We first tested the performance of the Laplace domain waveform inversion at different damping constant (s) values through a simple synthetic example. Then, a complex synthetic example was used to compare the inverted results of the time domain FWI, whose initial models are from the ray-based inversion and Laplace domain waveform inversion, respectively. Finally, the Laplace domain waveform inversion was successfully applied to field data and provides initials models for the time domain FWI.

#### 3.1. Synthetic Data 1: A Simple Model

Figure 1a,c show the permittivity and conductivity models for testing the performance of our Laplace domain waveform inversion. The model size was 6 m in both the horizontal and vertical directions. Two cylindrical anomalies with diameters of 0.5 m were placed in the homogeneous medium of relative permittivity

${\epsilon}_{r}=5.5$ and conductivity

$\sigma =0.005$ S/m. The upper left anomaly was located at the position of (2 m, 2 m), and had a relative permittivity of 7 and conductivity of 0.008 S/m; the relative permittivity and conductivity of the lower right anomaly at the position of (4 m, 4 m) were 4 and 0.003 S/m, respectively. The two 6 m deep boreholes were 6 m apart. Thirteen transmitter positions (circle indicated) were spaced at 0.5 m intervals in the left borehole and thirteen receiver positions (crosses indicated) were spaced at 0.5 m in the right borehole (

Figure 1).

The initial models in the ray-based inversion and Laplace domain waveform inversion have the same relative permittivity and conductivity values as the background medium of the true models. To weaken the near-field effect in the forward and backward modeling, a median filter [

35] was applied near the source regions [

18]. We used a stepped parameter updating approach because of the large gradient differences between the permittivity and conductivity. The permittivity was inverted with the fixed conductivity and then the conductivity was inverted with the fixed permittivity [

14]. In the inversion process, the relative permittivity and conductivity are updated in the logarithmic domain to ensure their positive values and improve convergence [

36].

Figure 1b,d are the inverted results of the ray-based inversion in which the two cylindrical anomalies cannot be distinguished because the diameter of the anomalies is smaller than the resolution of the ray-based method [

6,

7].

Figure 2 shows the results of Laplace domain waveform inversion at six different damping constants after 50 iterations. The two anomalies are well reconstructed when the Laplace domain waveform inversion is performed. However, the inverted results are smoother than the true models due to the long-wavelength features in the Laplace domain. The sizes of the reconstructed anomalies in

Figure 2 are larger than those of results from the time domain FWI [

18].

In

Figure 2a,b,g,h, the results of Laplace domain waveform inversion at the low damping constants of

s = 50 × 10

^{6} and

s = 100 × 10

^{6} fail to reconstruct the shape of the cylindrical anomalies. This is due to the inaccuracy of the numerical Laplace transform of the time domain wavefield at a low damping constant [

20]. When the damping constant is 300 × 10

^{6}, both the results of relativity permittivity and conductivity reconstruct the two anomalies (

Figure 2c,i). In the results of the Laplace domain waveform inversion at

s = 500 × 10

^{6}, 700 × 10

^{6} and 900 × 10

^{6} (

Figure 2d–f,j–l), the two cylindrical anomalies are well reconstructed. The oscillations in

Figure 2 are caused by the high wave-number updates in the models, and the high wave-number components dominate the wave-number information as the

s values are increasing.

Figure 3 shows cross sections of the relative permittivity and conductivity through the results obtained by the ray-based inversion, Laplace domain waveform inversion, and the true models. In the results of the ray-based inversion and Laplace domain waveform inversion at damping constants of

s = 50 × 10

^{6} and 100 × 10

^{6}, the positions and sizes of the anomalies are not well reconstructed. When the damping constants are 300 × 10

^{6}, 500 × 10

^{6}, 700 × 10

^{6} and 900 × 10

^{6}, the permittivity tomograms from the Laplace domain waveform inversion successfully reconstruct the positions of anomalies at a smooth scale (

Figure 3a). Similar to the relative permittivity, the conductivity tomograms of the Laplace domain waveform inversion at high damping constants (

s = 300 × 10

^{6}, 500 × 10

^{6}, 700 × 10

^{6} and 900 × 10

^{6}) depict the positions of the anomalies well (

Figure 3b). As the damping constant increases, the conductivity values of the anomalies are more accurate. This may be because the Laplace domain wavefield at high damping constants has higher contrast gradients than that at low damping constants [

20]. However, the higher the Laplace damping constant, the less the damped wavefield contains the time domain wavefield. In addition, the damped wavefield at high damping constants is very easily disturbed by noise [

20]. Therefore, the damping constant in the Laplace waveform inversion of cross-hole radar data should be chosen in a moderate range. We propose that approximately five times the value of the dominant frequency of the radar data is appropriate.

Laplace domain wavefields at different damping constants are also the zero frequency component of the damped wavefield, but at different values. The inverted results of relative permittivity are approximately the same. Since the first direct event always dominates the damped wavefield of the cross-hole radar data, multiple damping constants in the inversion would not help if an optimal damping constant is chosen. In this study, we used a single damping constant in our Laplace domain waveform inversion.

#### 3.2. Synthetic Data 2: A Complex Model

To further check the performance of the Laplace domain waveform inversion, we built a three-layered model containing three cylindrical anomalies (

Figure 4a,d). The horizontal and vertical sizes of the model were 6 m. The three anomalies with diameters of 0.5 m were placed at the positions of (1 m, 3 m), (3 m, 3 m) and (5 m, 3 m). The top layer and the bottom layer had the same relative permittivity of 5 and conductivity of 0.001 S/m. The three cylindrical anomalies of

${\epsilon}_{r}=7$ and

$\sigma =0.008$ S/m were buried in the middle layer of

${\epsilon}_{r}=5.5$ and

$\sigma =0.0028$ S/m. The initial models in the Laplace domain waveform inversion were the same as the middle layer. The locations of transmitters and receivers were the same as the simple example. In this example, the damping constant was 500 × 10

^{6} for the Laplace domain waveform inversion.

Figure 4 shows the relative permittivity and conductivity tomograms that result from the ray-based inversion and Laplace domain waveform inversion. The reconstructed anomalies in both relative permittivity and conductivity tomograms from the ray-based results (

Figure 4b,e) are mixed together, whereas the layered medium is delineated roughly. In the results of the Laplace domain waveform inversion, the layer interfaces are well reconstructed (

Figure 4c,f). However, the three anomalies in

Figure 4c,f cannot be distinguished in the horizontal direction because of the small interval between the anomalies. To further compare the results of the ray-based inversion and Laplace domain waveform inversion, we apply their results as the initial models in the time domain FWI [

18], and then compare their performance.

Figure 5a,c show the results of the time domain FWI based on the ray-based results, and

Figure 5b,d are the results of the time domain FWI based on the Laplace domain waveform results. For both relative permittivity and conductivity, the layered medium and three cylindrical anomalies are rebuilt well. The interface between the middle layer and the bottom layer in

Figure 5 agrees with that in the true model.

Figure 6a displays the curves of object function values during the 50 iterations of the time domain FWI in

Figure 5. We note that the time domain FWI based on the Laplace domain waveform inversion has slightly large errors in the first five iterations, after which the error curve (red line in

Figure 6a) decreases quickly and finally reach convergence at a lower value than the error curve (blue line in

Figure 6a) of the time domain FWI based on the ray-based inversion.

Figure 6b,c show cross-sections of relative permittivity and conductivity through the tomograms in

Figure 4 and

Figure 5. In the results of the time domain FWI, the positions and values of the cylindrical anomalies are clearly depicted. The relative permittivity values from the Laplace-based FWI are more accurate than those from the ray-based FWI (

Figure 6b). In

Figure 6c, FWI based on the ray-based inversion provides better values for the left and right anomalies. For the middle anomaly in

Figure 6c, we note that the FWI based on the Laplace domain waveform inversion performs better. Similar to the results shown in

Figure 3, inverted results of conductivity perform better than those of relative permittivity, as shown in previous studies [

15,

16,

18]. In this example, Laplace domain waveform inversion is slightly better than the ray-based inversion to provide initial models for the time domain FWI.

#### 3.3. Field Data Measured at Guizhou

Field data were collected in the Guizhou Province using radar systems (MALA Geoscience) with 100 MHz borehole antennas. The horizontal distance between the two boreholes was 18 m. Forty source positions with 1 m intervals were placed between depths of 9 m and 48 m. The receiver positions corresponding to each source position were different (

Table A1). The interval of receiver positions was 0.5 m. The raw data measured at the site is shown in

Figure 7.

The estimation of source wavelet takes an important role in the waveform inversion of field data because the source wavelet must be known for forward modeling. However, the source wavelet is commonly unknown. We followed the deconvolution method proposed by Ernst et al. [

15] to estimate the source wavelet, which has been presented in previous studies [

15,

18].

Before starting Laplace domain waveform inversion, the field data was subjected to careful preprocessing to improve the signal-to-noise ratio (SNR), including removal of the direct-current (DC) and band-pass (BP) filtering. In addition, careful muting of the signals before the first arrival event is needed due to the characteristics of the Laplace domain wavefield [

20]. The dominant frequency of the field data is approximately 50 MHz due to the water-saturated environment. Based on the conclusion presented in

Section 3.1, we chose 200 × 10

^{6}, which is approximately four times larger than the dominant frequency of the field data, as the damping constant in the Laplace domain waveform inversion.

Figure 8a,c show the inverted results of the ray-based inversion and

Figure 8b,d show the results of the Laplace domain waveform inversion after 83 iterations. From

Figure 8a,b, we find that the relative permittivity tomograms from the ray-based inversion and Laplace domain waveform inversion are similar. A tunnel or cave exists at a depth of ~25 m and runs through the two boreholes. The upper right and bottom left regions have high permittivity values, whereas the permittivity of the upper left and bottom right regions is low (

Figure 8a,b). However, the conductivity from the ray-based inversion (

Figure 8c) shows low values (<10

^{−3}), which are probably unrealistic due to the existence of underground water. The reason is that the ray-based inversion can only provide relative spatial distributions of conductivity [

7]. Conductivity results from the Laplace domain waveform inversion are significantly improved compared with that from the ray-based inversion.

Figure 9 shows the results of the time domain FWI by using the initial models from both ray-based inversion and Laplace domain waveform inversion.

Figure 10 shows the root mean square (RMS) error curves of the field inversions with the ray-based (blue) and Laplace-based (red) results as the initial models. The convergence curves of the ray-based FWI and Laplace-based FWI stop after 43 and 69 iterations, respectively. The increase at the 12th iteration in the RMS curve of the ray-based FWI is caused by an additional source estimation during the inversion. We find that the Laplace-based FWI converges to a little lower value than the ray-based FWI. Compared to the initial models, we find obvious improvements in the results of the time domain FWI in

Figure 9. At the depth of ~20 m, ~36 m and ~45 m, conductivity tomograms show horizontal layers with high values. We note that the tomograms from the Laplace-based FWI reveal more details than that from the ray-based FWI, especially for the conductivity tomograms.

Figure 11 shows the predicted receiver gathers based on the inverted results for the 36th source position.

Figure 11a,c are the predicted waveforms using the ray-based and Laplace domain waveform results. Though different initial models are used, waveforms of radar traces generated from the two FWI results agree well with the field waveforms (

Figure 11b,d). The comparisons of waveforms indicate that inversions using both ray-based and Laplace domain waveform results are successful in the time domain. Initial models from the Laplace domain waveform inversion improve their corresponding results of FWI with more details (

Figure 9b,d).