The Huaibei Coalfield (115°58′–117°12′E, 33°20′–34°28′N), located in the north of Anhui Province of China, was selected as the study area. It has a temperate monsoon climate with distinct seasons and mild climate. There are many coal mines in this region. Here, the terrain is flat, the surface vegetation is rich, and the farmland is numerous. The cumulatively proved coal reserves are nearly 13 billion tons. The large-scale and long-time exploitation of underground coal resources has resulted in a lot of ground subsidence in large area. It has seriously affected the local ecological environment and economic and social development. The study area is shown in Figure 1.

In order to evaluate the performance of the new phase unwrapping method proposed in this paper, 4 Sentinel-1A SAR data from ESA (European Space Agency) acquired from December 2017 to March 2018 were selected. Two interferometric pairs are constructed by the Sentinel-1A data which are acquired in the IW (interferometric wide) mode, C-band (with 5.6 cm wavelength), VV polarization, and ascending orbit. The Sentinel-1A radar data used in this study has an incident angle of 38.9° and a pixel size of about 2.33*13.90 m. The detail characteristics of the interferometric pairs are shown in Table 1.

Moreover, the SRTM (Shuttle Radar Topography Mission) DEM (digital elevation model) released by NASA (National Aeronautics and Space Administration) was applied to D-InSAR data processing in this study. The spatial resolution of DEM used in this study is about 90 m.

The data processing flow of InSAR can be referenced in (Massonnet and Feigl, 1998; Hanssen, 2001). The interferometric pairs were processed by D-InSAR (differential radar interferometry) and two interferograms of the whole area were obtained. The subsidence of the mining area will form a series of circular or elliptical interferometric fringes on the InSAR interferogram. Therefore, the subsidence basin is the object of our phase unwinding research to obtain the accurate land subsidence in the coal mining area. The subsidence basin is defined as the position where the subsidence is about 1 cm. The subsidence basins in study area were obtained by detection model based on HOG (histogram of oriented gradient) feature (Ronneberger et al., 2015) automatically. Then, the interferograms and coherence images including one complete subsidence basin were cut out from the original interferograms. Therefore, there are more than 25 subsidence basins within the coverage of a single Sentinel-1A.

Three subsidence basins with different interferometric quality were selected from interferograms generated by two interferometric pairs for phase unwrapping experiment, as shown in Figure 2. Figure 2A is a subsidence basin with good interferometric quality. It contains two interferometric fringes that resemble elliptical shapes. The interferometric fringe edges are characterized as smooth and continuous, without interruption or confusion, and the coherence is also very high, as shown in Figure 2D. Figure 2B is a subsidence basin with common interferometric quality. It contains approximately four interferometric fringes that resemble elliptical shapes. It is obvious that the interferometric fringes inside are partially interrupted or discontinuous, and the coherence value is poor, as shown in Figure 2E. Figure 2C is a subsidence basin with poor interferometric quality, in which the interferometric fringes are partially confused due to the massive noise. The coherence value is terrible, as shown in Figure 2F.

After underground coal mining, a series of goaf will be formed (Hu et al., 2013; Yang et al., 2018). In the InSAR interferograms, it is usually manifested as a series of concentric circles or concentric ellipses with similar shapes (Du et al., 2019; Wang et al., 2021). According to the principle of phase unwrapping (Chiglia and Pritt, 1998; Yu et al., 2019), the number of phase-jump (or wrap-count) is calculated by the interferometric fringes of subsidence basin and the phase unwrapping is realized through adding or subtracting 2N in the position corresponded to the interferogram.

The purpose of the edge detection algorithm is to achieve the boundary of subsidence basin in the interferogram. The wrap-count (integer jump of 2π) is determined by number of interferometric fringes. Thus, the phase values of subsidence basins are added or subtracted 2N based on some certain rules. The specific rules of phase unwrapping are that 2N is added to the phase value when there is the change from +π to −π or subtracted in the opposite condition.

In mathematical language, it can be expressed as follows: ϕ = φ + 2 π ⋅ N , N = 0 , ± 1 , ± 2 …, (1) where Φ represents the value after phase unwrapping, and φ and N are the wrapping phase and the number of phase-jump, respectively.

The swath of SAR image is large, such as Sentinel-1 image with the swath of 250 km, but no land surface deformation occurred in most areas. Only several subsidence basins will be formed in some areas, such as the underground coal mining area. Some factors, such as vegetation growth, surface water, temporal decorrelation, and spatial decorrelation (Zhou et al., 2009; Ronneberger et al., 2015), will bring a lot of phase noise into interferograms, which will result in great difference among interferograms in the mined-out area. For example, some interferometric fringes are clear and continuous, but some are partially interrupted or discontinuous. Moreover, some fringes are even indistinct or confused. Above on these problems, we will not obtain the correct results of phase unwrapping using the traditional phase unwrapping methods, which leads to unreliable monitoring results of land subsidence in the mining area.

To solve this problem, a novel phase unwrapping method based on U-Net convolutional neural network is proposed. Firstly, the U-Net convolutional neural network that has been trained according to some samples is used to extract the edge of the fringes. Then, it can automatically obtain the boundary information of the interferometric fringes in the region of subsidence basin. Secondly, an edge-linking algorithm is constructed based on edge growth and predictive search to connect the break of interferometric fringe automatically. Then, a complete and continuous edge of interferometric fringe is obtained. Finally, according to the rules of phase unwrapping, the number of phase-jump N or the wrap-count is calculated and the phase unwrapping is realized through adding or subtracting 2N at the corresponding position in the interferogram. The flow chart of the novel method is shown in Figure 3.

The phases of interferogram are wrapped (Hanssen, 2001; Wang et al., 2021). If the position of phase jump (that is wrap-count) in the interferometric fringe can be obtained, the true phase can be obtained. Therefore, the phase unwrapping can be turned into an image segmentation problem to a certain extent. For image edge extraction, many methods have been proposed. Although the FCN (fully convolutional network) can identify the spatial relationship between pixels and realize the segmentation at the semantic level, the process of restoring the image is relatively simple and the obtained fringe boundary is not accurate enough (Jonatnan et al., 2014; Liang et al., 2020). As for InSAR interferometric images, there are not a large amount of training data available for public use, so we can only collect and generate them by ourselves, and the number of samples generated is generally small. Due to the practicality of U-Net convolutional neural network and its learning ability with fewer samples (Ronneberger et al., 2015; Parente et al., 2019), the U-Net convolutional neural network is used to obtain the edge information (or the wrap-count at each pixel) of the interferometric fringes automatically, which is the first step of the new phase unwrapping method. The main role of U-Net convolutional neural network in new phase unwrapping method is to obtain the wrap-count (integer jump of 2π) at each pixel by edge detection.

The U-Net convolutional neural network is basically same as the FCN. Through multiple convolutions and pooling operations, the feature dimensions are improved and the images are compressed (Kando et al., 2019; Esch et al., 2020). In the process of image restoration, the U-Net convolutional neural network also convolved the fused images for many times (Qin et al., 2020), which enhanced the detailed information of images and achieved the better segmentation effect.

The training sample set of U-Net convolutional neural network came from 21 interferometric pairs composed of Sentinel-1A data corresponding to Huaibei Coalfield and Yanzhou Coalfield in China. A total of 150 training samples have been established and these samples have been uniformly resampled to 256 × 256×1. For the convolutional neural network, the number of samples is little. However, the U-Net convolutional neural network has a good generalization ability. This can compensate for shortage of samples to a certain extent.

As there are lots of surface vegetation and farmlands in the mining area, it will cause incoherence phenomena in monitoring the land subsidence by InSAR technology, which results in some discontinuous or interrupted interferometric fringes in the interferogram. Thus, it is impossible to extract the continuous and complete fringe boundary directly. Although the U-Net convolutional neural network can extract relatively clear interferometric fringe boundary, the edge thinning and edge linking are needed to carry out to obtain the closed and complete edge, especially in case of discontinuous, interrupted, or even confused interferometric fringes.

An improved algorithm is adopted for edge thinning (Xu et al., 2014), which is described as follows:

Step 1: the edge points in the image are traversed and the current edge point is recorded as P₀

The above three steps constitute a traversal and carry out as repeated iteration until that no points satisfy the rule to be marked.

In order to obtain the complete edge of the interferometric fringes, an edge linking operation is required. Based on edge growth and predictive search, a new algorithm is constructed by improving the edge growth method (Farag and Delp, 1995; Lin and Shi, 2000). The specific steps are as follows:

Step 1: search the endpoint P in the edge graph

The operation of edge thinning and edge linking is based on binary images. So, before the edge thinning and linking, the threshold segmentation of edge image is carried out to obtain the binary image of the result from the U-Net convolutional neural network.

Neural networks have been applied to InSAR phase unwrapping (Jonathan et al., 2015; Parente et al., 2019; Kando et al., 2019; Sica et al., 2020). As an important branch of machine learning, neural networks should be trained with sample data before edge extraction (Lin and Shi, 2000). Firstly, 60 targets of subsidence basins were extracted by HOG (histogram of oriented gradient) model (Wang et al., 2021) and their edge images were obtained to form the basic training samples. Secondly, in order to obtain more samples, the basic samples were uniformly resampled to 256 × 256 pixels by the resampling method of bilinear interpolation. Then, other 90 subsidence basins and corresponding edge images were achieved through transformation of rotation, translation, and zoom for the basic samples. Finally, a total of 150 training samples were obtained and some samples are shown as Figure 4. In the model established in this paper, the training group consists of 150 pairs of images, the test group consists of 10 images, and the validation group consists of 5 images.

The loss function is used to evaluate the difference between predicted and real values in the training process of neural network. The purpose of training or optimization operation is to minimize the loss function (Parente et al., 2019; Esch et al., 2020). The smaller the loss function value is, the closer the predicted value of the model is to the real value, and the result will be more accurate. The essence of edge extraction is a binary classification, so the binary cross entropy function is used as the loss function of U-Net convolutional neural network to represent the model accuracy. The overfitting problem is avoided by using the dropout function before and after the last convolutional layer and discarding some parameters randomly.

In order to further evaluate the performance of the new method quantitatively, the mean phase difference (M) and mean square error (MSE) are selected as assessment indexes to compare the results of new method with those of other methods quantitatively. The expressions are as follows: M = 1 mn ∑ i = 0 m − 1 ∑ i = 0 n − 1 ( I ( i , j ) − K ( i , j ) ), (2) MSE = 1 mn ∑ i = 0 m − 1 ∑ i = 0 n − 1 ( I ( i , j ) − K ( i , j ) ) 2 , (3) where m and n are the number of columns and rows of the image; I and K represent the phase unwrapping result of new method proposed in this paper and other methods, respectively. (i,j) represents the position of the pixel.

A subsidence basin with clear interferometric fringes from real interferometric data was selected to evaluate the performance of the new method. As shown in Figure 2A, the subsidence basin had good interferometric quality which is characterized with smooth and continuous interferometric fringe edges, and the coherence value is also very high in the coherence image. The results of edge thinning and linking after binarization process are shown in Figure 5 and the result of phase unwrapping is shown in Figure 6A. In order to verity the effectiveness of the new method proposed in this paper, the results are compared with other phase unwrapping methods, such as minimum cost flow (MCF) method, Goldstein’s branch-cut method, quality-guided phase unwrapping (QG) method, Flynn’s minimum discontinuity (MD) method, LP minimum norm method (LP), and so on. Goldstein’s branch-cut (Goldstein) algorithm (Yu et al., 2019) is the most representative of the residue-theorem-based methods, connecting the nearby positive and negative residues so that the residues are balanced. The minimum cost flow (MCF) PU method (Costantini, 1998; Liu and Pan, 2020) and Flynn’s minimum discontinuity (MD) method (Xu et al., 2016) can be classified as the LP-norm phase unwrapping method. Quality-guided (QD) phase unwrapping method assumes that pixels with high quality are less likely to cause PU error (Yu et al., 2019). The methods always attempt to first make the integration path that visits the highest-quality pixel provided by the quality map. The specific results are shown as Figures 6B–F. The quantitative comparisons are shown in Table 2.

It can be seen from Figure 6 and Table 2 that the result of new method proposed in this paper is generally consistent with those of other methods under the condition of clear interferometric fringes. The ranges of unwrapping phases from different methods are within the theoretical ranges. The values of M and MSE from different methods are all small. The results of unwrapping are highly consistent in images and values. In other words, in the case of clear interferometric fringes from real interferometric data, different phase unwrapping methods can achieve correct unwrapping. It also indicates that the new method can achieve correct phase unwrapping result in the case of clear interferometric fringes.

A subsidence basin with interrupted interferometric fringes from real interferometric data, as shown in Figure 2B, was selected to study. The interferometric quality of this subsidence basin is common, and the interferometric fringes inside are partially interrupted and discontinuous. Especially, the coherence value is poor, as shown in Figure 2E. The results of edge extraction and phase unwrapping obtained by the new method proposed in this paper are displayed in Figure 7 and Figure 8A, respectively. Moreover, in order to compare with other five methods, the subsidence basin is resolved by other methods, and the phase unwrapping results are displayed in Figures 8B–F. The quantitative comparison is shown in Table 3.

As shown in Figure 8, the result of new method proposed in this paper is obviously different between those of other five methods in the case of discontinuous interferometric fringes. The unwrapping result of new method in this paper is closer with that of Goldstein method than others. The results from MCF, QG, and LP minimum norm methods all have the regions which are not completely unwrapped, but the sizes of the regions are different. From the results of phase unwrapping, the result of new method in this paper is great and without any incompletely unwrapped regions. The detailed comparisons are shown in Table 3. Only the results of new method and Goldstein method are in the theoretical ranges. From the assessment indexes of M and MSE of different methods, the result obtained by LP minimum norm method is the worst among these methods, which had many incompletely unwrapped regions. The results of other four methods also have some obvious regions that are incompletely unwrapped, though the difference is small in M and MSE when compared with the method proposed in this study.

In order to validate the reliability of the method under the influence of extreme noise, a subsidence basin with confused interferometric fringes from real interferometric data is selected for the experiment. As shown in Figure 2C, the quality of the interferometric fringe of the subsidence basin is poor and the fringe is confused due to a large amount of noise. Especially, the coherence value is very poor, as shown in Figure 2F. The edge extraction results are displayed in Figure 9 and the phase unwrapping result is shown in Figure 10A. Moreover, in order to compare with other five methods, the subsidence basin is resolved by other methods and the phase unwrapping results are displayed in Figures 10B–F. The quantitative comparisons are shown in Table 4.

It can be seen from Figure 10 and Table 4 that the phase unwrapping result of the method proposed in this paper is relatively consistent with that of the MCF method in the case of confused interferometric fringes. In addition, two phase unwrapping results from the new method and MCF method are also within the theoretical ranges. Therefore, it can be considered that the phase unwrapping results of these two methods are correct. However, in terms of the results of Goldstein method and other four phase unwrapping methods, all of them have the regions that are not unwrapped correctly and they are not within the theoretical ranges. Moreover, there are many unwrapped pixels existing in the phase unwrapping results of Goldstein method and QG method, and the whole effect of the phase unwrapping is not good. Especially, the MSE of the result of Goldstein method is larger than others. Hence, under the condition of extreme noise, the result obtained by the new method proposed in this paper is better than others and it can greatly achieve the correct phase unwrapping results in the case of confused interferometric fringes of mining area.