3.1. SBAS-InSAR Algorithm
InSAR combines synthetic aperture radar technology with interferometric technology. Synthetic Aperture Radar (SAR) is an active microwave remote sensing that is used to record the scattering intensity information and phase information of the ground object. The former reflects the surface properties (water content, roughness, feature type, et al.), while the latter contains distance information between the sensor and the target. The basic principle of interference is the complex conjugate multiplication of SAR images in the same area to extract the topographic or deformation information of the target. The modes of radar interference include orbital interferometry, cross-track interferometry, and repeated orbital interferometry. This paper used repeated rail interference, and the interference phase expression is as follows:
In the equation, φflat represents flat phase; φtopo represents the topographic phase, which can be used to recover topographic information; φdef represents the phase caused by ground deformation between tA (start time) and tB (end time); φatmo represents the temporal atmospheric phase at different SAR acquisition tA and tB; φnoise represents the phase caused by observation noise (including the temporal decorrelation, orbital errors, and thermal noise).
In this paper, SBAS-InSAR technology by Bernardino [
45] was applied to calculate the ground deformation of Sentinel-1 SAR images in the research area, which greatly optimized the problems of decoherence and atmospheric effects caused by the excessive spatial baseline of D-InSAR technology, thus increasing the frequency of time sampling. The SBAS-InSAR method used an interferogram generated by SAR pairs with a short temporal–spatial baseline to improve coherence and reduce phase noise by the multi-view processing of differential interferograms to extract high-coherence pixels. This method inherited the advantages of the conventional D-InSAR method and could obtain the evolution of long-term slow surface deformation, which improved the time resolution of deformation monitoring. Thus, this method was selected as the main method to calculate the ground deformation of Sentinel-1 SAR images in the research area. The essential principles of SBAS-InSAR technology were as follows:
Suppose there are repeated orbital SAR images in time series t0, t1, …, tn. By setting the time and space baseline thresholds, we can get L small baseline sets. Within each set, the baseline of the interference pair is small, while the base of the interference pair is large between each set. Therefore, there are a total of M differential interferograms, where (N + 1)/2 ≤ M ≤ N (N + 1)/2. Suppose that the differential interference phase (tk, x, r) at any time (tk) and the initial time (t0) on any pixel (x,r) is unknown, and the differential interferograms δφk (x,r) (k = 1, …, M) is the observation.
For any differential interferogram, there is:
where
d(
ti,
x,
r) and
d(
tj,
x,
r) are the cumulative deformation variables relative to the start time, respectively. There are
M differential interferograms; therefore, M equations can be obtained according to Equation (2), and the equations are represented by a matrix:
where
A is an
M ×
N matrix consisting of –1, 0, and 1 elements; for example, Formula (1) in the matrix corresponds to line k, the value of the first element is 1, while the value of the last element is −1, and the other values are 0;
δφ(
x,
r) and
φ(
x,
r) are
N-order vectors. Matrix A is an approximate correlation matrix. If all the data belong to a baseline set, then
L = 1,
M ≥
N, where A is an N-order matrix. When
M =
N, the system of equations had a unique solution. When
M >
N, the system of equations is overdetermined. The least squares method could be used to solve the estimated value of
φ(
x,
r):
Usually, the data set was scattered in several subsets, L > 1, where was a singular matrix, and the rank of A is . The equations have an infinite number of solutions. In order to obtain a unique solution, SBAS-InSAR uses a Singular Value Decomposition (SVD) method to jointly solve multiple baseline sets, and find the least squares solution in the least norm sense to obtain the cumulative deformation result. The SARscape 5.2.1(Sarmap Company, Caslano, Switzerland) plugin was used, which was based on ENVI 5.3 software (Exelis Visual Information Solutions Company, Broomfield, America), and the detailed workflow of InSAR processing was given in the following parts:
The first step was to graphically output the generated interferograms. When the SAR images are entered alongside the spatial and temporal baseline threshold, the maximum logarithm that can be obtained was (
N × (
N−1))/2, and the tool would automatically select the best combination for pairing. In this study, a 50% spatial baseline of the critical baseline and time baseline was set to 365 days. These pairs were interfering and then used for SBAS inversion (
Figure 2). The program automatically selected a super master image, which was used as the reference image throughout the process, and all the pairs were registered to the super master image [
17].
The second step was to interfere with all the paired interference image pairs, including coherence generation, de-leveling, filtering, and phase unwrapping. All the data pairs were registered to the super-master image. What is important in this step was to obtain a satisfactory interferogram by adjusting the parameters and coefficients. Thus, the interferogram showed part of the low coherence area, and after the first screening, we chose to increase the intensity of the filter strength. The Goldstein filtering method is a commonly filtering method. In this study, the processing tools of the interference processing module were used to process some representative image pairs to debug the satisfactory processing parameters. After the second screening, the phase unwrapped graph was checked to determine if the coherence threshold needs to be adjusted until a satisfactory result was obtained. Finally, the unwrapping method this paper used was Delaunay Minimum Cost Flow (MCF), which was a good way to connect a set of high-coherence pixels and other isolated high-coherence regions, thus this method was recommended to minimize the impact of phase transitions.
The third step was to remove the constant phase and phase transitions that still exist in the unwrapped phase diagram, resulting in a re-leveling result. In SARscape, there were two optimization methods on the flattening parameter panel: orbit optimization and residual phase optimization. This paper used an orbital optimization method that utilized an orbital polynomial model to correct the orbital parameters of the secondary image to eliminate the ramp phase.
The next step was to estimate the deformation rate and residual topography, and reduce the atmospheric phase as much as possible by filtering. Atmospheric filtering played a role in the smoothing of the time series. The removal of the atmosphere was achieved by low-pass filtering and high-pass filtering. Finally, the deformation results in the time series were obtained. The software not only calculated the deformation rate of the Line-Of-Sight (LOS) direction, but also the vertical direction. This study used the deformation results in the vertical direction for analysis [
17]. Finally, the deformation results in the time series were obtained. The software not only calculated the deformation rate of the LOS direction, but also the vertical direction. This study used the deformation results in the vertical direction for analysis. The technical flow chart is shown in
Figure 3.
3.2. Acquisition of Building Area
The building area during the same period was acquired using the classification results of Landsat 8. The detailed information of workflow was divided into three parts. Firstly, the Landsat 8 OLI image that was achieved was pre-processed. Secondly, the land cover information was extracted by the object coverage classification method, which was verified with the traditional supervised classification and ground observations. Finally, the southern architectural coverage map along the Erhai region was obtained.
The original image was radiated and calibrated with the radiation calibration parameters of the OLI sensor, and the Digital Number (DN) value of the original image was transformed into the radiance by pixel scale. Then, the image was corrected using the Fast Line-of-sight Atmospheric Analysis of Hypercubes (FLAASH) model, and the radiance value was converted into the true reflectance of the surface [
46,
47]. In addition, geometric precision correction was performed on the image, making the correction error controlled within 0.1 pixel. Finally, the Gram–Schmidt algorithm was used to fuse band 2, band 3, and band 4 of the corrected image, and the resolution of the fused image was 30 meters, which was helpful to the extraction of urban building classification information. Supervised classification was based on prior knowledge and trained by selecting samples in order to establish statistical recognition function and classify categories according to probability rules. The application of this method of classifying land use could guarantee the reliability and representativeness of the results. In this study, ENVI 5.3 software was used as the operating platform, and the maximum likelihood algorithm was used to supervise and classify remote sensing images in the Erhai region. This research distinguished elements visually on the image and drew the polygon samples, and tried to ensure the purity of the elements in the sample. It also drew other samples of the feature in other areas, and the sample was distributed as evenly as possible across the image. According to the research needs, the selected land-use samples were building land, water, agriculture, forest land, and bare land. The training samples were selected according to the characteristics of the land-covering classification scheme and the sample description. The sample selection followed the principle of uniform distribution of the entire study area, and the region of interest was established through the visual interpretation in the Landsat 8 OLI image, combined with Google Earth high image optimization. In order to check the separability between each sample, the Jeffries–Matusita, Transformed Divergence parameters were used to investigate. When the values of those two parameters were between 0 and 2.0, greater than 1.9 indicated that the sample was separable and belonged to the qualified sample; less than 1.8 indicate a need to edit the sample or re-select the sample; less than 1 considered combining the two types of samples into a sample. The sample classification results were tested, and separation values of different types were greater than 1.9, which meant that the samples taken can be well differentiated. According to the classification of representative training samples, the maximum likelihood classification method was adopted in the southern Erhai region to obtain the building area. The classification accuracy was evaluated, and the classification results were compared with the sample points obtained in the field survey to verify the accuracy of the results. The data format of the urban building area extraction result was converted into the shape format, and the urban building area data was extracted in ArcGIS 10.4 to obtain the specific data of the building area of the Erhai region that belong to different times.
3.3. Predicting the Cumulative Ground Deformation Using Back-Propagation
Predicting the cumulative ground deformation was required in many fields associated with urbanization, environment protection, agricultural, and ecological applications [
48,
49]. However, the current prediction methods were either using linear regression or multiple linear regression. In other words, the current analysis or predicting of cumulative ground deformation was mainly dependent on the previous results of ground deformation [
50,
51]. The predicting was based on the analysis of the current condition; therefore, the relationships built from linear or multiple linear regression were so simple that their uncertainties of predicted results cannot be easily controlled. Then, the predicting results were not convincing or reliable. To face this challenge, a more reliable model with precise data should be built and adopted. BP is a supervised learning method of Artificial Neural Networks (ANN), which can be used to perform a given task [
52,
53,
54]. It requires a teacher who knows, or can calculate, the desired output for any input in the training set. BP requires that the activation function used by the artificial neurons (or “nodes”) be differentiable [
55,
56,
57]. Therefore, the ANN was adopted to predict the cumulative ground deformation using related independent variables such as temperature, precipitation, water level, or building area. For a given region, the related influencing factors are used as model input parameters, and the cumulative ground deformation is set as a model output parameter. The whole process needs to consider the input, output, network, node, study rate, et al.
In the equation, represents the number of hidden nodes; represents the number of input nodes; represents the number of output nodes, and represents the constant.
Indicators such as the root mean square error (RMSE) are selected to evaluate the precision of the simulated versus the observed data. The RMSE reflects the distance of the training sample, testing sample, and validating sample from the real data or true data. The RMSE is defined as follows [
58]:
In the equations,
represents the simulated results;
represents the observed results;
is the mean value of the observed data, and
n is the total number of comparisons. The basic process of BP was shown in
Figure 4, and the detailed workflow is defined as follows:
Normalization
For most machine learning and optimization algorithms, scaling the eigenvalues to the same interval makes it possible to obtain better performance models. For example, there are two different features. The first feature ranges from 1 to 10, and the second feature ranges from 1 to 10,000. In the gradient descent algorithm, the cost function was the least square error function. Therefore, when it comes to the algorithm, the second feature will be obviously chosen when using the gradient descent algorithm, because it has a larger range of values.
Propagation
The forward propagation of a training pattern’s input through the neural network was to generate the propagation’s output activations. Then, the BP of the propagation’s output activations through the neural network using the training pattern’s target was to generate the deltas of all the output and hidden neurons.
Weight Update
First, we multiply the output delta and input activation to get the gradient of the weight. Then, we bring the weight in the opposite direction of the gradient by subtracting a ration of it from the weight.