Definition and Invariance Properties of Structured Surface BRDF

Transcription

1 Definition and Invariance Properties of Structured Surface BRDF William C. Snyder to appear in IEEE Trans. Geoscience and Remote Sensing, 2002 May9,2002 Abstract. The bidirectional reflectance distribution function (BRDF) is used to characterize the directional properties of surface reflectance, but the most recent standardized definition fails to address some important aspects for the structured and varying surfaces encountered in remote sensing. This article proposes an updated definition of BRDF that is applicable for such surfaces. The updated definition then motivates an examination of the resulting measurement requirements and invariance properties. The conclusion is that structured surface BRDF can be defined in a consistent manner that is an intrinsic surface property, and it does exhibit the usual invariance properties associated with BRDF, such as reciprocity. 1 Introduction The study of bidirectional reflectance has become a critical part of classification and characterization of land surfaces and of the surface-atmosphere energy transfer. Advancement of theory and standardization were identified as priorities in the recent IFB conference [1], and in several of the papers in a recent special issue of Remote Sensing Reviews [2]. The bidirectional reflectance distribution function (BRDF) is intended to be a fundamental and intrinsic kernel for computing the directional reflectance properties of a surface, but it has not been defined for the structured and varying surfaces encountered in remote sensing, and so its use for that application is not standardized. The commonly-accepted remote sensing definition of the ratio of reflected radiance in one direction to in-plane irradiance from another direction [3], leaves undefined numerous important geometrical and radiometric parameters. The formal NBS definition [4] is not appropriate for surfaces with position-dependent radiance. A modified definition, presented here, addresses parallax and spatial averaging to ensure that the resulting surface characterization is consistent. Nicodemus and others introduced the BRDF as a reflectance kernel to be the theoretical basis for measurable reflectance quantities [5]. These are elements of radiometry, which is a convenient zero wavelength, geometrical optics approximation for the radiative transfer of incoherent light. For passive remote sensing in the solar and thermal infrared wavelengths these are good approximations which greatly simplify the analysis. BRDF in remote sensing usually is based on a model because a set of measurements over the full range of incident and reflected angles is practical only for laboratory instruments. Also, the BRDF was developed mainly to standardize laboratory characterizations of opaque, diffuse, flat materials. But the NBS definition gives a form with global illumination and a local field of view that handles uniform, isotropic materials with volumetric scattering. Even so, the form is not appropriate for natural, structured surfaces because they are neither uniform nor isotropic. Remote sensing usage of the term BRDF is often nonstandard. It appears that the usage has come to refer also to any bidirectional properties. There are cases where BRDF is defined to be the bidirectional reflectance factor, and where it is defined to be the average bidirectional reflectance. These are not interchangeable quantities. In addition, BRDF measurements are sometimes reported without making needed corrections and using non-standard symbols. There are recent efforts, however, to motivate standardization of the reflectance terminology in remote sensing [6]. BRDF characterization is an important tool in analysis, models, and measurements, often in combination. Planetary reflectance modeling has a long history, extending back at least as far as Minnaert s famous paper on Lunar reflectance [9]. Many more recent studies model the bidirectional effects of structured surfaces ranging from soils to forest canopies [10], [11]. These models, for example, are employed to

2 normalize measurements taken at a particular geometry to infer intrinsic properties of the surface. Such applications are made efficient by representing complex reflectance phase functions as compact linear and invertible expressions [12], [13]. Such invertible models are now operational and processing imagery from space-borne instruments to produce land surface BRDF as a data product [14], [15]. BRDF models have also been applied to synthesize hyperspectral, angular scene properties from component measurements [16], [17]. Although not all bidirectional models are written as BRDF, it is good practice because it makes the relations independent of collection parameters, and provides a direct way to compute other reflectance quantities. On the other hand, there are situations where BRDF and the associated reflectance quantities are not the best characterizations of the radiative transfer of a surface. This is true when the approximations associated with radiometry do not apply. For instance with the increasing use of lasers in remote sensing, coherence requires a more generalized treatment, such as that from Greffet [18]. Also, because of parallax, the BRDF for structured surfaces becomes an intrinsic property only in the limit as source and detector distances are increased. When either the source or detector are close to a structured surface, or when there is significant atmospheric scattering above the reference plane, the bidirectional properties will not be modeled well by BRDF and, in addition, will not exhibit the expected invariance properties. An accurate analysis, then, will require a more general radiative transfer approach. Practically all models for structured surface BRDF apply the global illumination and global field of view definition we propose. They usually neglect atmospheric effects, and the source and detector distances are taken to be infinite. The resulting modeled quantity corresponds to the ratio of spatially-averaged radiance to the global irradiance at the reference plane. Because of these ideal radiometric and geometric assumptions, most models exhibit the invariance properties that we show should hold for the BRDF, such as reciprocity and conservation of energy. Some structured surface models, however, do not exhibit the invariance properties, such as those from Eom [19], Walthall [20], and Wanner, Li and Strahler [21], [22]. In all cases these models make simplifying assumptions that violate reciprocity. Moreover, it has been shown that equally effective approximations were possible without the violations [12], [16]. A goal of the present study is to motivate structured surface BRDF models that are physically consistent by adhering to the invariance properties, and which give intrinsic surface characterizations. 2 Definition of the BRDF for Structured Surfaces The volumetric scattering version of BRDF from the NBS document applies to global illumination and local reflected radiance [4, page 5]. The derivation of this state-of-the-art version considers that for each point on the surface, the reflected radiance is the sum of incident irradiance contributions across some characteristic scattering region. The definition addresses this scattering by requiring a sufficiently large illumination area, and by considering the BRDF at one particular point on the surface to be the integration of the bidirectional scattering-surface reflectance-distribution function (BSSRDF). For a simplified final expression, the NBS definition takes the surface to be uniform and isotropic, with a position-independent radiance, and therefore a position-independent BRDF. We cannot invoke this simplifying assumption for natural, structured surfaces, and so, for one thing, the NBS definition will need to be modified to account for the spatial dependence of the BRDF. The idea is to go back to the point in the NBS derivation describing a deterministic but spatially varying point BRDF, and then definethebrdfofanareaatparticularincident and reflected angles to be the average of the point BRDF across the area. Proceeding, with the NBS definition, the BRDF is defined with respect to a horizontal reference plane at the surface. Fig. 1 shows the global illumination, local field of view definitions. An element of the incident flux from the direction (θ i,φ i )isdφ i, that is within solid angle dω i,andovertheareaelementda i at (x i,y i ). The portion of the reflected radiance which comes from this element of incident flux is dl r,in the direction (θ r,φ r ), and emitted from the point (x r,y r ). The linear relation that defines the BSSRDF, S, is the ratio of this radiance out to the flux in, dl r = S (θ i,φ i,x i,y i ; θ r,φ r,x r,y r )dφ i. (1) Note that in this and subsequent expressions we will not show the dependence on wavelength and polar- 2

3 Figure 1: illumination, local view form of BRDF. ization state. So far, the only assumption that is made about the scattering mechanisms is that they are linear at the wavelength of interest, and of course, that the basic assumptions associated with radiometric analysis are valid. Following the NBS derivation, take the incident flux to be uniform over the entire illumination area, A i, from which there is significant contribution to the reflected radiance at the exit point, (x r,y r ). Call this uniform irradiance de i (θ i,φ i )=dφ i /da i. Basically, this means that the illumination area is large enough so that any further increase in size would not change the radiance at the point (x r,y r ). Under these conditions we can integrate the BSSRDF over the illumination area, and we obtain the radiance at (x r,y r ) in the direction (θ i,φ i ), dl r (θ i,φ i ; θ r,φ r ; x r,y r ) Z = de i (θ i,φ i ) S(θ i,φ i,x i,y i ; θ r,φ r ; x r,y r ) da i. (2) A i The current NBS definition takes the function S to depend only on the distance, r, between (x i,y i )and (x r,y r ), and not on absolute position, and ends up with the current standard definition for BRDF, that uses the subscript r, Z f r = S(θ i,φ i ; θ r,φ r ; r) da i (3) A i Generally, we do not know S, but using Eq. (2) we can rewrite this as a ratio, which is the current global-local NBS definition for uniform irradiance over a large enough area of a uniform isotropic surface, f r dl r (θ i,φ i ; θ r,φ r )/de i (θ i,φ i ). (4) Here we have eliminated dependence on position, appropriate for a uniform surface. Next, we will modify the derivation to remove that assumption for structured surfaces, and then define the average BRDF. For the modification, we will return to Eq. (2) and retain the dependence of the BRDF on the position of reflected observation, (x r,y r ). In that case, dl r (θ i,φ i ; θ r,φ r,x r,y r ) is a spatially-dependent radiance pattern. Note that in general, the size and location of the required illumination area, A i, may depend on location also. In other words, different locations may have different multiple scattering properties. Proceeding, we write the NBS definition for the spatially dependent BRDF at a point for particular incident and reflected directions, Z f(θ i,φ i ; θ r,φ r ; x r,y r )= S(θ i,φ i,x i,y i ; θ r,φ r,x r,y r ) da i (5) A i 3

4 Figure 2: illumination, global view form of BRDF. which can be rewritten as the ratio of the radiance at that point to the uniform incident irradiance. f(θ i,φ i ; θ r,φ r ; x r,y r ) dl r (θ i,φ i ; θ r,φ r,x r,y r )/de i (θ i,φ i ). (6) But in remote sensing of natural surfaces, it is common to speak of the BRDF of a land cover type. So the BRDF at a point is usually not of interest and we want the value for an extended region. Thus to complete the definition for structured surfaces, consider Fig. 2, which shows an illumination area, A i, and a region of interest, A r. Let the global illumination, global fieldofviewdefinition of BRDF be the mean value taken over A r. Because the illumination is defined to be uniform, the mean need only be taken over the reflected radiance. We will use the subscript s (for structure) to distinguish this from the NBS version, f s (θ i,φ i ; θ r,φ r ) dl r (θ i,φ i ; θ r,φ r,x r,y r )/de i (θ i,φ i ). (7) For a complete BRDF definition, the region of interest, A r,mustbedefined. There are several ways to do this. The actual definition will depend on the nature of the application. Perhaps the most straightforward is to consider it to be a deterministic region. It is not necessary for it to be contiguous or have stationary statistics, but these properties are useful for measurements. The region could be a physical region on the Earth, such as from a segmentation of one land cover type, or it could be a modeled region. The BRDF is then a deterministic quantity which, in principle, can be measured by measuring the mean radiance. Another approach is to consider the region to be realizations of a random process. This is sometimes used in BRDF models. We still have a deterministic BRDF, however, which we will define to be the expectation of the mean BRDF of each realization. This expectation is not a random variable (but estimates of it will be). Finally, our goal is to make the BRDF of a region a unique value, and intrinsic to the surface. We accomplish this by including in the definition that the BRDF either is taken at an instant of time for an unchanging surface, or is a sufficient time average for a changing surface. Further, we define it at a particular wavelength and at an incident and detected polarization state, with infinitesimal solid angles and infinite source and detector distances, and with no atmospheric effects above the reference plane. Real measurements, on the other hand, require nonzero power, and we cannot usually measure without the atmosphere, so we must violate all of the ideal aspects of the definition to some extent. In addition, the BRDF is usually estimated by measuring a spatial sample of the region, and therefore the estimate will be a random variable. The measurement error is related to the properties of the radiance variability and the size of the sample. It turns out that the study of BRDF variability with scale and other factors is emerging 4

5 as a tool for surface characterization [23]. But by definition, the underlying average BRDF is intrinsic and deterministic. Next we will show that it also satisfies the usual invariance properties. 3 Invariance Properties 3.1 Reciprocity We would like to determine if our proposed definition of the BRDF of structured surfaces exhibits the invariance properties associated with the flat surface versions. Indeed, optical reciprocity of structured surfaces is a subject of some debate in the remote sensing community. There are a variety of reciprocal relations in optics most are attributed to an early statement of the principle by Helmholtz. Contemporary relations include reciprocity in radiative transfer [24], [25], obstacle scattering, or S-matrix reciprocity, where the positions of the source and detector are switched [26], [27], and radiance-intensity reciprocity through a pair of areas [9], [28]. There can be no general proof of any of these forms of reciprocity because there are known exceptions. For instance if the system is not invariant under time reversal, it can exhibit non-reciprocal behavior [29]. We can create a macroscopic time reversal asymmetry in an optical system using a magnetic field. An example is the Faraday isolator. We cannot, however, invoke reversibility to address invariance in surfaces with macroscopic structure. We must consider the geometric issues of BRDF as well. Structured surface BRDF is not a fundamental physical quantity, and its properties are closely linked to its definition. It turns out, however, that for structured surfaces it is the obstacle scattering reciprocity from S-Matrix symmetry that is the most direct and clear basis for BRDF reciprocity. That is because obstacle scattering reciprocity applies to objects with structure. In arguments similar to those connecting detailed balance to flat surface reciprocity, Shelankov and Pikus [26], Carminati, et al. [27], and Greffet, et al. [18] provide good reviews of the basis of obstacle scattering reciprocity. In brief, we can conclude that this arises from the symmetry of the scattering matrix (S-matrix) of an obstacle. That, in turn, results, as before, from materials that obey time reversibility which is also the basis for the Onsager statistical relations. In physical optics, obstacle scattering reciprocity applies for any point source and point detector position, but under the geometric optics assumptions of radiometry, reciprocity only holds for infinitely large source and detector distances. In brief, this is because at finite distances, diffraction affects the relation between amplitude and power. The objective is to connect obstacle scattering reciprocity in the geometrical optics regime to BRDF reciprocity by considering the finite region of interest of a structured surface to be a separate obstacle. What remains to make this connection remains an exercise in geometry and radiometry. Consider a finite area of definition, A r, of a structured surface to be the ground field of view (GFOV) of an distant detector, and take this portion to be uniformly illuminated by a distant source. We can consider the GFOV to be a separate obstacle with some sort of structure between the top and the bottom planes. In a sense, the detector field of view defines an object that is the region of interest. This concept is depicted in Fig. 3. To begin with, it turns out that it is useful to define an effective average radiance. One one hand, for diffuse surfaces, and a detector with a high spatial resolution, such as an imaging device, the radiance pattern exists near the reference plane, and when the detector is far enough away and focused on the plane, the average radiance is simply the area-weighted average of the point radiances in the recorded pattern. However, for detectors with a larger IFOV it is convenient to define the effective average radiance as the surface radiance that would have given rise to the same signal at the detector. Based on a ground field of view, A r, and the solid angle subtended by the detector as seen from the surface, ω d,wecanwriteforthe detector power, dφ d = dl r A r cos θ r ω d, (8) so rearranging, the effective average radiance is then dl r = dφ d A r cos θ r ω d. (9) 5

6 Figure 3: between obstacle scattering reciprocity and BRDF reciprocity. or equivalently, in terms of irradiance at the detector and the solid angle subtended by the projected surface area as seen from the detector, ω r,itistheratio dl r = de d. (10) cos θ r ω r This expression clearly is the same as averaging a radiance pattern but it will be useful in what follows. We will take the source and detector solid angles to be arbitrarily small and their distances at positions 1and2,D 1 and D 2, to be arbitrarily large with respect to the finite sized area. With a source of intensity I 1 at position 1 and an irradiance E 2 detected at position 2, we can compute the BRDF of the region as follows. The incident irradiance from the source in the plane of the surface is E i = I 1 D1 2 cos θ 1. (11) The cell is structured so it does not have a constant radiance. As derived earlier, we can define an alternate average radiance of the region, L r, through the irradiance it causes at the detector, obtained from E 2 = L ra cos θ 2 D2 2, (12) so the average radiance is L r = E 2D2 2. (13) A cos θ 2 Andsowithourdefinition, for the forward direction, the BRDF is given by f 12 = L r = E 2 D2D (14) E i I 1 A cos θ 2 cos θ 1 For the case where the source and detector positions are reversed, we have a source of intensity I 2 at position 2 and an irradiance E 1 detected at position 1. This time, the irradiance from the source in the plane of the surface is E i = I 2 D2 2 cos θ 2. (15) 6

7 Figure 4: Definitions for the edge effect analysis. As before, the average radiance from the area generating E 1 is L r = E 1D 2 1 A cos θ 1. (16) And so this time f 21 = L r = E 1 D1D (17) E i I 2 A cos θ 1 cos θ 2 But because obstacle scattering reciprocity implies E 2 /I 1 = E 1 /I 2, Equations (14) and (17) are equal, and so the BRDF is reciprocal, f 12 = f 21. (18) So we have a simple and direct proof of BRDF reciprocity for structured surfaces based on existing electromagnetic scattering theorems. These theorems apply to a wide range of materials including lossy dielectrics, metals, and so forth. The result also agrees with that of Snyder [8], who showed by a radiometric argument that a structured surface comprised of reciprocal facets in any configuration will have a reciprocal BRDF. In addition, this result agrees with the derivation by Greffet, et al. who studied the reciprocity properties of a generalized BRDF with coherent effects using an angularly distributed wave field [18]. Their result can be reduced to zero-wavelength BRDF, but they used a local definition of the BRDF and this caused some difficulties interpreting the results for surfaces that have volumetric scattering. A remaining issue is edge effects. For the obstacle scattering argument, the region of interest has exposed edges not representative of the BRDF definition. There are several cases to consider. First of all, we can take the surface to be a periodic array of cells that do not interact, and analyze a single cell as done by Li, et al. [30] and others. The cell walls can be black or reflective, depending on the nature of the modeled surface. On the other hand, if the surface cannot be accurately represented in this manner, we invoke the requirement that the edge scattering region have finite support shown as the hatched region, A,inFig.4. The area of the region of interest, A r, is proportional to the average radius, r 2, but the edge margin area is proportional to r for a fixed r. The ratio of the edge contribution to the total radiance diminishes at the rate 1/r. Therefore, in principle, we can replicate any region until the relative contribution from the margin to the average radiance, and thus to the BRDF, is arbitrarily small. Finally another edge treatment is to consider the hatched region, A, as a separate obstacle that must also exhibit reciprocal behavior. If the regions A i and A are reciprocal, then the region of interest, A r, is reciprocal if the scattering along its boundary into and out of A is symmetrical. Finally, given an arbitrary, passive surface structure, it is interesting and instructional to attempt to devise gedanken experiments which violate reciprocity and which challenge the definition of BRDF. There have been numerous configurations postulated to violate reciprocity but none appear to have survived a careful analysis. A recently proposed counter-example is cells comprised of folded telescopes [30]. It is argued that all of the light in the conventional forward direction passes though the system. When the direction is reversed little of it passes back the other way, so there will be a very large difference in BRDF 7

8 between the two directions. In fact, as with many similar proposals, there is no BRDF reciprocity violation [31]. 3.2 Kirchhoff s Law and Energy Conservation Another invariance property associated with the BRDF is Kirchhoff s radiation law, that equates hemispherical emissivity and absorptivity. Some arguments for directional equality have appeared that cite a thermodynamic form of detailed balance [32], but this form does not appear to be sufficient. For instance, a Faraday isolator does not exhibit the directional form of Kirchhoff s law. It can be shown by integration that surfaces with reciprocal BRDF, however, do exhibit the directional form of Kirchhoff s law [29], in other words, directional absorptivity equals directional emissivity, α (θ, φ) =ε (θ,φ). (19) ThismustbesoiftheBRDFisreciprocal,buttheconverseisnottrue. WedonotknowtheBRDFis reciprocal if the directional form of Kirchhoff s law holds. For structured surfaces the modified meaning of this law relates the average emissivity and absorptivity over the region for the directions specified, and by considering each point separately, it is clear that it holds. There is one invariance property that can never be violated by any surface or system. Conservation of energy constrains the integral of the BRDF over all reflected angles. For illumination from any particular incident angle, (θ i,φ i ), we have the inequality Z 2π Z π/2 0 0 f(θ i,φ i ; θ r,φ r )cosθ r sin θ r dθ r dφ r 1. (20) ThisistrueforthestructuredsurfaceBRDFforthesamereasonitistruefortheBRDFatapoint. Ifit were greater than unity, then more energy is leaving the system than is entering in a given time interval. 4 Discussion In the present study, a modification to the standard definition of BRDF was developed and we addressed the important geometrical and radiometric considerations. The resulting surface characterization was shown to be consistent and intrinsic. Most studies do use this definition implicitly. Some studies do not, and its adaptation will make any results more universal. One important conclusion is that it appears that structured surface BRDF has a unique definition that naturally follows from the existing standards. And with this definition the usual invariance properties hold. The invariance properties include reciprocity, equivalence of angular absorption and emission, and conservation of energy. These are important to consider as theoretical properties of the underlying, ideal BRDF. The use of invariance is well known in analysis [24], and these properties should be expected in models [8], but they can only be considered in measurements together with the measurement errors caused by sampling, detector noise and bias, inexact knowledge of the measurement conditions, such as solar irradiance, and inexact treatment and correction for the various geometrical constraints, such as parallax. Field measurements of BRDF are very difficult, and almost certainly will not exhibit invariance properties such as reciprocity without careful analysis and correction of these factors. The practice of filling in missing field data by the use of reciprocity is not recommended, and a better approach would be to fit the available data to a reciprocal model. References [1] S. Liang and A. H. Strahler, Summary of the international forum on BRDF, The Earth Observer, vol. 11, pp. 27, [2] Special issue on BRDF, Remote Sensing Reviews, vol. 18, pp ,

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