Underlying Topography Estimation over Forest Using Maximum a Posteriori Inversion with Spaceborne Polarimetric SAR Interferometry
<p>(<b>a</b>) Optical image of the test area in Google Earth. (<b>b</b>) L-band SAR image in the Pauli basis (R: HH - VV; G: HV; B: HH + VV) (<b>c</b>) Alos DEM projected to SAR coordinates. (<b>d</b>) The topographic phase estimated by Alos DEM.</p> "> Figure 2
<p>The DEM inversion flowchart for spaceborne PolInSAR. (<b>a</b>) Ground phase estimation based on MAPV. (<b>b</b>) DEM generation.</p> "> Figure 3
<p>The difference between Gauss distribution and Von Mises distribution. Up: In the Cartesian coordinate system. (<b>a</b>) Gauss distribution. (<b>b</b>) Von Mises distribution. Down: In polar coordinate. (<b>c</b>) Gauss distribution. (<b>d</b>) Von Mises distribution. The hollow black point represents the position of the topographic phase on the unit circle of the complex plane, and the asterisks correspond to the phases at the two peaks of the MLE.</p> "> Figure 4
<p>Illustration of the difference between MAPG and MAPV. The solid line, dotted line, and dashed line represent the normalized objective functions of the MLE method, MAPG method, and MAPV method, respectively.</p> "> Figure 5
<p>(<b>a</b>) Evolution of the CRLB with signal-to-noise ratio (SNR). (<b>b</b>) A detail extracted from (<b>a</b>).</p> "> Figure 6
<p>Illustration of the four-step optimization procedure for solving the ground phase.</p> "> Figure 7
<p>The convergence of the improved RMSprop. The top (<b>a</b>–<b>c</b>) and bottom (<b>d</b>–<b>f</b>) rows, respectively, depict the convergence of phase and objective function values when the global learning rate <math display="inline"><semantics> <mrow> <mi>ϵ</mi> <mo>=</mo> <mspace width="4pt"/> </mrow> </semantics></math>0.1, 0.4, and 0.7.</p> "> Figure 8
<p>Ground phase estimated by (<b>a</b>) MAPG. (<b>b</b>) MAPV.</p> "> Figure 9
<p>(<b>a</b>) MAPG-inverted DEM. (<b>b</b>) MAPV-inverted DEM.</p> "> Figure 10
<p>(<b>a</b>) Section elevation change curve of marked line segment in Area 1. (<b>b</b>) Section elevation change curve of marked line segment in Area 2.</p> "> Figure 11
<p>The geocoded DEM maps and the difference between the LiDAR DEM and the DEM from different sources. (<b>a</b>) Alos-30m. (<b>b</b>) SRTM-30m (<b>c</b>) TSI. (<b>d</b>) MAPV.</p> "> Figure 12
<p>Histogram of the difference between the DEM obtained through different methods and the LiDAR DEM.</p> ">
Abstract
:1. Introduction
2. Materials
3. Scattering Model
3.1. PolInSAR Data Description
3.2. The Random Volume over Ground (RVoG) Model
4. Maximum a Posteriori Estimation of the Ground Phase
4.1. A Prior Probability Model of the Ground Phase
4.2. MAP with Von Mises Distribution as Prior
4.3. The Cramer–Rao Lower Bound Analysis
4.4. Four-Step Optimized Solution for MAPV
- Initial phase: Due to the presence of two peaks in the objective function, a single initial point is prone to fall into the local maximum. Therefore, it is necessary to choose an appropriate strategy to ensure that the results converge to the global maximum.
- Global learning rate: Because of the large differences between the objective functions of different pixels, it is very important to choose the global learning rate so that the method can adapt to all objective functions.
4.4.1. Initial Phase
- Step 1: Gradient descent using as initial phase. Get the between the two peaks.
- Step 2: Give the a in the opposite direction to as a .
- Step 3: Perform gradient ascent separately using this and to obtain the two peak values and their respective phases.
- Step 4: Compare the magnitudes of these two peak values and select the phase corresponding to the larger peak value as the final result.
4.4.2. Global Learning Rate
Algorithm 1: Framework of improved RMSprop algorithm. |
Require: Global learning rate , Decay rate , Initial phase |
Initialize: , , , , , |
5. Results
5.1. Evaluation Indicator
5.2. Efficiency of FSO for MAPV
5.3. Results of DEM Inversion in the Test Area
5.4. Performance Assessment of the Proposed Method
6. Conclusions
- Compared to the traditional exhaustive search method, FSO significantly improves computational efficiency without compromising accuracy.
- The MAPV method effectively addresses the issue of elevation jumps in DEM caused by the discontinuity in ground phase solutions by MAPG.
- Using IceSat-2 data as a benchmark, the DEM of the test forest area is compared with the DEMs of Alos, SRTM, and TSI. The results show that MAPV has better estimation performance, with improvements in both mean error (ME) and root mean square error (RMSE).
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
Abbreviations
DEM | Digital Elevation Model |
PolInSAR | Polarimetric Interferometric Synthetic Aperture Radar |
Lidar | Light Detection and Ranging |
RVoG | Random Volume over Ground |
TSI | Three-Stage Inversion |
MAP | Maximum a Posteriori |
MAPG | Maximum a Posteriori with Gaussian distribution as prior |
MAPV | Maximum a Posteriori with Von Mises distribution as prior |
CRLB | Cramer–Rao Lower Bound |
FSO | Four-Step Optimization |
ME | Mean Error |
RMSE | Root Mean Square Error |
Appendix A. The Fisher Information of MAPV
Appendix B. Gradient of the MAPV Objective Function
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Size | Time (s) | Iterations | ||||
---|---|---|---|---|---|---|
ES | FSO | ES | FSO | |||
7015 × 2673 | 6880.0 | 1235.1 | 360 | 24.5 | 99.97% | 99.99% |
ID | ||
---|---|---|
MAPG | MAPV | |
Area 1 | 87.48% | 99.06% |
Area 2 | 89.13% | 99.14% |
DEM | ME (m) | RMSE (m) |
---|---|---|
TSI | −13.2407 | 73.7140 |
ALOS | 2.6370 | 7.7956 |
SRTM | 2.1311 | 7.8972 |
MAPV | 0.2111 | 5.9944 |
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Li, X.; Lv, X.; Huang, Z. Underlying Topography Estimation over Forest Using Maximum a Posteriori Inversion with Spaceborne Polarimetric SAR Interferometry. Remote Sens. 2024, 16, 948. https://doi.org/10.3390/rs16060948
Li X, Lv X, Huang Z. Underlying Topography Estimation over Forest Using Maximum a Posteriori Inversion with Spaceborne Polarimetric SAR Interferometry. Remote Sensing. 2024; 16(6):948. https://doi.org/10.3390/rs16060948
Chicago/Turabian StyleLi, Xiaoshuai, Xiaolei Lv, and Zenghui Huang. 2024. "Underlying Topography Estimation over Forest Using Maximum a Posteriori Inversion with Spaceborne Polarimetric SAR Interferometry" Remote Sensing 16, no. 6: 948. https://doi.org/10.3390/rs16060948