1. Introduction
Mercury, the planet closet to the sun, has the highest uncompressed density in the solar system [
1], which is related to its very large liquid metal core. Such a unique structure is the focus of Mercury exploration to understand its formation and development, even in the context of the solar system’s evolution [
2,
3,
4].
Due to the relatively limited amount of Mercury’s science data, the gravity field data calculated by the gravity perturbations from the orbit are the only means to study the Mercury gravity. The knowledge of gravity is the key to learn the internal structure of Mercury. Moreover, gravity data also contributes to the precise navigation of spacecraft, identifying buried craters and hidden faults, etc., providing geophysical support in global coverage.
The MErcury Surface, Space ENvironment, GEochemistry and Ranging (MESSENGER) spacecraft launched by National Aeronautics and Space Administration (NASA) in 2004 is the first orbiter of Mercury in 30 years, after the Mariner 10 [
5,
6,
7,
8]. The Mercury gravity data released by the MESSENGER team are applied and analyzed widely. Nevertheless, the HgM007, which is the first product based on the entire MESSENGER dataset, has stripes on the high-degree global gravity anomaly expansion grid [
9]. Such phenomena affect the geophysical analysis of Mercury. Improvement in the data quality can come from new missions or from additional observing campaigns, as in the case of the low-altitude MESSENGER campaign, which lead to the HgM008 gravity field. Here, we explore the possibility of improving the dataset, by taking advantage of other science data.
In addition to the gravity data, topographic data are another kind of important science data. The formation of the terrain is inextricably linked to the gravity field, and it is considered to be the main source of short-wavelength gravity field components of terrestrial bodies [
10,
11]. Consequently, the DEM (digital elevation model) can be employed to assist in the downscaling of the gravity field data in previous researches. By dividing the residual terrain model into rectangular prisms, Forsberg [
12] integrated the influence of topography on the short-wavelength gravity field. The rectangle prisms method with DEM has achieved good results in the gravity field refinement of Earth, the Moon and Mars [
13,
14,
15]. On this basis, Li et al. [
16] improved the gravity data of Venus by considering the global isostatic compensation correction [
17,
18,
19]. Such models can achieve a high degree and reach a relatively robust result, but they are complicated, time-consuming and do not take the spatial structure of the 2D grid data into account. More importantly, they are not applicable to Mercury, because the crustal density, which has not been measured on Mercury, is needed in the calculation process (the research of Venus is based on a hypothetical density, but inappropriate assumptions may cause subjective errors).
Therefore, from the perspective of not having to assume the crustal density and porosity, a deep learning approach is proposed, for the first time, to tackle Mercury’s gravity field data reconstruction. In our work, the internal hierarchical features of gravity field data and the spatial structure similarity with the DEM are fully exploited and extracted via the densely connected convolutional network, to avoid prior physical constraints. By directly establishing the mapping between low- and high-degree data, pretraining with lunar data and fine-tuning with Mercury’s samples, the original and stripe-contaminated HgM007 is reconstructed to a higher resolution and better spatial quality. Meanwhile, the injection of the high-resolution topographic spatial structure, which is fused with the high-degree gravity data, makes the subtle geographical details better restored.
The remainder of this paper is organized as follows. The data used and the cause of stripe noise are illustrated in
Section 2, and the whole downscaling network is presented in
Section 3. Next, experiments for verifying the effectiveness and performance of the proposed method are conducted, and the results are analyzed and discussed in
Section 4. Finally, the conclusion is given in
Section 5.
2. Data and Spatial Structure Similarity Analysis
Since the resolution of Mercury’s best gravity field data is not high enough, it is not able to provide sufficient samples for the proposed network. However, the resolution of lunar greatest gravity field data is 1200 degrees, which is 12 times that of Mercury’s. In addition, the Moon and Mercury are two celestial bodies in the inner solar system with similar appearances: Both of them have highly cratered old surfaces. More importantly, the lunar gravity data and DEM also have high spatial structure similarity. Therefore, the lunar data can be used as the training sets and then a small amount of Mercury’s data is utilized for fine-tuning, realizing the transfer learning between the terrestrial bodies. Additionally, the cause of stripe noises on the gravity data grid and the spatial similarity between gravity data and DEM are also introduced, as follows.
2.1. Mercury’s Datasets
All the Mercury datasets used in this paper come from the MESSENGER mission, which was launched in August 2004 and entered a highly eccentric orbit, with periapsis, in the northern hemisphere, around Mercury, in 2011. The acquisition time of the datasets is from 2011 to 2015 [
20,
21].
2.1.1. Gravity Field Data of Mercury
During the MESSENGER mission, the degree of Mercury gravity field model was improved markedly: Smith et al. [
22] released a 20-degree gravity field model, using the initial less-than-six-month orbit-tracking data. With the passage of time and the acquisition of new data, a 50-degree gravity field model [
23] and a 100-degree gravity field model were upgraded subsequently [
24].
However, the orbit of MESSENGER is fairly eccentric, with a periapsis located at high northern latitudes, which results in highly latitude-dependent data quality [
23]. The degree strength of the data in Reference [
24] reaches 100 in the northern hemisphere, but those in the southern hemisphere can only reach 40 degrees, while there are stripes in the gravity grid data in northern middle-latitude areas (in the red box of
Figure 1) due to the incomplete coverage of the orbital data.
2.1.2. Mercury’s Global DEM
The Mercury global DEM is obtained by using images collected by cameras onboard MESSENGER simultaneously from a least-squares bundle adjustment of common features in the Integrated Software for Imagers and Spectrometers (ISIS3), as shown in
Figure 2b [
25]. Since the topography is only available on mostly the northern hemisphere, the final DEM is the result of topography in the northern hemisphere and a co-registration of images from Mercury Dual Imaging System (MDIS) narrow-angle camera (NAC) and multispectral wide-angle camera (WAC) [
26,
27], and then inference of topography in the southern hemisphere. The spatial resolution of the global DEM raster data is 0.015625°/pixel. During the experiment, it is downsampled to the corresponding scale, as the high-degree gravity output.
2.2. Lunar Datasets
2.2.1. Gravity Field Data of the Moon
The lunar gravity field model has also undergone development from a low degree to a high degree [
28]. Prior to the Gravity Recovery and Interior Laboratory (GRAIL) mission [
29,
30,
31], the model released by the SELenological and ENgineering Explorer (SELENE) was only up to 100 degrees [
32]. The two probers of the GRAIL were launched in 2011, and each of them can send and receive signals with Earth or another probe. The gravity can be obtained by measuring the distance change between them. This novel measurement mode dramatically improves the resolution of the lunar gravity model up to 1500 degrees. The degree strength of the polar region is 890, and the far lunar surface central region reaches 680 [
33,
34] as shown in
Figure 3a. The spherical harmonic expansion of the GRAIL gravity field model shows that stripe noises will appear in some regions when the resolution exceeds 1200 degrees. By comprehensive consideration, data below 800 degrees are utilized as input data without stripe noise.
2.2.2. Lunar Global DEM
The Global DEM product of the Moon (
Figure 3b) is based on altimetry data acquired through mission phase Lunar Reconnaissance Orbiter (LRO)_ES_05 by the Lunar Orbiter Laser Altimeter (LOLA) [
35,
36]. The resolution is 256 pixels/degree [
37]. Furthermore, it is also downsampled to the corresponding scale, as the high-degree gravity output.
2.3. Spatial Structure Similarity Analysis of Gravity Data and DEM
Mercury is relatively flat, because the difference of the topographic lowest and highest points is less than 10 km [
38]. Likewise, the gravity anomalies of HgM007 in
Figure 2a are in the range of −270 to 200 mGal, which is reasonably minor and has a similar distribution trend to the DEM. The lunar datasets show the same pattern; however, the free-air gravity anomalies provide information on the mass anomalies at different depths from the topographic relief to Core-Mantle Boundary (CMB). In other words, they are not necessarily related to the surface features; these two kinds of data do have 2D spatial internal structure similarity from the data-driven perspective.
To quantitatively assess the spatial structure similarity, the spatial size of the 100-degree grid data is set to 200 × 100, the size of the 50-degree grid data is 100 × 50, and so on. The corresponding DEM datasets are downsampled to the same scale as the gravity datasets, for quantitative comparison. The Pearson correlation coefficient (
r) and the structural similarity index (SSIM) [
39] are employed to quantitatively assess the spatial relationship of them:
where
n denotes the total number of the grid points;
and
are the elements of the gravity and DEM data, respectively; and
and
are the mean values. The larger the
r is, the stronger the correlation will be.
where
and
stand for the mean values of the normalized gravity grid data and normalized DEM data, respectively;
and
are the variances;
is the covariance; and
and
are constants that prevent the denominator from being 0. The closer SSIM is to 1, the more similar are the spatial structures.
It can be seen in
Table 1 that the values of gravity and topography data are not strongly correlated with
r just in the range of 0.47 to 0.66, but the SSIMs reach above 0.96. Consequently, as heterogeneous data, there is no numerical correlation between them; it is not applicable for direct linear regression. However, they have strong spatial structure similarity, which means the structure information of high-resolution DEM can be extracted to help reconstruct the gravity data in the spatial domain.
3. Methodology
The flowchart of the proposed method is shown in
Figure 4. A nonlinear mapping between low-degree and coarse high-degree data is built. Simultaneously, the high-resolution topography information is employed for detail reconstruction, to get the final output. After training with a converged loss, the learned network can be applied to Mercury gravity downscaling and destriping. Details of our algorithm are interpreted below.
The entire structure of the proposed algorithm consisting of two parts is depicted in
Figure 5: the gravity field reconstruction and the DEM refining. For maintaining the original information as much as possible, the pooling layer, which performs well in describing geo-based features, is not employed in the whole architecture. We instead make the network deeper, to enlarge the receptive field, via using more context and exploiting higher nonlinearities, to strengthen the predictive power. It learns a nonlinear end-to-end mapping between different degrees of Mercury’s gravity field, which simultaneously employs the low-degree gravity field and the high-resolution DEM data.
3.1. Gravity Field Data Reconstruction Network
In this part, low-order degree data are inputted to get the preliminary high-degree result directly through the network, as shown in the upper part of
Figure 5. This portion is based on the DenseNet [
40], which can make full use of the hierarchical features from the original data, reducing the number of parameters substantially and stabilizing the training of deep network. It is composed of three modules: the feature extraction, the global feature fusion and the downscaling. One convolution layer
, is firstly used to extract low-level features
, from the low-degree input
,
Then
dense blocks are applied to learn the high-level features. The output
Fd of the
h dense block can be obtained as follows:
where
denotes the operations of the
-th dense block. Specifically, in each dense block, the feature maps of all preceding layers are concatenated as inputs for each layer. Denote
as the output features of the
-th layer in the
-th dense block:
where
is a composite function, including convolution and Rectified Linear Unit (ReLU), and
refers to the concatenation of the feature maps produced by the convolutional layers
in the
h dense block. If each function
generates
feature maps, it follows that the
-th layer has
input maps. The hyper-parameter
refers to the growth rate of the network.
The output of each dense block fully utilizes the corresponding convolutional layer within the block by short paths created between a layer and every other layer.
After extracting hierarchical features with a set of dense blocks, the feature fusion is further conducted in a global manner:
where
is a 1 × 1 convolution, and
denotes the fused features. The feature fusion can adaptively generate informative characteristics and improve computational efficiency.
Then one convolution layer for reconstruction followed by a deconvolution layer to get the high-degree gravity field data is used:
where
is the operation of the downscaling unit, and
is the output of the first part, which is considered as a coarse high-degree gravity field.
3.2. DEM Refining Network
The spatial correlation between the DEM and the gravity field was proved in
Section 2.3. Since the topography digitization project develops rapidly to achieve higher quality, the higher-resolution spatial structure of DEM can be integrated into the process of gravity field reconstruction to obtain a more-detailed high-precision gravity product.
Taking the spatial similarity between gravity field data and DEM into account, the high-resolution DEM is concatenated with the coarse output for further extracting informative features:
where
C is the concatenation operation, and
is the concatenated feature of coarse high-degree gravity field,
, and high-resolution DEM,
.
Since the coarse high-degree gravity-field data and the final output share the almost same distribution to a large extent, explicitly modeling the residual information by utilizing an identity skip-connection from the input to the end can achieve a faster convergence rate and superior performance [
41]. Therefore, a three-layer residual network is employed as follows:
where
and
are the first and the second convolution (including ReLU), respectively, to extra abundant informative characteristics that are already downscaled in the concatenated feature,
which are then leveraged by a convolution
to reconstruct the residual details. Ultimately, the residual information and the input coarse high-degree gravity field data are element-wisely added to get the final high-degree gravity field,
.
In addition, the proposed network is optimized via minimizing the mean absolute error (MAE) between the downscaled output and the input low-degree gravity field data.
5. Conclusions
In this paper, a novel convolutional-neural-network-based gravity-data downscaling and destriping algorithm is presented. This deep learning approach is applied to planetary gravity field data reconstruction for the first time, whereby input low-degree data can be reconstructed into higher-degree data through our network. First, the spatial structure similarity between DEM and gravity data is explored, which supports the idea that high-resolution topographic information can help the reconstruction of gravity data. Then, to tackle the lack of Mercury’s training samples, high-quality lunar samples are adopted for pretraining, based on the similarity between these two bodies. Three sets of experiments and power spectrum analysis elaborate the feasibility and reliability of the proposed network.
Compared with the HgM007, our estimation has a consistent distribution trend and shows more details. It can be used for the orbit determination of future missions, such as the BepiColombo, and the landing site selection. However, our method is data-driven and implicitly assumes a relation between gravity and topography at high degrees similar to the Moon. With this caveat in mind, it can still be used to provide a reference for fine-scale analysis and interpretation.
In the future, we plan to combine the data-driven approach with geophysical-constrained models, thereby exploring a more accurate and physically sound result.