DBPINet: A Physics-Informed Inversion Network for Martian Subsurface Radar Signal

Highlights

What are the main findings?

• A dual-branch physics-informed network (DBPINet) is proposed to achieve high-precision inversion of multiple subsurface parameters: layer thickness, permittivity, and loss tangent from dual-frequency radar signals.

What are the implications of the main findings?

• The proposed method significantly improves inversion precision in Martian lossy media, providing a reliable tool for Martian subsurface exploration.
• Numerical simulations and measured data experiments confirm the model’s practical applicability, with great potential for extension to more planetary radar exploration missions.

Abstract

Subsurface exploration of Mars is essential for understanding its geological evolution and potential water ice distribution. Subsurface radar sounding is an effective technique for detecting layered structure and physical parameters beneath the Martian surface. However, existing methods often neglect the influence of loss tangent and rely on data-driven approaches without physical constraints, limiting their accuracy in high-lossy environments and reducing their physical interpretability. To overcome these limitations, this paper proposes a dual-branch physics-informed network (DBPINet) for the joint inversion of layer thickness, permittivity, and loss tangent of Martian layered media. This method introduces a dual signal loss tangent branch (DSLT-Branch) to extract frequency-dependent attenuation features from dual-frequency radar signals and incorporates a physics-informed loss function based on the electromagnetic transmission-line model to embed physical laws into the learning process. Multiple numerical and measured experiments demonstrate the effectiveness of DBPINet. Compared with the MLP-based baseline and the more advanced LMPINet, DBPINet achieves significant improvements in different layered subsurface models. Specifically, on the three-layer models, the mean absolute percentage error (MAPE) for layer thickness, permittivity, and loss tangent is reduced by 4.793%, 3.600% and 4.559%, respectively. Meanwhile, DBPINet exhibits enhanced robustness under noisy conditions. When applied to real Mars Advanced Radar for Subsurface and Ionosphere Sounding (MARSIS) data acquired over the Medusae Fossae Formation (MFF) region, the inversion results reveal a three-layer subsurface structure (a volcanic ash surface layer, an ice-mixed basaltic middle layer, and a basaltic basement) that is consistent with existing geological interpretations.

1. Introduction

Martian subsurface preserves key records of climate, volcanism, and hydrology [1,2]. Orbital radar sounders, such as Shallow Radar (SHARAD) [3] and Mars Advanced Radar for Subsurface and Ionosphere Sounding (MARSIS) [4], enable subsurface sounding over large areas. By inverting key parameters (e.g., layer thickness, permittivity, and loss tangent) from radar echoes, researchers can infer composition, stratification, porosity, and water ice content of subsurface materials, thereby reconstructing subsurface evolution. For example, Plaut et al. [5] estimated the loss tangent of Martian south polar layered deposits (SPLD) as 0.001–0.005, supporting nearly pure water ice. Nerozzi et al. [6] interpreted the moderate loss tangent ($\tan \delta$ ≈ 0.016–0.019) in Hebrus Valles as evidence for basaltic lava flows. Watters et al. [7] estimated an average permittivity of ~3 in Medusae Fossae Formation (MFF) region, indicating an ice-rich material. Furthermore, Campbell et al. [8] estimated 300–600 m surface-sediment thickness there, consistent with periodic sedimentation [9]. Overall, subsurface parameter inversion is central to understanding planetary evolution and assessing ice resources.

High-precision inversion of Martian subsurface radar data (layer thickness, permittivity, and loss tangent) demands efficient and reliable algorithms. Layer thickness and permittivity control echo time delay and primary amplitude, while loss tangent controls signal attenuation, so classical workflows often treat them separately. The depth-correction method is widely used for thickness/permittivity [7,8,10] but depends strongly on prior geologic knowledge, limiting use in deep or complex settings. Full waveform inversion (FWI) [11,12] has been developed. For instance, Zhang et al. [11] used Bayesian genetic-optimization-based FWI for a two-layer SPLD model. However, FWI is computationally expensive, sensitive to initialization, and prone to local optima. Subsequently, planar layered inversion models were proposed. Fang et al. [13] employed a three-layer planar model to study buried craters in Elysium Planitia. Grima et al. [14] applied similar assumptions to SPLD. However, these methods typically require manual echo-power extraction of each layer, which is time consuming and hinders large-scale processing.

Loss tangent inversion is more challenging. Nerozzi et al. [6] and Watters et al. [7] estimated loss tangent from echo-power ratios, but this also requires manual power picking. Campbell and Morgan et al. [8,15] decomposed SHARAD’s wideband signal into multiple sub-bands and estimated loss tangent from inter-sub-band attenuation differences in polar regions and the MFF. However, this method cannot be directly applied to narrow-band MARSIS. Lauro et al. [16] instead utilized dual-frequency MARSIS attenuation difference to estimate the loss tangent of SPLD.

Recently, deep learning techniques have been widely applied in ground-penetrating radar (GPR) data inversion [17,18,19], primarily for detecting and inverting localized targets such as pipes and rocks. Representative models include GPRInvNet [19], DMRF-UNet [20], and GPRTransNet [21]. In addition, several generative adversarial network (GAN)-based inversion methods have also been proposed [22,23,24]. However, these studies primarily focus on anomaly detection tasks, such as localization and shape reconstruction. In contrast, Martian subsurface radar inversion aims at multiple macroscopic parameters of layered media with different physics and formulations. For multi-parameter layered-media inversion, Simsek [18] designed a multi-layer perceptron (MLP)-based network for one-dimensional parameter inversion, and Yang et al. [17] proposed LMPINet, which encodes radar echoes into pulse sequences and estimates subsurface parameters using a self-attention decoder. However, most existing methods invert only permittivity and layer thickness [17,18,19,20,21,22,23,24], neglect loss tangent, and are largely data-driven with limited interpretability.

Dual-branch and physics-informed architectures have emerged for complex inversion problems. For instance, Singh et al. [25] designed a dual physics-informed network for topology optimization. Li et al. [26] proposed a dual-branch convolutional neural network to learn complex scattering coefficients for 3D dielectric targets. Lu et al. [27] further optimized the dual-branch physics-informed network for wavefield inversion (a field analogous to electromagnetic inversion) by integrating multi-scale feature extraction in the dual-branch architecture and physical wave equation constraints. Zhou et al. [28] addressed the common multi-loss optimization challenge of this architecture by developing dual-balancing strategies, providing a general training solution for such frameworks. These studies suggest promise for physics-informed dual-branch design, but to our knowledge, they have not been applied to Martian subsurface radar inversion, where branch design and physics integration must match the specific sounding mechanisms.

In summary, current methods face three major limitations: (1) Physics-based inversions are often labor-intensive, costly, and susceptible to local optima; (2) Deep learning models often omit loss tangent, limiting their adaptability and reducing accuracy. (3) Data-driven models lack physical interpretability. To address these issues, this paper proposes a dual-branch physics-informed network (DBPINet) with two dedicated branches: an Eps-depth branch for joint inversion of thickness and permittivity and a dual signal loss tangent branch (DSLT-Branch) for loss-tangent inversion. We further develop a physics-informed loss based on an electromagnetic transmission-line model that reconstructs radar signals from predicted parameters and backpropagates the mismatch to regularize learning. This network is trained and tested on du-al-frequency MARSIS measurements, augmented with numerically simulated data generated using the instrument configuration parameters.

The main contributions are summarized as follows: (1) DBPINet with self-attention to capture dual-frequency time-delay, amplitude and attenuation features, enabling decoupled yet simultaneous inversion of thickness, permittivity, and loss tangent; (2) DSLT-Branch for accurate loss tangent inversion from attenuation cues; (3) a physics-informed loss based on transmission-line model that embeds electromagnetic constraints to improve reliability and interpretability.

The remainder of this paper is organized as follows: Section 2 introduces the radar echo modeling for planar stratified media and presents the proposed DBPINet for multi-parameter inversion. Section 3 reports experimental results on both simulated and measured datasets. Section 4 discusses the applicability and limitations of DBPINet under non-ideal Martian subsurface conditions (e.g., surface roughness and non-planar interfaces). Section 5 concludes the paper.

2. Methodology

2.1. Radar Echo Modeling for Planar Stratified Media

Martian subsurface is mainly shaped by climate cycles, volcanic activity, and water ice interaction, forming an approximately layered structure [11,13,14]. Such layering provides a suitable geophysical setting for subsurface detection via radar echoes. In this section, we establish a layered-media radar wave-propagation model for Martian subsurface exploration and derive the corresponding forward model for radar echoes, which forms the theoretical basis for subsequent inversion of medium parameters.

Each subsurface layer is assumed to be homogeneous and isotropic, with horizontal planar interfaces. As shown in Figure 1, $d_i$ is the thickness of the i-th layer, ${\acute{\epsilon }}_i$ is its permittivity, and $\tan {\delta }_i$ is loss tangent. The complex permittivity is expressed as ${\tilde{\epsilon }}_i={\acute{\epsilon }}_i-j \tan {\delta }_i {\acute{\epsilon }}_i$ [13]. In previous studies that employed narrowband signals (~3 MHz bandwidth) obtained by spectral decomposition of SHARAD data, $\tan {\delta }_i$ was often assumed to be frequency independent [8,15]. This paper adopts the same assumption for MARSIS, which operates with a bandwidth of 1 MHz.

Assuming a planar layered model and neglecting sidelobes, noise, and pulse-width effects, the received echo power $P\left(t\right)$ can be expressed as

$$S\left(t\right)\text{→}P\left(t\right)={\sum }_{\mathrm{j}=0}^{\mathrm{n}}{P}_{j+1} \mathrm{\delta }(t-{\tau }_{j+1})$$

(1)

where $S\left(t\right)$ represents the original radar signal.

Under low-loss condition and considering only the real part of permittivity, the Fresnel power reflection and transmission coefficients at the interface between the i-th and (i + 1)-th layers can be approximated as

$$r_{i,i+1}={\left({\displaystyle \frac{\sqrt{{\acute{\epsilon }}_{i }}-\sqrt{{\acute{\epsilon }}_{i+1}}}{\sqrt{{\acute{\epsilon }}_i}+\sqrt{{\acute{\epsilon }}_{i+1}}}}\right)}^2, t_{i,i+1}=1-r_{i,i+1}={\displaystyle \frac{4\sqrt{{{\acute{\epsilon }}_i\acute{\epsilon }}_{i+1}}}{{\left(\sqrt{{\acute{\epsilon }}_{i }}+\sqrt{{\acute{\epsilon }}_{i+1}}\right)}^2}}$$

(2)

Let $P_{1 }$ denote the surface echo power. The echo power at the (i + 1)-th layer interface can be expressed as

$$P_{i+1}=P_{1 }·r_{i,i+1}·\left({\prod }_{\mathrm{k}=0}^{\mathrm{i}-1}{\left(t_{k,k+1}\right)}^2\right)·\left({\prod }_{\mathrm{m}=1}^{\mathrm{i}}{\left(L_m\right)}^2\right)$$

(3)

Here, $L_m$ is the power attenuation factor for the single pass penetration of radar waves through the m-th layer [15],

$$L_m={\mathrm{e}}^{-2\pi {\displaystyle \frac{ f }{c}}\sqrt{{\acute{\epsilon }}_m }tan{\delta }_m{d}_m}$$

(4)

When performing inversion of Martian subsurface properties, multiple reflections are often neglected, and only single reflections are considered. Therefore, for a wave reflecting off the (i + 1)-th interface, it must undergo two-way transmission across all overlying interfaces and two-way attenuation within all overlying layers. Since the loss tangent is usually small, the reflection is dominated by the real part of the permittivity. Thus the echo power P_i+1 can be approximately decomposed into a lossless reflection component $P_{i+1}^{\left(0\right)}$ and cumulative two-way attenuation factor A_i+1, expressed as

$$P_{i+1}^{\left(0\right)}=P_{1 }·r_{i,i+1}·\left({\prod }_{\mathrm{k}=0}^{\mathrm{i}-1}{\left(t_{k,k+1}\right)}^2\right)=P_{1 }{\left({\displaystyle \frac{\sqrt{{\acute{\epsilon }}_i}-\sqrt{{\acute{\epsilon }}_{i+1}}}{\sqrt{{\acute{\epsilon }}_i}+\sqrt{{\acute{\epsilon }}_{i+1}}}}\right)}^2\left({\prod }_{\mathrm{k}=0}^{\mathrm{i}-1}{\left({\displaystyle \frac{4\sqrt{{{\acute{\epsilon }}_i\acute{\epsilon }}_{i+1}}}{{\left(\sqrt{{\acute{\epsilon }}_i}+\sqrt{{\acute{\epsilon }}_{i+1}}\right)}^2}}\right)}^2\right)$$

(5)

$$A_{i+1}\left(f\right)={\prod }_{\mathrm{m}=1}^{\mathrm{i}}{\left(L_m\right)}^2={\mathrm{e}}^{-4{\sum }_{m=1}^i{\alpha }_m{d}_m}=\exp \left(-4{\sum }_{\mathrm{m}=1}^{\mathrm{i}}{\displaystyle \frac{\mathrm{\pi } f \sqrt{{\acute{\epsilon }}_m }{d}_m}{\mathrm{c}}}\mathrm{t}\mathrm{a}\mathrm{n}{\delta }_m\right)$$

(6)

The time delay τ_i+1 represents the cumulative two-way travel times through all overlying layers:

$${\tau }_{i+1}={\displaystyle \frac{2}{\mathrm{c}}} {\sum }_{\mathrm{k} = 1}^{\mathrm{i}+1}{d}_k\sqrt{{\acute{\epsilon }}_k}$$

(7)

Observations at a single center frequency cannot distinguish whether differences in reflected power arise from interface reflections or attenuation within the medium. To address this issue, we introduce dual-frequency radar signals:

$$P_{f1}\left(t\right)={\sum }_{\mathrm{j}=0}^{\mathrm{n}}{P}_{j+1}^{\left(0\right)} A_{j+1}\left(f_1\right) \mathrm{\delta }(t-{\tau }_{j+1})$$

(8)

$$P_{f2}\left(t\right)={\sum }_{\mathrm{j}=0}^{\mathrm{n}}{P}_{j+1}^{\left(0\right)} A_{j+1}\left(f_2\right) \mathrm{\delta }(t-{\tau }_{j+1})$$

(9)

Here, the lossless reflection power $P_{j+1}^{\left(0\right)}$ and time delay ${\tau }_{j+1}$ are frequency independent, while the attenuation terms $A_{i+1}\left(f_1\right)$ and $A_{i+1}\left(f_2\right)$ are frequency dependent. The attenuation ratio between the two frequencies is

$$D_{i+1}(f_1,f_2)={\displaystyle \frac{A_{i+1}\left(f_1\right)}{A_{i+1}\left(f_2\right)}}=\exp \left(-4{\sum }_{m=1}^i{\displaystyle \frac{\pi (f_1-f_2)\sqrt{{\acute{\epsilon }}_m }{d}_m}{c}}\tan {\delta }_m\right)$$

(10)

The inversion process of DBPINet can be summarized as follows:

(1)

Extract frequency-independent components ($P^{\left(0\right)}, \tau$) as time delay and power amplitude features and frequency-dependent components ($A\left(f_1\right)$, $A\left(f_2\right)$ as attenuation features from the dual-frequency radar signals;

(2)

Calculate the interface reflection and transmission coefficients $r_{i,i+1},t_{i,i+1}$;

(3)

Calculate permittivity ${\acute{\epsilon }}_i$ and layer thickness $d_i$ via $r_{i,i+1}$, $t_{i,i+1}$, and τ. In this study, the surface permittivity ${\acute{\epsilon }}_1$ is fixed to a constant value in the training dataset and serves as a prior constraint during network training, and the surface echo power $P_{1 }$ is used as the normalization reference, which means each signal is divided by its corresponding surface echo power (in linear scale) to eliminate the influence of absolute power variations.

(4)

Calculate the loss tangent $\tan {\delta }_i$ for each layer via the dual-frequency attenuation ratio $D(f_1,f_2)$, along with the previously obtained ${\acute{\epsilon }}_i$ and $d_i$.

2.2. DBPINet for Multi-Parameter Inversion

Figure 2 illustrates the overall architecture of the proposed DBPINet. Given a predefined number of subsurface layers N, the model takes dual-frequency radar signals centered at ${Fc}_1$ and ${Fc}_2$ as inputs. A shared encoder first extracts features from the inputs. These features are then processed by two dedicated branches: the Eps-depth branch estimates the permittivities [${\acute{\epsilon }}_1$,${\acute{\epsilon }}_2$,…,${\acute{\epsilon }}_N$] and layer thickness [$d_1$,$d_2$,…,$d_N$], while the DSLT-Branch computes the loss tangents [$tan{\delta }_1$, $tan{\delta }_2$,…, $tan{\delta }_N$].

2.2.1. Dual-Frequency Radar Signal Feature Encoder

The input dual-frequency radar signals first undergo feature extraction through a shared encoder, which is responsible for capturing key time delay, power amplitude features, and attenuation features, and we refer to them as dual signal features. In the electromagnetic wave propagation model of layered media, the system responses are typically sparse in the time domain. Direct modeling with a fully connected neural network is prone to losing the time interval information between impulse responses. In contrast, the self-attention mechanism [29] can effectively capture the relative time delays between pulses by calculating the autocorrelation. Inspired by this, we design the encoder structure shown in Figure 3d: the input data first passes through a self-attention layer; subsequently, the self-attention outputs are fed into two fully connected layers, which enhance the learning capability of the model. A dropout layer is integrated after each fully connected layer, which effectively alleviates the overfitting of the model and improves its generalization performance by randomly deactivating part of the neuronal connections during training.

2.2.2. Design of Eps-Depth Branch and DSLT Branch

Upon extracting the dual signal features (i.e., time delay, power amplitude, and attenuation characteristics), we design distinct processing branches to invert the layer thickness, permittivity, and loss tangent, tailored to their specific physical meanings and inversion complexities.

To invert the layer thickness $d_n$ and permittivity ${\acute{\epsilon }}_n$, we design the Eps-depth branch, illustrated in Figure 3a. This branch concatenates the dual signal features to extract frequency-independent time delay and power amplitude features, which are then fed into a dedicated decoding layer. The detailed structure of the decoder is shown in Figure 3c: the input features first pass through a 1 × 3 convolutional layer for feature fusion and dimensionality reduction; then, a batch normalization layer is applied to stabilize the training process, and a ReLU activation function is used to introduce non-linearity; afterwards, a self-attention layer is employed again to refine the features and strengthen the capture of sparse structural information; finally, the features are mapped stepwise to the final inversion results through multiple fully connected layers. The output dimension of this branch is dynamically adjusted according to the number of subsurface layers N to be inverted, e.g., 3, 4, or 5 for three, four, or five layers, respectively.

To model the frequency-dependent attenuation of electromagnetic waves, we design a DSLT-Branch (Figure 3b). It first performs point-wise subtraction on the dual signal features (in dB scale) to obtain the power ratio of the two frequency signals. Since the loss tangent tanδ is physically correlated with permittivity $\acute{{\epsilon }_n}$ and thickness $d_n$, we introduce a cross-attention module. The predicted $\acute{{\epsilon }_n}$ and $d_n$ from the Eps-depth branch are projected via a fully connected layer to serve as the query (Q), while the dual signal features act as key (K) and value (V). These three vectors (Q, K, and V) are then used to generate cross-attention fusion features. Finally, these fusion features are concatenated and jointly fed into a decoding layer with a similar structure to the Eps-depth branch but independent parameters, which ultimately inverts the loss tangent of each layer.

2.2.3. Physics-Informed Loss Function

Inverting media parameters from radar echoes is a nonlinear and ill-posed problem. Relying solely on data-driven losses (e.g., mean squared error, MSE) can lead to physically implausible solutions. To address this, we introduce a physics-informed loss based on the electromagnetic transmission-line model, as shown in Algorithm 1 [30].

The core idea is to simulate radar echoes by calculating the frequency-domain reflection coefficients. Specifically, given the center frequencies $f_c$ of the dual-frequency radar signals and the model-predicted parameters ($\acute{{\epsilon }_n}$, $d_n$, and $tan{\delta }_n$) of the n-th layer, we compute the frequency-domain reflection coefficients under an ideal planar-layered assumption. The resulting frequency-domain reflection signals are then transformed to the time domain via inverse Fourier transform to generate simulated echoes. This same forward model is also employed to generate the training data in this study.

Algorithm 1 Multiple layer transmission-line model

Initialization:

$1: \mathrm{Initialize} \mathrm{Source} \mathrm{Wave} a_0\left(\omega \right)$$\mathrm{with} \mathrm{center} \mathrm{frequency} f_c$$, \mathrm{bandwith} B$$\mathrm{and} \mathrm{Match} \mathrm{Filter} a_0^{*}\left(\omega \right)$

$2: \mathrm{Initialize} \mathrm{layer} \mathrm{numbers} N$

$3: \mathrm{Input} \mathrm{parameters} d_{1\dots N}$$, {\epsilon }_{1\dots N}\prime$$, {\mathrm{t}\mathrm{a}\mathrm{n}\delta }_{1\dots N}$

$4: \mathbf{for} n=1$$\mathbf{to} N$ do

5 :${\sigma }_n \leftarrow 2\pi f_c{\epsilon }_0{\epsilon }_n\prime \tan {\delta }_n$

6 :   end for

$7: \mathbf{for} \mathrm{every} \omega$ in bandwidth $B$ do

$8: \mathbf{for} n=1$$\mathbf{to} N$ do

$9: k_n(\omega )\leftarrow \sqrt{\mathrm{j}\omega {\mu }_n({\sigma }_n+\mathrm{j}\omega {\epsilon }_n\prime )}$

$10: Z_n(\omega )\leftarrow {\displaystyle \frac{k_n\left(\omega \right)}{{\sigma }_n+\mathrm{j}\omega {\epsilon }_n\prime }}$

11 :         end for

$12: {\chi }_N\left(\omega \right){\leftarrow Z}_N\left(\omega \right)$

$13: \mathbf{for} n=N-1$ downto 1 do

$14: {\chi }_n(\omega )\leftarrow Z_n(\omega ){\displaystyle \frac{{\chi }_{n+1}\left(\omega \right)+Z_n\left(\omega \right)\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}\left(k_n\right(\omega \left)d_n\right)}{Z_n\left(\omega \right)+{\chi }_{n+1}\left(\omega \right)\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}\left(k_n\right(\omega \left)d_n\right)}}$

15 :         end for

$16: R(\omega )\leftarrow {\displaystyle \frac{Z_0(\omega )-{\chi }_1\left(\omega \right)}{Z_0\left(\omega \right)+{\chi }_1\left(\omega \right)}}$

$17: b_0(\omega )\leftarrow a_0\left(\omega \right)·R\left(\omega \right)$

$18: S_{\mathrm{out}}\left(\omega \right)\leftarrow b_0\left(\omega \right)·a_0^{*}\left(\omega \right)$

19 :  end for

$20: s_{\mathrm{out}}\left(\omega \right)=\mathrm{I}\mathrm{F}\mathrm{F}\mathrm{T}\left\{S_{\mathrm{out}}\right(\omega )\text{}}$

The physics-informed loss L_physical is defined as the MSE between the input ground-truth dual-frequency radar signals $S_{\mathit{true}}^{fc}$ and the reconstructed signals $S_{\mathit{recon}}^{fc}$:

$$L_{\mathrm{physical}}={\displaystyle \frac{1}{2}}\left(\mathrm{mean}\left({\left(S_{true}^{fc1}-S_{recon}^{fc1}\right)}^2\right)+\mathrm{mean}\left({\left(S_{true}^{fc2}-S_{recon}^{fc2}\right)}^2\right)\right)$$

(11)

The numerical inversion loss L_inv comprises MSEs for layer thickness, permittivity, and loss tangent:

$$L_{\mathrm{inv}}={\displaystyle \frac{1}{\mathrm{N}}}{\sum }_1^{\mathrm{N}}{\left(d_i-{\widehat{d}}_i\right)}^2+{\lambda }_1{\displaystyle \frac{1}{\mathrm{N}}}{\sum }_1^{\mathrm{N}}{\left({\acute{\epsilon }}_i-{\widehat{\acute{\epsilon }}}_i\right)}^2+{\lambda }_2{\displaystyle \frac{1}{\mathrm{N}}}{\sum }_1^{\mathrm{N}}{\left({\tan \delta }_i-{tan\widehat{\delta }}_i\right)}^2$$

(12)

where ${\widehat{d}}_i, {\widehat{\acute{\epsilon }}}_i, tan{\widehat{\delta }}_i$ denote ground-true values, and $d_i,{\acute{\epsilon }}_i,{tan\delta }_i$ are the predicted values.

The total loss is a weighted sum of the inversion loss and the physics-informed loss:

$$L=L_{\mathrm{inv}}+{\lambda }_{\mathrm{physical}} L_{\mathrm{physical}}$$

(13)

To stabilize optimization and mitigate loss oscillations caused by model complexity, we dynamically adjust the weight λ_physical within the interval [0.1, 1]. Specifically, λ_physical is decreased if $L_{\mathrm{physical}}$ drops in the current epoch and increased if it rises. Furthermore, to prevent the initially large physics loss from driving the model into erroneous local minima (e.g., all-zero outputs), we freeze its backpropagation for the first 10 epochs. This warm-up phase allows the model to learn the fundamental inversion mapping via $L_{\mathrm{inv}}$ alone, after which all loss terms are jointly optimized.

3. Results

3.1. Results on Simulated Data

We evaluate the proposed DBPINet through numerical experiments implemented in PyTorch 2.9.0. To ensure diversity and representativeness, training and evaluation datasets of multi-layered structures were generated via numerical simulation by randomly sampling parameters within physically plausible ranges. We compare DBPINet against two baseline inversion methods: MLP [18] and LMPINet [17], both reproduced and retrained on our datasets. Since the original MLP and LMPINet only invert permittivity and layer thickness, we adapted them by appending output neurons for the loss tangent. This modification enables the prediction of all three media parameters, ensuring a fair comparison.

To systematically assess inversion accuracy and robustness with increasing structural complexity, simulation experiments were conducted on datasets with three, four, and five layers. Independent models were trained for each dataset.

3.1.1. Simulated Dataset

The simulated dataset was designed based on typical geological characteristics of Martian sedimentary regions. The permittivity of the surface layer was fixed at 3 [7,8,31] as a prior, allowing the network to learn this constraint during training. Considering that deep echo signals in sedimentary region are strong and sediment thickness is large, subsurface media are expected to exhibit a downward compaction trend. Thus, the permittivity was set to increase with depth. The permittivity of the deepest layer was defined within the range of 6–9, corresponding to highly compacted sediments or dense basalt [7,8]. Intermediate transition layers were assigned permittivity between 3 and 6. Layer thickness was randomly sampled between 300 and 600 m for each layer. The loss tangent was randomly selected from 0.001 to 0.01, representing values ranging from pure ice to dust-rich or basaltic materials [6,15]. Since the deepest layer does not contribute to echoes, its loss tangent was fixed to 0.01.

For the four- and five-layer datasets, we incorporated two and three transition layers, respectively, with increasing permittivities in [3,6]. To ensure fair and credible performance comparisons across complexity levels, the same core network architecture was used for all datasets, adjusting only the output dimension to match the number of layers.

Because meaningful subsurface echoes in MARSIS data typically exceed −40 dB relative to the surface reflection, all input signal amplitudes were restricted to the [−40, 0] dB interval and normalized to eliminate scale effects. Datasets were generated using the transmission-line forward model (Section 3.2, Algorithm 1) with 4 MHz and 5 MHz LFM source waves. Each configuration (3-, 4-, and 5-layer) contains 10,000 training and 2000 validation samples.

Table 1. Parameters set in the simulation dataset.

Layer Number	Total	d (m)	$\stackrel{\mathbf{´}}{\mathit{\epsilon }}$	tanδ
3	12,000	300~600	3.0	0.001~0.01
		300~600	3.0~6.0	0.001~0.01
		2560-d1-d2	6.0~9.0	0.01
4	12,000	300~600	3.0	0.001~0.01
		300~600	3.0~6.0 ↑	0.001~0.01
		300~600	3.0~6.0 ↑	0.001~0.01
		2560-d1-d2-d3	6.0~9.0	0.01
5	12,000	300~600	3.0	0.001~0.01
		300~600	3.0~6.0 ↑	0.001~0.01
		300~600	3.0~6.0 ↑	0.001~0.01
		300~600	3.0~6.0 ↑	0.001~0.01
		2560-d1-d2-d3-d4	6.0~9.0	0.01

summarizes the key parameter settings, where ↑ denotes an increase relative to the overlying layer.

3.1.2. Evaluation Metrics and Training Settings

We employ two metrics to quantify parameter inversion and signal reconstruction errors. For the three inversion parameters (permittivity $\acute{\epsilon }$, loss tangent $\tan \delta$, and layer thickness d), we use the mean absolute percentage error (MAPE):

$$\mathrm{MAPE}={\displaystyle \frac{100\%}{\mathrm{N}}}{\sum }_{\mathrm{i}=1}^{\mathrm{N}}\left|{\displaystyle \frac{y_i - {\widehat{y}}_i}{{\widehat{y}}_i}}\right|$$

(14)

where $y_i$ denotes the predicted value and ${\widehat{y}}_i$ the ground truth. MAPE ranges from [0, +∞), with lower values indicating higher inversion accuracy.

For radar echo signal reconstruction, conventional MAPE is susceptible to bias due to zero-valued components and normalization. Therefore, the normalized absolute percentage error (NAPE) is defined:

$$\mathrm{NAPE}={\displaystyle \frac{100\%}{\mathrm{M}}}{\sum }_{\mathrm{j}=1}^{\mathrm{M}}\left|s_j-{\widehat{s}}_j\right|$$

(15)

Here, M is the number of sampling points of a single radar echo; $s_j$ and ${\widehat{s}}_j$ are the reconstructed and true signals, respectively. Lower NAPE percentages indicate higher reconstruction fidelity.

Models were trained using the Adam optimizer for 400 epochs with a batch size of 32. The learning rate was initialized at 1 × 10⁻⁴ and decayed to 1 × 10⁻⁶ via cosine annealing. Experiments were conducted on a single NVIDIA RTX 4090 GPU, requiring approximately 3.5 h per complete training run.

Training progression was monitored via the total and physics-informed losses on both training and validation sets. As shown in Figure 4, both losses converged steadily without signs of overfitting or underfitting. The physics-informed loss decreased gradually before converging to a low value, indicating that the model successfully learned and adhered to electromagnetic wave propagation laws to support accurate parameter inversion.

3.1.3. Hyperparameter Study

To systematically determine the optimal proportion of the loss function, a detailed loss weight tuning study was carried out in the hyperparameter experiments.

First, we observed that the physics-informed loss should not be introduced too early in training. Only the MSE loss $L_{\mathrm{inv}}$ was backward in the first 10 epochs After this warm-up phase, the physics-informed loss was added, and its weight ${\lambda }_{\mathrm{physical}}$ was dynamically adjusted between 0.1 and 1: if the validation physics-informed loss decreased in the current epoch, ${\lambda }_{\mathrm{physical}}$ was decreased; otherwise, it was increased to maintain adaptive balance during training.

Comparative experiments were performed to tune the loss weights for thickness, permittivity, and loss tangent. Initially, the loss weights for thickness and loss tangent (${\lambda }_2$) were fixed at 1, while the permittivity loss weight (${\lambda }_1$) was varied from 0.1 to 10. According to the NAPE results in Figure 5, either too low or too high permittivity loss weight can degrade overall inversion performance. The best NAPE was achieved when the permittivity–thickness loss weight ratio was 1:1, indicating their equivalent importance. An excessively high permittivity loss weight impairs thickness accuracy, whereas an excessively low loss weight weakens permittivity accuracy.

With the permittivity–thickness loss weight ratio fixed at 1:1, the loss tangent weight (${\lambda }_2$) was further tuned from 0.001 to 10. Figure 6 shows that an overly high ${\lambda }_2$ interfered with permittivity and thickness inversion, raising overall NAPE, while an overly low weight (e.g., 0.001 or 0.01) led to inaccurate loss tangent estimates, causing large echo-power deviations and also degrading NAPE. Based on these experiments, the optimal weight ratio for thickness, permittivity, and loss tangent was determined to be 1:1:0.1.

3.1.4. Comparative Experiments

To quantitatively evaluate the performance of the proposed DBPINet in the inversion of layered Martian subsurface media parameters, we selected the MLP [18] and LMPINet [17] as baseline methods and tested them under 3-, 4-, and 5-layer media scenarios.

Quantitative inversion results for each method under different layer numbers are presented in

Table 2. Quantitative comparison of inversion performance on numerical datasets.

Layer Number	Method	d MAPE	$\stackrel{\mathbf{´}}{\mathit{\epsilon }}$ MAPE	tanδ MAPE	Reconstructed Signal NAPE
3	MLP	6.501%	4.677%	9.123%	4.158%
	LMPINet	4.121%	1.971%	7.738%	3.0256%
	DBPINet	1.708% ↓	1.077% ↓	4.564% ↓	1.0558% ↓
4	MLP	11.804%	3.518%	15.297%	5.385%
	LMPINet	8.588%	4.347%	13.154%	3.335%
	DBPINet	2.721% ↓	1.609% ↓	7.744% ↓	0.923% ↓
5	MLP	13.842%	4.488%	34.837%	5.416%
	LMPINet	13.787%	4.931%	27.198%	6.269%
	DBPINet	6.886% ↓	2.321% ↓	10.665% ↓	1.342% ↓

, where “↓“ indicates that lower values represent better performance. The data clearly show that DBPINet achieves the best overall inversion performance in all test cases. Compared with MLP, DBPINet reduces the mean MAPE for layer thickness (d), permittivity ($\acute{\epsilon }$), and loss tangent ($\tan \delta$) by 6.94%, 2.56%, and 12.09%, respectively. Compared with the more advanced LMPINet, the corresponding reductions exceed 5.06%, 2.08%, and 8.37%.

Specifically, in terms of reconstructed-signal NAPE, DBPINet achieves 1.0558%, 0.923%, and 1.342% for the 3-, 4-, and 5-layer models, respectively—significantly lower than the results from MLP and LMPINet. Notably, as the number of media layers increases from 3 to 5, the inversion errors of all comparison methods rise, reflecting the increased problem complexity. However, DBPINet exhibits the slowest error growth across all metrics. For example, the reconstructed-signal NAPE of DBPINet increased by only about 0.29% from the 3- to the 5-layer configuration, whereas MLP and LMPINet showed increases of approximately 1.26% and 3.24. These results fully demonstrate the strong capability and robustness of DBPINet when handling more complex subsurface structures.

To further verify the inversion accuracy, representative test cases were presented below for different layer scenarios, encompassing strong, weak, and extremely weak reflection conditions. Inversion performance is comprehensively evaluated by comparing the depth profiles of the inverted permittivity and loss tangent, alongside the agreement between the reconstructed and input radar signals. All signals discussed in this section are generated based on a MARSIS LFM with a 5 MHz center frequency. In Figure 7, Figure 8 and Figure 9, the first two columns display the permittivity and loss tangent depth profiles, respectively, while the latter two columns illustrate the signal comparisons.

Figure 7 illustrates two typical 3-layer scenarios: a weak deep echo on the first line and strong echoes in on the second line. In these experiments, DBPINet demonstrated superior accuracy in inverting permittivity and loss tangent, as well as in signal reconstruction, yielding results that closely match the ground truth. Conversely, as seen in Figure 7(a-1,b-1), both the MLP and LMPINet exhibited significant errors in estimating the permittivity and thickness of the second and third layers, resulting in noticeable peak offsets in the reconstructed signals. By incorporating the physics-informed loss, DBPINet effectively mitigates these errors, achieving more precise permittivity and thickness inversions. Furthermore, as shown in Figure 7(a-2,b-2), the DSLT-Branch design enables DBPINet to achieve the highest accuracy in loss tangent inversion.

As shown in Figure 8, two representative 4-layer scenarios were selected: three distinct subsurface echoes on the first line and a weak echo with a small amplitude on the second line. In the 4-layer experiments, MLP and LMPINet still had obvious deviations in the inversion of deep-layer parameters, while the proposed DBPINet achieved the optimal inversion effect. As shown in Figure 8(a-1), the permittivity inverted by LMPINet was underpredicted and overpredicted in the second and third layers, respectively, which is reflected as a significant amplitude difference in the first two peaks in the reconstructed signal of Figure 8(a-4). As shown in Figure 8(b-1), the permittivity inverted by MLP was underpredicted in both the second and third layers, leading to a serious underestimation of the reflectivity between the two layers, which is reflected as a significant amplitude difference in the reconstructed signal of Figure 8(b-3). By introducing the physical constraint loss, the proposed DBPINet effectively suppressed such errors. Meanwhile, the DSLT-Branch further improved the model’s ability to invert the loss tangent, exhibiting stronger robustness in complex situations such as Figure 8(b-1–b-4) that require fine distinction between attenuation caused by media loss and reflectivity change caused by media permittivity variation.

Figure 9 evaluates 5-layer scenarios, featuring either distinct echoes on the first line or multiple weak echoes on the second line. In the distinct echo scenario, the baseline models struggle with localized errors. MLP misestimates deep-layer permittivity, leading to weak reconstructed echoes. Conversely, LMPINet underestimates the loss tangent in the second layer, resulting in artificially high echo amplitudes. DBPINet successfully eliminates these amplitude deviations and achieves high-precision inversion through its physics-informed loss and DSLT-Branch. The multiple weak echo scenario is highly challenging because it requires modeling minute interlayer reflectivities. Both MLP and LMPINet underestimate the first layer’s thickness, causing a severe overall shift (offset) in their reconstructed signals. DBPINet effectively overcomes this: its physics-informed loss accurately detects and corrects these signal offsets to yield precise permittivity and thickness values, while the DSLT-Branch ensures accurate loss tangent inversion even under extremely weak reflection conditions.

In summary, DBPINet’s superior performance stems from three key design innovations: First, the dual-frequency encoder utilizes a self-attention mechanism to effectively capture signal features, while the dedicated Eps-Depth Branch ensures highly accurate permittivity and thickness inversions. Second, the DSLT-Branch extracts critical attenuation and cross-attention fusion features, enabling precise loss tangent estimation. Finally, the physics-informed loss guarantees that the inversion results are not merely data-driven but strictly consistent with the physical laws of electromagnetic wave propagation, significantly boosting overall accuracy.

3.1.5. Ablation Study

To evaluate the core modules, we conducted ablation studies comparing the full DBPINet model against three variants: DBPINet_0 (all core modules removed), DBPINet_A (no physics-informed loss), and DBPINet_B (no DSLT-Branch). These were tested across 3-, 4-, and 5-layer scenarios, with key results summarized in

Table 3. Ablation experiments results.

Layer Number	Method	d MAPE	$\stackrel{\mathbf{´}}{\mathit{\epsilon }}$ MAPE	tanδ MAPE	Reconstructed Signal NAPE
3	DBPINet_0	2.800%	2.037%	10.17%	1.361%
	DBPINet_A	2.097%	1.168%	5.311%	1.427%
	DBPINet_B	2.059%	1.388%	7.113%	1.081%
	DBPINet	1.708% ↓	1.077% ↓	4.564% ↓	1.056% ↓
4	DBPINet_0	7.456%	2.189%	11.727%	1.762%
	DBPINet_A	3.733%	1.852%	8.370%	1.699%
	DBPINet_B	3.350%	1.689%	10.695%	1.114%
	DBPINet	2.721% ↓	1.609% ↓	7.744% ↓	0.923% ↓
5	DBPINet_0	12.471%	3.242%	20.693%	3.0581%
	DBPINet_A	8.062%	2.551%	11.008%	2.707%
	DBPINet_B	7.304%	2.389%	15.112%	1.756%
	DBPINet	6.886% ↓	2.321% ↓	10.665% ↓	1.342% ↓

, where ↓ indicates that lower values represent better performance.

The physics-informed loss embeds electromagnetic wave principles into the learning process, providing a strong physical prior for complex layered structures. For instance, in the challenging 5-layer scenario, adding this loss (DBPINet vs. DBPINet_A) significantly reduces the reconstructed signal NAPE by 1.365% (from 2.707% to 1.342%) and the thickness MAPE by 1.202% (from 8.062% to 6.886%). By comprehensively optimizing all metrics, the physics-informed loss effectively mitigates the ill-posed nature of the inverse problem and enhances the model’s adaptability to complex environments.

The DSLT-Branch significantly enhances $\tan \delta$ inversion accuracy. Compared to DBPINet_B (without this branch), the full DBPINet model reduces the $\tan \delta$ MAPE by 2.549%, 2.951%, and 4.447% in the 3-, 4-, and 5-layer scenarios, respectively. Furthermore, it simultaneously drives slight improvements in thickness and permittivity MAPEs, validating the effectiveness of the cross-attention mechanism.

In conclusion, the physics-informed loss ensures the overall accuracy of the inversion results, and the DSLT-Branch specifically improves the inversion ability of loss tangent. Together, they constitute the core of DBPINet.

3.1.6. Noise Robustness Analysis

In practical GPR applications, echo signals are inevitably corrupted by noise from system internals, environmental interference, and signal attenuation. To evaluate the noise robustness of our proposed method, we simulated real-world interference by adding additive white Gaussian noise (AWGN) at various signal-to-noise ratios (SNRs) to the validation sets. Specifically, AWGN with SNRs ranging from 0 to 50 dB was applied to the 3-, 4-, and 5-layer datasets. We evaluated overall inversion performance using the reconstructed signal NAPE, while isolating the loss tangent MAPE to specifically analyze its robustness.

Figure 10a–c presents the results. As expected, decreasing the SNR universally increases errors across all models. However, DBPINet consistently outperforms the baselines across all structures and noise levels, demonstrating superior anti-interference capabilities. At SNRs ≥ 20 dB, both LMPINet and DBPINet perform similarly to noise-free conditions. As noise further intensifies (SNR < 20 dB), the errors for MLP and LMPINet surge significantly, whereas DBPINet consistently maintains the lowest error margins. While loss tangent inversion error naturally increases with noise across all models—highlighting its inherent noise sensitivity—DBPINet exhibits a significantly lower MAPE growth rate at low SNRs (0–20 dB) compared to the baselines.

In summary, DBPINet demonstrates exceptional stability and accuracy in noisy environments. Specifically, at SNRs of 30 dB and above, its inversion results for both the loss tangent and overall media parameters remain highly reliable, underscoring its strong potential for real-world engineering applications.

3.2. Results on Measured Data

To verify the effectiveness of our proposed DBPINet for the inversion of real layered Martian subsurface data, we conducted measured experiments on the MARSIS with a specific focus on the data from the MFF GA region featuring typical stratification characteristics.

3.2.1. Measured Dataset

MARSIS has successfully detected subsurface structures in multiple Martian regions. In its subsurface sounding mode, it transmits linear-frequency-modulated (LFM) pulses with a 250 μs duration over a 1 MHz bandwidth, selectable across four frequency bands in the 1.3–5.5 MHz range (centered at 1.8, 3, 4, and 5 MHz). Data from two bands are acquired simultaneously per detection. Through onboard Doppler beam sharpening and coherent integration, MARSIS synthesizes multiple pulse echoes into a synthetic aperture with an along-track resolution of 5–9 km, which suppresses surface clutter and improves the signal-to-noise ratio (SNR) of weak subsurface signals [4]. Since its operating frequencies are near the ionospheric plasma frequency, MARSIS adopts interpolation [32] and autofocus [33] techniques to estimate and compensate for ionosphic phase distortion and group delay, ensuring the accuracy of subsurface echoes during range compression.

Among the numerous regions probed by MARSIS, the Medusae Fossae Formation (MFF), distributed near the Martian dichotomy boundary, is one of the most controversial non-polar sedimentary regions, with its genesis remaining unconfirmed to date. Sediments in this region have been interpreted as volcanic ash deposits [34], aeolian deposits [35], or ice-rich materials similar to polar layered deposits [7,8,9].

In this study, the MFF GA between Gordii Dorsum and Amazonis Mensa was selected as the study area (Figure 11), with a geographic range of approximately 143° to 150°W and 2°S to 7°N. This area consists of high-elevation lobate deposits extending to the Hesperian geologic unit in the southern Amazonian Planitia. The MFF GA is divided into two parts by a steep cliff with a height of about 300–400 m (marked by the dashed line in Figure 11). Previous SHARAD and MARSIS signals only penetrated the thinner MFF deposits on the northwest side of this cliff, with the thickness of this part gradually decreasing toward the Hesperian Plains [8]. Four MARSIS measured data tracks distributed from east to west were selected in this study, with the track IDs of 21969, 12456, 7177, and 2954, respectively.

The global Martian surface roughness coefficient map compiled by Campbell et al. [36] shows that most of the MFF GA Trough is located in a low-roughness area for MARSIS signals, which can be approximated as a parallel layered media structure [37]. To reduce the influence of ionospheric interference on radar data, the officially corrected Optimized MARSIS Radargram Data [38] products were adopted in this study. MARSIS provides data for multiple center frequencies (1.8 MHz, 3 MHz, 4 MHz, and 5 MHz). To balance high resolution and signal quality, dual-frequency combined data with center frequencies of 4 MHz and 5 MHz were selected for the study.

Figure 12a shows the radargram of MARSIS Track 7177 over the MFF GA, where two distinct subsurface reflectors can be observed. We introduce and train a deep unfolding super-resolution network [39]. Although originally developed for image super-resolution, this network incorporates physical priors and thus exhibits strong cross-domain applicability. It is trained on paired simulated SHARAD and MARSIS datasets. After training, the network effectively compresses the mainlobe width of the input MARSIS signals and extracts the peak positions of the major subsurface echoes. Subsequently, the data corresponding to the output peak positions was fed into the network as input for the measured experiment, aiming to suppress noise and mitigate the impacts of signal pulse width and sidelobes. The super-resolution results are presented in Figure 12b. An example of the super-resolution A-scan result is shown in Figure 12c, where the three-layer signals become significantly clearer after processing.

3.2.2. Performance on Measured Data

The three-layer DBPINet was adopted to deal with the MARSIS data acquired from four orbits (ID: 21969, 12456, 7177, 2954) passing through the MFF GA region, yielding the key media parameters of the three underground media layers in this region, namely, permittivity ($\acute{\epsilon }$), layer thickness (d), and loss tangent ($\tan \delta$), with the results presented in

Table 4. Quantitative inversion results from MARSIS measured dataset over the MFF GA region.

MARSIS ID	Layer	$\stackrel{\mathbf{´}}{\mathit{\epsilon }}$	d (m)	tanδ
21969	Layer 1	2.995 ± 0.014	351.65 ± 66.14	0.0031 ± 0.0013
	Layer 2	5.097 ± 0.727	423.22 ± 85.83	0.0061 ± 0.0019
	Layer 3	8.795 ± 0.318	/	/
12456	Layer 1	3.016 ± 0.022	360.84 ± 43.56	0.0024 ± 0.0014
	Layer 2	4.240 ± 0.913	410.92 ± 99.02	0.0048 ± 0.0021
	Layer 3	8.223 ± 0.464	/	/
7177	Layer 1	3.006 ± 0.015	397.66 ± 55.25	0.0032 ± 0.0011
	Layer 2	5.017 ± 0.422	569.80 ± 62.84	0.0049 ± 0.0020
	Layer 3	8.295 ± 0.477	/	/
2954	Layer 1	3.010 ± 0.011	415.06 ± 63.62	0.0030 ± 0.0004
	Layer 2	5.129 ± 0.723	491.86 ± 114.34	0.0045 ± 0.0022
	Layer 3	8.095 ± 0.237	/	/

. It should be noted that the third layer corresponds to the regional basaltic basement. No valid reflected signals from its underlying interface were detected in the radar echoes, leading to an absence of observational data to support the inversion of its layer thickness and loss tangent. In addition, the influence of the deep-layer loss tangent on the echo signals of the surface and subsurface layers is negligible. For these reasons, the inversion results of the layer thickness and loss tangent for the third layer are omitted. We summarize the results from this study alongside previous studies in

Table 5. Comparison of inverted parameters with previous studies.

Layer	Parameter	Our Study	Previous Studies	Reference
1	$\stackrel{\mathrm{´}}{\mathit{\epsilon }}$	~3.0	~3.0	[7,8]
	tanδ	~0.003	0.002–0.004	[7]
	d (m)	~350–~400	300–500	[8]
2	$\stackrel{\mathrm{´}}{\mathit{\epsilon }}$	4.0–5.0	/	/
	tanδ	0.0045–0.006	/	/
	d (m)	~410–~570	/	/
3	$\stackrel{\mathrm{´}}{\mathit{\epsilon }}$	~8	~8 (basalt)	[8]

.

The core characteristics of the measured data inversion results in Table 4 are summarized as follows:

(1)

Surface layer (Layer 1): The permittivity is stable around 3.0, which is consistent with the fixed prior of the surface layer permittivity in our model. When inverting for other regions, this prior can be adjusted by setting a different surface permittivity in the training dataset accordingly. Its loss tangent is approximately 0.003, highly consistent with the results of the overall electromagnetic properties of the MFF surface layer obtained by Watters et al. [7] and Campbell et al. [8]. As demonstrated in the above studies [7,8], this low loss tangent value can be interpreted as a volcanic ash cover layer with high porosity. The layer thickness increases gradually from west to east, from approximately 400 m to 500 m. This trend aligns with the depth correction results for the same region reported by Campbell et al. [8] using SHARAD data, validating the reliability of our inversion.

(2)

Subsurface layer (Layer 2): The permittivity ranges from 4.0 to 5.0, which is significantly lower than that of dry basalt (typically ~8) and higher than that of pure ice (approximately 3.1). The loss tangent is between 0.0045 and 0.006, slightly higher than that of the surface layer.

(3)

Basement layer (Layer 3): The permittivity is ~8, leading us to infer that this layer is the basaltic basement. This conclusion is consistent with the finding proposed by Campbell et al. [8] that the bottom reflective layer in the MFF GA region is continuous with the adjacent Hesperian basaltic plains, further confirming the rationality of our inversion results.

3.2.3. Discussion of Measured-Data Results

Combined with the geological evolution background of the MFF region and previous research findings, the genetic mechanism and scientific implications of the above-mentioned three-layer structure are discussed as follows:

Building on the conclusions of [7,8], the surface layer in the MFF region can be interpreted as a volcanic ash cover. Its high porosity results in the observed low permittivity and low loss tangent.

For the basement layer, from the perspective of regional geological evolution, the MFF region was formed contemporaneously with the adjacent Hesperian basaltic plains [10] and has undergone long-term geological processes such as erosion, transportation, and redeposition. The material composition of its basement is homologous to that of the Hesperian plains, which supports the inference of a “basaltic basement”.

For the subsurface layer, based on the experimental results of Mattei et al. [40], the electromagnetic properties of this layer (permittivity of 4.0–5.0 and loss tangent of 0.0045–0.006) are highly consistent with a 43% basalt–ice mixture, which is similar to that of the basal unit in the NPLD. We infer that the material composition of this layer is likely ice-mixed basalt. Regarding the formation of this layer, the glacial tectonic model established by Fastook et al. [9] provides a key geological context: multiple glacial activities have occurred in the equatorial region where the MFF GA is located. Glaciers not only shaped the regional surface morphology but also their carried glacial till or glacial meltwater infiltrated the underlying bedrock and modified its properties, ultimately leading to the formation of extensively distributed ice-mixed basaltic layers [41]. This geological process can reasonably explain the electromagnetic characteristics of this layer, and also echoes the “ice-rich sediment hypothesis” for the MFF region proposed by Watters et al. [7], the surface volcanic ash deposits act as an insulating layer, which can effectively inhibit the sublimation of ice in the underlying ice-mixed materials, thus providing favorable conditions for the preservation of subsurface ice-rich structures.

If the three-layer structure revealed in the MFF GA (especially the hundreds-of-meters-thick ice-mixed basalt layer) can be reasonably extrapolated to the entire MFF region, it will significantly enhance the contribution of this region to the water ice inventory in the Martian mid-latitudes. Combined with results in Mattei et al. [40], we can estimate that the ice content of this ice-mixed basaltic layer is approximately 40–50%. It is known that the overall estimated thickness of the MFF region is about 3700 m [7]. Based on the above ice content, the volume of water ice stored in the entire MFF region will increase by approximately 10% or more compared with previous estimates. This will significantly alter our understanding of the Martian global water cycle history and the potential for future resource utilization.

In summary, this study realizes the high-precision inversion of the underground media parameters in the MFF GA region via the DBPINet model and proposes a three-layer structure consisting of a volcanic ash surface layer, an ice-mixed basaltic subsurface layer, and a basaltic basement. This finding not only supports previous geological hypotheses but also provides new measured data support for the assessment of water ice inventory in the Martian mid-latitudes.

4. Discussion

The proposed DBPINet is developed based on a planar-layered media assumption and has demonstrated promising performance in both numerical simulations and measured data experiments. However, real Martian subsurface structures are often geologically complex, exhibiting topographic slopes, surface roughness, and non-planar interfaces that may deviate from this idealized assumption.

Surface roughness can significantly affect radar sounding performance in three main aspects [37]: attenuation of nadir echoes, enhanced cutter interference, and reduced penetration depth and resolution. Furthermore, it is important to note that the severity of these roughness effects is highly dependent on the radar signal wavelength relative to the scale of the [36]. MARSIS operates in the 1.3–5.5 MHz frequency band, corresponding to free-space wavelengths of approximately 60–166 m. This long-wavelength characteristic makes the planar-layer assumption reasonable for most Martian lowland and plains regions, as long-wavelength signals are inherently less sensitive to small-scale surface roughness and can effectively penetrate to deeper structures. According to the Martian surface roughness map compiled in [36], the MFF GA study area investigated in this study lies within a low-roughness zone, thereby validating the applicability of our method.

However, caution is warranted when applying our method to high-roughness terrains, such as the Amazonis Mensa in Figure 12. In such regions, direct application of the current DBPINet framework may lead to inaccurate inversion. Future work should focus on enhancing the model’s robustness by integrating roughness correction algorithms and advanced clutter suppression techniques.

5. Conclusions

Martian subsurface exploration is a core approach to revealing its geological evolution, water ice distribution, and potential habitable environments for life. As a key technology for the non-destructive detection, subsurface radar inversion currently faces challenges such as the neglect of the loss tangent’s influence and over-reliance on data-driven approaches with poor physical interpretability. To address these issues, this study proposes DBPINet, which achieves high-precision inversion of layer thickness, permittivity, and loss tangent for layered Martian media, thus providing a novel method for planetary subsurface exploration. The core innovations and contributions of this study are mainly reflected in three aspects:

(1)

DBPINet is designed with a dedicated dual-branch structure (the Eps-depth branch and the DSLT-Branch) and integrates self-attention. This design effectively enhances the model’s ability to capture dual signal features in dual-frequency Martian subsurface radar data, thereby enabling the accurate inversion of multiple media parameters.

(2)

To address the challenge of low inversion accuracy for loss tangent, we innovatively construct the DSLT-Branch. This branch extracts the attenuation features and introduces prior information on permittivity and layer thickness via cross-attention mechanisms, which significantly improves the inversion accuracy of this parameter.

(3)

To enhance the physical interpretability, a physics-informed loss function based on the electromagnetic wave transmission-line model is constructed. This mechanism ensures that the physical law of electromagnetic wave propagation is embedded in model training.

Numerical simulation and measured data experiments have verified the superiority of the DBPINet method: in the three-layer, four-layer, and five-layer media scenarios, the MAPEs of the inversion results are significantly lower than that of baseline models such as MLP and LMPINet. Ablation experiments confirm that the physics-informed loss and the DSLT-Branch module are the key components for improving the overall inversion performance. Noise experiments demonstrate that DBPINet maintains stable inversion accuracy even under moderate signal-to-noise ratio (SNR) conditions, exhibiting excellent anti-interference capability. The trained model is applied to the measured MARSIS data of the MFF GA region, and the inversion results reveal a three-layer subsurface structure consisting of a volcanic ash surface layer, an ice-mixed basaltic subsurface layer, and a basaltic basement. This inference is highly consistent with existing geological understandings, which not only verifies the effectiveness of DBPINet in practical exploration missions but also provides new quantitative evidence for understanding the water ice storage mechanism in the MFF region.

In conclusion, DBPINet significantly improves the accuracy, reliability, and physical interpretability of Martian radar parameter inversion. This study not only provides a powerful new tool for Martian subsurface exploration but also its methodological framework can offer valuable technical references for subsurface radar exploration missions in other research fields, such as the Earth’s polar regions and the Moon.

Future research work can be carried out in the following five aspects: First, explore the extension of the proposed method to the inversion of radar data from more frequency bands to further enhance the inversion capability for more complex media. Second, explore an adaptive layer number inference mechanism in the network to address the potential lack of prior information on subsurface structures, and develop a network architecture capable of automatically determining the number of media layers and inverting parameters synchronously. Third, explore a complex geological structure modeling mechanism in the network to address the inversion adaptability issues of non-horizontal stratification and anisotropic media. Fourth, future efforts will investigate methodologies to move beyond the current assumption of a frequency-independent loss tangent and to incorporate the effects of multiple reflections. Last but not least, promote the practical application of this method in the processing of more planetary radar data (e.g., lunar subsurface rada) to verify its engineering applicability across different scenarios and missions.