Multiscale 3-D Stochastic Inversion of Frequency-Domain Airborne Electromagnetic Data

Abstract

In mineral, environmental, and engineering explorations, we frequently encounter geological bodies with varied sizes, depths, and conductivity contrasts with surround rocks and try to interpret them with single survey data. The conventional three-dimensional (3-D) inversions significantly rely on the size of the grids, which should be smaller than the smallest geological target to achieve a good recovery to anomalous electric conductivity. However, this will create a large amount of unknowns to be solved and cost significant time and memory. In this paper, we present a multi-scale (MS) stochastic inversion scheme based on shearlet transform for airborne electromagnetic (AEM) data. The shearlet possesses the features of multi-direction and multi-scale, allowing it to effectively characterize the underground conductivity distribution in the transformed domain. To address the practical implementation of the method, we use a compressed sensing method in the forward modeling and sensitivity calculation, and employ a preconditioner that accounts for both the sampling rate and gradient noise to achieve a fast stochastic 3-D inversion. By gradually updating the coefficients from the coarse to fine scales, we obtain the multi-scale information on the underground electric conductivity. The synthetic data inversion shows that the proposed MS method can better recover multiple geological bodies with different sizes and depths with less time consumption. Finally, we conduct 3-D inversions of a field dataset acquired from Byneset, Norway. The results show very good agreement with the geological information.

1. Introduction

The airborne electromagnetic (AEM) method is a rapid geophysical exploration technique that has been widely used in mineral, environmental, and engineering groundwater explorations [1,2,3,4]. With the increasing demand for high-accuracy interpretation of geological targets varying in size, depth, and conductivity, it has become crucially important to rapidly obtain the electrical structure in the underground from an AEM survey with a multi-scale resolution. The conventional three-dimensional (3-D) inversions are mainly conducted in the space domain, where an Lp-norm is generally used as constraints to directly obtain the underground conductivity updates by solving an inversion equation such as in Gauss–Newton (GN) method [5,6]. These inversions are efficient, but they have limitations, which is that the grid size should be less than the smallest target. This will result in a large amount of grids from fine discretization and a lot of unknowns to be solved. This brings big challenges to 3-D AEM inversion, especially for large datasets. Many attempts, like the Octree or local mesh methods, etc., have been made. Yang et al. [5] and Haber et al. [7] have been proposed a method to deal with this problem, but they all need to have prior information on the target to grid. In recent years, the multi-scale (MS) analysis has been proposed and found wide application in fields like image processing, digital signal processing, computer vision, applied mathematics, and medical imaging [8,9,10]. An MS analysis method illustrates the features of an image at different scales and analyzes the information contained in the image via MS decomposition in deep structures, so that it can improve the accuracy in describing the image features.

AEM multi-scale inversion is a potential means to obtain multiscale information of the subsurface. The mainstream MS methods include the wavelet transform [11], the curvelet transform [12], and the shearlet transform [13,14]. The wavelet transform is considered as an effective time–frequency analysis method based on the Fourier transform. It possesses the characteristics of MS analysis and time–frequency localization. By performing scaling and translation operations, a wavelet transform gradually analyzes signals at multiple scales, allowing for both high- and low-frequency components to enter and enabling the localization of arbitrary signal details. Liu et al. applied the Daubechies (DB) wavelet to 3-D frequency-domain (FD) AEM data inversions and obtained multi-scale inversions [15]. Deleersnyder et al. [16] investigated one-dimensional inversions of FD EM data and effectively improved the resolution of the inversion results by applying different constraint weights to wavelet coefficients at different scales. Although the wavelet transform exhibits MS characteristics and provides nearly optimal sparse representation for the one-dimensional signal, the wavelet basis functions are isotropic and have limited directionality that is difficult to deal with in complex images. Candès and Donoho proposed a curvelet transform that is multi-scale, multi-directional, and anisotropic [17]. It provides an optimal sparse representation for the edge features of a target object [12]. Wang et al. [18] applied the curvelet transform to denoise AEM data and significantly improved the signal-to-noise ratio. Gao et al. [19] used a curvelet transform in AEM leveling and effectively removed the strip-type artifacts by exploiting the multi-directional nature of the curvelet transform. Except for the curvelet method, a shearlet transform with shearing parameters can also enable the selection of multiple directions [13,20]. It has a mother shearlet function that can be adjusted through scaling, shearing, and translations. Compared to the wavelet transform, the shearlet transforms not only allow for MS analysis but also possess the properties of multi-directionality, localization, and compact support, which can enhance the description to the objects with curved edges. Meanwhile, compared to the curvelet transform, the shearlet transform has more complete mathematical theory and can offer greater flexibility in choosing the number of directions [17]. The improved flexibility in selecting directions and the mathematical completeness make the shearlet transform a good tool for MS analysis of sparse representations of high-dimensional signals. Su et al. [21] implemented a 3-D inversion of AEM data using the sparse regularization algorithm based on shearlet transform.

However, an AEM survey can generally cover hundreds of line kilometers and create huge amounts of data (even after processing); this will bring big challenges to 3-D AEM inversions due to its computational cost both in time and memory, which can be prohibitive. Although dozens of acceleration methods such as the moving footprint, direct solver, preconditioned technique, etc., have been proposed [22,23,24,25], we still need further consideration on how to run a fast multi-scale analysis for the interpretation of the underground structure at high precision. A stochastic optimization uses stochastic datasets to perform inversions that enables fast computations for problems entailing massive training samples [26,27,28,29,30,31]. The stochastic inversion method has been widely used in signal processing and machine learning [32,33]. Although this method only randomly uses the samples selected as the training set, it can deliver good results in realizing the target process, which makes it possible to run a fast computation. However, the stochastic method converges slowly sometimes as the gradients are generally not accurately calculated. To solve this problem, scholars have proposed methods like the classic momentum method and various step-size-control strategies, etc., to improve the convergence [34,35,36,37,38,39]. Ren et al. [2] proposed a fast preconditioned stochastic inversion for time-domain AEM data. To obtain the random samples in stochastic optimization, an appropriate sampling method needs to be used. Fichman et al. [40]; Donoho et al. [41,42]; and Candès et al. [17,43] pointed out that when a signal is sparse or can be sparsely represented in a transform domain, it can be projected into a low-dimensional space through a measurement matrix for compression. The key to achieving compressed sensing (CS) relies on the random sampling, the sparse expression, and the reconstruction methods. Hennenfent and Herrmann [44] introduced the Jitter sampling method in seismic wave field reconstruction and achieved good results by controlling the maximum interval of sampling points to make their distribution more random and uniform. The Fourier [45], wavelet [46], and curvelet transform [47] are also being implemented in the CS sparse expression. Additionally, the most commonly used optimization algorithms are base tracking and focusing algorithms [48]. Abma and Kabir [49] used a convex set projection algorithm to reconstruct 3-D seismic data and achieved good results. Wang et al. [50] proposed to apply the CS in time-domain AEM 3-D forward modeling, where the responses of the entire survey area can be quickly and accurately reconstructed using a small amount of forward calculations. This greatly improves the efficiency of 3-D forward modeling in AEM. Ren et al. [2] adopted the CS and preconditioned stochastic approximation method to achieve a fast time-domain AEM 3-D inversion, but they cannot provide any multi-scale information for a detailed interpretation.

In this paper, we use a shearlet transform as sparsity constraints to conduct multi-scale 3-D stochastic inversion for frequency-domain AEM data. In the sequence, we first introduce the principle of compressed sensing and provide an example of FD AEM data reconstruction. Then, we introduce the theory on sparsely constrained inversion of shearlet transform based on compressed sensing. Following that, we show the preconditioned stochastic inversion with sparse regularization. Finally, we test our method both on synthetic and field survey data.

2. Theory

In 3-D forward modeling, we use the CS algorithm to perform random sampling and reconstruct from the responses of sampled points to non-sampled points and obtain the responses of all points, with the aim to reduce the computational complexity and improve the efficiency. For the inversion, we transform the model into the sparse domain to extract the multiscale information, then we take the shearlet coefficients as parameters in the optimization of the objective function. Meanwhile, the sensitivity matrix with null columns is obtained by calculation at random sampling points, and then the gradient and Hessian matrix are calculated. To enhance the convergence, we apply a preconditioner to the stochastic inversion equation. During each iteration, a random resampling is performed to ensure that there is no significant error in the results due to bias in a particular sampling. By updating the shearlet coefficients in each iteration, and then transforming them into model domain to update the model parameter for calculating AEM responses and data misfits, we can iteratively achieve a fast 3-D EM inversion at high resolution. The workflow is given in Figure 1. The theory for our inversion can be described in three aspects: the compressed sensing, the multiscale sparse regularization, and preconditioned stochastic optimization.

2.1. Compressed Sensing

The basic idea of CS is to use the sparsity of the signal and recover signals $\mathit{f}\in {\mathbb{R}}^M$ from incomplete samplings $\mathit{y}\in {\mathbb{R}}^N$ (with N ≫ M). The sparsity of a signal refers to the fact that most of the signal values or their counterparts in a transformation domain are zero or close to zero, so that the signal is compressible within the transformation domain. The sparsity of the signal indicates that a small number of large coefficients in the transformation domain can be used to sparsely approximate the original signal, while still retaining the main information of the signal. We use the sparse transform to convert the signal f into sparse coefficients in the curvelet domain, which can be written as

$$\mathit{f}={\mathbf{\Psi }}^H\mathit{x},$$

(1)

where x is the sparse coefficient of signal f, which is the coefficient in the curvelet domain. Here, the symbol Ψ^H is defined as sparse inverse curvelet transpose. From the above equation, we can see that f and x characterize the signal, respectively, in the spatial and curvelet domains.

Furthermore, we assume that y is the sampling of signal f via a sensing matrix S. Then, the compressed sensing requires that the sensing matrix S and the sparse inverse transform matrix Ψ^H be as uncorrelated as possible. This non-coherence can prevent the errors introduced by any matrix from being amplified and prevent us from missing important information when filtering out a large number of coefficients close to zero. The degree of coherence between S and Ψ can be expressed as

$$\mu (\mathbf{S}, \mathbf{\Psi })=N^{1/2}\mathit{max}|<S_k, {\Psi }_j>|,$$

(2)

where k and j represent the column numbers, while N represents the number of columns. $\mu (\mathbf{S}, \mathbf{\Psi })$ describes the degree of correlation between any two columns of S and Ψ. If there is a strong correlation between two columns, the coherence factor μ is large, otherwise it is small. Herrmann stated that a random sampling matrix should be incoherent with most sparse transform matrices [47]. It is precisely the non-correlation between the random sampling matrix and the sparse representation basis that makes it possible to cover all information with only a small sampling rate, thus making sparse data reconstruction of incomplete data possible.

Based on the sparsity of signal x, the non-coherence of sampling matrix S, and the sparsity of the basis Ψ, we receive

$$\mathit{y}={\mathbf{S}\mathbf{\Psi }}^H\mathit{x}.$$

(3)

According to the theory of sparsity of signal and non-coherence, the coefficient x in the sparse domain can be recovered by using a small amount of survey data y, and the original signal f can be obtained by using the inverse curvelet transform.

We adopt the Poisson disk sampling method to obtain random data, which can make the sampling more uniform and random by adjusting the distance between adjacent sampling points [51]. The signal reconstruction is a process of solving optimization problems, i.e.,

$$\mathit{min}{\left|\right|\mathit{x}\left|\right|}_1 s.t. {\left|\right|\mathit{y}-{\mathbf{S}\mathbf{\Psi }}^H\mathit{x}\left|\right|}_2\le \epsilon$$

(4)

where ε represents the data noise. By continuously searching for sparser coefficients x, we use the convex set projection algorithm to solve the above equation [45].

When we receive the randomly distributed “compressed data”, the high-precision 3-D responses can be directly obtained by using a reconstruction algorithm, so that we can have high-quality data for subsequent inversions. In our 3-D forward modeling, we only need to calculate the responses at the randomly sampled data points, use the compressed sensing to reconstruct the sparse solution of the forward responses at all observation points, and then we can obtain the forward responses of all observation points via an inverse sparse transform. Therefore, the data reconstruction of the entire survey area can be achieved through a small amount of forward calculations. Figure 2 shows the CS process.

To obtain an optimal sampling rate before 3-D forward modeling and inversions, we first use a series of sampling rates to sample the geophysical survey data. Then, with the sampled data, we can receive the response at all survey stations via a CS reconstruction. Next, we calculate the relative errors between the geophysical survey data and the reconstructed ones. Finally, we obtain the optimal sampling rate by judging the reconstruction error as less than 1%, which can balance the accuracy and efficiency well.

2.2. Multiscale Sparse Regularization

For the objective function of the sparsely constrained regularized inversion, the form of the data fitting term remains unchanged, while a new regularization term is constructed using coefficients in sparse domain instead of model parameters in space domain. Thus, unlike the model constraints in the spatial domain, the sparsely constrained regularization inversion transforms the model in the spatial domain into coefficients in the sparse domain, which can be expressed as

$$\stackrel{\mathrm{~}}{\mathit{m}}={\mathbf{W}}_s\mathit{m},$$

(5)

where W_s is a sparse transformation operator, which is a 3-D shearlet transform here. $\stackrel{\mathrm{~}}{\mathit{m}}$ is a 1-D sparse coefficient vector obtained by rearranging the sparse coefficient matrix. Since most of the sparse coefficients are close to zero, the main features and boundary information of the model are stored in large coefficients. Thus, the purpose of sparse regularization inversion is to find a solution that fits the observed data while ensuring the sparsity of the coefficients as much as possible. Research has shown that using L1-norm constraints on sparse coefficients can yield sparse solutions [42,52]. Thus, we apply L1-norm constraints on coefficients in the sparse domain to construct the regularization term. The regularization term for sparse coefficients can be written as

$${\phi }_s={‖(\stackrel{\mathrm{~}}{\mathit{m}} {-\stackrel{\mathrm{~}}{\mathit{m}}}^{\mathit{ref}})‖}_1.$$

(6)

Based on the above discussion, the objective function for sparse regularization inversion can be defined as

$$\Phi ={\phi }_d+ \lambda {\phi }_s,$$

(7)

where φ_d and φ_s are the data misfit and coefficient regularization terms, respectively. Finally, we have

$$\Phi ={‖{\mathbf{W}}_d({\mathit{d}}^{\mathit{obs}}-{\mathit{d}}^{\mathit{prd}})‖}_2^2+\lambda {‖(\stackrel{\mathrm{~}}{\mathit{m}}-{ \stackrel{\mathrm{~}}{\mathit{m}}}^{\mathit{ref}})‖}_1.$$

(8)

We use the Lp-norm proposed by Ekblom [53] to represent the sparse regularization term φ_s that can be written as

$${\phi }_s(\mathit{x})={({\mathit{x}}^2+{ \epsilon }^2)}^{1/2},$$

(9)

where $\mathit{x} =\mathit{m} -{ \mathit{m}}^{\mathit{ref}}$. The objective function of the n^th iteration can be expressed as

$$\Phi ={\phi }_d^n(u)+{\lambda }^n{\phi }_s^n(v) ,$$

(10)

where, u and v are given by Farquharson, [54]

$$\mathit{u}={ \mathbf{W}}_d({\mathit{d}}^{\mathit{obs}}-{\mathit{d}}^{n-1}-\mathbf{J}\delta \mathit{m}),$$

(11)

$$\mathit{v} ={ \mathbf{W}}_s({\mathit{m}}^{n-1}+\delta \mathit{m} -{\mathit{m}}^{\mathit{ref}}).$$

(12)

Since we update the sparse coefficients in our sparse inversion, we need to calculate the derivatives with respect to these sparse coefficients. We use the diagonal matrix R_s to represent the derivative of the sparse regularization term with respect to the coefficient updates [21] that can be expressed as

$$\partial \phi (\mathit{x})/\partial \delta \stackrel{\mathrm{~}}{\mathit{m}}={\mathbf{B}}^T\mathbf{R}\mathit{x}.$$

(13)

For the data misfit term, the matrix B can be written as

$$\mathbf{B} ={\displaystyle \frac{\partial \mathit{u}}{\partial \delta \stackrel{\mathrm{~}}{\mathit{m}}}}=-{\displaystyle \frac{\partial {\mathbf{W}}_d\mathbf{J}\delta \mathit{m}}{\partial \delta \mathit{m}}} ·{\displaystyle \frac{\partial \delta \mathit{m}}{\partial \delta \stackrel{\mathrm{~}}{\mathit{m}}}}=-{\mathbf{W}}_d\mathbf{J}{\mathbf{W}}_s^{-1}.$$

(14)

We can further obtain the derivative of the data misfits with respect to the coefficient updates, i.e.,

$${\displaystyle \frac{\partial {\phi }_d^n(\mathit{u})}{\partial \delta \stackrel{\mathrm{~}}{\mathit{m}}}}={{(-{\mathbf{W}}_d\mathbf{J}{\mathbf{W}}_s^{-1})}^T\mathbf{R}}_d{\mathbf{W}}_d({\mathit{d}}^{\mathit{obs}}-{\mathit{d}}^{n-1}-\mathbf{J}\delta \mathit{m}),$$

(15)

where W_s⁻¹ is the sparse inverse transform operator. According to the orthogonal properties of the sparse transformation, we have W_s⁻¹ = W_s^T [15].

For the sparse regularization term in the objective function, the matrix B can be calculated by

$$\mathbf{B} ={\displaystyle \frac{\partial \mathit{v}}{\partial \delta \stackrel{\mathrm{~}}{\mathit{m}}}}=-{\displaystyle \frac{\partial {\mathbf{W}}_s\delta \mathit{m}}{\partial \delta \mathit{m}}} ·{\displaystyle \frac{\partial \delta \mathit{m}}{\partial \delta \stackrel{\mathrm{~}}{\mathit{m}}}}=\mathbf{I}.$$

(16)

Thus, the derivative of the sparse regularization term with respect to the coefficient update can be written as

$${\displaystyle \frac{\partial {\phi }_s^n(\mathit{q})}{\partial \delta \stackrel{\mathrm{~}}{\mathit{m}}}}={\mathbf{R}}_s{\mathbf{W}}_s({\mathit{m}}^{n-1}+\delta \mathit{m} -{\mathit{m}}^{\mathit{ref}}) ={\mathbf{R}}_s\delta \stackrel{\mathrm{~}}{\mathit{m}}-{\mathbf{R}}_s({\stackrel{\mathrm{~}}{\mathit{m}}}^{\mathit{ref}}-{\stackrel{\mathrm{~}}{\mathit{m}}}^{n-1}).$$

(17)

In summary, combining Equations (13) and (16), we obtain the linear equation solved in the n^th inversion iteration, i.e.,

$${(\mathbf{W}}_s{ \mathbf{J}}^T{\mathbf{W}}_d^T{\mathbf{R}}_d{\mathbf{W}}_d\mathbf{J}{\mathbf{W}}_s^{-1}{{+\lambda }^{ n}\mathbf{R}}_s)\delta \stackrel{\mathrm{~}}{\mathit{m}}={\mathbf{W}}_s{ \mathbf{J}}^T{\mathbf{W}}_d^T{\mathbf{R}}_d{\mathbf{W}}_d({\mathit{d}}^{\mathit{obs}}-{\mathit{d}}^{n-1}){+\lambda }^{ n}{\mathbf{R}}_s({\stackrel{\mathrm{~}}{\mathit{m}}}^{ \mathit{ref}}-{\stackrel{\mathrm{~}}{\mathit{m}}}^{n-1}),$$

(18)

where R_d and R_s are given by

$${\mathbf{R}}_d=2,$$

(19)

$${\mathbf{R}}_s=1/\sqrt{{({\stackrel{\mathrm{~}}{\mathit{m}}}^{\mathit{ref}}-{\stackrel{\mathrm{~}}{\mathit{m}}}^n)}^2+{\epsilon }^2)}.$$

(20)

The iterative linear equations system given in Equation (18) can be simplified as Ax = b and solved using the conjugate gradient (CG) method. The pseudo code (Algorithm 1) for the sparse regularization inverse algorithm based on sparse transform is given below.

Algorithm 1. Shearlet-based CS inversion

Set up: the starting model m0, the reference model mref, and the invert dataset dobs, the sampling matrix S.

Set up: the initial λ = 100, the reduction factor k = 0.5, and the thresh = 1.0

while rms not reduce to thresh do

1: Forward modeling for random under-sampling data by CS:

2: Compute the gradient from reconstruction data by

${\widehat{\mathit{g}}}^{ n}={\mathbf{W}}_s{\widehat{\mathbf{J}}}^{ T}{\mathbf{W}}_d^T{\mathbf{R}}_d{\mathbf{W}}_d({\mathit{d}}^{\mathit{obs}}-{\widehat{\mathit{d}}}^{n-1}){+\lambda }^{ n}{\mathbf{R}}_s({\stackrel{\mathrm{~}}{\mathit{m}}}^{\mathit{ref}}-{\stackrel{\mathrm{~}}{\mathit{m}}}^{n-1})$

3: Compute Hessian matrix from reconstruction data by

${\widehat{\mathit{H}}}^{ n}={ \mathbf{W}}_s{\widehat{\mathbf{J}}}^{ T}{\mathbf{W}}_d^T{\mathbf{R}}_d{\mathbf{W}}_d{\widehat{\mathbf{J}}\mathbf{W}}_s^{-1}{{+\lambda }^{ n}\mathbf{R}}_s$

4: Calculate the coefficient updates by CG:

${\widehat{\mathit{H}}}^{ n}·\delta {\stackrel{\mathrm{~}}{\mathit{m}}}^n={\widehat{\mathit{g}}}^{ n}$,

5: Update the model in the spatial domain:

${\mathit{m}}^n={\mathit{m}}^{n-1}+{s\mathbf{W}}_s^{-1}\delta {\stackrel{\mathrm{~}}{\mathit{m}}}^n$,

6: Calculate data misfit rms.

end while

Output: final results m, rms.

2.3. Preconditioned Stochastic Optimization

Given that only random sampled data are used in the model updates, the gradient is not as accurate as that calculated from the full dataset. This will influence the convergence in the inversion process. Here, we use a preconditioner to build a mapping relationship between the local gradient calculated from the under-sampling points to the full-batch gradient [2]. The preconditioner ${\mathbf{P}}_{\mathit{pre}}$ is a symmetric positive-definite preconditioning matrix that considers both the sampling rate and the gradient noise caused by random sampling. Therefore, we can rewrite Equation (18) with the preconditioner as

$${(\mathbf{Q}}_{\mathit{pre}}+{\mathbf{P}}_{\mathit{pre}} \widehat{\mathbf{H}} ) \delta \stackrel{\mathrm{~}}{\mathit{m}}=-{\mathbf{P}}_{\mathit{pre}} \widehat{\mathit{g}},$$

(21)

with

$${\widehat{\mathbf{H}}=\mathbf{W}}_s{ \mathbf{J}}^T{\mathbf{W}}_d^T{\mathbf{R}}_d{\mathbf{W}}_d\mathbf{J}{\mathbf{W}}_s^{-1}{{+\lambda }^{ n}\mathbf{R}}_s,$$

(22)

$$\widehat{\mathbf{g}}={\mathbf{W}}_s{ \mathbf{J}}^T{\mathbf{W}}_d^T{\mathbf{R}}_d{\mathbf{W}}_d({\mathit{d}}^{\mathit{obs}}-{\mathit{d}}^{n-1}){+\lambda }^{ n}{\mathbf{R}}_s({\stackrel{\mathrm{~}}{\mathit{m}}}^{ \mathit{ref}}-{\stackrel{\mathrm{~}}{\mathit{m}}}^{n-1}).$$

(23)

In Equation (21), the preconditioner ${\mathbf{P}}_{\mathit{pre}}$ can be expressed as

$${\mathbf{P}}_{\mathit{pre}}=\mathrm{diag}[\alpha {\mathbf{\Gamma }}^{-1}]\mathbf{\Pi }(\theta ),$$

(24)

where $\mathbf{\Gamma }$ is the sampling rate matrix, $\alpha$ is the weight coefficient that is always set to a value between 1 and 2, and $\mathbf{\Pi }(\theta )$ is an operator matrix that accounts for the gradient distribution and truncates the small gradient values. In addition, the ${\mathbf{Q}}_{\mathit{pre}}$ in Equation (21) can be expressed as

$${\mathbf{Q}}_{\mathit{pre}}={\displaystyle \frac{\partial {\mathbf{P}}_{\mathit{pre}}}{\partial \stackrel{\mathrm{~}}{\mathit{m}}}} \widehat{\mathit{g}}.$$

(25)

The MS stochastic inversion Equation (21) is solved by the conjugate gradient method. Therefore, when running the stochastic inversion with the Gauss–Newton method, we have to perform preconditioning both for the Hessian matrix and gradient (Equation (21)) to improve the efficiency. We choose to use the preconditioner proposed by Ren et al. [2] to establish the frequency-domain Gauss–Newton inversion equation. The sampling rate and weights are parametrized in the diagonal matrix in Equation (24), and we use $\mathbf{\Pi }(\theta )$ to truncate the gradient where the values less than the threshold are taken as noise.

3. Numerical Experiments

We first test our inversion algorithm on a synthetic model shown in Figure 3. The resistivity of an irregularly shaped anomalous bodies are 5 Ω·m and the background resistivity is 100 Ω·m. We set the initial model to be a uniform half-space of 100 Ω·m. All 3-D models in this paper are meshed using regular rectangular elements. In the horizontal direction, the middle calculation area has equidistant grids. In the vertical direction, the grid thickness increases by a certain multiplier (>1) to accommodate the diffusion of electromagnetic waves at low frequencies. We divide the model into 64 grids in the x- and y-directions, with a size of 15 m. This results in a total length of 960 m in the x- and y-directions. Further we divide the model into 30 layers in the z-direction. The first layer has a thickness of 2 m. After that, each layer has a thickness 1.1 times of its above layer until the 20th layer and then by 1.4 times until the 26th layer. The 27–30 layers are extension layers with the thicknesses of 100, 200, 400, and 600 m, respectively. The total thickness is 1617 m in the z-direction.

A total of 17 survey lines are designed, with each having 17 measuring points. The spacings between survey lines and points are both 45 m, and a total of 289 survey locations are evenly distributed within the central area of the synthetic model. The layout of measuring points is shown in Figure 4a. A 3% Gaussian random noise is added to the forward modeling data as the input data for our inversion. The cooling method was used to update the regularization factor in the inversion. We use a 3-D shearlet transform to achieve sparse regularized inversion, where the shearlet system contains 4 different scales and 49 directions at each scale.

We take the frequency-domain Helicopter Electromagnetic (HEM) system of the Norway Geology Survey as an example. This system has three horizontal coplanar coil pairs. The transmitting frequencies are 880, 6606, and 34,133 Hz, while the corresponding T-R separations are 6, 6.3, and 4.9 m, respectively. This system also has two vertical coaxial coil pairs, the frequencies are 980 and 7001 Hz, and the T-R separations are 6 and 6.3 m, respectively. Figure 4a shows the horizontal slice of the model at z = 70 m. Figure 4b–d show the vertical slice along the yellow, red, and blue lines in Figure 4a, respectively.

We first calculate AEM responses at all measuring points and show in Figure 5a the real parts at 880 Hz. Then, we use a Poisson disk random sampling [56] and perform forward modeling on those points that are 50% randomly sampled (Figure 5b). After that, we adopt the CS method to reconstruct the responses at non-sampled points to obtain the complete responses of all points (Figure 5c). Figure 5d shows the relative error between the forward responses calculated at all the survey points and the CS reconstructed ones. It is seen that most of the errors are less than 5%.

We implement 3-D L2-norm inversion and our shearlet-based regularized inversion to this model and compare the results of full-batch data with those of CS reconstructed data. Figure 6a,b show the results, respectively, with full-batch data and 50% random samplings, using the L2-norm inversion method; while Figure 6c,d show corresponding results with the shearlet-based multiscale method. It can be seen that the resolution of shearlet-based regularized inversions is higher than that of L2-norm inversions. Obviously, the anomalous bodies at different scales are better resolved by our shearlet-based method, especially the two small targets that are recovered at high resolution. Comparing the full-batch data inversion with 50% sampling data inversion, we find that the L2-norm method in Figure 6b creates fake anomalies, while the shearlet-based method in Figure 6d has the almost same performance as the full data inversion in Figure 6c. Thus, we conclude that for multiple complex geological targets in the underground, our shearlet-based MS inversion can recover the underground electrical structures well. We further determine statistics for the relative errors between the true model and the recovered ones. The results show that the average recovery relative errors for Figure 6a–d are, respectively, 24.97%, 30.97%, 22.84%, and 15.26%, implying that the sparse regularization stochastic inversion has the best fitting to the true model.

We further analyze the results of L2-norm and shearlet-based regularized inversions. Figure 7a,b and Figure 7c,d show, respectively, the horizontal and vertical slice results of the full-batch data and 50% random samplings for L2-norm inversion. Figure 7e,f and Figure 7g,h show the horizontal and vertical slices of inversion results of full-batch sampling and 50% random sampling data for shearlet-based regularization inversion, respectively. It is seen that both inversions recover the model well. However, for the smallest block of 5 Ω·m, the L2-norm inversions show obvious larger resistivity recovery (~15 Ω·m in Figure 7a,c), and the sparse inversion shows better, but relatively larger, resistivity (~7.17 Ω·m in Figure 7e; ~13 Ω·m in Figure 7g). This is probably because of the lack of enough high-frequency information. The conventional sparse inversion with full-batch data shows some fake anomalies in the background (Figure 7e), but our stochastic inversion based on CS performs better with both clean background and good anomaly recovery.

Figure 8 shows the parameters versus iterations for different inversion methods. It is seen that the inversions with full-batch data achieve convergence with RMS decreasing to around 1, while the inversion with 50% sampling has slightly slower convergence. This is due to the inaccurate gradient in each inversion iteration, although the preconditioner helps improve it.

Table 1. Test on synthetic data.

Inversion Methods	L2-Norm with Full Data	L2-Norm with 50% Sampling	Shearlet-Based with Full Data	Shearlet-Based with 50% Sampling
Iterations	11	12	14	15
Data misfit (rms)	1.06	1.22	1.01	1.23
Time (h)	8.7	4.65	15.3	9.22

shows the inversion parameters versus iterations and calculation time for two inversion methods. For the L2-norm inversions, the RMS for full-batch data decreases to 1.06 after 11 iterations, and the total inversion time is 8.7 h, while the RMS for 50% sampling data inversions decreases to 1.22 after 12 iterations. The total inversion costs 4.65 h. By comparison, the inversion efficiency is improved by about 46.5%. The RMS of only 14 iterations for shearlet-based with full-batch data inversion decreases to 1.01, with a total time cost of 15.3 h. The RMS of 15 iterations for shearlet-based inversion with 50% sampling is 1.23, with a total time of 9.22 h. Therefore, the MS stochastic inversions with a 50% sampling rate can achieve comparable results with the full-batch data inversion, but it can save about 6.08 h for this model experiment.

To perform a multiscale analysis, we show in this section the inversion results at different scales. The shearlet system used in this paper includes 4 scales. Figure 9a shows the first scale (coarse scale) inversion model that mainly reflects the outline of the model. Thus, we can see the general distribution of the targets without enough details. For instance, the smallest anomalous body cannot be well recovered, and the boundary details of the largest anomalous body are not well recovered. From Figure 9b–d, it is seen that with the addition of the 2nd and 3rd scale coefficients into the inversion, more information about the boundary and resistivity is gradually revealed; in particular, the smallest block is revealed more clearly, and the detail of the largest geological target is also obtained. Meanwhile, from the 3-D result in Figure 6 and the horizontal and vertical slice results in Figure 7, it is also seen that the detailed boundary of the anomalous body becomes clearer with the addition of other three scale coefficients. With statistics of the relative errors between the true model and recovered ones, we see that the average relative errors of Figure 9a–d are, respectively, 35.05%, 26.69%, 23.32% and 15.26%, which also demonstrate a multi-scale feature and the effectiveness of our sparse regularization stochastic inversion.

4. Field Data Inversion

To further test the effectiveness of our MS stochastic inversion algorithm based on compressed sensing, we invert an AEM dataset from the Byneset region of Norway that was acquired by the Geological Survey Norway (NGU). The system parameters can be found in previous synthetic studies. Figure 10 shows the Quaternary geological map of the Byneset area, where the blue part in the figure shows the marine sediments from the period of glacial retreat about 10,000 years ago. Due to the balanced rebound of glaciers, marine sediments are now exposed on present-day land [57]. Some regions also preserve ancient seabed. The bedrock in this area is mainly composed of chlorite slate and phyllite, which may contain well-conductive graphite. There are approximately 100 landslides marked in the central part of the Byneset area [58]. The green line in the figure indicates the location of a landslide that occurred in 2012. The black lines in Figure 10 represent AEM survey lines, with a total of about 60 lines spaced approximately 100 m apart and measuring points spaced approximately 50 m apart. The horizontal and vertical coordinates of the location map of the survey area and the inversion results in this section are all presented at the WGS-84 coordinate system.

We select 38 survey lines in the red box of Figure 10 for our inversions, with 58 survey points for each line selected. This creates a dataset with a total of 2204 survey points. To invert the data within the red box in Figure 10, we use 128 × 128 × 30 grids for our inversion. We divide the model into 128 units in the x- and y-directions, with a grid size of 30 m in the x-direction and a total extension of 3840 m, and 36 m in the y-direction with a total extension of 4608 m. In the z-direction, the model is divided into 30 layers, with an initial layer thickness of 4 m. Subsequently, the thickness of each layer increases by a factor of 1.1 until the 26th layer. The 27–30 layers are edge-expanding layers with the thicknesses of 60, 120, 240, and 500 m, respectively. Totally, we have an extension of 1356 m in the z-direction.

We set the initial model to be a half-space of 100 Ω·m and assume the initial regularization factor to be 1000 that is reduced to 0.1 times of its previous value in each iteration until it reaches 0.001. Then, we keep it unchanged in subsequent iterations. To validate our algorithm, we compare the inversion results of 50% sampling MS stochastic method with those of the full-batch MS inversions. For a better analysis, we also provide a CS-based L2-norm inversion results with 50% sampling.

Figure 11 shows the resistivity distribution at different depths. Figure 11a–c provide the results of L2-norm inversion with full-batch data in slices. Figure 11d–f provide the results of L2-norm inversion with 50% sampling in slices. The inversion results of shearlet-based regularized inversion from full-batch data and 50% sampling ones are, respectively, shown in Figure 11g–i and Figure 11j–l. We select three horizontal slices at depths of z = 18, 32, and 48 m and compare the results of the three inversion methods. From the figure, we can see that the L2-norm inversion with 50% sampling method shows a smooth variation in resistivity. The resolution of the inversion results from the shearlet-based regularized inversion is higher than the CS-based L2-norm inversion, which can provide more detailed information in the horizontal direction. Figure 10 shows the Quaternary geological map of the Byneset area. The inversion results in Figure 11 correspond to the area marked in red box in Figure 10. By comparing Figure 10 and Figure 11 in the north and west parts of the survey area, there are marshes (the brown area in Figure 10) with high resistivity. The central part of the survey area with conductive structures corresponds to the marine sediments (the light blue area in Figure 10). All these inversions have shown good recovery to the high-resistivity anomalies in the northern and western parts of the survey area and good recovery to the low-resistivity of the marine sediments in the central and southern regions.

Figure 12 shows the inversion results of the vertical slice at y = 7,030,191 m in Figure 10. The green box area in Figure 10 corresponds to the position of the red box area in Figure 12. Figure 12a shows the laterally constrained inversion (LCI) results using the Aarhus code from Baranwal et al. [59], Figure 12b shows L2-norm inversions with 50 percent sampling, while Figure 12c,d show the shearlet-based MS inversions from the full-batch data and the reconstructed data with 50% sampling, respectively. It is seen that the three inversion methods have consistency in characterizing the main low-resistivity anomalous structures and the location of the landslide faults marked in the red box in Figure 12, which is also consistent with the previous research (Figure 12a). This verifies again the effectiveness of the inversion methods presented in this paper. We also note that at the right side of the landslide area, Figure 12a–d show a conductive anomaly, while at the left side they show a quasi-layered distribution.

We make statistics to the inversion parameters and show them in Figure 13. From the figure, it is seen that the full-batch data inversion converges a litter faster than the 50 percent sampling.

Table 2. Test on field survey data from Norway.

Inversion Methods	L2-Norm with Full Data	L2-Norm with 50% Sampling	Shearlet-Based with Full Data	Shearlet-Based with 50% Sampling
Iterations	12	11	10	10
Data misfit (rms)	1.47	1.38	1.72	2.23
Time (h)	38.4	25.2	51.1	34.7

shows the iteration parameters and calculation time for the inversions. The L2-norm inversion with full-batch data takes 12 iterations and costs a computational time of 38.4 h, the L2-norm inversion with 50% samplings takes 11 iterations and costs a computational time of 25.2 h, the shearlet-based inversion with full-batch data takes 10 iterations and costs a computational time of 51.1 h, while our shearlet-based MS stochastic inversion with 50% sampling takes 10 iterations and costs a total of 34.7 h. Thus, the MS stochastic inversion is slower than the L2-norm method with 50% sampling; this is because the number of shearlet coefficients is much higher than the parameters in the space domain and because the forward/inverse transform takes time. Our method is faster than the shearlet-based regularized inversion with full-batch data although they both provide comparable good results.

5. Conclusions

Based on sparse regularization stochastic inversion algorithm, we combine the compressive sensing and preconditioner to achieve a high-resolution and efficient inversion of 3-D AEM data. By using a shearlet transform, we have illustrated the underground electrical resistivity in different scales and we can invert the models from a coarse to fine scale to achieve a high-resolution inversion with random data. During this process, we use a preconditioner to improve the convergence of the stochastic inversion. We find the following results from the synthetic and field data inversions:

(1)

The sparse regularization inversion has a higher resolution than the conventional L2-norm inversion but it also takes a bit more time;

(2)

The sparse regularization with random CS data has a comparable inversion result with that based on full-batch data, while it reduces the time consumption by 30~40%;

(3)

The sparse regularization stochastic inversion is more beneficial for exploring multiple geological bodies with different sizes and large areas that have a large amount of data to be inverted.

In the future, we will focus more on how to further improve the inversion efficiency such as new preconditioners, better sampling methods, etc.