well as extending the analy- 75

96 sis to include vegetation covered surfaces and receive channel cross-correlation measurements. This paper therefore presents a simulation study of CP soil moisture retrievals in these cases, based on an underlying forward model [20] for scattering from low vegetation covered surfaces. 4.3 Review of Compact Polarimetry CP received signals can be expressed in terms of the scattering matrix for an equivalent fully-polarimetric measurement as follows: E h E v = S hh S hv S hv S vv = 1 S hh + eiδs hv 2. (4.1) S hv + e iδ S vv e iδ In Equation 4.1, the general form (ĥ + eiδˆv)/ 2 is used for the transmit signal (where ĥ and ˆv are horizontally and vertically polarized unit vectors for the incident field), so that either 45 degree linear or circularly polarized incident fields are included as options. The effects of system gains and losses (e.g. antenna gain, propagation loss, etc.) have been left out here for simplicity. The ensemble-averaged backscattered cross sections observed on H and V receive channels (and their cross-products) are then: 76

97 σ H = (S hh + e iδ S hv )(S hh + e iδ S hv ) = σ hh + σ hv + 2Re{S hh S hve iδ }, σ V = (S hv + e iδ S vv )(S hv + e iδ S vv ) = σ hv + σ vv + 2Re{S hv S vve iδ }, and (4.2) σ cross = (S hh + e iδ S hv )(S hv + e iδ S vv ) = S hh S hv + S hv S vv + e iδ S hh S vv + e iδ σ hv. For surfaces with reflection symmetry, terms involving co-/cross-polarized crossproducts in Equation 4.2 will be small and can be ignored. This study assumes that soil moisture estimates at spatial resolutions of one or more kilometers are of interest. It is typically expected that geophysical media at such scales will satisfy the reflection symmetry assumption, except for unusual cases with oriented structures (e.g. the row structures of very large, tilled fields) remaining at such resolutions [79, 80]. Assuming the observed scene exhibits reflection symmetry, CP measurements reduce to: σ H = σ hh + σ hv σ V = σ hv + σ vv (4.3) σ cross = e iδ (σ hv + e i2δ S hh S vv). 77

98 Assuming an RHCP (e iδ = i) transmitted wave, σ RH = σ hh + σ hv σ RV = σ vv + σ hv (4.4) σ cross = i(σ hv S hh S vv). The RH and RV cross sections are thus exactly the sums of the corresponding co- and cross-polarized returns from a reflection symmetric target. Since copolarized returns are on the order 7 db or more higher than cross-polarized for most cases of retrieval interest (i.e. agricultural scenes where vegetation has a relatively small influence on backscattered signals compared to surface returns), the RH and RV returns are expected to be close to those of HH and VV for a fully polarimetric mode, as in [78]. The RH-RV correlation term contains a combination of both cross-polarization information and the HH-VV correlation. 4.4 Reconstruction Techniques One apparent disadvantage of compact polarimetry as compared to fully polarimetric operation is the fact that the HV return is not directly measured. Since HV is often used as a vegetation classifier when using a fully-polarimetric radar, this can be a significant concern. FP reconstruction algorithms [76, 77] offer the potential of recovering HV information from CP measurements under assumptions about the relationship between observed cross sections. Souyris 78

99 et al. [76] describe a reconstruction algorithm under the assumption that S hv 2 S hh 2 + S vv 2 = (1 ρ ) 4 (4.5) where ρ := S hh S vv Shh 2 S vv 2. is the co-pol correlation coefficient. Studies of the Souyris reconstruction approach using the data cubes described in Section 1.7 (40 degrees incidence angle, L-band) show that accurate retrieval of HV returns is difficult. Figure 4.1 plots reconstructed HH and VH NRCS values as compared to the true values for 252 truth cases from the grasstype data cube described in Section 1.7. These 252 cases are combinations of six soil moisture values, six roughness values, and seven vegetation water content values. The reconstruction was performed without the inclusion of any noise effects (either thermal or speckle noise) in the simulation. While copolarized returns are reconstructed fairly accurately, the cross-polarized term cannot be accurately reconstructed for many cases, particularly cases where the surface roughness is small compared to the wavelength. Similar simulations including speckle and thermal noise effects show further degradations in performance. 79

100 Figure 4.1: Reconstructed and truth values for reconstructed HH and VH returns for 252 data cube cases. The results in Figure 4.1 indicate that the model used in generating the data cubes does not satisfy the assumptions Equation 4.5 invoked in the reconstruction process. The results also suggest that HV reconstruction from CP in general should be expected to be challenging, and that alternative methods for terrain classification should be sought. Reference [81] has demonstrated, however, that an HV return reconstructed from CP measurements can serve as 80

101 a suitable biomass estimate, at least in certain cases. It should be noted that from CP imagery alone, it cannot be discerned whether reflection symmetry holds, and thus whether the scattering models described here are appropriate. An alternative terrain classifier for CP has been proposed [78] through a parameter µ, defined as: µ = 2Im{σ cross} (σ RH + σ RV ) = 2Re{S hhs vv} 2σ hv σ hh + σ vv + 2σ hv (4.6) It is shown in [78] that this classifier ranges from -1 to 1, with larger positive values indicating a dominance of surface scattering, negative values indicating a dominance of double-bounce scattering, and values near zero indicating a dominance of volume scattering. Although these properties are distinct from HV backscatter returns, which primarily reflect volume scattering alone, the ability to separate the mechanisms of backscattered returns may provide additional advantages in practice. In addition, it is shown that µ is unaffected by Faraday rotation effects. An illustration of the properties of µ for the grass data cube is provided in Figure 4.2. The upper left plot illustrates µ obtained from the data cube at a VWC value of 2.37 kg/m 2. The black contour in the plot marks the boundary beneath which µ is less than The other plots in the figure illustrate the fraction of the total HH return due to surface (upper right), volume (lower left), and double bounce (lower right) contributions. These contributions were calculated by separating the appropriate terms in the forward model. The 81

102 general capabilities of µ in distinguishing these mechanisms (surface returns have µ large and positive, volume returns have µ near zero, double bounce returns have µ negative) are apparent. Similar capabilities are observed at other VWC levels of the data cube. It is noted however that RH and RV contributions due to surface, volume, and double bounce effects are distinct, so that µ alone is insufficient to distinguish any differences in the mechanisms of RH and RV scattering due to volume, surface, and double-bounce effects. 82

103 Figure 4.2: (Upper left) µ from the grass data cube at VWC=2.37 kg/m 2. The black contour marks the region beneath which µ < (Upper right) Fraction of corresponding HH return due to surface contributions. (Lower left) Fraction of corresponding HH returns due to volume scattering. (Lower right) Fraction of corresponding HH returns due to double bounce scattering. 4.5 Retrieval Studies Beyond the issue of classification, soil moisture retrieval performance using a time series approach and including vegetated surfaces and field phase information is of interest. The time-series data cube approach described in Section 1.6 is utilized for this purpose. In this method, the RMS height of the sur- 83

104 face is assumed to remain constant while soil moisture and vegetation water content may vary through a time series of N backscatter measurements. It is further assumed that ancillary information on the vegetation water content (for example through relationships with optical sensor measurements [82]) is available to a specified level of accuracy. True VWC values are corrupted by a fractional error that is Gaussian distributed and whose standard deviation is taken as equal to 10%. Thus, for CP measurements, there are 2N complex valued observations (E Rh and E Rv at each time series point) and N + 1 unknowns consisting of a single RMS height parameter and N soil moisture (dielectric constant) values. The time-series data are in effect used to reduce the influence of measurement noise during the estimation of surface roughness Monte Carlo Simulation of Fields Simulated single-look CP observations for a given truth scene were generated using T Rh = E Rh + Noise h = E hh + e iδ E hv + Noise h (4.7) and T Rv = E Rv + Noise v = e iδ E vv + E hv + Noise v. (4.8) Similarly, single-look FP measurements for the truth scene were generated using T hh = E hh + Noise h, (4.9) 84

105 T vv = E vv + Noise v (4.10) and T hv = E hv + Noise h /2 + Noise v /2. (4.11) Field (E αβ ) and noise (Noise α ) quantities above are realizations of complex Gaussian random variables, as follows: E hh = σhh 2 (X 1 + iy 1 ) (4.12) E vv = σvv 2 [ρ (X 1 + iy 1 ) + (X 2 + iy 2 ) 1 ρ 2 ] (4.13) E hv = σhv 2 (X 3 + iy 3 ) (4.14) Noise h = σn 2 (X 4 + iy 4 ) (4.15) Noise v = σn 2 (X 5 + iy 5 ) (4.16) where X i, Y i are all independent, standard Gaussian random variables and ρ is the complex correlation coefficient. The above process ensures that speckle is correlated appropriately in the simulated RH and RV fields. It should be emphasized that Equation 4.7 through Equation 4.14 will properly simulate RH and RV fields only when the observed scene is reflection symmetric. For a given truth scene, the specified cross section and HH-VV correlation values are available from the forward model, and the noise equivalent normalized radar cross section σ n is a parameter in the simulation. A final parameter 85

106 in the Monte Carlo simulation is the number of independent looks, L, available to the radar system. For a given simulated observation, L independent realizations of the measured quantities, labeled T (l) Rh produced. for example, can then be CP observations appropriately averaged over these independent looks are then computed as U CP = U CP 1 U CP 2 U CP 3 U CP 4 1 L T (l) Rh 2 L l=1 1 L T (l) Rv 2 = L l=1 1 L. (4.17) Re{T (l) Rh L T (l) Rv } l=1 1 L Im{T (l) Rh L T (l) Rv } l=1 The Monte Carlo simulation process generates many realizations of U CP for a given state, so that RMS retrieval errors can be investigated. In the simulations performed, 300 realizations of U CP were created, each averaged over 200 independent looks. A noise equivalent NRCS of -32 db was also assumed. This noise equivalent NRCS value also takes into account the 3 db reduction in signal power using a CP configuration. A similar Monte Carlo simulation can be performed for fully polarimetric observations; in this case speckle and thermal noise corrupted multi-look averaged measurements of E hh, E hv, and E vv are produced, and the vector corresponding to U has 5 entries (HH, VV, and HV power measurements and real and imaginary parts of the HH-VV correlation.) FP observations appro- 86

107 priately averaged over independent looks are computed as U F P = U F P 1 U F P 2 U F P 3 U F P 4 U F P 5 1 L T (l) hh 2 L l=1 1 L T (l) vv 2 L l=1 = 1 L T (l) hv 2 L. (4.18) l=1 1 L Re{T (l) hh L T vv (l) } l=1 1 L Im{T (l) hh L T vv (l) } l=1 A noise equivalent NRCS of -35 db was used for the FP retrieval simulation (as opposed to -32 db in the CP mode, where the noise floor is effectively higher due to the transmit power being split between H and V channels) Maximum-likelihood Retrieval The time-series data cube retrieval for either CP or FP data is based on a maximum-likelihood method: retrieved parameters are determined as those that maximize the likelihood that the observations obtained were drawn from the retrieved truth state [14, 15]. An algorithm for implementing this method can be determined by considering the statistics of the thermal and speckle noise corrupted measurements. Begin by considering a single-look CP measurement: T CP = T Rh T Rv (4.19) 87

108 which is a complex Gaussian vector with a covariance matrix T Rh 2 T Rh TRv Σ t,cp = (4.20) TRh T Rv T Rv 2 where, if the observed scene is reflection symmetric, T Rh 2 =σ hh + σ hv + σ n T Rv 2 =σ vv + σ hv + σ n T Rh T Rv =e iδ σ hv + e iδ ρ σ hh σ vv. The thermal noise contribution, σ n, is assumed to be the same for both receive channels. The probability density function (PDF) of T CP is ( ) 1 f T CP T CP 0 = π 2 Σt,cp e T H CP 0 Σ 1 t,cp T CP 0 (4.21) where T H CP 0 denotes the conjugate transpose of T CP 0. Note that Σ 1 T Rv 2 T Rh TRv t,cp = 1 TRh T Rv T Rh 2 T Rh 2 T Rv 2 T Rh T. Rv 2 88

109 To simplify notation in what follows, define Σ 1 t,cp = α CP β CP β CP δ CP (4.22) such that Σ 1 t,cpt CP = α CP T Rh + β CP T Rv (4.23) βcp T Rh + δ CP T Rv and T H CP Σ 1 t,cpt CP = α CP T Rh 2 + β CP T RhT Rv + β CP T Rh T Rv + δ CP T Rv 2 = α CP T Rh 2 + δ CP T Rv 2 + 2Re{β CP }Re{T RhT Rv } + 2Im{β CP }Im{T RhT Rv }. (4.24) For L independent observations (looks) of T CP, the joint PDF of T 1 CP,T 2 CP,...,T L CP is f ( T 1 CP, T 2 CP,..., T L CP = ) = L l=1 1 exp ( (1)H π 2L Σt,cp L T CP Σ 1 t,cpt (1) 1 π 2 Σt,cp exp ( T (l)h CP Σ 1 t,cpt (l) ) CP CP T (2)H CP Σ 1 t,cpt (2) CP... T (L)H CP Σ 1 t,cpt (L) CP ) = 1 π 2L Σt,cp L exp{ L(α CP U CP 1 + δ CP U CP 2 + 2Re{β CP }U CP 3 + 2Im{β CP }U CP 4 )} (4.25) 89

110 where U CP i refers to the corresponding elements of the U CP vector. Equation 4.25 is exactly the Wishart similarity measure in component form [83]; data represented by a similar covariance matrix has also been considered in [84]. In the retrieval process, a measurement of the U vector (or time series of such measurements) is available, but the covariance matrix Σ t,cp (or matrices in the time series case) and the associated α CP, β CP, and δ CP values are not. Given a set of possible truth cases (or classes) in the data cube (each considered equally likely to arise), the available Σ t,cp s are all assessed for the likelihood that the observed data was drawn from each class, and the maximum likelihood class selected as the retrieved class. Because maximizing the probability density function also maximizes the logarithm of the probability density function, the retrieval can be performed by finding the class that minimizes ln Σ t,cp + αcp U CP 1 + δ CP U CP 2 + 2Re{β CP }U CP 3 + 2Im{β CP }U CP 4 (4.26) for the observed U CP vector. Note that this distance function includes field phase effects by incorporating the cross-correlation terms. For time series measurements, a distinct class is retrieved for each time series point through use of the cost function summed over measurements (because the speckle in each observation remains independent). The retrieval process is described further in [38], and results in the single RMS height and N soil moisture values that minimize the cost function. A similar derivation was applied to determine 90

111 the maximum likelihood cost function for FP time series measurements so that CP and FP performance can be compared in what follows. Begin by considering a single-look FP measurement: T F P = T hh T vv T hv (4.27) which is a complex Gaussian vector with a covariance matrix T hh 2 T hh T vv 0 Σ t,fp = Thh T vv T vv 2 0. (4.28) 0 0 T hv 2 The probability density function (PDF) of T F P is ( ) 1 f T F P T F P 0 = π 2 Σt,fp e T H F P 0 Σ 1 t,fp T F P 0 (4.29) where T H F P 0 denotes the conjugate transpose of T F P 0. Note that Σ 1 t,fp = 1 Det(Σ t,fp ) T hv 2 T vv 2 T hv 2 T hh T vv 0 T hv 2 Thh T vv T hv 2 T hh T hh 2 T vv 2 T hh Tvv 2 91

112 where Det(Σ t,fp ) = 1 T hh 2 T vv 2 T hv 2 T hv 2 T hh T vv 2. (4.30) To simplify notation in what follows, define Σ 1 t,fp = α F P β F P 0 βf P β F P δ F P (4.31) such that and Σ 1 t,fpt F P = α F P T hh + β F P T vv βf P T hh + γ F P T vv δ F P T hv (4.32) T H F P Σ 1 t,fpt F P = α F P T hh 2 + γ F P T vv 2 + δ F P T hv 2 + β F P T hht vv + β F P T hh T vv = α F P T hh 2 + γ F P T vv 2 + δ F P T hv 2 + 2Re{β F P }Re{T hh T vv} +2Im{β F P }Im{T hh T vv}. (4.33) For L independent observations (looks) of T F P, the joint PDF of T 1 F P,T 2 F P,...,T L F P is: 92

113 f ( T 1 F P, T 2 F P,..., T L ) L 1 F P = π 2 Σt,fp exp ( T (l)h F P Σ 1 t,fpt (l) ) F P = l=1 1 exp ( (1)H π 2L Σt,fp L T F P Σ 1 t,fpt (1) F P T (2)H F P Σ 1 t,fpt (2) F P... T (L)H F P Σ 1 t,fpt (L) F P ) = 1 π 2L Σt,fp L exp{ L(α F P U F P 1 + γ F P U F P 2 + δ F P U F P 3 + 2Re{β F P }U F P 4 + 2Im{β F P }U F P 5 )} (4.34) therefore the cost function to be minimized in the FP case is +αf ln Σ t,fp P U F P 1 +γ F P U F P 2 +δ F P U F P 3 +2Re{β F P }U F P 4 +2Im{β F P }U F P 5. (4.35) Results The Monte Carlo simulation of the retrieval was performed for each case in a grid of six surface RMS heights by seven VWC truth values (i.e. 42 cases.) For each of these points, a time series of soil moisture was utilized that consisted of six soil permittivity values (i.e. six time series observations; one observation for each soil permittivity value). The soil permittivity values of the time series are identical for each of the 42 truth cases, and correspond to a uniformly increasing soil moisture condition over the time series from very dry to very wet (though these soil moisture conditions are idealized, the temporal 93

114 order of the soil permittivity does not affect the retrieval performance). Given the truth time series, 300 realizations of speckled observations (averaged over 200 independent looks) and thermal noise (-35 db noise equivalent NRCS for FP and -32 db for CP) were produced, and the corresponding retrieved soil moistures determined. The simulation incorporates a 10% error in the ancillary VWC estimate; these corrupted estimates are generated independently at each step in the time series. Figure 4.3 illustrates retrieval errors in RMS height (upper) and soil moisture (lower) for truth RMS height 0.99 cm; soil moisture retrieval errors are determined by inverting the dielectric mixing model [32] under assumed soil temperature and texture conditions identical to those used in generating the truth data. The left side plots illustrate the compact polarimetric retrieval errors, while the right side contains the ratio of errors obtained from compact polarimetric to those of the fully-polarimetric approach. The results show a degradation in soil moisture retrieval using the CP approach when compared to FP. This degradation generally increases with higher VWC and RMS height values (not shown). For a VWC of 3 kg/m 2, the degradation in RMSE of soil moisture predictions is less than 0.02 cm 3 /cm 3. Figure 4.4 further illustrates retrieval errors using a snapshot approach (only one element in the time-series), and shows a significantly increased degradation in CP performance as compared to FP. 94

115 Figure 4.3: (Upper left) RMS error in retrieved surface RMS height for compact polarimetric (CP) approach versus VWC (vegetation water content) and surface RMS height. (Upper right) Ratio of CP RMS height retrieval error to that obtained by fully-polariemtric (FP) method. (Lower left) RMS error in retrieved soil moisture for CP approach at surface RMS height 1 cm. (Lower right) Ratio of CP soil moisture retrieval error to that obtained by the FP method. 95

116 Figure 4.4: Same as Figure 4.3, but for a snapshot (N = 1) approach. Note roughness is retrieved at each time step in this case. Figure 4.5 displays RMS errors averaged over the six soil moisture and six RMS height truth values in both the CP and FP cases for the six-point time series retrieval. The results again show only a modest degradation of retrieval performance caused by the use of CP measurements. It should be noted that the RMS errors shown in Figure 4.5 are intended to represent differences among retrieval strategies rather than absolute error levels to be encountered in practice, primarily due to the limitations of the grass cube utilized. These 96

117 results also are not representative of globally averaged statistics, as they are uniform averages over the grid of truth conditions in soil moisture and surface RMS height. Figure 4.5: Comparison of CP and fully polarimetric (FP) RMS retrieval errors as a function of VWC (averaged over surface RMS height and soil moisture). 4.6 Conclusions The results in this chapter show first that existing methods for reconstructing FP information from CP data may be limited with respect to vegetation covered soil surfaces. The behavior of the alternate µ classification parameter however appears reasonable for the forward model utilized. Second, simula- 97

118 tions of time series soil moisture retrieval performance over a grass-like surface showed only modest degradations as compared to FP observations. While these results are subject to the assumptions utilized (primarily that an accurate forward model is available for the observed scene), the results nevertheless provide insight into the relationships of FP and CP measurements for future soil moisture sensing systems. Applying the method to measured data is of interest, but requires that a robust polarimetric forward model be available for a variety of terrain classes. Data cube generation and tuning is ongoing, and future versions of the data cubes could potentially be used to implement the retrieval methods described here on experimental datasets. This study was originally motivated by the consideration of a compact polarimetric radar mode for the SMAP radar. Although a CP mode was not adopted for SMAP, these studies will prove useful in the design of future radar remote sensing mission which consider the use of a CP mode. The next chapter presents another soil moisture retrieval algorithm using L-band radar, which departs from the data-cube-based estimation method discussed in this chapter and in Chapter 1. 98

119 Chapter 5 A New Time-Series Soil Moisture Inversion Algorithm for Vegetated Surfaces Using L-band Radar Backscatter 5.1 Motivation Though there are many documented studies of soil moisture estimation for bare surfaces using radar, the subject of radar-derived soil moisture estimates for vegetated surfaces remains a subject of continued research. The presence of vegetation significantly increases the complexity of electromagnetic scattering in the observed scene. Scattering from a vegetation canopy often obstructs contributions from the underlying soil surface, thus complicating the problem of soil moisture estimation in the presence of significant vegetation biomass. Existing approaches to estimating soil moisture under vegetation using radar observations often rely on the use of a forward model to describe the backscattered signal and often require that the vegetation biomass of the ob- 99

120 served scene be provided by an ancillary data source. Ancillary vegetation information may not be reliable or available during the radar overpass of the observed scene (e.g. due to cloud coverage if vegetation information is derived from an optical sensor). The approach described in this chapter is a modified version of an existing change-detection method for soil moisture estimation which does not require the use of ancillary vegetation information, nor does it make use of a complicated forward scattering model. Modifications to the existing algorithm include a new technique for bounding soil moisture estimates based on climate models, extension to multiple polarizations, and a correction factor which subtracts additive contributions to the backscattered signal due to vegetation. Results from simulation and measured data sets show that the algorithm can consistently achieve RMS errors of about 7-8% for most land cover types, even in the presence of significant vegetation. 5.2 History and Basic Formulation The so-called alpha approximation method (or simply the alpha method ) was first described in [85]. This single-channel, time-series method has been applied to field campaign data to derive soil moisture estimates from C- and L- band radar backscatter data from agricultural scenes. The algorithm relies on several simplifying assumptions about the electromagnetic scattering physics of the observed scene. The alpha approximation method is effectively a change detection proce- 100

121 dure which maps changes in radar backscatter signatures to changes in surface soil moisture (surface permittivity). This simple method offers several advantages over other active soil moisture retrieval algorithms. One such advantage is that a complicated forward electromagnetic scattering model is not required, thus reducing the computational complexity of the model. The major advantage of the alpha method is that it requires no ancillary information about the vegetation biomass present in the observed scene, which is a requirement for many model-based soil moisture retrieval techniques. As mentioned in Chapter 1, the backscattered NRCS from a vegetation canopy above a soil surface is often represented as the sum of three components, representing surface, volume, and double-bounce contributions to the total backscattered signal, i.e. σ t pq = σ s pq exp( τ) + σ v pq + σ sv pq (5.1) where σ t pq, σ s pq, σ v pq, and σ sv pq are the total backscattered signal for pq-polarization and its contributions by surface, volume, and surface-to-volume scattering mechanisms respectively. exp( τ) represents a modification to the surface contribution due to attenuation by two-way propagation through the vegetation canopy. The alpha approximation method makes several simplifying assumptions about the observed scene. As a low-order approximation, the original alpha method assumes that the second two terms in Equation 5.1 can be neglected, so 101

122 that vegetation contributions are treated as strictly multiplicative. While this is generally not true, the assumption can hold for certain land cover types, such as cereal crops. It is furthermore assumed that the contributions due to vegetation and surface roughness do not change significantly between consecutive radar measurements of an observed scene. The assumption that vegetation and roughness are temporally stable generally holds given that the latency between consecutive radar measurements is roughly one to three days [19]. If the aforementioned assumptions hold, then the ratio between two consecutive, co-polarized radar backscatter measurements (from a time series of such measurements) can be represented as σ 0(t 2) P P σ 0(t 1) P P α P P (ɛ (t 2) r, θ i ) α P P (ɛ (t 1) r, θ i ) 2 (5.2) where ɛ (t i) r is the relative permittivity of the soil surface at time t i and θ i is the angle (measured from nadir) at which the incident wave impinges upon the observed scene. According to first-order Small Perturbation Method (or the Small Slope Approximation) for rough surface scattering [29, 86], the magnitudes of these alpha coefficients are given by (ɛ r 1) α HH (ɛ r, θ i ) = (cos θ i + ɛ r sin 2 θ i ) 2 (5.3) and (ɛ r 1)[sin 2 θ i ɛ r (1 + sin 2 θ i )] α V V (ɛ r, θ i ) = (ɛ r cos θ i + ɛ r sin 2 θ i ) 2. (5.4) 102

123 Since the alpha coefficients defined in 5.3 and 5.4 are solely a function of the incidence angle and the permittivity of the surface, the magnitudes of these alpha coefficients can be used to obtain surface permittivity from which a soil moisture estimate can be made for each element of a time-series of radar acquisitions. Soil moisture can be inverted from α P P by comparing α P P with a look-up table of alpha magnitudes versus soil moisture conditions. This look-up table is generated using knowledge of the soil texture and the Dobson/Ulaby/Peplinski dielectric model to map soil moisture values to a set of complex permittivity values, which are then used to calculate the alpha coefficient magnitude for each soil moisture value considered. Figure 5.1 shows the dependence of α HH and α V V for a fixed incidence angle and varying soil moisture conditions. 103

124 Figure 5.1: Plots of HH- and VV-polarized alpha coefficient magnitudes versus surface soil moisture. The sand and clay fractions are 0.4 and 0.2 respectively. Surface soil moisture is mapped to soil permittivity using the Peplinski/Ulaby/Dobson dielectric model [32]. The incidence angle is fixed at θ i = 40. Consider a time-series of radar backscatter coefficients represented by σ 0(t i) P P for i = 1...N where i designates the index of the measurement within a timeseries of N measurements. In this case, a matrix equation can be constructed to obtain the unknown alpha coefficient magnitude for each element of the time series. For example, if three radar acquisitions are available (N = 3) for 104

125 the target scene, a matrix equation can be constructed as follows: σ 0(t 1) P P 1 0 σ 0(t 2) P P σ 0(t 2) P P 0 1 σ 0(t 3) P P α (t 1) P P α (t 2) P P α (t 3) P P = 0. (5.5) 0 In this matrix equation, the unknown alpha coefficient magnitude at a given time t i (which corresponds to permittivity information) is related to the ratio of backscatter measurements at times t i+1 and t i. Note that Equation 5.5 is easily extensible to multiple polarizations or an arbitrary time-series length. As an example for the case of multiple polarizations, consider a time series of three HH- and VV-polarized NRCS measurements for an observed scene. The matrix equation of interest becomes: σ 0(t 1) HH σ 0(t 2) HH σ 0(t 3) HH σ 0(t 3) HH σ 0(t 1) V V σ 0(t 2) V V σ 0(t 2) V V σ 0(t 3) V V α (t 1) HH α (t 2) HH α (t 3) HH α (t 1) V V α (t 2) V V α (t 3) V V 0 0 =. 0 0 (5.6) Since the matrices in Equations 5.5 and 5.6 are not of full rank, a bestfit solution must be found. To this end, a linear, least-squares optimization approach has been implemented [87]. The algorithm has been found to be 105

126 sensitive to the allowable range of alpha coefficients during the optimization process. For this reason, a bounded optimization procedure is implemented to derive a best-fit solution [87]. The bounds of the alpha coefficients are derived from the dynamic range of soil moisture for the observed scene, which is given by ground sampling or climate modeling. Once the magnitudes of alpha coefficients have been estimated using the least-squares technique, soil moisture can be inverted from the α (t i) values by comparing α (t i) with a look-up table of P P alpha magnitudes versus soil moisture conditions. This look-up table is generated using knowledge of the soil texture and the Dobson/Ulaby/Peplinski dielectric model to map soil moisture values to a set of complex permittivity values, which are then used to calculate the alpha coefficient magnitude for each soil moisture value. As the sliding window of the time series shifts (i.e. when new radar measurements are made), multiple soil moisture estimates will be made for the same day. However, only the most recent soil moisture estimate is retained, which corresponds to the N th time series element. The single-channel alpha approximation method has been previously applied to several field campaign data sets [85, 88 91]. In these data sets, airborne and/or satellite-borne SAR data was collected concurrently with ground measurements of soil texture, soil moisture, surface roughness, and vegetation biomass. In general, the algorithm is found to perform well for certain crop types during periods of low to moderate vegetation biomass [85]. Under these conditions, the underlying assumptions of the algorithm (namely that vegetation contributions are strictly multiplicative) are most applicable. In the 106 P P

127 following sections, the alpha method was further tested using measured and simulated data sets. New modifications to the alpha method were developed using these same data sets. The most extensively-used data sets in algorithm refinement were the SMAPVEX12 field campaign data set and the GloSim SMAP simulation tool. 5.3 Analysis and Refinement Tools SMAPVEX12 Field Campaign The Soil Moisture Active / Passive Validation Experiment 2012 (SMAPVEX12) field campaign was carried out in the regions surrounding Winnipeg, Manitoba, Canada [35]. Data was collected during a six-week time period from June to July 2012, which corresponded to the growth season of many local crops. SMAPVEX12 included airborne synthetic aperture radar (SAR) acquisitions accompanied by ground sampling of soil moisture, surface roughness, and vegetation biomass. The SAR instrument used during the SMAPVEX12 campaign was the UAVSAR (Uninhabited Aerial Vehicle SAR). UAVSAR provides fully-polarimetric strip-map SAR images at L-band (1.26GHz). SAR acquisitions were accompanied by extensive ground sampling during the course of the campaign. On days where the UAVSAR instrument was flown, ground teams sampled soil moisture conditions in pre-selected locations (usually agricultural fields) using dielectric probes. On off-days, where the airborne instruments were grounded, vegetation biomass was sampled destruc- 107

128 tively and non-destructively for the same fields in which soil moisture was sampled. Surface roughness for each of the sampled areas was characterized with the use of a pin-profiler. In-situ sensors including rain gauges and soil moisture sensing networks were also in place during UAVSAR acquisitions. The SMAPVEX12 dataset was engineered to foster the development of time-series soil moisture retrieval algorithms. A time-series of SAR data and ground sampling data was available for widely-varying soil moisture and vegetation conditions. Over the course of the campaign, many of the fields sampled experienced significant changes in soil moisture as well as vegetation growth. This extensive dataset allowed soil moisture estimation techniques to be rigorously tested over a long time-series with a wide spectrum of ground conditions. Figure 5.2 provides a map showing the approximate area of the SMAPVEX12 study region which was ground-sampled imaged by UAVSAR. 108

129 Figure 5.2: A map of the main SMAPVEX12 study region around Elm Creek, Manitoba, Canada [35]. The orange areas represent the approximate areas imaged by UAVSAR during the campaign. 109

130 5.3.2 GloSim SMAP Simulator GloSim2014 was developed by NASA-JPL as a simulation of the SMAP satellite, its orbit, and its data acquisition/processing [19, 92, 93]. The simulation tests the framework of SMAP software from the transmission and reception of radar pulses (Level-0 processing) up to development of a 3-km gridded SAR map for HH, VV, and HV polarizations (Level-1 processing [31]), and processing the SAR data into a soil moisture map (Level-2 through Level- 4 processing [19]). GloSim2014 also includes simulated climate data ( nature run data) including soil moisture conditions, soil texture, soil surface temperature, freeze/thaw states, vegetation biomass, etc. The soil moisture generated by the nature-run is treated as truth data in the analyses that follow. A modified version of the Level-2 radar code for GloSim2014 was developed to implement and test the modified alpha method. This modified version of the Level-2 Soil Moisture Active (L2SMA) processor can produce a global map of soil moisture estimates based on a simulation of SMAP-like SAR imagery. The processor retrieval uses ancillary data only for land cover information and the sand/clay fraction of the soil surface in order to reduce the complexity of the retrieval process. 110

131 5.4 Novel Modifications to the Alpha Approximation Method Selection of Bounds for Least-Squares Optimization Since matrix equations such as Equations 5.5 and 5.6 are not of full rank, best-fit solutions to these equations must be derived. A bounded, linear leastsquares optimization is implemented to estimate the alpha coefficients. While previous studies [85] have shown improved error performance through trialand-error adjustment of the alpha coefficient bounds, this section considers the inclusion of ancillary information to adjust these bounds. If the bounds are not chosen properly, the accuracy of the algorithm degrades significantly. The studies outlined in this section, using both simulated and measured data, show that it is prudent to constrain this best-fit solution carefully, preferably using climate data for the observed scene. Figure 5.3 shows scatter plots of estimated versus measured soil moisture of the alpha method for a sample of SMAPVEX12 / UAVSAR data. These data are for wheat fields; the assumptions of the alpha method were found to be most applicable to cereal crops. For the wheat fields considered in these plots, vegetation biomass is relatively high, ranging from 1.83 kg/m 2 to 3.69 kg/m 2. In the upper plot, the bounds of the alpha coefficient magnitudes have not been fixed, i.e. the bounds were not adjusted before least-squares 111

132 optimization. These bounds are fixed, regardless of the field being examined. In the lower plot, the least-squares optimization for the alpha coefficient magnitudes is constrained such that the minimum and maximum possible values for the estimated alpha magnitudes correspond to the minimum and maximum soil moisture values observed over the time series. The tuned alpha magnitude bounds are calculated using Equations 5.3 and 5.4 according to ground-sampled soil moisture and soil texture data (assuming 10% fractional error in a-priori soil moisture information), where the Peplinski/Ulaby/Dobson model [32] is used to derive soil permittivity from soil moisture. The minimum and maximum un-tuned alpha magnitude bounds are 0.45 and 0.8, which are subsequently tuned to 0.61 and 0.72 respectively. Note that for both cases the maximum possible retrieved soil moisture value is 0.5 cm 3 /cm

133 Figure 5.3: Performance of the alpha method when applied to a sample of SMAPVEX12 field campaign data, with a focus on wheat fields. This version of the alpha method uses a time-series of HH-polarized backscatter data (a) Retrieved versus measured soil moisture using un-tuned alpha coefficient bounds. (b) Retrieved versus measured soil moisture using alpha coefficient bounds which are tuned in accordance with surface soil moisture values obtained from dielectric probe measurements. 113

134 It is recommended that this algorithm be provided with a set of estimated minimum/maximum soil moisture values for each season of the calendar year. These estimates may come from forward climate models or from a collection of satellite data. On a local scale, minimum and maximum soil moisture information may be available through the use of in-situ soil moisture monitoring network of meteorological data. In this sense, the algorithm is similar to Wagner s change detection approach [94], which requires knowledge of the radar backscatter signatures corresponding to the driest and wettest possible conditions for the observed scene. This constraint presents a trade-off: while the algorithm described herein does not require that ancillary vegetation information be provided, knowledge of the seasonal dynamic range of soil moisture in the observed scene is required such that the soil moisture estimates are bounded A Simulation Study Including Additive Vegetation Effects It is well-known that in general, a vegetation canopy provides both additive and multiplicative contributions to the total backscattered signal from a natural scene. Except in the case of specific vegetation structures, it is an over-simplification to consider only multiplicative vegetation contributions as with the traditional alpha method proposed in [85]. The studies in this section describe the consideration of additive vegetation contributions in a modified 114

135 version of the alpha method. While many approaches to characterizing vegetation using backscattered radar signatures have been implemented, many of these approaches require fully coherent measurements. These approaches usually take the form of some polarimetric decomposition model which attempts to separate contributions due to the soil surface from contributions due to vegetation. The inclusion of phase information often serves to better estimate the amount of vegetation biomass present in the observed scene, since larger amounts of vegetation biomass will generally result in multiple scattering interactions. Since SMAP is incapable of making such measurements and only records the magnitude of radar backscatter for HH, VV, and HV polarizations, a coherent polarimetric decomposition cannot be applied to SMAP data. The following simulation study uses the SMAP data cube (described in Chapter 1) as a forward model. The data cube considered for this study is a grass-type data cube, the model for which is described in Motofumi Arii s dissertation [20]. In this model, surface scattering is described by the Small Perturbation Method. The vegetation canopy is represented by a collection of randomly-oriented dielectric cylinders. Scattering from the cylinders is described by an infinite cylinder approximation, in which the fields internal to the cylinders are approximated as those of an infinite cylinder. The effects of the cylinders end-caps are included via integration of the finite cylinder. To simulate increasing VWC, only the number density of cylinders is increased, rather than their dielectric constant or physical dimensions. Predictions made 115

136 by Dr. Arii s grass data cube model are based on the assumption that contributions from surface, volume, and double-bounce scattering mechanisms are incoherently summed when calculating the total backscattered power. A visualization of these scattering mechanisms has been provided in Chapter 1, in Figure 1.1. The procedure by which this data cube is implemented in a preliminary simulation study of the alpha method is outlined in the flowchart in Figure 5.4. First, the data cube is loaded and a look-up table for the HH-polarized alpha coefficients is constructed (such that the retrieved alpha coefficients can be compared with the look-up table later in the retrieval process). In the Monte Carlo simulation procedure, surface soil moisture values are randomly generated for each step of the time series, with vegetation biomass and surface roughness fixed the length of the time series. A new series of random soil moisture values is generated for each Monte Carlo trial. The alpha coefficients are derived from the time series of radar back scatter measurements. Error sources such as speckle and thermal noise are not considered in this simulation, which only seeks to compare the original alpha method with the modified version. The alpha coefficient bounds are derived assuming the upper and lower limits of truth soil moisture are known to within a 10% fractional error. Once the alpha coefficients are estimated, they are inverted to obtain a soil moisture value for each step of the time series. This process is repeated for every (VWC, ks) pair represented in the data cube. 116

137 Figure 5.4: Flow diagram of data cube simulation procedure for the alpha method. This process is used to examine the benefit of various methods of considering a correction factor for additive vegetation contributions. 117

138 The result shown in Figure 5.5 shows the error performance of the original alpha method algorithm applied to the grass-like data cube in accordance with the diagram in Figure 5.4. The RMSE map in Figure 5.5 shows the RMSE of the alpha method, averaged over soil moisture conditions, as a function of vegetation biomass (VWC) and roughness (ks; the electromagnetic wavenumber multiplied by surface RMS height). The simulation is for the grass-type data cube with surface, volume, and double-bounce components included. Figure 5.5: A color map of RMSE (averaged over soil moisture conditions) versus VWC and ks (roughness), when the original alpha method is applied to the grass-like data cube in accordance with the simulation procedure outlined in Figure

139 The error performance shown in Figure 5.5 is unacceptably high (RMSE in excess of 10% volumetric soil moisture) for most vegetation and roughness conditions, the exception being where vegetation levels are very small. This suggests that for some land cover types, additive contributions to the backscatter coefficient should be considered when significant vegetation is present. For low surface roughness values, surface returns are minimized and double-bounce can only occur when the ground and the cylinders behave as corner reflectors. The volume contribution is significant in this region, hence the poor retrievals. When no vegetation is present, a perfect retrieval is possible for all roughness conditions. In theory, it is desirable to remove only the volume and doublebounce component before applying the alpha method estimation technique. The Freeman-Durden polarimetric decomposition provides a simple method for separating surface and volume contributions to the backscattered signal [95]. According to the Freeman-Durden decomposition, σ 0 HH,volume = 3σ 0 HV,total. (5.7) Therefore, to subtract the volume contribution to the backscattered signal, the following equation is applied: σ 0 HH,surface+doublebounce = σ 0 HH,total 3σ 0 HV,total. (5.8) Substituting the expression for σ 0 HH,surface+doublebounce in Equation 5.8 into 119

140 the matrix in Equation 5.5, the same data cube simulation procedure described before is applied again, this time using the Freeman-Durden decomposition to consider additive vegetation effects (i.e. direct scattering from the vegetation canopy). The results from the simulation are shown in Figure 5.6. Figure 5.6: A color map of RMSE (averaged over soil moisture conditions) versus VWC and ks (roughness), when the alpha method (with a modification inspired by the Freeman-Durden decomposition) is applied to the grass-like data cube in accordance with the simulation procedure outlined in Figure 5.4. Figure 5.6 shows a modest improvement in error performance for some (VWC, ks) pairs. The applicable regime is not extended by much in VWC, ks space. Performance for bare surfaces is degraded slightly. The Radar 120

141 Vegetation Index described in [96] is considered in what follows. The Radar Vegetation Index (RVI) parameter developed by Kim and van Zyl [96] provides an attractive approach to vegetation quantification in that it only requires instantaneous measurements of the incoherent radar backscatter for HH, VV, and HV polarizations (a time-series is not needed to estimate vegetation parameters). The RVI is a measure of the randomness of scatterers present in the observed scene, and is given by: RV I = 4σ 0 HV σhh 0 + σ0 V V +. (5.9) 2σ0 HV It has been found that the best vegetation correction term is based on RVI. The RVI-based correction factor of the time series of co-polarized backscatter measurements generally leads to an improvement in algorithm performance. The proposed vegetation correction modifies the HH-polarized backscatter to form σ 0 HH which is given by σ 0 HH = σ 0 HH(1 γrv I) (5.10) where γ is a tunable constant which can be varied according the land-cover/crop type. γ is set to 1 in the simulation analysis in this section. If a time series of σ 0 HH and RVI information is available for an observed scene, then a matrix equation can be constructed. Through this matrix equation, an alpha coefficient can be estimated for the observed scene for each step 121

142 of the time series. For example, if three radar acquisitions are available for the target scene, the matrix equation can be constructed by substituting Equation 5.10 into Equation 5.5: σ 0 (t 2) HH 1 0 σ 0 (t 1) HH σ 0 (t 3) HH 0 1 σ 0 (t 2) HH α (t 1) HH α (t 2) HH α (t 3) HH = 0. (5.11) 0 A bounded linear least-squares optimization is applied to solve Equation 5.11 for the unknown alpha coefficient magnitudes. As the sliding window of the time series shifts (i.e. when new radar measurements are made), multiple soil moisture estimates will be made for the same day. However, only the most recent soil moisture estimate is retained, which corresponds to the N th time series element. As demonstrated in subsequent sections, it is best if the bounds of the alpha magnitudes are derived using knowledge of the seasonal climate of the observed scene. A data-cube-based simulation study was once more performed using this modified version of the alpha method. Simulation results similar to those shown previously are depicted in Figure

143 Figure 5.7: A color map of RMSE (averaged over soil moisture conditions) versus VWC and ks (roughness), when the alpha method (with the RVI-derived modification expressed in Equation 5.10 and Equation 5.11) is applied to the grass-like data cube in accordance with the simulation procedure outlined in Figure 5.4. Figure 5.7 displays the results of the data cube simulation study, where the alpha method is assessed with the use of the RVI parameter. Figure 5.7 shows that the RVI parameter successfully isolates the surface return over most vegetation biomass and roughness conditions, resulting in a drastically improved error performance (compared with Figure 5.5) for most (VWC, ks) 123

144 pairs. The algorithm still performs relatively poorly when surface, multiplebounce, and volume contributions are all of roughly the same magnitude; this is the case for the vegetation and roughness conditions corresponding to the bottom left region of Figure Studies of Polarization Effects on the Alpha Method Though the alpha approximation as proposed by Balenzano et al. was originally formulated as a single-polarization method [85], the procedures outlined in previous sections are easily extensible to multiple polarizations. Traditionally the method has been applied to HH-polarized data in the presence of crops with a regular, vertically-aligned structure. However, recent studies have shown that when the alpha coefficient bounds are carefully chosen as outlined in previous sections, the inclusion of VV-polarized NRCS data can improve retrieval performance. Two single-polarization versions and a dual-polarization version of the alpha method were applied to SMAPVEX12 data. Figure 5.8 shows scatter plots of estimated versus measured soil moisture for several versions alpha method for a sample of SMAPVEX12 wheat fields observed by UAVSAR. The variations of the algorithm (corresponding to the different data markers) use different polarization configurations to apply the generalized matrix equation in Equation 5.5. The RVI correction factor is applied to HH-polarized backscatter, but not to VV-polarized backscatter. Similar analyses (not shown here) 124

145 have shown a degradation in error performance when the RVI term is applied to VV-polarized backscatter. After experimenting with the SMAPVEX12 data it was concluded that γ = 0.5 was optimal for RVI correction of HH-polarized data. A sliding-window time-series length of six is implemented in the analysis. During this sliding-window implementation, multiple soil moisture estimates are made for some days, but these estimates are treated as independent for the purposes of this particular error analysis. In this analysis, seven wheat fields are focused upon because the underlying assumptions of the alpha approximation have been found to be most applicable to cereal crops. A total of thirteen radar measurements were made for each wheat field over the course of five weeks (from June 7 to July 14, 2012). Each field is ground-sampled at 16 discrete points, concurrently with radar overpasses [35]. Radar backscatter values and ground truth measurements are averaged over each agricultural field being examined. The upper and lower bounds of the HH and VV-polarized alpha coefficient magnitudes are derived from the ground-sampled soil moisture data as described in Section

146 Figure 5.8: Error performance of the alpha method when applied to a sample of SMAPVEX12 field campaign data, with a focus on wheat fields. The RMSE values displayed in the lower right corner are fractional errors in volumetric soil moisture, expressed as percentages. The data markers correspond to three different versions of the aforementioned alpha method. Single-polarization versions, using only HH-polarized backscatter or only VV-polarized backscatter are shown in blue and red respectively. The purple markers correspond to a version using a combination of HH- and VV-polarized backscatter values. HH-polarized NRCS data is modified using the RVI correction term with γ = 0.5. Although reference [85] encourages the use of HH-polarization only, the results shown in Figure 5.8 clearly demonstrate that the use of VV-polarization can improve error performance, even for vertically-oriented crops such as wheat. This is because the direct surface return tends to be stronger for VV-polarization 126

147 than for HH, even when considering attenuation through the vegetation canopy. Inclusion of both polarizations in the retrieval is recommended to help ensure that polarization-dependent vegetation effects will not degrade retrieval performance. Note that since these data were collected during wet conditions where the crops were mature, a decreased sensitivity of radar backscatter to changes in soil moisture is expected. Further, a relatively high amount of vegetation biomass was present during some radar acquisitions. Despite these unfavorable conditions, it is demonstrated here that the algorithm is well-suited to tracking changes in soil moisture, which is sometimes of more interest than estimation of absolute soil moisture [19]. In this analysis, the modified, dual-polarized alpha method exhibits an RMSE of 5.6% volumetric soil moisture where the alpha coefficient bounds are optimized Further Simulation Studies of the Modified Alpha Method Figure 5.9 displays a global map of soil moisture estimates for late November to early December, generated using the modified alpha method (used in Section 5.4.3) within the modified GloSim2014 SMAP simulator (described in Section 5.3.2). A time-series of six radar acquisitions was used for this purpose (corresponding to a time window length of about 18 days). Note that the frozen or snow-covered areas of the globe are flagged out and the retrieval 127

148 is not attempted in these regions. Figure 5.10 displays a corresponding error map for this sample of simulation data. Figure 5.11 shows the soil moisture truth data generated by the climate simulation. The γ parameter in Equation 5.10 was set to either 0 (for shrublands, savannas, and bare soil) or 0.5 (for all other land cover types). It was found that selecting a larger value for γ would often create a disproportionately large RVI correction term in the GloSim data. Both the HH- and VV-channel were used when applying the alpha method, but the RVI correction was only applied to HH-polarized backscatter. The retrieved alpha coefficient magnitudes were bounded in accordance with the minimum and maximum soil moisture values encountered by the GloSim nature-run climate model over the three-month period spanning from October 2014 to December 2014, on a pixel-by-pixel basis. The processor uses ancillary data only for land cover classification, sand/clay fraction of the soil surface, and the climate model used to estimate minimum/maximum soil moisture for each pixel. 128

149 Figure 5.9: Results from the modified SMAP L2SMA processor, where the modified alpha method has been applied to 20 days of simulated SMAP radar data (descending half-orbits only). This is a global map of soil moisture values obtained by the modified alpha method. Retrievals are generated using a history of radar backscatter data into which a time-series is organized for each multi-looked and spatially-averaged radar pixel (3 km resolution). Grey regions correspond to areas where the retrieval was not attempted (frozen soil, ocean, no data radar data available, etc.). 129

150 Figure 5.10: Results from the modified SMAP L2SMA processor, where the modified alpha method has been applied to 20 days of simulated SMAP radar data (descending half-orbits only). This is a global map of the method s RMSE, with the GloSim nature-run climate model having generated truth soil moisture data, shown in Figure

151 Figure 5.11: Soil moisture truth generated by the GloSim2014 SMAP simulator, corresponding to the regions and time intervals for which the retrieval data in Figure 5.9 was generated. The RMSE shown in Figure 5.10 is calculated by comparing retrievals with the soil moisture truth data shown here. 131

152 The error rates shown in Figure 5.10 are dependent on land cover classification. A break-down of RMSE versus land cover class has been provided in Figure Since the only simulation data available was generated for the months of November and December, data for wheat and corn are not available since these crops are out-of-season. The RMSE varies drastically with land cover type because of the different scattering effects exhibited by different vegetation structures. Figure 5.12: Results from the modified SMAP L2SMA processor, where the modified alpha method has been applied to 20 days of simulated SMAP radar data. This is a break-down of the RMSE for the modified alpha method as a function of land cover classification. 132

153 Since error rates shown in Figure 5.10 also depend on vegetation biomass conditions, a breakdown of RMSE versus land cover and VWC is shown in Figure In general, error rates increase significantly with increasing vegetation biomass. Note that for most land cover types and vegetation biomass conditions, the algorithm provides an RMSE of 8% volumetric soil moisture or better. Figure 5.13: Results from the modified SMAP L2SMA processor, where the modified alpha method has been applied to 20 days of simulated SMAP radar data. This is a break-down of the RMSE for the modified alpha method as a function of land cover classification and vegetation biomass (VWC). 133

154 5.4.5 Application of the Modified Alpha Method to SMAP Radar Data The modified alpha method described in Section was applied to SMAP radar data measured between April 28 and May 31, A time-series of six radar acquisitions was used for this purpose (corresponding to a time window length of about 18 days). Comparisons have been made with ground data provided by soil moisture sampling networks used for the purpose of calibrating and validating SMAP soil moisture products [93]. Figure 5.14 displays a global map of soil moisture estimates generated using the modified alpha method with measured SMAP radar data from May, As with the GloSim simulation data, the γ parameter in Equation 5.10 was set to either 0 (for shrublands, savannas, and bare soil) or 0.5 (for all other land cover types). Both the HH- and VV-channel were used when applying the alpha method, but the RVI correction was only applied to HH-polarized backscatter. The retrieved alpha coefficient magnitudes were bounded in accordance with the minimum and maximum soil moisture values encountered by the GloSim nature-run climate model over the three-month period spanning from April 2014 to June 2014, on a pixel-by-pixel basis. The processor uses ancillary data only for land cover classification, sand/clay fraction of the soil surface, and the climate model used to estimate minimum/maximum soil moisture for each pixel. Figure 5.14 provides a qualitative assessment of the modified alpha method 134

155 soil moisture estimation algorithm. Soil moisture conditions predicted by the algorithm are consistent with predictions for this time of year, which corresponds to the dry season for much of the Northern hemisphere. Note the data gap over China and the Indochina Peninsula is caused by the radar having switched to a calibration state. 135

156 Figure 5.14: Results from the modified SMAP L2SMA processor, where the modified alpha method has been applied to 5 days of measured SMAP radar data (descending half-orbits only, May 27-31). Retrievals are generated using a history of radar backscatter data into which a time-series is organized for each multi-looked and spatially-averaged radar pixel (3 km resolution). This is a global map of soil moisture values obtained by the modified alpha method. Grey regions correspond to areas where the retrieval was not attempted (frozen soil, ocean, no data radar data available, etc.). A data gap exists in half-orbits over China and the Indochina Peninsula because the radar had been switched to a calibration state for this orbit. 136

157 Figure 5.15 shows comparisons between modified alpha method retrievals using SMAP radar data and two reference areas within an in-situ soil moisture sampling network: the Soil moisture Sensing Controller and optimal Estimator (SoilSCAPE) in Tonzi Ranch, California, US [97]. Figure 5.16 shows similar comparisons with in-situ soil moisture sampling data from the Texas Soil Observation Network (TxSON) in Texas Hill Country, Texas, US [98]. Data from both sites is upscaled to SMAP resolutions (3 km for radar and 36 km for radiometer) such that ground measurements can be compared with SMAP soil moisture products for validation and calibration purposes [99]. Figure 5.15 shows that comparisons with data from SoilSCAPE exhibit a low RMSE (0.021 m 3 /m 3 and m 3 /m 3 for the two 3 km reference pixels within the study area) between soil moisture estimated from SMAP radar data and ground-sampled/upscaled soil moisture data. The data from SoilSCAPE, however, represent soil moisture conditions which are stable over time and do not exhibit a wide dynamic range. Conversely, Figure 5.16 shows the ground-measured data from TxSON exhibit a wide dynamic range with several precipitation and dry-down conditions. Comparisons with TxSON data are still lower than predictions from GloSim simulation data, with an RMSEs of m 3 /m 3 and m 3 /m 3 for the respective reference pixels within the study area. 137

158 Figure 5.15: Comparisons between retrieved soil moisture obtained from the modified alpha method applied to SMAP radar data and ground-measured, upscaled volumetric soil moisture (VSM) for two SMAP radar reference pixels within the SoilSCAPE calibration/validation site. The error metrics displayed in these plots are un-biased root mean square error (ubrmse), RMSE, bias, and correlation coefficient (R). The different data markers correspond to flags in the radar data. Green data markers correspond to days on which radar data was taken, but a long enough time series (six radar acquisitions) had not been acquired to build the modified alpha method s matrix equation for soil moisture estimation. Pink data markers correspond to days on which the radar data acquired was not reliable (usually due to a lower-than-usual number of looks used for speckle noise reduction) and use of the soil moisture product is not recommended. Black data markers correspond to days on which the soil moisture estimation from radar data succeeded and is recommended for use. 138

159 Figure 5.16: Comparisons between retrieved soil moisture obtained from the modified alpha method applied to SMAP radar data and ground-measured, upscaled volumetric soil moisture (VSM) for two SMAP radar reference pixels within the TxSON calibration/validation site. The error metrics displayed in these plots are un-biased root mean square error (ubrmse), RMSE, bias, and correlation coefficient (R). The different data markers correspond to flags in the radar data. Green data markers correspond to days on which radar data was taken, but a long enough time series (six radar acquisitions) had not been acquired to build the modified alpha method s matrix equation for soil moisture estimation. Pink data markers correspond to days on which the radar data acquired was not reliable (usually due to a lower-than-usual number of looks used for speckle noise reduction) and use of the soil moisture product is not recommended. Black data markers correspond to days on which the soil moisture estimation from radar data succeeded and is recommended for use. 139

160 5.5 Conclusions A method of estimating surface soil moisture from a time-series of L-band radar backscatter measurements has been presented. The algorithm was assessed using recent field campaign data including airborne SAR measurements. The choice of polarization was found to significantly influence retrieval performance in ways not previously documented. The RVI parameter, designed to estimate scattering from the vegetation canopy, was introduced in a modified version of the alpha method. The γ parameter acts as a land-cover-specific variable which varies the degree to which the RVI parameter is considered in the retrieval process. A simulation study using a grass-like data cube has shown the potential benefits of using γ and RVI to subtract volume and doublebounce effects and isolate backscatter from the soil surface. The modified version of the alpha method was also applied to a large-scale simulation tool designed for testing the SMAP radar soil moisture retrieval. Error performance varied with respect to land cover type but was generally found to be within 8% RMSE. The modified alpha method presented here is attractive for soil moisture remote sensing using L-band radar due to its ability to track changes in soil moisture even in the presence of significant vegetation. It was found that although this algorithm does not need to be supplied with a-priori vegetation information, since it is effectively a change-detection approach, it is required that it be supplied with a climate model capable of estimating the dynamic range of soil moisture for the area/time of interest. Finally, the mod- 140

161 ified alpha method was directly applied to SMAP radar data. Soil moisture estimates using the modified alpha method with SMAP data were compared with two SMAP validation sites in which ground-sampled soil moisture data was collected. Comparisons with upscaled, ground-sampled data consistently yielded error performance below 7% RMSE. 141

162 Chapter 6 Inland Water Body Detection using L-band Radar 6.1 Motivation The subject of discerning between water and land surfaces using radar is an important topic in remote sensing of soil moisture, since water bodies should be flagged before a soil moisture retrieval is attempted. Bare land surfaces and water bodies both exhibit low backscatter returns which are often below the noise floor of the radar system. As such, applying a simple threshold to NRCS returns will not necessarily predict the presence of a water body accurately. Tracking of transient water bodies serves to further motivate the study of water body detection using radar. In other words, it is important to discern between flooded and dry regions of land, particularly when applying a soil moisture retrieval algorithm to radar backscatter signatures. This chapter first presents the original baseline SMAP algorithm for detecting inland water bodies, which involves applying a threshold to the ratio 142

163 between HH- and VV-polarized NRCS values. This algorithm performs poorly where the signal-to-noise ratio is low due to small backscatter returns. Two major improvements to the original algorithm are outlined here, which improve water body detection rates in the presence of system noise and low backscatter measurements. 6.2 Algorithm History Kim et al. [100] proposed using the ratio between HH- and VV-polarized NRCS values to determine whether a radar pixel is located within a water body. For inland water bodies, HH and VV NRCS values tend to be similar. Thus, Kim et al. propose applying a threshold to HH/VV in order to classify a pixel as either water or land. Results from applying the algorithm to inland, static water bodies are favorable, but strongly dependent on system noise. The original Kim et al. algorithm is illustrated in the flow-chart in Figure

164 Figure 6.1: A flowchart for the original water body detection algorithm proposed by Kim et al. This is a simple threshold place on the ratio of HH- and VV-polarized NRCS values for each radar pixel. 6.3 Algorithm Assessment In order to assess the performance of water body detection algorithms using L-band radar, data from airborne sensors were artificially noise corrupted to simulate data observed from a satellite platform. Specifically, data from UAVSAR, a polarimetric, L-band strip map SAR instrument, were considered for analysis of water body detection performance. To simulate a SMAP-like noise environment, UAVSAR data was artificially corrupted on a pixel-by-pixel basis. This artificial noise corruption was injected before the algorithms were applied. The UAVSAR imagery was injected with simulated SMAP-like noise due to speckle, calibration errors, and radio-frequency interference (RFI). Although 144

165 calibration errors and radar RFI are sometimes treated as having a non-zero mean (which may change with time, or drift [31]), calibration issues were lumped with speckle errors for simplicity. These three error sources were collectively treated as complex Gaussian random variables with zero mean and a standard deviation which scaled with the uncorrupted NRCS value, in the same manner as speckle noise. Corruptions due to these artificial noise sources were mitigated to some degree using an incoherent multi-looking technique, where a number of NRCS measurements (magnitude only) were averaged. To study the performance of the water body detection algorithm, L-band UAVSAR imagery from airborne field campaign data over two water bodies were examined. The first scene considered was a section of the Assiniboine River, west of Winnipeg, Manitoba, Canada. A sample radar image of the river is displayed next to a corresponding map in Figure 6.2. Radar data over Yellowstone Lake in Wyoming, US was also incorporated into this study. A sample radar image of the lake is displayed next to a corresponding map in Figure

166 Figure 6.2: Radar backscatter (left) and optical (right) imagery of the Assiniboine River in Manitoba, Canada. The black line overlayed on the radar data is the approximate region where the SAR incidence angle is 40 degrees. The red box overlayed on the optical imagery is the approximate area which corresponds to the radar image on the left. Figure 6.3: Radar (left) and optical (right) imagery of Yellowstone Lake. The black line overlayed on the radar data is the approximate region where the SAR incidence angle is 40 degrees. The red box overlayed on the optical imagery is the approximate area which corresponds to the radar image on the left. 146

167 6.4 Subtraction of Noise-Equivalent NRCS Given the poor performance of the water body detection algorithm described above in the noise-limited case, a method of quantifying and subtracting noise contributions from NRCS values was sought. Y. Kim and J. J. van Zyl have shown that a performing a thermal noise subtraction based on an estimate of the system SNR yields better vegetation classifier predictions using the Radar Vegetation Index (RVI) parameter [96]. In this procedure, an average of noise-only measurements from the radar system are taken and converted into an estimated noise-equivalent NRCS (NEσest). 0 The noise-equivalent NRCS is a way of quantifying system noise, and is the effective NRCS observed by the radar system when no radar pulses are being received. Figure 6.4 shows a modified version of the water body detection algorithm, where the noiseequivalent NRCS is subtracted from HH- and VV-channel NRCS data before a threshold is applied. 147

168 Figure 6.4: A flowchart for a modified version of the water body detection algorithm where the noise-equivalent NRCS is subtracted. This is the same procedure as illustrated in Figure 6.1, the noise-equivalent NRCS, estimated from noise-only radar measurements, is subtracted before the -3 db threshold is applied. Using data collected over the Assiniboine River and Yellowstone Lake, the thermal noise correction method illustrated in Figure 6.4 was assessed for the purpose of water body detection. Table 6.1 shows the performance of Kim s algorithm with and without the implementation a mean thermal noise correction. Note that a time-series of data was collected over the Assiniboine 148

169 River, but Table 6.1 contains data for only the first day of the time series. Table 6.1: Radar Water Body Detection Algorithm Performance Before and After Thermal Noise Subtraction Yellowstone Lake No Noise subtraction With Noise subtraction Missed detection 62.62% 39.72% False alarm 0.54% 0.64% Assiniboine River No Noise subtraction With Noise subtraction Missed detection 7.72% 2.28% False alarm 2.62% 4.45% Table 6.1 shows a clear advantage to performing mean thermal noise subtraction prior to applying a threshold to the co-pol ratio. However, there is significant room for improvement to the algorithm, as shown by the poor error performance for Yellowstone Lake particularly. The remainder of this section will describe a modification made to this algorithm to extend its applicability to inland water body detection. 149

170 6.5 Adjustment for Calm, Open Water The significant error which arises from Kim s original algorithm when observing Yellowstone Lake is caused by the very low backscattered signal generated by a calm water surface. The mirror-like surface of a calm lake produces a predominantly specular reflection and generates very little diffuse scattering. A radar signal which impinges on such a surface at an oblique incidence angle will therefore generate a very small backscattered signal, which is often below the noise floor of the observing system. In the noise-limited case, where the backscattering of a water body is below the noise floor, the algorithm in the previous section produces a significant number of missed detections. This section describes the inclusion of a threshold on HH-polarized backscatter in the detection algorithm and presents results based on this new method. Figure 6.5 shows example radar images of Yellowstone Lake, zoomed in to the area around 40 degrees incidence within the strip map SAR imagery. Note that the lake itself is clearly visible in the the HH-polarized backscatter image, generally corresponding to the blue or blue-green areas (areas where σ 0 HH is less than about -17 db). In the bottom image, most of the eastern portion of the lake is well above the -3 db threshold for HH-VV. This is due to the calmer waters on the eastern side of the lake, which are shielded from westerly winds (which would otherwise increase the roughness of the water s surface) by nearby mountains. This presents a noise-limited case for the water 150

171 detection algorithm since the radar backscatter produced by the calm waters is well below the noise floor. Figure 6.5: Zoomed-in radar imagery of Yellowstone Lake where the incidence angle (within the strip map SAR image) is approximately 40 degrees. The top image shows HH-polarized backscatter and the bottom image shows the ratio between HH- and VV-polarized backscatter, expressed in db. The lake itself generally corresponds to the blue or blue-green areas in the HH image. The images in Figure 6.5 suggest that the use of a threshold on HHpolarized backscatter itself is merited. Figure 6.6 illustrates a new water body 151

172 classification procedure including the use of such a threshold. This is the same as the procedure illustrated in Figure 6.4, but with a new threshold placed on the noise-subtracted HH NRCS. The value of the new threshold, T HH, is -25 db. Note that any radar pixel for which the noise subtracted HH NRCS is below T HH will be classified as water. Placing a universal threshold on HH NRCS values will tend to falsely classify bare, dry soil as water, since dry, bare soil tends to have a very low co-polarized NRCS. It is assumed, however, that inland, transient water body detection for a dry, barren scene will not generally be an issue given that flooding is not likely to occur in such a climate. 152

173 Figure 6.6: A flowchart for the modified water body detection algorithm, including a threshold of HH-polarized backscatter [101]. T HH is set to -25dB. Table 6.2 shows the performance of the method outlined in Figure 6.6. The algorithms with and without use of an HH threshold are compared for imagery over Yellowstone Lake. Since much of the lake is well below the threshold, this algorithm performs better than for only applying a threshold to the HH/VV ratio. The land pixels used in this analysis only include the areas immediately surrounding the lake, which are dominated by evergreen forest. 153