# GAN-Based Inversion of Crosshole GPR Data to Characterize Subsurface Structures
## 2 Methodology  
In this paper, we propose a GAN-based inversion network to transform crosshole GPR data into defect reconstruction images, as depicted in Figure 1. Instead of using a single GAN to invert crosshole GPR data, we adopt a two-stage strategy, namely, low- and high-resolution GANs. The low-resolution GAN extracts global features from crosshole GPR time-domain waveforms and produces a low-resolution defect reconstruction image. The high-resolution GAN enlarges the size of the above predicted map and modifies it so that the final results can be presented with high resolution. Both the low- and high-resolution GANs consist of a generator and discriminator. The generator encodes input crosshole GPR images to feature maps and then decodes the feature maps to the corresponding defect reconstruction image. The discriminator combines the input data with the generated defect reconstruction image to judge the verisimilitude of the predicted image, assigning label \"\(1\)\" if the result is a true defect reconstruction image or label \"\(0\)\" if it is a fake defect reconstruction image, and subsequently feeds the result back to the generator to update its weights. With the training process, the inversion network is optimized when the discriminator no longer differentiates the simulated defect reconstruction image from the training images [35]. The objective function of the network is defined as  
\[\underset{\text{G}}{\text{min}}\underset{\text{D}}{\text{max}}V(\text{D}, \text{G})=E_{x\sim P_{\text{data}}(x)}[\log(\text{D}(x))]+E_{z\sim P_{z}(z)}[ \log(1-\text{D}(\text{G}(z)))]. \tag{1}\]  
Since network training in one epoch can be divided into two steps, which are discriminator training and generator training, Equation (1) can be disassembled into the following two equations:  
\[\underset{\text{D}}{\text{max}}V(\text{D},\text{G})=E_{x\sim P_{\text{data}}( x)}[\log(\text{D}(x))]+E_{z\sim P_{z}(z)}[\log(1-\text{D}(\text{G}(z)))] \tag{2}\]  
and  
\[\underset{\text{G}}{\text{min}}V(\text{D},\text{G})=E_{z\sim P_{z}(z)}[\log(1 -\text{D}(\text{G}(z)))], \tag{3}\]  
where \(\underset{\text{G}}{\text{min}}V(\text{D},\text{G})\) is the discriminator's performance value function, representing the discriminator's ability to distinguish between true and false images. The smaller the function value is, the better the discriminator performs. \(\underset{\text{D}}{\text{max}}V(\text{D},\text{G})\) denotes the generator's performance value function, which indicates the difference between the virtual image generated by the generator and the real image. The greater the function value is, the better the generator performs. \(x\) is a real sample that obeys the distribution of the real dataset. \(z\) is an input sample of the generator that follows the distribution of the generator's input data. \(E_{x\sim P_{\text{data}}(x)}[\log(\text{D}(x))]\) is a mathematical expectation constructed according to the logarithmic loss of the discriminator identifying the real samples. When the performance of the discriminator is optimized, the discriminator gives the judgment of \"\(1\)\" for all images in the real dataset. \(E_{z\sim P_{z}(z)}[\log(1-\text{D}(\text{G}(z)))]\) is the logarithmic loss of virtual samples made based on the discriminator's judgment. Similarly, in the optimal state, the discriminator labels all images in the virtual dataset with \"\(0\)\", that is, \(\text{D}(\text{G}(z))=0\). Figure 1: Structure of the GAN-based crosshole GPR data inversion network. In summary, the network first optimizes the performance of the discriminator to maximize the mathematical expectation of the two logarithmic losses in Equation (1). Then, it optimizes the performance of the generator by producing virtual samples to obtain real labels from the discriminator, that is, \(\mathrm{D}(\mathrm{G}(z))=1\). Therefore, the optimization process of the generator's performance is to minimize the mathematical expectation of the logarithmic loss in Equation (3).