Interferometric synthetic aperture processing: a comparison of sonar and radar

Transcription

1 Interferometri syntheti aperture proessing: a omparison of sonar and radar M. Hayes and P. T Gough University of Canterbury, Private Bag 4800, 80 Christhurh, New Zealand mihael.hayes@anterbury.a.nz 6303

2 Interferometri aperture synthesis is an inverse problem that attempts to form an elevation map of the earth in the ase of radar or a bathymetri map of the seafloor in the ase of sonar. In both ases, a pair of transduers is onfigured as an interferometer. After aperture synthesis is performed to produe a pair of images, the height of eah resolvable satterer an be estimated using time delay estimation between the image pairs and knowledge of the system geometry. While interferometri syntheti aperture sonar InSAS seems like an obvious extension of the methods of interferometri syntheti aperture radar InSAR, the height estimation algorithms are surprisingly different. In this paper we start with the priniple of generalised orrelation for optimal time delay estimation. This filters the signals to maximise their oherene sine the auray of the time delay estimates, and thus the height estimates, strongly depends upon the signal oherene. We then onsider the fundamental differenes between InSAR and InSAS; namely the relative signal bandwidth, aperture sampling rate, and geometry and show how appliation of generalised orrelation time delay estimation leads to the differenes in how InSAS and InSAR signals are proessed. 1 Introdution Interferometri aperture synthesis is an inverse problem that attempts to form an elevation map of the earth in the ase of radar or a bathymetri map of the seafloor in the ase of sonar. With single-pass interferometri syntheti aperture radar InSAR a pair of antennas are onfigured as an interferometer, where one of the antennas or a separate antenna is used to illuminate the sene either from a spaeborne or airborne platform. Alternatively, a single transmitter/reeiver pair is flown on a similar but displaed trak; a tehnique known as repeat pass interferometry. After pulse ompression and syntheti aperture reonstrution taking are to preserve the phase two images are obtained d 1 y, r and d y, r, where y is along-trak position and r is slant range. The goal of interferometri proessing is to reonstrut a height image zx, y where x denotes ross-trak position. In priniple, interferometri syntheti aperture sonar InSAS is similar to InSAR but with a towed or autonomous sonar platform having a transduer projetor to ensonify the seafloor and a pair of hydrophone arrays to reord the sattered ehoes. The arrays of hydrophones are required to inrease the along-trak sampling rate sine the speed of sound propagation is many orders of magnitude slower than eletromagneti propagation. The ehoes from eah hydrophone array are pulse ompressed and ombined using syntheti aperture reonstrution algorithms to produe a pair of images. While InSAS height estimation seems like an obvious equivalent to the more mature InSAR, the height estimation algorithms are surprisingly different. In this paper we start with a simple sattering model and using this model onsider the orrelation oeffiient between the two signals. The height estimation problem is then posed as an optimisation problem where both the time delay and time saling need to be onsidered. We show that by ignoring the time saling that the height reonstrution problem is equivalent to time delay estimation followed by a geometri mapping to infer height. We then onsider the priniple of generalised orrelation for optimal time delay estimation. This filters the signals to maximise their oherene sine the auray of the time delay estimates, and thus the height estimates, strongly depends upon the signal oherene. Finally, 1 z d ξ r 11 r β β 1 α 1 Figure 1: xaggerated interferometer geometry for two reeivers labelled 1 and with orresponding footprints labelled 1 and, and grazing angles β 1 and β. The interferometer baseline is d, the interferometer tilt is ξ, and the loal terrain slope is α. we onsider the fundamental differenes between InSAR and InSAS; namely the relative signal bandwidth, aperture sampling rate, and geometry and show how appliation of generalised orrelation time delay estimation leads to the differenes in InSAS and InSAR proessing. Sattering model Consider an interferometer shown in Figure 1 formed by a pair of reeivers loated at x 1, z 1 and x, z with a transmitter at x i, z i. The transduers are side-looking in the +x diretion so sattering from the x diretion an be ignored. At a given instant t after a transmission, the pulse-ompressed eho signals an be modelled as the superposition of the satterers at a range t/. Thus, ignoring trivial amplitude fators and approximating the -D sattering by a line integral 1], the baseband pulseompressed omplex envelope signals sattered from a nominally flat seafloor have the form: 1 d h t = axp t r h + r i 1 r h r i exp jk 0 r h + r i dx. x 6304

3 This is a time-domain simplifiation of the Kirhhoff sattering model where the subsript h denotes the reeiver number, is the speed of propagation, pt is the baseband pulse-ompressed signal, k 0 = πf 0 / is the angular wavenumber of the entre frequeny f 0, r i is the range from the soure to a sattering point at x, zx, while r h is the range from the sattering point to reeiver h: r i = x x i + z z i, r h = x x h + z z h. 3 Assuming a quasi-monostati geometry with the transmitter loated near the reeivers; that the grazing angles β h are large so that the Kirhhoff approximation is appliable ]; that the projeted pulse length is small ompared to the range so that the Fraunhofer approximation is valid; that the surfae roughness is small ompared to the range resolution; that the loal terrain tilt is negligible this an be modelled by a loal oordinate rotation; and ignoring beam pattern effets these an be modelled as a time-domain onvolution sine the hange in grazing angle over eah footprint is small ompared to the broad vertial beam pattern, 1 beomes d h t 4 t exp jπf 0 t x h η xh ax exp jk 0 hxη zh xh x p η xh exp jk 0 xη xh dx. 4 Here hx denotes the loal height variation, x h, z h denotes the footprint entre for the h th reeiver at eah instant t, βh = β h t are the sattered grazing angles for eah footprint, os β h = x h x h r hh, 5 β ih = β ih t are the inident grazing angles, os β ih = x h x i r ih, 6 and η xh = η xh t and η zh = η zh t are the wavenumber saling fators given by η xh = os β ih + os β h, 7 η zh = sin β ih + sin β h. 8 Despite the many approximations in the derivation 4, it is worth noting that for a given time t after pulse transmission, the eho signals have been sattered from slightly different regions or footprints, a phenomenon known as the footprint shift. In addition, due to the differene in grazing angle, the footprint sizes the length of the projetions of pt are also slightly different. 3 Correlation oeffiient A ommon measure of the similarity between two signals is the orrelation oeffiient. The orrelation oeffiient at eah instant t as a funtion of delay τ between the signals is given by ρ 1 t, τ = d td 1t τ] d 1 t τ ] 9 ]. d t Assuming that the satterers are unorrelated and uniformly distributed, then the orrelation oeffiient for a small delay τ an be expressed as ρ 1 t, τ = exp jφt M f0 η x1 η x, τ where Φt is the interferometri phase given by, 10 Φt = exp jk 0 x 1 η x1 x η x, 11 and M = Mf x, τ, where { F x p x x η x p x 1 x η x1 τ } M = p x x η x dx p x1 x. η x1 dx 1 Here F x {.} denotes a Fourier transform over x evaluated at spatial frequenies f x. quation 10 shows that to maximise the orrelation oeffiient, the footprint shift x 1 x needs to be ompensated exept for the sattering point on the interferometer axis where the footprint shift is zero. It also shows that the orrelation is never perfet due to the slight differene in look angle resulting in η x1 η x. This slight saling of the footprints results in baseline deorrelation. For example, if the ompressed pulse has the form pt = ret t/t p then the maximum orrelation an be approximated by 1] D max {ρ 1 } t exp jk 0 sin β ξ 13 f0 T p D sin H sin β tan β + α os β ξ, where ξ is the tilt of the interferometer axis, D = d/ is the distane between the interferometer phase entres, H is altitude of the interferometer, and β is the mean grazing angle. This expression desribes the baseline deorrelation assuming registration of the two signals so that the footprints overlap 3, 4] i.e., when τ is hosen so that the interferometer axis is steered to point at the ommon footprint. Note, in the radar literature it is more ommon to model the ompressed pulse as a sin funtion so that the sin in 13 is replaed with a triangular funtion 5 7]. A differene see Table 1 between InSAS and InSAR espeially spaeborne InSAR is the grazing angle β measured from the horizontal. InSAS is ommonly deployed in shallow waters suh as harbours and thus the grazing angles are small, typially 5 30 degrees, where the Kirhhoff approximation is poor 8]. Airborne In- SAR uses a similar geometry but for spaeborne InSAR the grazing angles are steeper, typially degrees and thus the deorrelation is greater. Another key differene is that the relative bandwidth of InSAS systems is muh greater than InSAR apart from a few VHF airborne systems suh as CARABAS 9]. This results from 6305

4 InSAR spae air InSAS the time delay between the two signals that maximises the orrelation oeffiient: Propagation speed fast fast slow Inidene angle small large large Relative bandwidth small small large Noise reeiver reeiver environment Footprint shift small small signifiant Wavenumber shift signifiant signifiant insignifiant Table 1: Key differenes between InSAR and InSAS. The footprint shift is ompared to the range resolution while the wavenumber shift is ompared to the wavenumber bandwidth. the desire of high range resolution given the slow speed of aousti signals in water, the availability of broadband transduers, and no restritions on usage of the underwater aousti spetrum. Indeed, it is not unommon to have a sonar with an otave bandwidth; thus the produt f 0 T p is muh smaller for InSAS than InSAR. However, this has little effet on the baseline deorrelation sine the baseline D is hosen to minimise baseline deorrelation. The disadvantage of a large relative bandwidth is that aurate o-registration is required to redue deorrelation due to the footprint shift. To see this, let s assume that the look angles are equal so η x1 = η x = η x and thus there is no baseline deorrelation. The resulting orrelation oeffiient due to the footprint shift is ρ 1 t, τ = exp jφt M 0, τ, 14 where M0, τ = p x x η x p x 1 x η x τ dx p x η x. 15 dx 4 Height estimation Interferometri height estimation is a super-resolution tehnique that makes the assumption that there is only a single dominant sattering path for eah slant range thus multipath and layover are assumed negligible. ssentially the task is to estimate the terrain height z that maximises the orrelation oeffiient between the eho signals. For eah height guess z at a slant range r = t/ the footprint entres x h, grazing angles β h, and wavenumber sale fators η xh an be determined. The eho signals an then be time saled, shifted, and orrelated using ρ 1 t, τ, η = d ηtd 1t τ] d 1 t τ ] 16 ], d ηt where η = η x1 /η x. The height estimate ẑ, is the height guess that maximises the orrelation oeffiient. This an be expressed as a searh: ẑt = arg max ρ 1 t, τz, ηz. 17 z Sine this is omputationally expensive, a simpler approah is to ignore the time saling or to make a orretion based on the expeted height and to just estimate ˆτt = arg max ρ 1 t, τ, 18 τ d 1td t τ] = arg max τ d 1 t ] 19 ], d t τ where.] is the expetation operator. The delay estimate is then onverted to a height estimate through a geometri mapping. quation 19 requires averaging over an ensemble of independent ehoes. While a number of along-trak samples an be averaged, this degrades the along-trak resolution. Additional averaging an be obtained at the expense of range resolution by assuming that the signals are ergodi and performing a time-orrelation over an observation time T : ˆτt = arg max τ where u t, t u 1t, t τdt u t, t dt, u 1t, t dt 0 t u h t, t = d h t + t ret. 1 T The orrelation an be performed in the frequeny domain using U t, fu 1 t, f exp jπfτ df ˆτt = arg max τ U t, f df, U 1t, f df where U h t, f = u h t, t exp jπft dt. 3 Time delay estimation theory shows that the variane of the time delay estimate about the true delay an be redued using generalised orrelation 10]. Here eah signal is filtered by a filter response W h f or equivalently one of the signals an be filtered by W f = W 1 fw f to minimise the variane of the time delay estimates. The optimal filter response to minimise the variane is 11] γ 1 f W f = 4 ], S 1 f 1 γ 1 f where S 1 f is the ross power spetral density between u 1 and u, and γ 1 f is the ross spetral oherene: S 1 f γ 1 f = S1 fs f, 5 U fu1 f] U 1 f ] 6 ]. U f Here S 1 f and S f are the power spetral densities of u 1 and u. Note if both hannels have unorrelated 6306

5 additive noise with the same power spetrum S nn f, the ross spetral oherene beomes γ 1 f = S 1 f S1 f + S nn f S f + S nn f. 7 Assuming a rough sattering surfae with stationary statistis, the power spetral density of the pulse ompressed ehoes are proportional to the energy spetral density of the ompressed pulse signal pt weighted by the beam patterns and the footprint response. Sine the vertial beam patterns are wide to span a broad swath, the slight differene in look angle results in minimal differene in beam pattern response. The differene in look angle does, however, result in a small variation in the footprint response, known as the wavenumber shift 1]. The wavenumber shift is a narrowband approximation of the saling of the sattered wavenumber spetrum. This results in a temporal frequeny shift f between equivalent features in the eho power spetra, given by f fd H sin β tan β + α os β ξ, 8 where α is the loal terrain tilt for the footprint of interest. The wavenumber prefiltering 1] suggested to redue the InSAR phase variane is essentially a form of generalised orrelation. However, sine the frequeny shift is so small otherwise the baseline deorrelation would be unaeptably large, wavenumber prefiltering is only useful for narrowband InSAR. It is more important for a broadband InSAS system to ompensate for the wavenumber saling. Care is required with the interpolation so that the small gain is not lost by interpolation noise during the time saling. Using the spetral weighting given by 4, the Cramér- Rao lower bound CRLB for the time delay variane is given by 11] ] σ τ T πf γ 1 f 1/ 1 γ 1 f df. 9 This shows that the greater the signal bandwidth and thus the broader γ 1 f, and the longer the observation time T, the smaller the variane of the delay στ and thus the depth estimates. However, the longer the observation time T the extent of the orrelation window, the lower the time and thus range resolution. Thus there is a tradeoff between resolution and the variane of the bathymetry estimates. This translates to a trade-off between aross-trak resolution and height auray. InSAR systems are primarily narrowband and thus the CRLB of the estimated time delay using the signal phase is almost idential to the CRLB obtained using a time domain orrelation. Thus the time delay an be estimated from the phase of the Hermitian produt of the omplex baseband signals. While the footprint shift is not signifiant for narrowband signals, it should be minimised by o-registering the signals before phase estimation. However, this requires an initial estimate of the terrain height or an initial delay estimate. stimating the phase has a omputational advantage sine it avoids omputing the orrelation and a searh for the orrelation peak. The disadvantage is that the time delay estimate is ambiguous sine the phase an only be estimated modulo π and thus phase unwrapping tehniques are required to redue the ambiguity. The unambiguous time delay interval an be extended using a lower frequeny but with a degradation in auray or by employing additional reeivers multiple baseline 13]. It an also be extended by inreasing the signal bandwidth 14]. One the signal bandwidth is omparable with half the entre frequeny, ambiguities an be avoided and phase unwrapping is unneessary. The time delay estimates an be improved by ombining multiple estimates. As mentioned before, the variane of the estimate of τ is redued by averaging the time orrelations over neighbouring along-trak positions, although this redues the along-trak resolution. This is equivalent to a maximum likelihood phase filter 15] when the signals are narrowband or aurately o-registered to ompensate for the footprint shift 16]. Sine eletromagneti propagation is so muh faster than aousti propagation, InSAR systems are better sampled in the along-trak diretion than InSAS. Thus it is ommon for InSAR to average over more along-trak looks than InSAS and onversely for InSAS to average over more range samples than InSAR. When there are more than two reeivers the time delay estimate an be further improved by averaging the signal orrelations, inversely weighted by the expeted varianes 17]. An interferometer with additional reeivers also allows additional arrival angles to be estimated for eah slant-range and thus an be used to mitigate the effets of layover 18] and multipath 19]. The CRLB of the time delay estimate using orrelation depends on the the seond moment of the normalised eho energy density spetrum: β 0 = 0 f P f df 0 P f df. 30 This is related to the rms signal bandwidth β and entre frequeny f 0 through the identity 0]: β 0 = β + f So if we define the signal quality fator Q as the ratio of the entre frequeny to rms bandwidth, Q = f 0 β 3 then the seond moment an be expressed in terms of the entre frequeny and Q: β 0 = f Q. 33 Thus for narrowband systems with Q >> 1 then β 0 f 0 and hene there is little improvement performing a orrelation over narrowband phase estimation using the phase of the Hermitian produt of the eho signal omplex envelopes. This assumes that the SNR is suffiiently high so that the phase does not wrap. The disadvantage of narrowband phase estimation is that the delay estimate is ambiguous and phase unwrapping is required to redue the ambiguities. 6307

6 With InSAS it is ommon to orrelate the omplex envelopes of the eho signals rather than the real signals. This is more omputationally effiient but the delay estimate using the orrelation envelope is poor the CRLB is inversely proportional to β instead of β 0. The estimate an be improved using the phase at the orrelation peak 1]; the resulting CRLB of the delay estimate is inversely proportional to f 0. This is lose to optimal for narrowband systems but for broadband systems with low Q the eho signals should be orrelated after remodulating the omplex envelopes to the entre frequeny. 5 Conlusions The key differene between InSAR and InSAS is the relative bandwidth. As a result, the narrowband InSAR algorithms are not appliable unless the InSAS signals are sub-banded and treated as a number of a narrowband estimates. A small improvement in the orrelation oeffiient an be obtained using generalised orrelation, where the signals are filtered to maximise the oherene at eah frequeny. A slightly higher orrelation oeffiient an be obtained by time saling one of the two signals equivalent to remapping the slant range data to a ommon ground plane and thus ompensating for the wavenumber spetrum saling. However, unless a searh is performed, this requires an estimate of the terrain height. Referenes 1] G. Jin and D. Tang, Unertainties of differential phase estimation assoiated with interferometri sonars, I J. Oean. ng., vol. 1, pp , Jan ] D. Jakson and M. Rihardson, High-frequeny seafloor aoustis. Springer, ] X. Lurton, Swath bathymetry using phase differene: Theoretial analysis of aoustial measurement preision., I J. Oean. ng., vol. 5, pp , July ] J. S. Bird and G. K. Mullins, Analysis of swath bathymetry sonar auray, I J. O. ng., vol. 30, pp , Apr ] F. K. Li and R. M. Goldstein, Studies of multibaseline spaeborne interferometri syntheti aperture radars, I Trans. Geosi. Remote Sens., vol. 8, pp , Jan ]. Rodriguez and J. M. Martin, Theory and design of interferometri syntheti aperture radars, I Pro. Part F, vol. 139, pp , Apr ] R. Bamler and P. Hartl, Topial review: Syntheti aperture radar interferometry, Inverse Problems, vol. 14, pp. R1 R54, Aug ] J. M. Restrepo and S. T. MDaniel, Two models for the spatially ovariant field sattered by randomly rough pressure-release surfaes with Gaussian spetra, J. A. So. Am., vol. 87, pp , May ] L. M. H. Ulander and P. Frölind, Ultra-wideband SAR interferometry, I Trans. Geosi. Remote Sens., vol. 36, pp , Sept ] C. H. Knapp and G. C. Carter, The generalised orrelation method for estimation of time delay, I Trans. Aoust., Speeh, Signal Proess., vol. 4, pp , Aug ] A. H. Quazi, An overview on the time delay estimate in ative and passive systems for target loalization, I Trans. Aoust., Speeh, Signal Proess., vol. 9, pp , June ] F. Gatelli, A. M. Guarnieri, F. Parizzi, P. Pasquali, C. Prati, and F. Roa, The wavenumber shift in SAR interferometry, I Trans. Geosi. Remote Sens., vol. 3, pp , July ] G. Corsini, M. Diani, F. Lombardini, and G. Pinelli, Simulated analysis and optimization of a three-antenna airborne InSAR system for topographi mapping, I Trans. Geosi. Remote Sens., vol. 37, pp , Sept ] R. Lanari, G. Fornaro, D. Riio, M. Migliaio, K. P. Papathanassiou, J. R. Moreira, M. Shwäbish, L. Dutra, G. Puglisi, G. Franeshetti, and M. Coltelli, Generation of digital elevation models by using SIR-C/X-SAR multifrequeny two-pass interferometry: The tna ase study, I Trans. Geosi. Remote Sens., vol. 34, pp , Sept ] D. C. Ghiglia and M. D. Pritt, Two-dimensional phase unwrapping: Theory, algorithms, and software. New York, NY, USA: John Wiley and Sons, ] T. O. Sæbø, R.. Hansen, and H. J. Callow, Height estimation on wideband syntheti aperture sonar: xperimental results from InSAS-000, in MTS/I OCANS 007, ] M. P. Hayes, Results of a multiple-baseline interferometri syntheti aperture sonar in shallow water, in IVCNZ 06, Image and Vision Computing New Zealand, Great Barrier Island, ] F. Lombardini, M. Montanari, and F. Gini, Refletivity estimation for multibaseline interferometri radar imaging of layover extended soures, I Trans. Signal Proess., vol. 51, pp , June ] M. P. Hayes, Multipath redution with a three element interferometri syntheti aperture sonar, in Pro. Seventh uro. Conf. Underwater Aoust. CUA, Delft, The Netherlands, pp , CUA, July ] A. W. Rihazek, Priniples of High-Resolution Radar. New York: M Graw-Hill, ] T. O. Sæbø, R.. Hansen, and H. J. Callow, Height estimation on wideband syntheti aperture sonar: xperimental results from InSAS-000, in MTS/I OCANS 05 USA, Washington DC, USA, Sept