2.1. Geometric Calibration of Spaceborne SAR
Geometric Calibration of Spaceborne SAR is to find out and calculate the main error sources leading to geometric positioning system errors. Sensor instability, platform instability, signal propagation delay, terrain height, and processor error are the main factors affecting geometric positioning accuracy of spaceborne SAR [
21]. For error characteristics, those that may affect the geometric positioning accuracy of spaceborne SAR can be classified as fixed system errors, time-varying system errors, and random errors, as described in the following subsections.
(1) Fixed System Error. The ranging signals of SAR system mainly depend on precise time measurement, including fast time (range direction) and slow time (azimuth direction). The two-dimensional time error is mainly affected by the time delay error of the SAR system and the azimuth time synchronization error, which is the main error source for the geometric positioning of spaceborne SAR. The radar signal through each component of the signal channel is the main cause of the time delay error. Time delay is mainly caused by the pulse-width and bandwidth of radar signal. The time delay errors of different pulse-width and bandwidth remain unchanged during SAR satellite operation. The error of time control unit of system equipment is the main factor leading to the azimuth time synchronization error. This error is relatively stable and does not change due to changes in the imaging modes for the same spaceborne SAR.
(2) Time-Varying System Error. Some of the error sources that affect geometric positioning accuracy are affected by time. These mainly include the atmospheric propagation delay error and the imaging processing error. The main factors affecting the atmospheric propagation delay of radar signals are atmospheric pressure intensity, temperature, water vapor content, ionospheric electron density, and the emission frequency of radar signals. Therefore, the atmospheric propagation delay error is a systematic error related to the incident angle of the radar beam and the imaging time of the SAR image.
(3) Random Error. In general, eliminating random error effectively by ground treatment methods is very difficult. Therefore, random error is the main factor affecting the theoretical limit of geometric positioning accuracy in the spaceborne SAR system. The random errors include predominantly satellite position error, SAR system delay random error, SAR antenna dispersion error, ground control point error, and atmospheric propagation delay correction model error [
19].
In these error sources, the main error sources of spaceborne SAR are two-dimensional time errors, which mainly causes the geometric positioning error of SAR image in the range and azimuth direction. Thus, the geometric calibration model for spaceborne SAR is constructed as
where
tf and
ts are, respectively, the fast time in range and slow time in azimuth;
tf0 and
ts0 are, respectively, the measured value of the starting time in range and azimuth;
tdelay is the atmospheric propagation delay time; Δ
tf and Δ
ts are the system delay time errors;
fs is the sampling frequency,
fprf is the pulse repetition frequency; and x and y are pixel coordinates.
The starting time of the satellite record is the time of radar signal received. The starting time in the azimuth direction will be affected by the imaging processing of GF-3 satellite. The intermediate time between transmitting and receiving time is the approximate equivalent SAR imaging time [
22]. Therefore, it should be compensated for as:
where
is the echo receiving time recorded on the satellite,
N is the number of times the radar signal is transmitted from the transmitter to the receiver, and
tsample_delay is the sample time delay in the satellite record.
The Range Doppler (RD) model is a rigorous geometric model for spaceborne SAR that establishes a rigorous relationship between the object space coordinate and the image space coordinate [
19]. For the geometric calibration of SAR images, based on the RD model, earth ellipsoid equation and geometric calibration Equations (1) and (2), which is shown as Equation (3), N GCPs are used to calculate the geometric calibration parameters by using the least square method [
23]. The GCPs are obtained from the corner reflector points or the central points of the cross road in the SAR image and are corrected according to the influence of solid earth tides (SET), which are calculated using the International Earth Rotation Service (IERS) Conventions 2003.
where
Rs = [
Xs Yx Zs]
T and
Vs are the orbit vector;
Rt = [
Xt Yt Zt]
T and
Vt are the position vector and velocity vector of the target point;
fD is the Doppler centroid frequency;
λ is the radar wavelength;
R is the slant range;
X is the column number of the target point in the SAR image;
c is the speed of light;
is the mean equatorial radius; and
is the polar radius with a flattening factor of
.
For the geometric calibration algorithm, based on the error equation of Equation (1), N control points are used to calculate the geometric calibration parameters by using the least square method, as shown in
Figure 1. Then, the geometric calibration parameters are compensated to Equation (1). Based on the updated Equation (1), the geometric positioning accuracy after calibration is evaluated by Equation (3).
2.2. General Geometric Processing Model of Spaceborne SAR
Quite a lot of research teams have done the work of Block adjustment for optical imagery with RPCs. To facilitate the subsequent generalized and scaled processing, it is necessary to convert the compensated RD model parameters to the RPCs with terrain independent method. In this way, it is not necessary to model each satellite separately, and many optical imagery processing programs can also be directly used to process SAR images.
The related literature has shown that the RPC model is a generalized geometric model, which can be used to replace the RD model. The RPC model also establishes the relationship between the ground coordinates and the corresponding image coordinates. It makes full use of the auxiliary parameters of satellite images to create a general model, and can then be fitted to a RD model, and its fitting accuracy is better than 0.05 pixels [
24]. In addition, in order to improve the numerical stability, we offset the 2D image coordinates and 3D ground coordinates, and scale it to the range of −1.0–1.0 by regularization parameter. The RPC model can be defined as follows [
25]:
where
,
, and
are the normalized latitude, longitude, and height, respectively,
indicates the line,
indicates the sample,
is the normalized line number,
is the normalized sample number, and
,
,
, and
are the terms of the third-order polynomial of
.
For example, the form of the polynomial
is as follows:
where
aij (
i = 1, 2, 3, 4;
j = 0, 1, …, 19) are RPCs; there is a total of eighty parameters for RPCs.
The RPC model can be used as an alternative to the RD model, which is a traditional geometric SAR model. When the RD model is available, it is always possible to solve the parameters in a terrain-independent manner [
24,
25,
26,
27].
The proposed estimation process, using a least-squares approach, requires only the RD model and the maximum and minimum heights in the image area, which can be extracted from the global DEM supplied by the United States Geological Survey. As shown in
Figure 2, this method involves three main steps:
- (1)
Determination of an image grid and establishment of a 3D object grid of points using the RD model;
- (2)
RPC fitting; and
- (3)
Accuracy checking.
Experiments with different kinds of SAR data were carried out to verify this method, which show that RPC was able to replace the RD model [
24].
2.3. Planar Block Adjustment Based on RPC
The RPC adjustment model uses an affine transformation to represent these two categories of difference between the calculated and the measured image-space coordinates. Similar to the traditional RPC-based block adjustment, the planar block adjustment does not correct RPCs, but merely corrects their affine transformation parameters, which is defined as follows [
24]:
where
are the affine transformation parameters and
are values used to compensate for systematic errors of the image point. The errors that can be eliminated by the relevant parameters have been introduced in detail in document [
24].
Using Equations (5) and (6), the affine transformation parameters
and
of the image space compensation can be set as unknowns and be solved together with the plane coordinates
and
of the ground point. Additionally, the area covered SAR images are often encountered the weak convergence geometric problem, which cannot be solved by traditional block adjustment, so a planar block adjustment based on RPC is carried out [
15]. The elevation coordinates of Tie Points (TPs, playing as the corresponding points between images) are obtained by interpolating a DEM of the area, which primarily serves as a height constraint [
28]. This method has been improved and validated in different test areas with Ziyuan-3 (ZY-3) optical satellite images [
29]. The planar block adjustment error equation based on the RPC model is as follows:
where
v is the residual vector of the image coordinate observation.
r and
c are the coordinates of the image point as measured manually;
and
are the coordinates of the image point as calculated by the RPC and affine transformation parameters; Δ
e0, Δ
e1, Δ
e2, Δ
f0, Δ
f1, and Δ
f2 are corrections of affine transformation parameters; Δ
X and Δ
Y are plane vectors containing increments of the ground point; and
p is the weight of the observation equation. Due to the observation of equal weight, the value of
p is 1. Equation (7) can be written in the matrix form:
where
is the residual vector of the image coordinate observation,
is the incremental vector of the affine transformation parameters,
is the incremental vector of the object space coordinates of the target point, A and B are coefficient matrices containing partial derivatives of the unknowns, and
is the discrepancy vector. P is the unit matrix. As all the coordinates of image points are observations of equal precision, the initial value of P is an identity matrix.
The normal equation can be established from Equation (8) according to the principle of least-squares adjustment:
After each adjustment, the plane coordinates of a tie point (TP) in the object space were refreshed, and an auxiliary DEM was used as the height constraint. Elevation Z in Equation (4) of the TP was interpolated from the DEM instead of the intersection of multiple SAR images. Along with the plane coordinates X and Y, Z was set as a new ground coordinate value of the TP and was subsequently substituted into the adjustment system for the next iterative calculation until the entire adjustment process converged.
Finally, the digital orthophoto map (DOM) of GF-3 image is generated by ortho-rectification. It is a classical method of remote sensing image processing [
27]. The image is changed from the image coordinate system to the geodetic coordinate system according to the orientation parameter solved by the block adjustment. The planar block adjustment and orthorectification solution procedure for GF-3 imagery is shown in
Figure 3 as follow steps:
- (1)
Adjustment input file preparation. Including GCP File, TP File, and RPC File. GCP File and TP File are obtained by manual measurement.
- (2)
Weak convergence determined. Calculating the intersection angle of two SAR images. If the angle is less than 10°, the planar block adjustment process will be executed. Otherwise, the stereo block adjustment will be executed.
- (3)
Planar block adjustment. First, the initial values of TPs are obtained by forward intersection. Considering the situation of weak intersection. Second, the elevations of TPs are obtained by DEM interpolation. Third, adjustment calculation. Fourth, the posteriori weight method is used to assign weights again. Fifth, if adjustment convergence, it can go to step 4. Otherwise, continue the implementation step 3.
- (4)
Ortho-rectification.