Tomographic formulation of interferometric SAR for terrain ...

726 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 33, NO. 3, MAY 1995

Tomographic Formulation of Interferometric SAR for Terrain Elevation Mapping

Nick Marechal, Senior Member, IEEE

Abstract-Topographic mapping with spotlight Synthetic Aper- ture Radar (SAR) using an interferometric technique is studied. Included is a review of the equations for determination of terrain elevation from the phase difference between a pair of SAR images formed from data collected at two differing imaging geometries. This paper builds upon the systems analysis of Li and Goldstein in which image pair decorrelation as a function of the “baseline” separation between the receiving antennas was first analyzed. In this paper correlation and topographic height error variance models are developed based on a SAR image model derived from a tomographic image formation perspective. The models are general in the sense that they are constructed to analyze the case of single antenna, two-pass interferometry with arbitrary antenna line of sight, and velocity vector directions. Correlation and height error variance sensitivity to SAR system parameters and terrain gradients are studied.

I. INTRODUCTION HIS PAPER concerns topographic mapping with Syn- T thetic Aperture Radar (SAR) using an interferometric

technique requiring two complex valued images formed from data collected at differing imaging geometries. The so-called INSAR or IFSAR technique has been studied since the mid 1970’s and currently is a subject area of considerable interest and activity. This is because such systems have a unique, dayhight, all weather data collection capability from which accurate elevation maps over large swaths are formed via automated data processing. As an early example [2] presented INSAR derived contour maps from an airborne two X-band antenna SAR system. As another example [3], published results of an airborne two L-band antenna system in which reasonable agreement with USGS topographical data was demonstrated. In 1988, [4], presented INSAR results from SIR-B, the single L-band antenna shuttle imaging radar, for data collected from so-called crossed orbits. Included in that work is a discussion of imaging geometry constraints and image cross correlation. In 1990, [l], published a study of spacebome INSAR, including a system error analysis, image cross-correlation modeling and data analysis results using data collected in multiple passes by SEASAT, a single antenna spaceborne L-band system. Reference [ 5 ] presented an IN- SAR system error analysis which includes consideration of a multitude of error sources. In 1992, [6] augmented the correlation model developed in [ 11, gave consideration to

general scattering surfaces, and presented a K,-band spaceborne INSAR system design example. In the same year, [7] published results obtained with TOPSAR, which is a two C-band antenna INSAR, integrated into the NASNJPL DC- 8 AIRSAR system. That paper includes a discussion of the elevation estimation errors observed and an analysis which decomposed those errors into those attributed to unmodeled aircraft motion and those statistical. In 1993, [8] published a paper regarding multiple-pass interferometry with a Convair 580 airborne SAR operating at C and X bands. That paper reported experimental results which demonstrate the potential for determination of subtle scatterer displacements on the order of millimeters occumng between data collection passes.

The theoretical development of the INSAR equations contained in [ll-[8] is based on a conventional strip map SAR perspective. In this paper we review the derivation of the equations for determination of terrain elevation from the phase difference between two registered spotlight SAR images from an alternative tomographic imaging perspective. Generally speaking, tomography’ is a generic term used in the context of object reconstruction (image formation) from projections or slices (data). It appears not only in the context of medical imaging, for example [ 121 regarding X-ray tomography, but in other areas such as reflection seismology [13], and physical oceanography [14]. In the case of medical imaging with X- rays, the data derived from measurements are the line integrals of the attenuation coefficient of the imaged tissue between the X-ray source and detector. Whereas in radar imaging, the data are fundamentally the integrals of the terrain reflectivity density over the wavefronts incident upon the illuminated scene. In the far-field approximation the wavefronts are planar and the measurements made corresponding to a single transmitted pulse are a sampling of the reflectivity density integrated over planes orthogonal to the line-of-sight direction. In either example, the data derived from the measurements are projections of the objecthmage which is to be reconstructed. In that sense the generic term tomography is applicable to radar imaging as well as other areas.

An explicit tomographic formulation of spotlight SAR was published in [ 151 and is intimately related to the polar format processing approach to image formation introduced by [ 161. The polar format processing approach is known to include a solution to the range cell migration problem and the ap-

Manuscrjpt received June 25, 1993; revised July 5,1994, This research was plicability Of that technique t’ strip map SAR’s via supported by the Air Force under contract F04701-88-C-0089. subpatch processing is discussed in the survey article published

The author is with The Aerospace Corporation, Radar and Signal Systems

IEEE Log Number 9409905. section. Department, Mail Stop M4/927, El Segundo, CA 90245-4691 USA. ‘Tomography is derived from the Greek word tomes meaning slice or

0196-2892/95$04.00 0 1995 IEEE

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MARECHAL: INTERFEROMETRIC SAR FOR TERRAIN ELEVATION MAPPING 121

in [17]. It is also related to the so-called Fourier reconstruction method, found in medical imaging literature [12], in the sense that both methods are based on similar theoretics and each involves a polar-to-rectangular interpolation to exploit the computational efficiency of the fast Fourier transform. Another approach to radar image formation known as convolution (or filtered) backprojection is explored in [IS] and is a common reconstruction method used in X-ray tomography [12]. The [19], [20], respectively, on radar imaging also contain similar fundamental results and observations to those mentioned above. In 1992, [21], [22] presented a three4imensional (3D) tomographic formulation of spotlight SAR in which it is observed that the twdimensional (2D) complex-valued radar image is essentially the line integral of the 3D reflectivity* in the direction normal to the slant plane, a key fact which is exploited in this paper.

With respect to INSAR, the results presented here are most closely related to those published in [l], [6]. In particular, analogous equations concerning image pair registration, cross correlation, and topographic height error variance are derived and dependencies on SAR system parameters and terrain are explored. Local image registration is a key data processing step resulting in a conjugate image pair product whose phase contains the topographic height information. Image cross correlation is significant as it bears upon the local signal-to- noise ratio of the image pair product from which the phase is estimated. Image pair decorrelation as a function of the baseline separation between the receiving antennas was studied in [I]. The correlation model developed in [ l ] is extended in [6] to include terrain slope dependencies. In this paper we build upon those analyses, although we choose a 2D Fourier domain characterization of decorrelation in terms of the support (Le., nonzero region) intersection of the associated SAR frequency response functions. It is also worth mentioning that most recently, a similar spotlight INSAR formulation has been independently written in [9]. That paper is similar to this in that a 2D Fourier domain correlation characterization is also adopted and the analysis supports general two-pass imaging geometries. This paper differs from [9] in that special attention is given to terrain gradient modeling and the effects of imaging mountainous regions are studied in detail.

Regarding image pair correlation, [l], [6], [9], and this paper are related to [23], [24], concerning range resolution enhancement with multiple-pass SAR. In [23] and [24] it is shown that the range frequency response support associated with one image will in general be “shifted” from the support of the other in the case that the two passes occurred at differing “off nadir” angles. In that case, the union of the two range frequency response supports spans more range bandwidth than either support alone. Having estimated the pixel to pixel phase difference between the image pair, which is possible when the range frequency response supports intersect (i.e., the image pair is correlated), resolution improvement is achieved from a coherent sum of the two images. The “spectral shift” can alsa be deduced from the image domain characterization of decorrelation given in [l], [6] following representation of

*That is, the complex reflectivity density as a function of 3D position

the associated image impulse response functions in terms of the inverse Fourier transform of the corresponding frequency response functions. The 2D Fourier domain characterization of image pair correlation described here and in [9] is a natural extension of the 1D Fourier model published in [23], [24]. One concludes that the fundamental idea proposed in [23] applies and azimuth as well as range resolution would be improved by a coherent summation of two complex images when, for example, one of the two associated slant planes is an azimuthal as well as grazing angle rotation of the other. Part of what differentiates this paper from the earlier ones

is that the correlation model is derived to accommodate general two-pass imaging geometries such as the crossed orbit (variable baseline) case and in that respect is generalization of the corresponding analysis published in [ll, [6], 1231, and [24l. The generalization is relevant because single antenna radar data, suitable for interferometry, are collected by spaceborne systems (e.g., ERS- 13), despite the fact that complex-valued image pairs from such systems are subject to varying degrees of decorrelation due to time-dependent changes in scatterer dielectric properties, for example. Furthermore, not all future spaceborne SAR systems (e.g., SIR-C4) include a two- antenna-on-a-single-platform concept. In any case, the use of multiple-pass INSAR for the purposes of monitoring subtle environmental change is very likely to grow in the future [25]. Therefore, an image cross-correlation model which supports multiple-pass SAR systems for general imaging geometries and terrain should continue to be relevant.

The correlation and topographic height error variance pre- diction models derived here are a function of SAR imaging geometry, system parameters, and terrain gradients. With the models, spatially variant correlation and height error variance maps are computed as a function of position on a prototypical mountain. One observes that the previously mentioned spectral shift of the two-frequency response supports can vary significantly over a mountain, even for a two-antenna, single-pass SAR system. This sensitivity study is another aspect in which this paper differs from [I], [6]. With respect to INSAR, there are two interesting implications when imaging mountainous regions, one regards correlation restoration via bandpass filtering and the other concerns coherent multilooking. Each are discussed briefly below.

References [9], [31] have developed a technique to restore correlation by bandpass filtering each image prior to forming the conjugate image product. The filtering operation amounts to nulling the Fourier domain data in the wavenumber regions of each image which do not belong to the support intersection. Restoration of correlation, which is possible only when the frequency response support intersection is nonempty, max- imizes the signal-to-noise ratio of the image product with a corresponding loss of resolution. The effectiveness of the technique depends on the stability of the support intersection as a function of terrain imaged. Strictly speaking, the observed sensitivity of the frequency response support intersection to terrain gradients, consistent with mountainous regions, implies

3Earth Remote Sensing Satellite-] of the European Space Agency. 4Shuttle Imaging Radar-C of the National Aeronautics and Space Admin-

istration.



that a priori knowledge of the local terrain gradient is required to define the filters’ passbands for the technique to be fully effective. An example of support intersection variability is presented in Fig. 2(a)-(d) and discussed in Section IV-C. However, at the same time, correlation is improved by filtering from each image those wavenumbers which are never part of the support intersection, under expected imaging geometry and terrain gradient conditions.

The coherent multilooking technique introduced in [I] is an ad hoc method intended to improve the estimate of the 2D phase difference between the image pair. It is motivated by the desire to obtain a weighted sum of independent measurements. A conjugate image product is formed for each look pair. Then the image products are coherently summed over the number of looks and the phase difference is extracted from the sum. However, the azimuth component of the spectral shift vector varies proportionally with the azimuth component of the terrain gradient. Again refer to Fig. 2(a)-(d) and the discussion in Section IV-C. Therefore, depending on the azimuth bandwidth of each of the looks, the steepness of the terrain, and the radar viewing geometry, the look pairs may not correspond to the same azimuth wavenumber region in which case some degree of decorrelation occurs. This effect can be mitigated to the extent that the frequency response support intersection is known a priori, as then each look of an associated pair can be computed from the same azimuth wavenumber band.

The error variance model is viewed not only as a predictor, but as an objective functional to be minimized as a function of geometry and/or system parameters. Given fixed system parameters such as center-frequency, resolution and power, objective function minimization as a function of the baseline separating the receiving antennas is a method to balance the competing requirements of image correlation and system sensitivity to terrain elevation. This “optimal baseline” concept first emerged in the literature in [l] and subsequently in [6]. The presence of terrain gradients in the correlation model implies that a baseline which is optimal for ground-plane- located scatterers is suboptimal for those located on mountain slopes and conversely. The height error variance model is studied analytically in the special parallel path (nonvariable baseline) case, as considered in [l], 161. That analysis suggests that the optimal baseline (or grazing angle differential between the associated slant planes) be defined alternatively as that which minimizes the error corresponding to the steepest terrain expected to be viewed by the SAR. As in [6], optimization of the INSAR with respect to system waielength for a fixed grazing angle differential is explored as well.

Ultimately, the reflectivity (spatial) correlation and image models are the key ingredients which determine the image cross correlation and elevation error variance models. As in [l], [6], a delta correlation model is assumed for the reflectivity. The analysis approach in [I], [6] is to begin with a given image model and then proceed with the correlation calculation. Here the image model is derived first, then the correlation calculation follows. Although the image model derivation is fundamentally similar to those in the previously published journal articles [15]-[20], the emphasis here on

the 3D position dependency of the reflectivity gives rise to an image model which is tailored for the subsequent Correlation and elevation error variance analysis. The 3D emphasis begins with recognition of the fact that the radar is essentially measuring the Radon transform of the reflectivity5 and therefore the Projection Slice Theorem applies [26], [27]. These attributes are often associated with the generic term tomography as discussed earlier and is the motivation for the title of this paper.

The remainder of this paper is organized as follows: In Section 11, the spotlight SAR image equations are reviewed. In particular, the image phase dependency on 3D scatterer location is derived. Section I11 contains a brief review of the conceptual basis for topographic height determination from the phase of a conjugate image pair product as preparation for subsequent analysis. In Section IV, the image pair cross- correlation model is developed based on the image model derived in Section 11. The topographic height error variance model is derived and studied as a function of geometry, image cross correlation, and image signal to noise in Section V. A summary is given in Section VI.

11. IMAGE MODEL REVIEW In this section we review the equations leading to an

image model expressed in terms of an integrated complex reflectivity density and phase dependhg on the collection geometry for a spotlight SAR, The derivation provides a theoretical foundation and is a means to define the notation used in the subsequent sections. Some of the essential elements of the derivation are found in [17], [19], and are based on the polar format processing approach to SAR image formation introduced in [16]. An attractive feature of the derivation which follows is that it applies to rather general radar signals, in particular a linear FM assumption is not required. As in the tomographic formulation of spotlight SAR published in [ 151, one will find reference to the Projection Slice Theorem, for example refer to [27]. The theorem states a relation involving Fourier and Radon transforms. An extension of [ 151 is presented in [21], [22], in which it is shown that the 2D complex-valued radar image is essentially the line integral of 3D reflectivity in the direction normal to the slant plane. The same result will be observed here as part of the review and it is used in the subsequent image cross-correlation model derivation.

For the moment consider reception of a single transmitted pulse s ( t ) denoted by

g ( t ) = 1 s(t - T)T(T) d7- (1)

where T ( T ) is the reflectivity density 0 integrated over all illuminated scatterers at a distance corresponding to a round- trip signal propagation time T. Fig. 1 shows the collection geometry with the SAR plaJform at position p’ with the origin of the coordinate system 0 at the image scene center. The unit line-of-sight vector for the transmitted pulse is defined by

5That is, the reflectivity integrated over the wavefronts incident upon the imaged scene.


MARECHAL INTERFEROMETRIC SAR FOR TERRAIN ELEVATION MAPPING 129

transform of the Radon transform (projection) of a function is equal to the 3D Fourier transform of the function evaluated along a line (slice) in the vector (line-of-sight) direction associated with the Radon transform. Specifically, the equality is written as I

G ( k ) = 6(ku') (10)

and k has units of wavenumber. With that theorem and a change of variables one computes the Fourier transform

Fig. 1. Slant and ground-plane geometry with a scatterer located at Z = (2, y, z ) and S A R positioned at $for a given transmitted pulse.

u' = -fl/llfl\ and the round-trip time to the scene center is denoted by TO = 2IIfll/c with c the speed of light, and I ~ T J ~ denoting the length of the vector g. For the purpose of defining where o is integrated, let S and P denote the sphere and plane

s = {?I 112 - fll = CT/2} P = {zlzG'. u'= C ( 7 - 7 0 ) / 2 } .

(2) (3)

Given this notation the integrated reflectivity density is given by the following:

T ( T ) = o(2) dS

1: CT(C(T - 70)7?/2 + 3) dP (5)

= R,a(C(T - 7 0 ) / 2 ) (6)

with dS and dP above denoting d (surface area) for the spherical and planar wavefronts, respectively. Equations (5) and (6) define the Radon transform of o in a given direction u', denoted by &,I&). The radial distance p = C(T - 70)/2 is measured from the scene center. The planar wavefront or far-field approximation above is implicit to equations above involving the Radon transform. Also we have absorbed into the definition of the reflectivity density o both antenna illumination shading and signal attenuation due to signal energy spreading over spherical wavefronts.

Consider the received pulse after range compression which is the operation of convolving the received data with the matched filter f ( t ) = S ( - t ) . The Fourier transform of the range compressed data is expressed as

fY9(4 = P ( W M 4 (7) = 1 S ( W ) l 2 i ( W ) (8)

and the range of frequencies sampled is dependent on the transmit center frequency fo and bandwidth BW

(9)

At this point, we compute i ( w ) explicitly in terms of the 3D Fourier transform of the reflectivity density denoted 6. The Projection Slice Theorem is used in the calculation. An informal 3D statement of the theorem is: the 1D Fourier

]W - 2 ~ f o l 5 2 s B W / 2 .

& ~ ( c ( T - 7 0 ) / 2 ) e - ~ ~ ~ d7 (12)

- - e--iwro -&,cJ(2w/c) 2

- - e--iwro - 2 6( 2wu'/c).

(13)

(14)

Combining (7)-(14) we observe the relationship between the 1D Fourier transform of the range compressed pulse f * g and the 3D Fourier transform of the reflectivity density o

C

f i g ( u ) e - i w ~ o = 210(w) l2~(2wi l /c ) . (15)

For convenience it is assumed that the magnitude of the transmit signal frequency response [.4(w)l is unity over the interval of frequencies defined in (9) and is zero elsewhere.

Equation (15) is arranged so that the Fourier transform of the range compressed data is multiplied by the conjugate of the complex sinusoid whose phase is a function of the round- trip time 70 to the image scene center. That multiplication step is known as motion compensation [17]. Setting k = 2w/c, the equation above shows that the range-compressed, Fourier- transformed, and motion-compensated data, corresponding to reception of a single pulse, is the 3 0 Fourier transform of the reflectivity density evaluated in the direction u' over the wavenumber segment (k- 4r/XI I 27rBW/c [16], [19]. The wavelength X = c / fo is defined in terms of the transmit center frequency fo and the speed of light c.

Operating in spotlight mode, the SAR steers the radar beam keeping it centered on the scene being imaged. Thus as the pulses are collected, the radar :he-of-sight vector u' varies according to the SAR [email protected] motion as a function of the pulse reception time sequence {tp}. Then having applied the prescribed data processing scheme on a pulse-to-pulse basis, one obtains 6(ku'(t,.)). Therefore, the 3D Fourier transform of the reflectivity density is evguated on a surface in 3D wavenumber space defined by k = ku'(t,) with the scalar IC confined to the interval defined above. That transform is written as

(4)

and R3 denotes the 3D vector space of real numbers. The image is formed by computation of a 2D inverse Fourier

transform with respect to azimuth and range wavenumbers which are defined together with the slant plane as follows. Let t , denote the center of the pulse collection period and define



the following unit vectors based on the line of sight vector u'(tc) to the scene center and SAR platform velocity v'(tc):

Define the SAR (slant-plane) image impulse response h to be the inverse 2D Fourier transform of a function which is unity (or some prescribed window shape) over the wavenumbers (kx, k,), defined by the SAR motion and transmit bandwidth for the corresponding measured radar data. Then the image

(17) GX = Z(tc) range

u'* =Gz x v'(tc)/llu', x .'(tc)II slant plane normal (18) formed is expressed as

(50, yo) = h( 20 - 2, YO - Y ) O ~ (z, Y) dz dZI. (30) J Zy =u', x Zx azimuth. (19)

Our intent is to write the Fourier integral (16) in terms of a coordinate system defined by the unit vectors above so let The significance of (27), (28), or its smoothed counterpart

3 = (2. Zz)Z, + ( 2 . u',)Z, + (3. &)Z*

k = ( k . Zz)Gz + (i. u',)CY + (L. Z*)Z*

(20) (21) (22)

(23)

In the equations above we have tacitly taken the range wavenumber variable to be a deviation about 47r/X for subsequent convenience. For simplicity, it is assumed that the SAR platform motion is such that the line-of-sight vector u' belongs to the slant plane defined by 2. u'* = 0 which implies k , = ku' . i& = 0. In most applications this assumption is reasonable. In the absence of such an assumption additional phase terms are modeled which are scatterer-location-dependent. Given the coordinate system definitions and the motion assumption, the vector inner product is expressed by

=xCz + yZy + zu'* + +

= (F + k X ) G + kYCY + k,u'*.

(30), is that the phase of an image pixel at (5 , y) depends on both the projection z = Z. GX of the scatterer location 3 onto the range unit vector Gx and the integral of the reflectivity density normal to the slant plane. Ideally, in topographic mapping applications, the reflectivity u(zu', + yu', + z&) is concentrated about some z = [(z, y) for each (z, y) so that scattering centers are not superpositioned onto the same image pixel6 as described by (27) and (28). Over sufficiently steep terrain or tree stands this assumption may not be valid. Setting superposition aside, the phase difference between two registered complex valued images will be attributed to: data collection geometry differences, changes over time in the dielectric properties or positions of the scatterers (single antenndtwo-pass system), or differing realizations of receiver noise. For topographic mapping the underlying assumption is that the image phase differences are due primarily to known collection geometry differences and system noise.

(24) 111. IMAGE PHASE DIFFERENCE AND ELEVATION

The principal objective of this section is to briefly review -~

and the integral (16) is rewritten as the conceptual basis for topographic mapping based on the image phase difference measurement given in (32) below, and provide some error variance relationships which are used later in this paper. In the previous section we observed, (27), (28), and (30), that for a scatterer at location 3 the phase at slant-plane image location (z, y) = ( 2 . u', ,Z. GY) contains a smoothed version of the phase

e ( k , , IC,, 0) = u(z, y, z)e--i4.rrxlx s,, (25)

(26)

. e- i (zkz + Y h ) d z dy d3:

= s,, e - i ( z k z + y k ~ ) ~ s ( z , y) dy dz

r (31) .on 47r

$I(.'; GX) = --3. x Zx.

Then given two registered complex-valued images, the image phase difference is a smoothed version of

(27)

(28) -e-i41rx/x - u(z& + yZ, + zGZ) d z .

is useful for the image cross-correlation analysis of Section IV. The notation of (28) facilitates calculation of the statistical expectation of the product of two such line integrals in the case that the corresponding imaging geometries and thus the unit vectors Zz . . . are different.

Setting aside the finite bandwidth or wavenumber limits, the image ui (z, y ) is obtained by inversion of the 2D Fourier transform (26) with respect to the wavenumbers (kx, ky) in which case

where the unit vectors Zjx , u'j,, G j 2 , j = 1 , 2 are defined as in (17)-( 19) for the two collections.

The approach taken in this paper is to express the scatterer location 3 in terms of the orthonormal basis GlX, ill,, u'l, and for each (z1,yl) = (3.u'1,,3.Z1,) determine z1 = [ ( z l , y ~ ) . This gives the relative 3D location of the scatterer in terms of the slant-plane coordinate system as the ordered triple [zl , yl, z1IT whose origin is at the scene center. A coordinate transformation is then applied to express the scatterer location

a-i(z,y) = us(z,y) (29)

and the image is the integral (27) and (28) of the complex reflectivity density over the line orthogonal to the slant plane.

in terms of any desired ground-plane system defined by the orthonormal basis C3,, &,, u'3* giving the triple [z3, y3, z3IT

6The effect is often called layover.


MARECHAL: INTERFEROMETRIC SAR FOR TERRAIN ELEVATION MAPPING 73 1

from which relative topographic height z3 and correct ground- plane location (a3,ys) are obtained. This approach to com- puting a scatterer's ground-plane location compensates for a nonlinear image distortion known as foreshortening due to the nonlinear variation in terrain elevation.

Using the orthogonality of the unit vectors one computes 4w

4(Zl,Yl) = - +(I - u'zz . u'lZ>.l

- (u'zz . u'1y)Yl - (u'ZZ . u'l*)zlI. (33)

With the phase function above, the scatterer's displacement z1 = <(XI, y1) normal to the slant plane is given according to

1 21 = -4(Z1,Y1) + (1 - u'zz . u'l,)Zl - (u'h . u'1y)Yl

. (ZZz . u'lz)-l. (34)

[L Conceptually, determination of z1 follows from an estimate of the phase function 4(q ,y1) . In practice, the phase function is determined by a 2D phase unwrapping algorithm applied to the conjugate image product uliazi, where the pixel locations from the respective images are assumed to correspond to the same scatterer. Examples of phase unwrapping techniques are found in (281-[30], and references cited therein.

Notice that the coefficient of z1 in (33) is 4w(u'zz . u'lZ)/X and when it is small there is little phase sensitivity to scatterer displacement normal to the slant plane. That coefficient tends to zero as the two imaging geometries become identical. Another perspective is that the coefficient becomes a divisor in (34) which amplifies the error component e6 in the estimate of the phase function 4(z1, yl). Let e,l denote the the error in estimate of the slant-plane displacement zl. Then from (34) one observes the fundamental error variance relation

(35) x 2

var(ez1) = [ 4Tu'2r . u'lz] var(e4).

A change of coordinate transformation provides a mapping of errors in the slant-plane coordinate triple [q, y1, z1IT into the ground-plane triple [x3, y3, z3IT. Of particular interest is the error variance of the topographic height estimate z3 as a function of the error variance of the phase estimate. The topographic height is expressed in terms of the slant-plane coordinates [q, y1, z1IT by

z3 = (zlu'lz + ylu'ly + zlu'lz) ' u'3z. (36)

Therefore, from (35) and (36) the variance of the error e23 in topographic height is given by

where in summary

= slant plane unit normal vector, image 1 (38) (39) 22, = slant plane unit range vector, image 2

u'3* = ground plane unit normal vector (40)

also refer to (17)-(19). This relation will be used in Section V where elevation error variance as a function of image cross correlation is studied.

Iv. IMAGE PAIR CROSS CORRELATION

In this section a general cross-correlation model is derived and studied. The equations, developed in Section IV-A, quan- tify correlation dependencies on the SAR system parameters, imaging geometry differentials, and terrain. Development of the correlation model here follows the same path as that in [l], [6]. Given an image model, for example (27), (28), and (30), one computes the cross correlation assuming that the complex-valued terrain reflectivity is spatially delta-correlated. A discussion of the physical assumptions underlying the delta- correlation model is given in [6]. While the path is the same, the details of the development are different. As an example, the image model developed here is expressed in terms of the slant-plane coordinate system defined by the unit vectors Gz, ZY , Zz , (17)-(19), whereas a ground-plane system is used in [l], [6]. The choice of the slant-plane system is motivated by the image model (27) and (28) being essentially the line integral of the reflectivity normal to the slant plane, a fact which is used in the correlation model development. Other differences are attributed to the two-pass, crossed-track imaging geometries which are supported in this spotlight SAR oriented analysis. In Section IV-B we study the special case of parallel passes, where the associated azimuth unit vectors are equal. A reduction in the number of terms in the correlation model is observed and results in an expression which is consistent with the correlation models developed in [l], [6]. As an example, the general correlation model developed in Section IV-A is evaluated over a prototypical mountain to produce a spatially variant correlation map in Section IV-C for a two-pass, crossed-track case.

A. Correlation Model Derivation and Discussion

Referring to (30), the cross correlation between two slant- plane images oli(zol, yol) and u~i(x02, yo2) is defined by the expectation

(W(ZO1, Y01)azi(xo2, Yo2))

= s,, s,* qzo1 - 21, yo1 - Yl)h(Z02 - x 2 , Yo2 - YZ)

x ( ~ i s ( ~ i , y i ) ~ z s ( ~ z , y 2 ) j dxzdyzdxi d ~ i . (41)

Define the vectors

z1 = x. il, + Ylu'l, + ZlCl* 2 2 = x2u'zx + y/au'zy + Z2u'ZZ

(42) (43)

and for the purpose of analytic modeling assume the following clutter correlation model:

(44)

In the 6-correlation model (44), the clutter variance ~3~ has units of reflectivity squared per unit volume squared and lzlyZz represents the product of the physical correlation lengths of the reflectivity. The product of the clutter variance and the correlation lengths has units of reflectivity squared per unit volume which is denoted by

(o(zl)a(&)) = u3c(z l )L~yzz~(z l - &).

u'(z1) = (T3c(z1)121ylz. (45)



Given that we have chosen to parameterize the terrain surface in terms of (21,yl). the integration (41) with respect to the variables xz and y2 is done first. In what follows we use the definitions (27) and (28) of as, the definition (32) of the phase function #I corresponding to the conjugate image product, and the clutter correlation model (44)

most of its mass is concentrated in a neighborhood where its argument is zero. That observation is the basis for the following local approximations to the terrain surface function z1 = c(s1,yl) in a neighborhood of (aol,y01) and subsequently the phase function 4 ( ~ , 91, c) as well as ~ ~ ( 2 1 , yi, C ) and ~ ~ ( 2 1 , y1, C ) to facilitate analytic calculation of (54). The local linearization of the surface is

with the abbreviated notation defined

(56) aco ac - = -(~01,Y01). dY dY = ei+ L, (T1(3l)h(Z02 - ~Z(%), YO2 - Y2(21)) dz1.

(47) In the calculation involving the S function above, the

equality 3 2 = Zl holds when the following linear system of

Next define the difference A4 as follows:

A ~ ( Q , y1) = 4 ( q , y1, [(XI, 91)) - ~ ( Z O I , Yoi, c(Zoi , 2/01))

equations (change of coordinates) holds:

"2(31) = (Xlu ' l , + Ylu ' ly + Zlu'l,) . u'2z

YZ(31) = (nu'1, + Ylu'ly + Zlu'l*) . u'2y

z2(31) = (Zlu'l, + Y l u ' l , + Zlu'l,) . z2*.

Given the calculation (46) and (47), the cross-correlation expression of (41) is reduced to

(mi(2o1, Yo1)*2i(aoz, Y02))

. h(Z02 - 22(31), Yo2 - YZ(21))

x ei~(z1~y11z1)a'(31) dzl dzl dyl. (51)

At this point in the analysis it is assumed that the clutter brightness density a' is concentrated in a neighborhood of the the terrain surface and is zero elsewhere. The surface is parameterized as a function of the slant-plane image variables z1 = c(z1,yl). Those considerations and (45) lead to the model

(52) 4 2 1 ) = ao(zl,Yl)S(zl - C(Z1,Yl))

where ao(z1 y1) is defined as the integrated quantity

ao(zl,yl) = zZzylz /03c(21~yl:zl)dzl (53)

and has units of reflectivity squared per unit area.

of integrations in (51) by one giving The S-function reflectivity model (52) reduces the number

( ~ l i ( ~ O 1 , Yo1)Fzi(ao2, Y02))

= L2 h(zo1 - 21, YO1 - Y1)

.h(Z02 -22(21,Yl,c),Y02 -Y2(a,Y17e))

x ei4(zlrYlrC)aO(Z1 , 91) dXl dYl. (54)

With unity weighting of the system frequency response over a rectangular bandwidth, the corresponding impulse response h(z , y) has a separable [sin ( z ) / x ] x [sin (y)/y] character. So

where the constants a and by

in the equation above are defined

a = - - 1 - u'2,. ill, - (&,. u'lr)- a"] dX 4x [ 4T x [ dY

(60)

wy) - (ZZ, - u ' ~ , ) ~ ] (61) p = - - -(G2,. - x

and are the key components of the "spectral shift" of one frequency response support relative to the other, discussed later in this subsection.

The linearizations for the arguments x ~ ( q , y 1 , < ) and y2(21,y1, c) are determined by the approximation (55) and the system of (48) and (49)

2 2 (u'1, . 22,)Zl + (u'ly . u'2$)Y1 + (Zl, . u'2,)

(62)

(63)

- $01) + -(Y1 - YOl) 1 aY - - a(z1 - 201) + b(y1 - 901) + p

Yz (u'lz . u'2y)z1+ (ZlY . u'2y)y1 + (GI, . u'2y

= 4 x 1 - 201) + d(y1 - Y O l ) + v and the constants a, b, c, d, p, v are defined as

X O

X O

X O X O

a = (u'l, . u'2x) + ( u ' l z . u ' 2 z ) K

b = (u'l, . u'2,) + (GI, . &,)- dY

c = (u'l, . u'2y) + (GlZ . Gy)- 32

d = ply . u'2y) + (ill, . u'zy)- aY



P = (x0lGlz + Y O l G y + COGZ) .Gz, (70) (71)

Notice that ( p , v) in (70) and (71) is the scatterer's 3D location (zoi,yo1, Co) projected onto the second slant-plane image coordinates.

Given the local approximations above, we continue the calculation of the integral in the correlation (54)

integrals analogous to (81) are computed explicitly giving an image domain correlation characterization. A virtue of the image domain characterization is that decorrelation resulting from image misregistration, as well as support shifting, is given by an evaluation of either integral in (8 1). The frequency response supports are defined by

v = (xolGlz + YOlGl, + C0Glz) * Gzy.

N e ~ b ( ~ o l , Y o l , C o ) e-i(ar+oY)h(x, ?/) J,. -

x h(xoz + a x + by - p,yoz + CZ+ dy - v) x ~ ~ o ( x o i - 2, YOI - y) dx dy. (73)

Next the interpretation and computation of the correlation equation is improved with the following matrix and vector definitions:

(74)

?7 = [x, YIT (75) Go = [XOl, y01IT (76) 20 = [xoz - p, yo2 - .IT a = [a, PIT (78)

(77)

then the correlation equation is rewritten compactly as

( m ( x 0 1 , Y O l ) ~ Z i ( ~ 0 2 , YOZ))

N ei4(z013Y017c0) L2 e-i".ah(f)h(Ay'+ ZO)CO(& - y') dy'

(79)

N - ao(go)ei4(zo1 ? Y O l , C O ) l2 e-i"'ah(gh(Ay'+ 20) dG.

(80)

In the equation above, the CTO is taken to be constant in the neighborhood of GO. Using the convolution theorem, an appropriate change of variables, and the assumption that the impulse response h is real-valued, one derives the identity

At this point, we observe from the identity that the maximum correlation depends on the integra) of the prodtct of SAR frequency response functions h(J) and &(ATE + a). For simplicity, we assume that the frequency response i(0 is unity over its rectangular support (i.e., nonzero region). Then the cross correlation is proportional to the area a, of the intersection of the frequency response supports. As discussed in the Introduction, similar Fourier domain characterizations have been observed in [9], [23], [24] whereas in [ll, 161

where an 2 or y subscript on a given vector in (82) and (83) denotes the vector's component in that direction, and the range and azimuth slant-plane resolutions Ax and A y are defined as the peak to first null distance pf thejmpulse response function h. In general, the support of h(A*E + 6') is a scaled, sheared, and shifted version of the support of A((). In the limiting case of no difference in the respective imaqng geometries, the supports are identical ( A = I and 6' = 0) and the cross correlation is unity. Examples of frequency response supports are depicted in Fig. 2(a)-(d) which are discussed in the next section.

Fro? (83) the support center of the frequency response k(ATJ + 6') is given approximately by the spectral shift vector [-ala, -P/dlT as the matrix A is nominally diagonally dominant. Examination of the definitions of a, p, a, d given in (60), (61), (66), and (69) shows that the spectral shift vector varies as 1/X. Thus smaller wavelength systems (e.g., K-band) are more sensitive to decorrelation than the longer wavelength systems (e.g., L-band) for a given imaging geometry and local terrain gradient. Total decorrelation occurs when the magnitude of the range or azimuth component of the shift vector exceeds the available SAR range or azimuth bandwidth, respectively. The range component of the spectral shift vector -a /a depends mainly on the range component of the terrain gradient vector and G2z-G1z as can be observed from (60). The azimuth component of the spectral shift vector -P/d contains the term Gzz . GlY which accounts for crossed-track imaging geometries. In the "parallel path" case, discussed in Section IV-B, the azimuth unit vectors are equal GI, = G2,, so that 22, . GI, = 0 and the azimuth component of the spectral shift vector is dependent only on the azimuth component of the terrain gradient vector and Gzz GlZ, as can be observed from

Let c, denote the support intersection area a, normalized (61).

by the system 2D bandwidth

Ax A y c, = a,- (2r)Z . (84)

Then by definitgn, c, 5 1. Local image pair registration implies Zo = 0 as can be verified with (70), (71), and the definition (77) of 20. Then from (79)-(81) and (84) the correlation is expressed as


134

1.0

0.5

5 : f i g 0.0

.: = - I

E: -0.5

-1.0

IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 33, NO. 3, MAY 1995

-

-

: -

- -

2D SAR frequency response supports 1 S L ' " ' I " ' ' I ' ' " 1 " ' " ' ' ' " " ' ' 4

0.5

5 :

,: Y ; r -

a 0.0

I

-0.5

-1.0

-1.5

1.0 -

g 0.5 - 5 : P i

.E I

a 0.0 r - I

B -0.5 -

-1.0-

- ! !

! ! ! !

! !

! ! !

-

I ! -

: ! - -

- -

, I , . , . I , . . . I . , , 1 . * . .

-1.5 -1.5 -1.0 -0.5 0.0 0.5 1 .o 1.5

range wave number

(a)

2D SAR frequency response supports 1.5 " " I " " I " " I " " ~ " " I " " 4

! !

-1st.. , * I I S , , 1 , . I * , I . , , a I " "

-1.5 -1.0 -0.5 0.0 0.5 1 .o 1.5 range wave number

2D SAR frequency response supports 1 . 5 1 ' ' " I ' ' ' ' I ' * ' ' I ' ' ' ' I ' ' ' . I ' ' ' ' 4

l.o[

g 0.5

5 : J l g 0.0

.: 1

-

r - - -0.5 -

-1.0 -

! ! I

-1.5 -1.5 -1.0 -0.5 0.0 0.5 1 .o 1.5

range wave number

2D SAR frequency response supports 1.5 , . . . . I ' " ' " ' . ' ! , ' . ' I . . ' . 1 ' ' ' .

l.ok

Fig. 2. (a)-(d). SAR frequency response support intersection as a function of terrain gradients relative to the slant plane for a two-pass crossed-track case. A 6-m resolution, C-band system imaging at a 45' grazing angle relative to the ground plane is assumed. The slant planes are separated by differential grazing and azimuthal angle rotations of 0.065 and ().lo, respectively, which causes the 2D spectral shift of one support relative observed in Fig. 2(a). Fig. 2(b) and (c) corresponds to scatterers located on the side of a mountain having a 30° slope relative to the ground plane. The azimuth components of the terrain gradients are fl for (b) and (c), respectively, which causes the supports to shift in opposite azimuth wavenumber directions. Fig. 2(d) shows the support intersection for scatterers located on the near side of the mountain as imaged by the S A R where the decorrelation is greatest. The variability of the support intersection depicted in these figures can occur within a single image scene.

As a matter of bookkeeping we mention again that the quantity 00 has units of reflectivity squared per unit area. The coeffi-

conjugate image product. With regards to noisy imagery, the

same value as the right side of (85) , provided that the noise A x 7 (alAz a realizations are independent with zero mean.

A tractable analytic expression for the support intersection area giving a, is obtained as follows. Again, the matrix A is nominally diagonally dominant. For the purpose of area

and a calculation gives the lengths B, and By

cient c, is observed to be a decorrelation factor in the expected

expectation of the noisy conjugate image product has the

Iy

-ma-{-- n- n-

- . -ma{ _ _ 7T

By =min - - - E } { l y ' IdTAy d

estimation, we approximate it by zeroing its off-diagonal AY' IdlAY d elements _which then eliminates the shearing of the support of h(AT[ + .'). Then the intersection is a rectangle with area

with the imaging geometry, terrain gradient, and wavelength- dependent variables a1 P, a, d defined in (60), (61), (66), and (69). In the equations above the lengths are taken to be zero

a, 2: B,B,

. , - in the case that the support intersection set is empty. The presence of the factors (a( and Id( in the expression above (86)



accounts for the scaling of one support relative to the other. C. Correlation as a Function of Terrain Gradients

Ax Ay c, N B,By-

(27r)Z .

B. The Parallel-Path Case

Consider the special case in which the velocity vectors associated with each image have the same direction. Further, assume that range unit (line-of-sight) vectors GI, and Zzx are orthogonal to the respective velocity vectors. Then the azimuth unit vectors are equal, Zly = ZzY and in this "parallel-path" case the slant planes are a differential grazing angle7 y rotation of each other. The parameter d = 1 and we further assume that la1 N 1 as well. The following vector inner products are expressed in terms of the grazing angle differential as:

Gl, . Zz, = sin (y) N y GIx . Gzx =cos (7) N 1 - y2/2.

(90) (91)

With the approximations above and the orthogonality of the unit vectors, one obtains for the parallel-track case, the first- order approximations for the lengths B, and By

(92)

(93)

Referring to (89) the normalized support intersection area is approximately

(94)

The above model for correlation compares favorably with those found in [l], [6] (assuming that the images are registered). The above model will be revisited in Section V when height error variance and the concept of an optimal baseline or grazing-angle differential y are studied.

It can be shown that the special case correlation model (94) as compared to (87)-(89) can overestimate the support intersection area if used in the crossed-path case. This is because the projection of the range unit vector of image 2 onto the azimuth unit vector of image 1, ZZ, . ZlY, contained in (61), is in general not zero as assumed in the special-case model and therefore the magnitude of azimuth component of the spectral shift vector can be underestimated.

0.057m) with a grazing angle relative to the ground plane of 45'. For convenience, the second slant-plane unit basis vectors are defined as a grazing and azimuthal differential angle rotation of the first slant-plane reference basis. The differential grazing and azimuthal angles are 0.065 and 0.1", respectively.* Fig. 2(a) represents the frequency response supports for scatterers residing in the ground plane in which case OC = [I, 01. The change in azimuthal viewing angle (= X/2Ay) to support 6-m resolution at C-band is 0.27" and the differential azimuthal angle causes a nominal azimuth spectral shift equal to 37% of the azimuth bandwidth. In Fig. 2(b) one observes diminished support intersection due to a local terrain gradient of OC = [2, 11 while in Fig. 2(c) v< = [2, -11, and the support intersection is greater. The gradients for those figures correspond roughly to that which exists on a mountain whose geometry is described below.

Correlation restoration [9], [3 13, discussed in the Introduc- tion, is an operation in which each image is bandpass-filtered such that only those wavenumbers belonging to the support intersection are retained. Given such filtering, the expectation calculation of Section IV-A shows that the correlation is unity e, = 1, provided that the support intersection is not empty. It is also pertinent to note that an a priori terrain gradient estimate is needed to define spectral shift vector and thus the bandpass filters. In Fig. 2(a)-(d) observe that the support intersection varies significantly with the local terrain gradient. One concludes that filtering to accommodate the largest gradients (i.e., the smallest support intersection) can reduce the useful resolution bandwidth in those image regions where the corresponding support intersection is relatively large. At the same time one observes that for each image there is a region of wavenumbers which do not belong to the support intersection for any of the associated terrain gradients. The filtering of those wavenumbers from each image can improve, although not necessarily fully restore, correlation even in the absence of precise terrain gradient information.

Using the approximations above we study the correlation e, as a function of terrain gradients. Consider a cone with a slope of 30" relative to the local horizontal as a prototypical mountain model. A 6-m resolution, C-band system and a 45' grazing angle relative to a ground plane are assumed. The slant planes have the same differential grazing and azimuthal angles described above for Fig. 2(a)-(d). Fig. 3 depicts the correlation e, as a function of slant-plane coordinates spanning a 1200-m range by 1200-m azimuth region. The correlation is computed every 60 m in each coordinate direction. As expected, decorrelation is greatest in a neighborhood of the near side of the mountain as viewed by the SAR, also refer to the support intersection Fig. 2(d). The asymmetry of

sThe 0.065' differential grazing angle is reasonable for a two-pass, single- antenna spacebome S A R such as ERS-1 where the baseline separation can be on the order of 1 km, and the range to the scene imaged is roughly 850 km.

'The grazing angle is defined as the acute angle between the slant and ground planes pictured in Fig. 1.



Fig. 3. Correlation or normalized support intersection area c, over a prototypical mountain as a function of slant-plane location. The mountain is modeled as a cone with a slope relative to the ground plane of 30'. As in Fig. 2(aHd), a 6-m resolution, C-band system is assumed with the same differential grazing and azimuthal rotation angles. Samples are spaced 60 m apart in each direction. The decorrelation is greatest on the near side of the mountain as viewed by the SAR. Asymmetries in the azimuth coordinate direction are due to the azimuthal angle rotation which can occur under crossed-track imaging geometry conditions.

correlation with respect to the azimuth coordinate is consistent with the support intersections depicted in Fig. 2(b) and (c).

V. SNR AND HEIGHT ERROR VARIANCE

In this section, the relation between topographic height error variance as a function of imaging geometry differential and radar system parameters is studied. With our sights set on topographic mapping, the right side of (85) is the expected signal whose phase C#I is sought. Thus we define the signal to noise SNR as the magnitude squared of the expected signal divided by the variance of the noise in the conjugate image product

where n denotes the noise in the conjugate image product. The noise n contains terms involving the products of the individual image signals and noises

where nl and 122 are the noise random variables corresponding to each of the two images and are assumed to be independent with zero mean and equal variances, denoted var (ni). Then the mean of the image product noise is zero with variance

v.r(n) = (I.1iI2)var(nz) + (I(72212)var(n1) + var (711) var (n2). (97)

Our aim is to express the signal to noise of the conjugate image product S N R in terms of the signal to noise of a single complex-valued image, denoted SNRi, which is assumed to be the same for each of the two images. The same statistical calculation as in Section IV gives the signal to noise of a single

complex-valued image at slant-plane location &

As an aside, we mention that the image noise variance is given by the product of the (white) noise power spectral density level with the S A R image bandwidth var (n;) = N/Ax Ay so that SNR; N oo(&)/N. Combining the SNR, var(n), and SNR; relations in (99, (97), and (98) gives

(99) SNRi 1

2+SNRa1 ' 2 S N R N ~ 1 ! -SNR;C;.

For the purpose of analytic modeling we use the following approximation for the phase error variance in terms of the SNR:

1 1 2SNR - SNR;CZ

var(e+) N - - ___

[32]. This approximation is derived by consideration of two complex numbers; one representing the noise, the other representing the signal whose argument (phase) is sought. Each complex number represents a vector with the noise having a relatively small modulus as compared to the signal. The phase error variance versus SNR relation (100) follows from two assumptions. The first is that the error is dominated by the projection of the noise vector onto the direction orthogonal to the signal, which is reasonable for signal-to-noise ratios on the order of 8 dB or more. The second assumption is that the real and imaginary parts of the noise are statistically independent with zero mean and equal variances. Incidentally, the variance is for the phase or argument modulo 2n and thus does not reflect the potential errors emerging after a 2D phase unwrapping algorithm has been applied.

A. Height Error Variance Model

Now we return to (37) for the topographic height error variance and express it in terms of the image signal to noise and correlation using (100)

The terrain elevation error variance (101) is a function of the system wavelength and 2D bandwidth, the imaging geometry associated with each receiving antenna, and signal to noise. It will be utilized to study expected terrain elevation errors which depend on terrain gradients for a given imaging geometry and system parameters. Fig. 4 corresponds to a numerical evaluation of the height error standard deviation, that is, the square root of (101) for the prototypical mountain whose slope is 30" relative to a local ground plane described previously. The same 6-m, C-band system and crossed-track pass are assumed as described for the correlation plots given in Figs. 2(a)-(d) and 3. The signal to noise is assumed to be a uniform 12 dB over the scene. Because of illumination shading this assumption is properly questioned. However, we are interested in this study to examine terrain gradient effects alone and thus signal to noise is held constant. Relative to Fig. 3, the elevation error surface of Fig. 4 is rotated to avoid


MARFCHAL: INTERFEROMETRIC SAR FOR TERRAIN ELEVATION MAPPING 137

Fig. 4. Terrain elevation estimation error standard deviation as a function of slant-plane coordinates over the prototypical mountain, described for Fig. 3. The same 6-m resolution, C-band system with the same crossed-track imaging geometries is assumed as in Figs. 2(a)-(d) and 3. Samples are spaced 60 m apart in each direction. A uniform SNR; = 12 dB is assumed to study the error sensitivity to the varying terrain gradient alone. The largest error (roughly 10 m) is located on the near side of the mountain as viewed by the SAR which is where the decorrelation is greatest. This figure has been rotated relative to Fig. 3 to avoid hidden lines.

Conversely, for a given baseline (and range) one can also prescribe an optimal system wavelength A,.

Let F be the height error variance (101) scaled by the ratio of the image signal to noise and the square of the inner product of the slant- and ground-plane unit normal vectors. With the correlation model (94), derived under the parallel path assumption, the scaled error variance to be minimized is

where by definition 77 = y/X. Notice that the scaled error variance F is independent of X when expressed as a function of the grazing angle differential-wavelength ratio 71. Local extrema are defined by F = 0 and are given as the solutions to

(103) 3BCq2 - 2(B + C ) / Q ~ + 1 = 0

where B and C are defined in terms of resolution and terrain gradients

When aCo/ay # 0 the optimal ratio vo is obtained from (103) as

hidden lines. The maximum error is on the order of 8 m over those slant-plane locations corresponding to the near side of the mountain as viewed by the SAR. Further note that B + C - J(B+C)'-33BC . (105)

In the case that la<O/aYI is small as compared to l a b / a x l y

which can occur for example over flat terrain, the equation above is approximated with a Taylor expansion giving

3BC because this is a two-pass crossed-track case, the errors are 710 = f asymmetric with respect to azimuth location and are consistent with the asymmetries in the correlation depicted in Fig. 3 and the support intersection plots of Fig. 2(b) and (c).

B. Optimization

From a system analysis viewpoint, the terrain elevation error variance (101) represents a functional to be minimized as a function of system and/or imaging geometry parameters. As stated previously, the concept of an optimal baseline or grazing angle differential? which balances the competing requirements of image correlation and sensitivity to terrain elevation, was introduced in [l]. In [6] optimization is pursued further and one finds the height error minimized as a function of various parameters such as cross correlation and number of coherent looks. Given the correlation model (94) one observes that the error variance model (101) is a function of the grazing- angle differential y, a terrain gradient relative to the slant plane, resolution, and wavelength A. Here the focus is on optimization of height error variance model as a function of y or X under fixed terrain gradient conditions for the parallel- path case described in the beginning of Section IV-B. In what follows, an optimal ratio y / X is determined analytically via differentiation of (101) for a fixed terrain gradient. The other parameters in (101) such as SNRi are also taken to be fixed. Then for a given X one prescribes an optimal yo or baseline.

'The differential grazing angle is essentially the baseline over the range to the scene imaged.

710 E f AX - + 4Ay - . (106) ( 1:y One observes from the relation above that the optimal grazing angle differential-wavelength ratio rl, decreases as the magnitude of the gradient increases. With 71, defined in either (105) or (106), Jm is the minimum (scaled) elevation error standard deviation corresponding to (102).

In [6], a two-antenna, 30-m ground resolution, spacebome INSAR system design example is provided in which a 12- m baseline, an orbit altitude of 400 km, and a look angle 0 = 30" is assumed from which an optimal system wavelength is computed to be approximately 1 mm. An equivalent result is obtained for a two-pass single-antenna system" with the model for the optimal ratio (106) as follows. The values for the baseline, altitude, and look angle translate to a (fixed) grazing-angle differential y 11 3 x x cos (e). Assuming a nominally flat image scene ~ C O / ~ X = tan (90 -e), aCo/ay = 0, and one computes the optimal system wavelength A, = y/qo N 2.7 mm which is consistent with the result given in [6] given the antenna concept difference.

'OThere is a factor-of-two difference in the phase difference equations which results in each factor of 4 appearing in (106) being replaced by 2, for example refer to [l] where both antenna concepts are modeled.



As another simple example, consider the case when X is fixed, a<o/ay = 0, and the optimal yo = Avo is given by (106), then

and is seen to be proportional to the resolution and to the range component of the terrain elevation gradient relative to the slant plane. The relation above is motivation to prefer imaging geometries for which the slant and ground planes are close to each other so as to minimize la<oo/axl. However, a loss of signal to noise can result when the planes are arbitrarily close, as the area of shadowed regions can become large. Equation (107) provides a simple "best case" analytic terrain elevation error model when scaled by Z3z . GI,/-.

Fig. 5 depicts the (scaled) elevation standard deviation as a function of grazing angle differential at C and L bands, with wavelengths X = 0.057 and 0.25 m, respectively. In this comparison the pair of dashed curves correspond to a terrain gradient of 0 5 = [tan(45"), 01 and are associated with scatterers located in the ground plane with the SAR looking at a grazing angle of 45' relative to the ground plane. The pair of solid curves correspond to a terrain gradient of 0 5 = [tan(45' + 30°),0] which can be associated with imaging the near side of a mountain with a 30' slope relative to the ground plane. For each given wavelength, the minimum (107) is attained at the optimal grazing angle differential yo defined in (106). Observe that the L-band is less sensitive to grazing angle differential than the C-band. Also observe that the optimal grazing angle differential varies with terrain gradient. This suggests an optimization strategy in which one chooses a yo which minimizes the error variance (102) over the steep terrain where large errors are expected. Indeed, it can be shown that"

provided that

and yo = Avo defined in (106) is optimized for the largest gradient la<o/axl. Again, a<o/ay = 0 is assumed in this simple example. The inequality (108) implies that, although yo is suboptimal for the less steep terrain, the elevation errors over such regions will be no more than those over the steep terrain for which yo is optimal. Fig. 5 also shows that if the grazing-angle differential is optimized for relatively small terrain slopes the errors over steep terrain can be large.

VI. SUMMARY

In this paper, fundamental equations related to terrain elevation mapping using interferometry were reviewed for spotlight SAR. An alternative tomographic image formation perspective was adopted principally as a tractable method for modeling

"The dependency of the scaled elevation error variance F on the range component of the terrain gradient is expressed explicitly in (108) whereas in (102) it is suppressed.

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 grazing angle diff (degrees)

Fig. 5. Scaled terrain elevation estimation error as a function of grazing-angle differential between slant planes. A grazing angle of 45' is assumed relative to the ground plane. The dashed pair of curves corresponds to scatterers located in the ground plane for 6-m, C- and L-band systems. The solid pair of curves corresponds to scatterers located on the near side of a mountain, as viewed by the SAR, whose slope is 30' relative to the ground plane. Error sensitivity to grazing-angle differential, system wavelength, and terrain slope are depicted.

general two-pass imaging geometries, such as crossed-orbit cases. This paper builds on concepts such as baseline decorrelation, resolution enhancement, and optimization introduced in previous papers as discussed. Image cross correlation was characterized in terms of the intersection of the 2D SAR frequency response supports with dependencies on local terrain gradients, imaging geometry, and SAR system parameters. Analogous to previous 1D treatments, it was observed that pairs of 2D frequency response supports can span greater 2D bandwidth than either support alone which implies a potential for improving the azimuth as well as range resolution by a coherent sum of the two images. It was also shown that the support intersection can vary significantly over mountainous regions and the implications regarding correlation restoration and coherent multilooking were discussed. The special case of parallel-antenna paths was considered leading to a correlation model which is consistent with those developed in previously published papers.

A general topographic height error variance model was developed as a function of correlation, imaging geometry, image signal to noise, and wavelength. Both the correlation and height error models were evaluated as a function of location on a prototypical mountain to observe their sensitivities to terrain gradient variation relative to the slant plane. Finally, the concept of optimizing an INSAR with respect to differential grazing angle (baseline) and system wavelength, to minimize the height error variance, was explored analytically, in the special case of parallel-antenna paths.

ACKNOWLEDGMENT

This author wishes to thank his colleagues J. Avrin, S. Hov- anessian, G. Sassenrath, and M. Whitt for useful conversations and suggestions concerning this paper. Appreciation is also expressed to the reviewers for their helpful comments.


MARECHAL INTERFEROMETRIC SAR FOR TERRAIN ELEVATION MAPPING 739

REFERENCES

F. K. Li and R. M. Goldstein, “Studies of multibaseline spaceborne interferometric synthetic aperture radars,” IEEE Trans. Geosci. Remote Sensing, vol. 28, pp. 88-97, Jan. 1990. L. C. Graham, “Synthetic interferometer radar for topographic mapping,” Proc. ZEEE, vol. 62, pp. 763-768, June 1974. H. A. Zebker and R. M. Goldstein, “Topographic mapping from interferometric synthetic aperture radar observations,” J. Geophys. Res., vol. 91, no. B5, pp. 49934999, Apr. 10, 1986. A. K. Gabriel and R. M. Goldstein, “Crossed orbit interferometry: Theory and experimental results from SIR-B,” Znt. J. Remote Sensing, vol. 9, no. 5, pp. 857-872, 1988. I. Cumming, D. Hawkins, and L. Gray, “All-weather mapping with interferometric radar,” in Proc. 23rd Int. Symp. on Remote Sensing Environment, Bangkok, Thailand, Apr. 18-25, 1990, pp. 1249-1262. E. Rodriguez and J. M. Martin, “Theory and design of interferometric synthetic aperture radars,” Proc. Inst. Elec. Eng., pt. F, vol. 139, no. 2, pp. 147-159, Apr. 1992. H. A. Zebker, S. N. Madsen, J. Martin, K. Wheeler, T. Miller et al., “The TOPSAR interferometric radar topographic mapping instrument,” IEEE Trans. Geosci. Remote Sensing, vol. 30, pp. 933-940, Sept. 1992. A. L. Gray and P. J. Farris-Manning, “Repeat-pass interferometry with airborne synthetic aperture radar,” IEEE Trans. Geosci. Remote Sensing, vol. 31, pp. 186191, Jan. 1993. P. H. Eichel, D. C. Ghiglia, C. V. Jakowatz, Jr., P. A. Thompson, and D. E. Wahl, “Spotlight SAR interferometry for terrain elevation mapping and interferometric change detection,” preprint, Sept. 2, 1993. C. V. Jakowatz, Jr., D. E. Wahl, P. H. Eichel, and P. A. Thompson, “New formulation for interferometric synthetic aperture radar for terrain mapping,” to appear in Proc. SPIE Conf: Algorithms for Synthetic Aperture Radar Imagery, Orlando, FL, Apr. 4-8, 1994. A. M. Cormack, “Representation of a function by its line integrals, with some radiological applications,” J. Appl. Phys., vol. 34, pp. 2722-2730, 1963. R. A. Brooks and G. DiChiro, “Principles of computer-assisted tomography in radiographic and radioisotopic imaging,” Phys. Med. Biol., vol. 21, pp. 689-732, 1976. T. N. Bishop, K. P. Bube, R. T. Cutler, R. T. Langan, P. L. Love etal. , “Tomographic determination of velocity and depth in laterally varying media,” Geophys., vol. 50, no. 6, pp. 903-923, June 1985. B. D. Comuelle, “Acoustic tomography,” IEEE Trans. Geosci. Remote Sensing, vol. GRS-20, pp. 326332, 1982. D. C. Munson, J. D. O’Brien, and W. K. Jenkins, “A tomographic formulation of spotlight-mode synthetic aperture radar,” Proc. IEEE,

J. L. Walker, “Range doppler imaging of rotating objects,” IEEE Trans. Aerosp. Electron. Syst., vol. AES-16, pp. 23-52, Jan. 1980. D. A. Ausherman, A. Kozma, J. L. Walker, H. M. Jones, and E. C. Poggio, “Developments in radar imaging,” IEEE Trans. Aerosp. Electron. Syst., vol. AES-20, no. 4, pp. 363-399, July 1984. M. D. Desai and W. K. Jenkins. “Convolution backuroiection imarre

V O ~ . 71, pp. 917-925, Aug. 1983.

[20] D. L. Mensa, S. Halevy, and G. Wade, “Coherent Doppler tomography for microwave imaging,” Proc. IEEE, vol. 71, pp. 254-261, Feb. 1983.

1211 C. V. Jakowatz, Jr. and P. A. Thompson, “Tomographic formulation of spotlight-mode synthetic aperture radar extended to three dimensions,” in Proc. IEEE Digital Signal Processing Workshop, Sept. 13-16, 1992, pp. 5.6.1-5.6.2.

1221 -, “A new look at spotlight mode synthetic aperture radar as tomography: Imaging three dimensional targets,” preprint, Aug. 19, 1992.

1231 C. Prati and F. Rocca, “Range resolution enhancement with multiple SAR surveys combination,” in Proc. ZEEE Geoscience and Remote Sensing (IGARSS’92) Conf: (Houston, TX, May 1992), pp. 1576-1578.

1241 -, “Improving slant-range resolution with multiple SAR surveys,’’ IEEE Trans. Aerosp. Electron. Syst., vol. 29, no. 1, pp. 135-143, 1993.

[25] R. M. Goldstein, H. Engelhardt, B. Kamb, and R. Frolich, “Satellite radar interferometry for monitoring ice sheet motion: Application to an Antartic ice stream,” Science, vol. 262, pp. 1525-1529, Dec. 3, 1993.

Boston, M A Birkhauser, 1980. [26] S. Helgason, The Radon Transform. [27] F. Natterer, The Mathematics of Computerized Tomography. New

York Wiley, 1986, pp. 9-11. [28] R. M. Goldstein, H. Zebker, and C. Wemer, “Satellite radar interfer-

ometry: Two-dimensional phase unwrapping,” Radio Sci., vol. 23, pp. 713-720, JulyfAug. 1988.

1291 D. C. Ghiglia and L. A. Romero, “Direct phase estimation from phase differences using fast elliptic partial differential equation solvers,” Opt. Lett., vol. 14, no. 20, pp. 1107-1109, Oct. 15, 1989.

I301 -, “Robust two-dimensional weighted and unweighted phase unwrapping that uses fast transforms and iterative methods,” J. Opt. Soc. Amer. A, vol. 11, no. 1, pp. 107-117, Jan. 1994.

13 11 Rocca, personal communication, Interferometric SAR Technology and Applications Symp., Ft. Belvoir, VA, Apr. 13-14, 1993.

[32] J. R. Jameson, personal communication, Feb. 1993.

1988 he became a membc

Nick Marechal (M’87-SM’91) received the B.A. degree from the University of California at Los Angeles (UCLA) in 1980, and the Ph.D. degree in mathematics, applied program, also from UCLA in 1986.

While at UCLA his research concerned inverse problems associated with wave propagation through inhomogeneous lossy media, having applications in areas such as reflection seismology. In 1986 he joined Arete Associates as a staff scientist and worked on S A R ocean imaging experiments. In

:r of the technical staff at The Aerospace Corporation, _ _ reconstruction for spotlight mode synthetic aperture radar,” IEEE Trans. Image Processing, vol. 1, no. 4, pp. 505-517, Oct. 1992.

I191 W. M. Brown, “Walker model for radar sensing of rigid target fields,” IEEE Trans. Aerosp. Electron. Syst., vol. AES-16, pp. 104-107, Jan. 1980. Mathematics (SIAM).

El Segundo, CA. His current research interests include nonparamesc phase error estimation techniques for S A R autofocus, topographic mapping with SAR, and aliasing mitigation techniques for ground-based Doppler radars.

Dr. Marechal is a member of the Society for Industrial and Applied


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