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A novel estimation approach of dynamic and coupling baseline for distributed satellite SAR.


Distributed satellite synthetic aperture radar (DS-SAR) is a new earth observation system, in which several satellites work cooperatively. The system has three major applications: high-resolution wide-swath imaging, producing global high-precision digital elevation model (DEM) data and ground moving target indication (GMTI) [1,2].

Baseline determines performance of interferometric SAR (InSAR) system and is also an important parameter in elevation reversion. Therefore, baseline estimation is very important in data processing of InSAR system [3-5].

For conventional spaceborne InSAR system, baseline can be estimated from satellite state vector [6], interferometry calibration (using Ground Control Points (GCPs) [7-9], coarse DEM data [10] or sea surface data [11]) or interferometric phase information [12,13]. The precision of satellite state vector method is usually low due to its dependence on satellite positioning. In the method based on GCPs, known altitude and interferometric phase are used to calculate the baseline parameters based on geometry relationship in InSAR system. But it needs high precise layout and measurement of GCPs [14-16], which is hard to fulfill. The method based on DEM cannot obtain precise estimation due to the poor quality of reference DEM data. The calibration method based on sea surface is proposed in literature [11] for spaceborne single-pass dual-antenna interferometry without along-track baseline. The correlation of signals from different SAR antennas is low due to movement of sea surface. The estimation metod based on interferometric phase information [12] uses typical two-dimensional (2-D) geometry of InSAR to deduce the relationship between interferometric fringe frequency and cross-track baseline, without considering curvature of the earth. The method is used widely, because there is no need for satellite orbit ephemerides or accurate information of GCPs.

Satellites in DS-SAR system formate according to specific rules, so that baselines can be configured expediently [2], which are usually coupled for formation flying [17], that is, along-track baseline and cross-track baseline exist simultaneously [18,19]. The coupling baselines induces coupling problem in the interferometric phase [4]. Furthermore, the baseline is three-dimensional (3-D) and also changes dynamically during the data-take time due to formation flying of satellites in the system [19]. Therefore, it is hard to obtain high precise DS-SAR baseline data.

Xu separates along-track baseline and cross-track baseline through image registration, and estimates the two kinds of baseline errors by image registration and subspace decomposition respectively [20]. However, the method requires very high precision of image registration. Another method [21] is to measure the relative state vector between satellites based on differential Global Positioning System (DGPS), laser and radio. But in DS-SAR, measured baseline is different from interferometric baseline. It may introduce errors when converting the measured baseline.

The precision of baseline estimation method, based on interferometric phase information in literature [12], is low because of unaccurate geometry relationship and low-precision fringe frequency estimated through Fourier analysis. A novel method to achieve DS-SAR baseline estimation based on interferometric phase information is proposed in this paper. The method makes improvements from five aspects. Firstly, accurate 3-D geometry relationship is set up for high resolution DS-SAR system. Secondly, a new method to decompose coupling interferometric phase into two parts related to along-track baseline and cross-track baseline respectively is proposed as well. Thirdly, accurate mathematical relationship between the coupling baseline and the 2-D interferometric fringe frequency is deduced under the ellipsoid earth model. Fourthly, several segments of interferometric phase data are selected to acquire the dynamic variation model of the baseline via a linear filtering process with estimated instantaneous baselines. Furthermore, Semi-Newton iterative method is applied to estimate high-precision fringe frequency to improve baseline estimation precision.

This paper is structured as follows. Section 2 investigates the mathematical model of DS-SAR interferometric phase and decomposition of the coupling interferometric phase. In Section 3, theory of coupling baseline estimation is presented. Section 4 depicts the framework and implementation method of baseline estimation, while emphatically explaining fringe frequency estimation method based on Semi-Newton method. In Section 5, simulation validates the high-precision baseline estimation method. Conclusion is given in the final section.


To present interferometric signal model, the spatial geometry of two satellites formation DS-SAR is shown in Fig. 1. A 3-D coordinate system is adopted. The system is fixed to centroid of the master satellite, where Z-direction is along the line between the earth and the master satellite, positive earth deviation. The X-direction is the master satellite velocity direction, and Y-direction is orthogonal in the right hand system to X.

[S.sub.m] and [S.sub.s] denote the master and slave satellite, respectively. Radar signals are transmitted from SAR antenna on the master satellite, while being received by both the master satellite SAR antenna and the slave satellite SAR antenna. [S'.sub.s] is the projection of [S.sub.s] in the YZ plane. [V.sub.m] represents the velocity of master satellite and [V.sub.s] represents the velocity of slave satellite. Here, [V.sub.m] and [V.sub.s] are parallel to the X axis. [H.sub.m] is the height of the master satellite. N denotes the nadir of the master satellite. O is the geo-center and [R.sub.e] denotes the local earth radius. B is the baseline, which can be expressed as the vector ([B.sub.a],[B.sub.y],[B.sub.z]) in the 3-D coordinate system. [B.sub.a] is along-track baseline, [B.sub.y] is the horizontal component of cross-track baseline [B.sub.r] and [B.sub.z] is the vertical component. Here,

[B.sub.y] = [B.sub.r] * sin [[theta].sub.0] (1)

[B.sub.z] = -[B.sub.r] * cos [[theta].sub.0] (2)

[[theta].sub.0] is the angle between cross-track baseline [B.sub.r] and the negative Z-direction. [theta] represents the look angle of master satellite SAR. [r.sub.m] is the slant range between the phase center of master satellite SAR antenna and the ground target T, and [r.sub.s] is the slant range between the phase center of slave satellite SAR antenna and T. [r'.sub.s] is the projection of [r.sub.s] in the YZ plane.


The single look complex (SLC) SAR data pair can be obtained through range pulse compression and azimuth processing [22] on echoes from the master and slave satellite SAR. The model can be expressed as [22,23]:

[s.sub.m](r,x) = [sigma](r,x)[cross product]exp {-j 4[pi] / [lambda] [r.sub.m]} * exp{-j2[pi][] * x / V} [cross product[[w.sub.r](r)([w.sub.a](x)] + [n.sub.m] (3)

[s.sub.s](r,x) = [sigma](r,x)[cross product]exp{-j 2[pi] / [lambda] ([r.sub.m] + [r.sub.s])} * exp{j[pi]([] + [f'.sub.ds]) * x / V} [cross product[[w.sub.r](r)[w.sub.a](x)] + [n.sub.s] (4)

where [sigma](r,x) is the backscatter coefficient, r and x respectively are coordinates in the slant range direction and azimuth direction. [w.sub.r](r) and [w.sub.a](x) respectively represent the impulse response functions of point target in range and azimuth direction. [n.sub.m] and [n.sub.s] denote the systematic thermal noises. [] and [f.sub.rm] respectively are the Doppler centroid frequency and modulation frequency of master satellite, while [f'.sub.ds] and [f'] represent that of salve satellite under assumption that it is in the transmitting and receiving model. Here, the slave SAR is in the receiving-only model, so that the overall contribution of Doppler of slave satellite is [24].


Through formula (3) and (4), the interferometric phase can be given as following:

[PHI] = arg([s.sub.m](r,x) * [s.sup.*.sub.s](r,x)) = 2[pi] / [lambda]([r.sub.s] - [r.sub.m]) + [pi]([f'.sub.ds] - []) x / V (6)

In Equation (6), the first term is related to slant range, and the second to doppler information. The first phase component is mainly introduced by cross-track baseline information, denoted as [[PHI].sub.r], namely:

[[PHI].sub.r] = 2[pi] / [lambda]([r.sub.s] - [r.sub.m]) (7)

The second phase component of formula (6) is introduced by along-track baseline information denoted as [[PHI].sub.a], namely:

[[PHI].sub.a] = [pi]{[f'.sub.ds] - []) X / V = 2[pi]([f.sub.ds] - []) x / V (8)


This section deduces the relationship between coupling baseline and 2-D interferometric fringe frequency in DS-SAR.

3.1. Instantaneous along-track Baseline Estimation

Along-track baseline in DS-SAR causes different squint angles of master satellite and slave satellite, which leads to different Doppler centroid frequencies. Therefore, the along-track baseline can be calculated via the difference of Doppler centroid frequencies.

Along-track geometry of DS-SAR is given in Fig. 2, where [[phi].sub.m] and [[phi].sub.s] are squint angles of master satellite and slave satellite, respectively. [S".sub.S] is the projection of [S.sub.s] on the X axis, [B.sub.a] is the along-track baseline, [R".sub.s] is the slant range between S'S and the target T and [[phi].sub.'s] is the angle from [r".sub.s] to the X axis. Based on the geometry, [] and [f.sub.ds], which represent Doppler centroid frequencies of master satellite and slave satellite, are obtained [25]:

[] = - 2V / [lambda] cos [[phi].sub.m] (9)

[f.sub.ds] = - V / [lambda](cos [[phi].sub.m] + cos [[phi].sub.s]) (10)


From Equations (9) and (10), the difference of Doppler centroid frequencies is calculated as:

[DELTA][f.sub.d](m,s) = [f.sub.ds] - [] = V / [lambda](cos [[phi].sub.m] - cos [[phi].sub.s]) = 2V / [lambda] sin ([[phi].sub.s] - [[phi].sub.m] / 2)sin([[phi].sub.m] + [[phi].sub.s] / 2) [approximately equal to) V[B.sub.a] sin[phi] / [lambda][r.sub.m] (11)

where we assume that

[phi] = ([[phi].sub.m] + [[phi].sub.s])/2 (12)

[[phi].sub.s] [approximately equal to] [[phi]'.sub.s], ([V.sub.s]//X axis,[B.sub.r] [much less than] [r.sub.s]) (13)

Sin([[phi].sub.s] - [[phi].sub.m] / 2) [approximately equal to] [[phi].sub.s] - [[phi].sub.m] / 2 [approximately equal to] [[phi]'.sub.s] - [[phi].sub.m] / 2 [approximately equal to] [B.sub.a] / 2[r.sub.m], ([[phi].sub.s] - [[phi].sub.m] / 2 [much less than] [pi]) (14)

It is indicated from Equation (11) that the relationship between difference of Doppler centroid frequencies [DELTA][f.sub.d](m,s) and along-track baseline [B.sub.a] is linear.

Based on Equation (8), we have:

[[PHI].sub.a] = 2[pi] ([f.sub.ds] - []) x / V = 2[pi] * [DELTA][f.sub.d](m,S) * [t.sub.a] (15)

where [t.sub.a] = x/V represents azimuth time. It is indicated from Equation (15) that the relationship between the difference of Doppler centroid frequencies [DELTA][f.sub.d](m,S) and the azimuth component of interferometric phase [[PHI].sub.a] is linear in short time as well. By analyzing [[PHI].sub.a], along-track baseline can be solved via Equation (16), as follows:

[B.sub.a](m,s) [approximately equal to] [lambda][r.sub.m][DELTA][f.sub.d](m,s) / V sin[phi] (16)

where [DELTA][f.sub.d](m,s) is obtained through interferometric fringe frequency estimation in azimuth direction. Then along-track baseline [B.sub.a] can be estimated by Equation (16).

3.2. Instantaneous Cross-track Baseline Estimation

Based on geometry in Fig. 1, cos [theta] can be denoted as a function f([r.sub.m]) with regard to [r.sub.m] [12]:

f([r.sub.m]) = cos [theta] (17)

In flat terrain, frequency of range interferometric fringe is




It can be proved that frequency of interferometric fringe [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] in flat terrain is varying with the slant range of master satellite [r.sub.m] monotonously. Because in flat terrain,


As [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is a function upon length and tilted angle of cross-track baseline from Equation (18), cross-track baseline can be acquired according to information on frequency of range interferometric fringe. Once instantaneous frequencies of range interferometric fringe corresponding to two slant range coordinates [r.sub.m]'s are estimated repectively, cross-track baseline [B.sub.r] and tilted angle [[theta].sub.0 can be solved.

Two functions F([r.sub.m]) and G([r.sub.m]) with regard to the slant range of master satellite are defined as:


In which, [r.sup.min.sub.m] and [r.sup.max.sub.m] represent the nearest and farthest slant range of master satellite SAR data selected, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are the frequencies of interferometric fringe corresponding to [r.sup.min.sub.m] and [r.sup.max.sub.m]. Equation as follows can be deduced from Equations (18) and (21):


Thus, the cross-track baseline can be obtained.


Deducing the calculation formula of along-track and cross-track baseline, as Equations (16) and (22), the high-precision estimation method of dynamic coupling baseline is proposed in this section, with implementation steps provided as well.

Besides coupling problem, there is dynamic variation of DS-SAR baseline in along-track direction during the whole data-take time. Since it can be considered stable during short interval, several interferometric phase data segments can be selected with short interval in along-track direction. As long as there is no drastic topographic relief of each segment, instantaneous baseline of each segment can be estimated with high precision, and dynamic baseline variation rule in long time can be obtained via interpolation and fitting. This section focuses on the framework of dynamic baseline estimation and implementation in detail. Then, in order to improve the estimation accuracy of interferometric fringe frequency, which is an important technology in the baseline estimation, we introduce the Semi-Newton iterative method.

4.1. Implementation Steps for Dynamic and Coupling Baseline Estimation

Figure 3 shows the flowchart of dynamic and coupling baseline estimation of DS-SAR. The whole estimation process includes three main steps:

(1) Performing SLC Imaging [26-28] and interferomery, which can be implemented by three additional steps.

(a) Getting SLC SAR data pair using DS-SAR imaging algorithm.

(b) Achieving registration of both master and slave SAR images via coarse registration, interpolation and precise registration [29,30].

(c) Generating interferometric phase data via interferometry and phase filtering [31].

(2) Selecting N segments of interferometric phase, which correspond to relatively flat terrain, to estimate the instantaneous baseline in both along-track and cross-track direction of each segment. As seen from Fig. 3 inside the blue pane, it can be performed by three additional steps.

(a) Obtaining the Doppler centroid frequency difference of master satellite and slave satellite [DELTA][f.sub.d](m,s) by interferometric fringe frequency estimation of ith data segment to calculate along-track baseline [B.sub.a] via Equation (16).

(b) Acquiring the minimum and maximum value of frequencies [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], which correspond to the nearest and farthest slant range of the data segment, by estimating the interferometric fringe frequency in range direction. Then calculate the two components [B.sub.y] and [B.sub.z] of cross-track baseline with known [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and other parameters via Equations (18), (21)-(22).

(c) Obtaining instantaneous baseline vector [B.sub.i] = ([B.sub.a],[B.sub.y],[B.sub.z]), i = 1,...,N of the ith data segment.

(3) Concluding 3-D baseline variation rule with azimuth time B([t.sub.a]) = [[B.sub.a]([t.sub.a]),[B.sub.y]([t.sub.a]),[B.sub.z]([t.sub.a])], by interpolating and fitting N baseline vectors [B.sub.i](i = 1,...,N), where ta denotes the azimuth time (in along-track direction).

4.2. Interferometric Fringe Frequency Estimation Based on Semi-Newton Iterative Method

The 2-D (azimuth and range direction) interferometric fringe frequency estimation method is proposed in this section. The estimation accuracy of traditional interferometric fringe frequency estimation method, based on Fourier transform, cannot satisfy the DS-SAR system perfectly. In order to improve estimation precision, Semi-Newton iterative method, applied in the implementation steps (2a) and (2b) shown in Fig. 3, is introduced to estimate fringe frequency.

It is assumed that there are N phase data segments selected in azimuth direction, denoted as PhaseDatai(i = 1,...,N), with [] pixels in azimuth direction and [N.sub.ri] pixels in range direction. The ith phase data segment is shown in Fig. 4. The interferogram is filtered before unwrapping. The data in red rectangle is used to estimate azimuth interferometric fringe frequency. The data in blue rectangle is used to estimate range interferometric fringe frequency.


The number of pixels in red rectangle is [N.sup.AT.sub.a] x [N.sup.AT.sub.r] (the superscript AT represents 'along-track'), with data in azimuth direction and [N.sup.AT.sub.r] data in range direction. The phase data are filtered in range direction to reduce phase noise. [N.sup.AT.sub.a] x 1 filtered phase data are used to estimate the azimuth frequency [] (i = 1,...,[N.sup.AT..sub.a]). The average of [](i = 1,...,[N.sup.AT.sub.a]) will be used as [DELTA][f.sub.d](m,s) in Equation (16).


The number of pixels in blue rectangle is [N.sup.CT.sub.a] x [N.sup.CT.sub.r] (the superscript CT represents 'cross-track'), with [N.sup.CT.sub.a]data in azimuth direction and [N.sup.CT.sub.r] data in range direction. The phase data are filtered in azimuth direction. 1 x [N.sup.CT.sub.r] filtered phase data are used to estimate the cross-track frequency [k.sub.i](i = 1,...,[N.sup.CT.sub.r]). The estimated frequency [k.sub.1], related with the left-most pixel in the blue rectangle, is corresponding to [r.sup.min.sub.m] in Equation (22). Frequency [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], related with the right-most pixel, is corresponding to [r.sup.max.sub.m] in Equation (22). Here, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Azimuth frequency and range frequency will be estimated separately. Assume that the filtered phase data sequence is ([p.sub.1],[p.sub.2],...,[p.sub.L]) and the ideal frequency sequence relatively is ([f.sub.1],[f.sub.2],...,[f.sub.L]). L is the sequence length, which is equal to [N.sup.AT.sub.a]for azimuth frequency estimation and [N.sup.CT.sub.r] for range frequency estimation.

The complex sequence ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]), where ([p.sub.1],[p.sub.2],...,[p.sub.L]) are phases, can be modeled as:


It is centralized with the ith data [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. T is the sample interval, [phi] is the phase noise.

The frequency can be obtained using the maximum likehood estimator of [f.sub.i][32],


Based on Equation (24), the frequency can be estimated through two main steps as follows:

1. The coarse frequency estimation([f.sup.0.sub.1],[f.sup.0.sub.2],...,[f.sup.0.sub.L]), used as the initial value in the iterative method, is obtained through Fourier transform and interpolation.

The frequency of [p.sub.i](1 [less than or equal to] i [less than or equal to] L) is estimated through the FFT to the finite data sequence ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]) centralized with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where 2M + 1 < L. It is assumed that the corresponding data sequence in the frequency domain is ([F.sub.i-M],...,[F.sub.i],... [F.sub.i+M]) and the maximum of the sequence ([absolute value of [F.sub.i-M],...,[absolute value of [F.sub.i]],... [absolute value of [F.sub.i+M]]) is [absolute value of [F.sub.i]](i - M [less than or equal to] j [less than or equal to] i + M). To improve the frequency resolution, the number of data performed Fourier transform must satisfy certain demand [13]. Then the sequence ([F.sub.j-n],...,[F.sub.j],..., [F.sub.j+n]) (n < M) can be interpolated to further increase the frequency resolution. The relevant frequency of the maximum of new sequence obtained by interpolation is the coarse estimated frequency [f.sup.0.sub.i].

2. Semi-Newton iterative method is employed to improve the frequency estimation precision.

From Equation (24), the instantaneous frequency can be acquired using the maximum likelihood estimator,


G(f) = Re{[M.summation over (m=-M)] [] x [e.sup.-jmTf-j[phi]]} (26)

Equation (25) is equivalent to solving the zero point of G'(f), which is the differentiator of G(f). Because [phi] is unknown, Semi-Newton iterative method is introduced.

[f.sup.n+1.sub.i] = [f.sup.n.sub.i] - Re{H'([f.sup.n.sub.i)] / H"([f.sup.n.sub.i])} (27

H(f) = [N.summation over (m=-N)] [][e.sup.-jmTf] (28)

where n is the iterative time and the coarse estimation [f.sup.0.sub.i] in the first step is set to be the initial value in the iterative operation until the demand is satisfied.


Table 1 shows simulation parameters, where the data-take time is 3 minutes. In this simulation [33-35], ten data segments for even area are selected in along-track direction, and each data segment includes 64 pulses and 1024 range cells.

By modelling the satellite-ground geometry of DS-SAR, SLC images of master and slave satellite SAR are simulated. The simulation results corresponding to the third segment are shown in Fig. 5, where

(a) is the image of master satellite SAR and (b) is the image of slave satellite SAR. Fig. 5(c) shows the interferogram given by coarse registration, interpolation, precise registration and interferometric process on both images. To refine the interferogram, phase filtering is performed, and result is shown in Fig. 5(d).


Table 2 shows the estimation results of baseline vector errors of ten segments of interferometric phase data, which include errors of along-track baseline (Ba), errors of horizontal component of cross-track baseline(By) and errors of vertical component Bz. From Table 2, precision of along-track baseline estimation achieves millimeter level, while the others achieve centimeter level.




Figure 6 is the comparison between estimation and theory value of along-track baseline ([B.sub.a]). Fig. 7 and Fig. 8 present the comparison results between estimation and theory values of horizontal component [B.sub.y] and vertical component [B.sub.z] of cross-track baseline. In all the three figures, the solid line denotes the theory value, while the asterisk is the estimation value. It is shown that estimation values match with theory values well, which validates the proposed approach.


This paper proposes a novel DS-SAR dynamic and coupling baseline estimation approach based on interferometric phase. Firstly, the mathematical model of interferometric phase is established, and interferometric phase decoupling is proposed. Secondly, the theory of instantaneous baseline estimation is presented, and accurate baseline calculation is deduced considering the earth curvature. Then, the whole framework and detailed implementation of DS-SAR baseline estimation are presented. Furthermore, in order to improve baseline estimation precision, Semi-Newton iterative algorithm is applied to estimate high-precision interferometric fringe frequency. Combining accurate fringe frequency with deduced baseline calculation formula, the baseline estimation achieves centimeter level. Finally, simulation results validate the approach proposed in this paper.


The authors would like to thank all the anonymous reviewers for their careful work on this paper. This work was supported in part by the Commission on Science, Technology, and Industry for National Defense of China under Grant No. A2120060006.

Received 31 August 2011, Accepted 15 November 2011, Scheduled 6 January 2012


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L. Feng *, H. P. Xu, C. S. Li, S. Li, and H. Gao

School of Electronic and Information Engineering, Beihang University, Beijing, China

* Corresponding author: Liang Feng (
Table 1. Simulation parameters for
DS-SAR system.

        Parameter             Value

  Satellite height / km      514.000
Satellite velocity / m / s   7600.00
     Look angle / deg         30.00
      Wavelength / m          0.031
  Signal bandwidth / MHz     100.00

Table 2. Estimation errors of baseline
vector / cm.

Phase data   Error of    Error of    Error of
  No. in     [B.sub.a]   [B.sub.y]   [B.sub.z]

    1          0.34         3.7        -4.2
    2          0.27         4.3         8.1
    3          -0.81       -0.9        -0.5
    4          -0.86        3.0         5.9
    5          -0.23       -2.2         4.6
    6          0.52        -0.8         1.7
    7          -0.19        3.5        -5.1
    8          0.83        -6.9        -7.2
    9          0.56         2.2        -4.8
    10         0.27         4.3         6.7
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Author:Feng, L.; Xu, H.P.; Li, C.S.; Li, S.; Gao, H.
Publication:Progress In Electromagnetics Research
Article Type:Report
Geographic Code:9CHIN
Date:Jan 1, 2012
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