A Scalable and Consistent Method for Multi-Component Gravity-Gradient Data Processing

Abstract

We demonstrate the potential of using the convolutional equivalent layer to jointly process large gravity-gradient datasets. Based on the equivalent-layer principle, we assume a single fictitious physical property distribution on a planar layer can approximate all components of the gravity-gradient tensor. Estimating this distribution using the classical technique ensures physical consistency among components. However, the classical approach becomes computationally prohibitive for large datasets due to the need to solve a large-scale inversion with a massive sensitivity matrix. To overcome this limitation, we exploit the block-Toeplitz Toeplitz-block structure of the sensitivity matrix for data on a regular horizontal grid. This structure significantly reduces computational cost—by orders of magnitude—compared to the classical method. Applications to synthetic and real datasets show that our method offers a computationally efficient alternative for processing large gravity-gradient data from various acquisition systems (AGG and FTG), even when data are irregularly spaced or flight lines are misaligned. On a standard laptop configuration, our method processed over 290,000 AGG data points in a few tens of seconds. It also handled between 726,000 FTG and 1,250,000 AGG data points within seconds to a couple of minutes, demonstrating practical computational efficiency for large-scale datasets.

1. Introduction

The advancement of airborne gradiometry allows the use of gravity-gradient data to facilitate the interpretation of gravity sources in mining and hydrocarbon explorations (e.g., [1,2,3,4,5,6,7,8]). The equivalent-layer and Fast Fourier Transform (FFT) techniques are feasible approaches for de-noising and/or transforming between conventional gravity and gravity-gradient components. For example, refs. [9,10] developed techniques to calculate the full gravity-gradient tensor from pre-existing vertical components of the gravitational attraction ($g_z$-component) by using FFT and a fast equivalent-layer technique, respectively. Ref. [11] used a data-adaptive 1D wavelet filtering technique to de-noise gravity-gradient data. Ref. [12] also computed the tensor components from the $g_z$-component; however, using the cosine transform technique shows it is less affected by noise than the FFT approach. Ref. [13] presented the first method to assess and highlight the internal consistency of airborne gravity-gradient components by representing all measured data through a common potential function that accounts for both lateral and vertical variations. Ref. [14] introduced a de-noising method for full-tensor gravity-gradiometer data using translation-invariant wavelets with mixed and adaptive Bayesian thresholding to effectively suppress noise while preserving high-frequency details and minimizing artifacts. Ref. [15] presented a joint inversion algorithm for gravity and magnetic data based on a sequential strategy and cross-gradient structural coupling, deriving a novel formula for minimizing the structural similarity term by combining an alternating strategy with the randomized singular value decomposition algorithm to improve computational performance. Ref. [16] proposed a real-time joint filtering method for gravity and gravity-gradient data that uses the derivative relationship between the two to construct the system and measurement equations of a Kalman filter. An improved Kalman filter is then applied, enabling simultaneous noise reduction in both datasets while ensuring that the filtered outputs satisfy the Laplace equation constraint. By using variational mode decomposition and the Hilbert transform to extract time-frequency features of airborne gravity-gradient data, ref. [17] applied a time-frequency feature threshold to suppress noise while preserving high-frequency features of the data.

The equivalent layer is a well-known method based on the potential theory. Following this theory, a set of potential-field data produced by a 3D source can be reproduced by a 2D infinite layer. In practice, ref. [18] showed that this layer can be represented by a finite and discrete physical-property distribution (e.g., point masses). The classical approach consists of estimating the physical-property distribution by solving a linear system subject to fitting the observed data. Nevertheless, the processing of a large potential-field dataset by using the classical equivalent-layer method may lead to costly computational schemes. As a result, some authors search for ways to increase the computational efficiency of the equivalent-layer technique, (e.g., [5,19,20,21,22,23,24,25,26,27,28,29,30,31,32]).

From the above-mentioned authors, there are just a few works for processing measured gravity-gradient measurements. Ref. [24] were the first to use the equivalent-layer technique for processing gravity-gradient data. Ref. [27] compared the equivalent-layer technique and FFT for processing airborne full-tensor gravity-gradient data. Ref. [5] used the equivalent-layer technique for de-noising gravity-gradient data and estimating the errors associated with the de-noised data. Ref. [29] combined the Gauss–FFT with Landweber’s iteration to propose a fast equivalent layer technique for processing two-component of the gravity-gradient data simultaneously.

Recently, refs. [33,34] developed an efficient and rapid equivalent layer technique for processing gravity and magnetic data by using an equivalent source to carry the calculations of the forward problem of iterative methods to estimate the physical property distribution through a 2D discrete convolution. This method reduces the memory RAM usage taking advantage of the symmetry in Block-Toeplitz Toeplitz-block (BTTB) contained in the sensitivity matrices. However, it is important to note that neither [33] nor [34] explicitly mentioned that this specific structure, the BTTB form, emerges only when the sensitivity matrix is defined in the coordinate system of the observed data. Consequently, all computational operations in this study were performed in the data reference frame, which is aligned with the acquisition geometry axes.

We demonstrate an efficient equivalent layer technique for jointly processing gravity-gradient data by exploiting the block Toeplitz with Toeplitz blocks (BTTB) structure of the sensitivity matrices. Our approach combines the conjugate gradient least squares (CGLSs) algorithm with 2D FFT, following a structure similar to the method proposed by [34]. By estimating a single equivalent layer, our method is capable of handling all available gravity-gradient components simultaneously. Although inspired by the same general objective as [34], our formulation differs in key aspects. In [34], the sensitivity matrix is constructed as a weighted sum of directional components derived from magnetic data. In contrast, our method uses gravity-gradient observations and constructs the sensitivity matrix by stacking contributions from each measured component. Additionally, unlike the method proposed in [34], which assumes acquisition geometries aligned with north–south and east–west directions, our approach is designed to remain effective even when the flight lines follow rotated geometries. Furthermore, it can handle data distributed over irregularly spaced points or uneven observation surfaces. Tests on both synthetic and real datasets confirm the robustness and flexibility of the proposed method under a variety of acquisition configurations.

2. Methodology

2.1. Gravity-Gradient Data

Consider a right-handed Cartesian coordinate system with the x and y oriented horizontally, pointing north and east, respectively, and the z-axis pointing vertically downward. Gravity-gradient data are commonly defined in terms of second derivatives $g_i^{\alpha \beta }$, $\alpha ,\beta \in \{x,y,z\}$, of the gravitational potential at the ith observation point $(x_i,y_i,z_i)$, $i\in \{1,\dots ,N\}$. In the source-free region, these second derivatives form a symmetric $3\times 3$ gravity-gradient tensor given by

$${\mathbf{G}}_i=\left[\begin{array}{ccc}{g}_i^{xx}& g_i^{xy}& g_i^{xz}\\ g_i^{xy}& g_i^{yy}& g_i^{yz}\\ g_i^{xz}& g_i^{yz}& g_i^{zz}\end{array}\right],$$

(1)

with trace equal to zero, i.e., $g_i^{xx}+g_i^{yy}+g_i^{zz}=0$, due to Laplace’s equation. This means that there are five linearly independent components $g_i^{xx}$, $g_i^{xy}$, $g_i^{xz}$, $g_i^{yy}$, and $g_i^{yz}$ [35].

The two most common airborne systems that can provide gravity-gradient data are the Full Tensor Gradiometry (FTG) system and the single axial Falcon airborne gravity gradiometer (AGG) system. The FTG system measures five linearly independent components of the gravity-gradient tensor (i.e., $g_i^{xx}$, $g_i^{xy}$, $g_i^{xz}$, $g_i^{yy}$ and $g_i^{yz}$). The Falcon AGG system measures the two horizontal differential curvature gradients $g_i^{xy}$ and $g_i^{uv}=(g_i^{yy}-g_i^{xx})/2$ [36].

2.2. Classical Equivalent Layer

Let ${\mathbf{g}}^{\alpha \beta }$ be an $N\times 1$ vector whose ith element is a gravity-gradient measurement of the $\alpha \beta$-component $g_i^{\alpha \beta }$, $\alpha ,\beta \in \{x,y,z\}$, at the observation point $(x_i,y_i,z_i)$, $i\in \{1,\dots ,N\}$. According to the equivalent-layer principle, the observed data vector ${\mathbf{g}}^{\alpha \beta }$ can be approximated by

$${\mathbf{d}}^{\alpha \beta }\left(\mathbf{m}\right)={\mathbf{H}}^{\alpha \beta }\mathbf{m},\alpha ,\beta \in \{x,y,z\},$$

(2)

where ${\mathbf{d}}^{\alpha \beta }$ is the predicted gravity-gradient data vector of size $N\times 1$ due to a discrete set of M point masses located below the observation points, at the coordinates $(x_j,y_j,z_c)$, $j\in \{1,\dots ,M\}$, on a horizontal plane with constant vertical coordinate $z_c>z_i$, $\forall i\in \{1,\dots ,N\}$. In Equation (2), $\mathbf{m}$ is the $M\times 1$ parameter vector whose jth element is the mass of the jth point mass and ${\mathbf{H}}^{\alpha \beta }$ is an $N\times M$ sensitivity matrix whose element $ij$ is defined as

$$h_{ij}^{\alpha \beta }=\left\{\begin{array}{cc}{c}_g\gamma {\displaystyle \frac{3{({\alpha }_i-{\alpha }_j)}^2-r_{ij}^2}{r_{ij}^5}}\hfill & \mathrm{if}\alpha =\beta \hfill \\ c_g\gamma {\displaystyle \frac{3({\alpha }_i-{\alpha }_j)({\beta }_i-{\beta }_j)}{r_{ij}^5}}\hfill & \mathrm{if}\alpha \ne \beta \hfill \end{array}\right.,$$

(3)

where $\alpha ,\beta \in \{x,y,z\}$, $c_g={10}^9$ is a transformation constant from 1/${\mathrm{s}}^2$ to eotvos, $\gamma$ is the gravitational constant in ${\mathrm{m}}^3$/(kg ${\mathrm{s}}^2$), and $r_{ij}$ is the Euclidean distance between the ith observation point and the jth point mass given by

$$r_{ij}={\left[{(x_i-x_j)}^2+{(y_i-y_j)}^2+{(z_i-z_c)}^2\right]}^{{\displaystyle \frac{1}{2}}}.$$

(4)

The classical equivalent-layer technique applied to gravity-gradient data involves estimating a single mass distribution $\widehat{\mathbf{m}}$ that simultaneously fits all the $\alpha \beta$-components of the gravity-gradient data. It can be accomplished by minimizing the data-misfit function

$$\mathsf{\Phi }\left(\mathbf{m}\right)={\displaystyle \frac{1}{N_t}}{\parallel {\mathbf{g}}^o-\mathbf{H}\mathbf{m}\parallel }_2^2,$$

(5)

which is subjected to some constraint, where $N_t$ is the total number of observations, ${\parallel ·\parallel }_2$ denotes the Euclidean norm, ${\mathbf{g}}^o$ is an observed data vector containing all measured $\alpha \beta$-components of the gravity-gradient data, and $\mathbf{H}$ is a sensitivity matrix depending on the specific airborne system. For example, if we consider gravity-gradient data measured by the FTG system, ${\mathbf{g}}^o$ is a $N_t\times 1$ vector given by

$${\mathbf{g}}^o={\left[{{\mathbf{g}}^{xx}}^{\top },{{\mathbf{g}}^{xy}}^{\top },{{\mathbf{g}}^{xz}}^{\top },{{\mathbf{g}}^{yy}}^{\top },{{\mathbf{g}}^{yz}}^{\top }\right]}^{\top },$$

(6)

and $\mathbf{H}$ is the $N_t\times M$ matrix composed by the sub-matrices ${\mathbf{H}}^{\alpha \beta }$, $\alpha ,\beta \in \{x,y,z\}$ (Equation (3))

$$\mathbf{H}={\left[{{\mathbf{H}}^{xx}}^{\top },{{\mathbf{H}}^{xy}}^{\top },{{\mathbf{H}}^{xz}}^{\top },{{\mathbf{H}}^{yy}}^{\top },{{\mathbf{H}}^{yz}}^{\top }\right]}^{\top },$$

(7)

where $N_t=5N$. Otherwise, if we consider the two components measured by the Falcon AGG system, ${\mathbf{g}}^o$ is a $N_t\times 1$ vector given by

$${\mathbf{g}}^o=\left[\begin{array}{c}{\mathbf{g}}^{xy}\\ {\mathbf{g}}^{uv}\end{array}\right],$$

(8)

and $\mathbf{H}$ is the $N_t\times M$ matrix

$$\mathbf{H}=\left[\begin{array}{c}{\mathbf{H}}^{xy}\\ {\mathbf{H}}^{uv}\end{array}\right],$$

(9)

where $N_t=2N$ and

$${\mathbf{H}}^{uv}={\displaystyle \frac{1}{2}}({\mathbf{H}}^{yy}-{\mathbf{H}}^{xx}).$$

(10)

The estimated mass distribution $\widehat{\mathbf{m}}$ minimizing the data-misfit function $\mathsf{\Phi }\left(\mathbf{m}\right)$ (Equation (5)) without any constraint can be obtained by solving the normal equations

$$\left({\mathbf{H}}^{\top }\mathbf{H}\right)\widehat{\mathbf{m}}={\mathbf{H}}^{\top }{\mathbf{g}}^o.$$

(11)

Notice that, for large datasets, there exists a very large computational demand to store the required matrices and vectors, solve the normal equations (Equation (11)), and perform matrix–vector operations. Our approach to overcome this computational demand is adapting the convolutional equivalent-layer methods proposed by [33] and [34] to take advantage of the particular structure of matrix $\mathbf{H}$ (Equations (7) and (9)).

We call attention to the fact that the Equations (2)–(10) highlight key differences between our method and the convolution-based formulation proposed by [34]. Unlike their approach, which constructs the sensitivity matrix as a weighted sum of directional components derived from magnetic data, our formulation is based on gravity-gradient observations and builds the sensitivity matrix by vertically stacking the individual contributions from each measured component (Equations (7) and (9)). Moreover, whereas the method proposed in [34] presumes data-acquisition geometries strictly aligned with the north–south and east–west directions, our approach is designed to maintain effectiveness when flight lines are oriented along rotated geometries that do not conform to these alignments.

2.3. Convolutional Equivalent Layer in Rotated Coordinates

Consider that the observed gravity-gradient data are distributed over a horizontal plane with constant vertical coordinate $z_c=z_i,i=1,\dots ,N$, at a regularly spaced grid of points along the x and y axes of our Cartesian coordinate system, which we refer to as the north-east system in which the x-axis points north and the y-axis to the east. In practice, however, gravity-gradient data are often referenced to a different coordinate system whose horizontal axes are not aligned with geographic directions. We refer to this as the data system, which is aligned with the acquisition geometry, with local coordinates $\tilde{x}$-axis and $\tilde{y}$-axis (Figure 1). The data system is rotated relative to the north-east system by an angle $\psi$ from north. To relate these two systems, we apply a coordinate rotation. The gravity-gradient tensor $\tilde{\mathbf{G}}$ defined in the data system can be obtained from the tensor $\mathbf{G}$, which in turn is defined in to north-east system by Equation (1) through the transformation:

$$\tilde{\mathbf{G}}=\mathbf{R}\mathbf{G}{\mathbf{R}}^{\top },$$

(12)

where $\mathbf{R}$ is a rotation matrix given by

$$\mathbf{R}=\left[\begin{array}{ccc}cos\psi & sin\psi & 0\\ -sin\psi & cos\psi & 0\\ 0& 0& 1\end{array}\right].$$

(13)

Let us consider an equivalent layer located on a horizontal plane below the grid of observations, with constant vertical coordinate $z_c>z_i$, having one point mass directly below each observation point, so that the number of point masses M is equal to that of observation points N. In this case, ref. [34] described the particular structure of the sensitivity matrix $\mathbf{H}$ (Equations (7) and (9)); however, they did not mention that this specific structure, namely, the block-Toeplitz Toeplitz-block (BTTB) structure, emerges only when $\mathbf{H}$ is defined in the coordinates of the data system (Figure 1). For this reason, all computational operations in this work were performed in the data system, which is aligned with the $\tilde{x}$- and $\tilde{y}$-axes (the acquisition geometry). These operations include the calculation of the BTTB matrix eigenvalues of $\mathbf{H}$, parameter estimation, prediction of gravity-gradient tensor components, and gravity-gradient data transformations.

After completing all computational operations in the data system, the results should be transformed back to the north–east system, which is aligned with geographic directions. This is conducted using the following transformation:

$$\mathbf{G}={\mathbf{R}}^{\top }\tilde{\mathbf{G}}\mathbf{R}.$$

(14)

We stress that the north-east system was used solely for visualizing the results, including the predicted gravity-gradient data, estimated parameters, and transformed data, since it corresponds to the spatial arrangement of the original observations.

To estimate the mass distribution $\widehat{\mathbf{m}}$, we solve the normal equations (Equation (11)) iteratively by using the standard conjugate gradient least squares (CGLSs) method ([37], p. 166) that requires three matrix-vector products: one out of the loop and two per iteration. These products can be written generically as

$$\mathbf{H}\mathbf{v}={\mathbf{H}}_1{\mathbf{v}}_1+\cdots +{\mathbf{H}}_L{\mathbf{v}}_L,$$

(15)

and

$${\mathbf{H}}^{\top }\mathbf{v}={\mathbf{H}}_1^{\top }{\mathbf{v}}_1+\cdots +{\mathbf{H}}_L^{\top }{\mathbf{v}}_L,$$

(16)

in terms of a generic partitioned matrix

$$\mathbf{H}={\left[{\mathbf{H}}_1^{\top },\dots ,{\mathbf{H}}_L^{\top }\right]}^{\top },$$

(17)

and a generic partitioned vector

$$\mathbf{v}={\left[{\mathbf{v}}_1^{\top },\dots ,{\mathbf{v}}_L^{\top }\right]}^{\top }.$$

(18)

The matrix–vector product ${\mathbf{H}}_{\ell }{\mathbf{v}}_{\ell }$ or ${\mathbf{H}}_{\ell }^{\top }{\mathbf{v}}_{\ell }$ in Equations (15) and (16), $\ell \in \{1,\dots ,L\}$, is computed using the fast discrete convolution approach introduced by [33]. It is important to note that L (Equations (15)–(18)) represents the number of measured gravity-gradient components. In the case of gravity-gradient data obtained from the FTG system, for example, the matrix $\mathbf{H}$ (Equation (17)) consists of $L=5$ sub-matrices, as defined by Equation (7). Otherwise, when the data is measured by the AGG system, the matrix is defined with $L=2$ sub-matrices, as specified in Equation (9). Our convolutional equivalent-layer method can be schematically defined as follows:

The convergence criterion of the algorithm (step 3) is based on the invariance of the normalized data-misfit function (Equation (5)) between successive iterations. The algorithm stops when

$$\delta ={\displaystyle \frac{\left|\mathsf{\Phi }{\left(\mathbf{m}\right)}^{\left(i\right)}{-\left(\mathbf{m}\right)}^{(i+1)}\right|}{\mathsf{\Phi }{\left(\mathbf{m}\right)}^{\left(i\right)}}}\le ϵ,$$

(19)

where $ϵ$ varies from ${10}^{-2}$ to ${10}^{-5}$, $\mathsf{\Phi }{\left(\mathbf{m}\right)}^{\left(i\right)}$ and $\mathsf{\Phi }{\left(\mathbf{m}\right)}^{(i+1)}$ are the data-misfit function at the ith and $(i+1)$th iterations, respectively.

2.4. Computational Performance

To quantify the computational performance of our method, we use the total number of flops associated with the required matrix operations ([38] p. 12). We also refer the reader to ([39] Section 7) for a detailed explanation of the flops computation presented below. Previously, we established that the total number of observations and the total number of equivalent sources are given by $N_t=NL$, where L is the number of measured gravity-gradient components (see Equations (15)–(18)) and N the number of observation points of each component.

We now determine the total number of flops required to solve the normal equations (Equation (11)) using the classical Cholesky decomposition approach ([38] p. 262); see also ([39] Equation (27)):

$$f_{\mathrm{classical}}={\displaystyle \frac{7}{3}}{\left(NL\right)}^3+4{\left(NL\right)}^2.$$

(20)

The number of flops associated with the standard CGLS algorithm without exploring the structure of matrix $\mathbf{H}$ (see ([38] Section 11.3) and ([37] p. 165); see also ([39] Equation (28)) is:

$$f_{CGLS}=L\left(2N^2+it(4N^2+4N)\right),$$

(21)

where “$it$” is the number of iterations.

If we now define the number of flops associated with the CGLS algorithm by considering that the matrix–vector products are efficiently computed as fast discrete convolutions, and the resulting flops for our method are given by:

$$f_{our}=L\left(\kappa \left(16N\right){log}_2\left(4N\right)+26N+it(\kappa \left(16N\right){log}_2\left(4N\right)+58N)\right),$$

(22)

where $\kappa =5$. We refer the reader to ([39], Section 7.7, Equation (67) and Algorithm 7) for a detailed explanation of the flops computation underlying the method, which has been adapted in this work to handle gradient-gravity data. It is important to note that we did not consider the flops associated with data transformations (Algorithm 1).

Algorithm 1 Convolutional equivalent-layer for gravity-gradient data in rotated coordinates.

Input: ${\mathbf{g}}^o$ (defined in the data system), angle $\psi$ (Figure 1) 1: Transform the observed data from north-east system to data system (Equation (12)). 2: Interpolate the rotated data to produce a dataset closer to a regular grid (optional). 3: Estimate the discrete equivalent layer from the interpolated data by combining the Algorithm standard Conjugate Gradient Least Squares (CGLS) method ([37] p. 166) and the fast discrete convolution approach. 4: Compute the gravity-gradient data produced by the equivalent layer (predicted data). 5: Transform the observed and predicted data from data system to north-east system using Equation (14).

Figure 2 shows the number of flops using the classical (black line), standard CLGS (red line), and our method (blue line). The latter two methods were computed considering $it=50$ and 5 components. This result demonstrates the excellent efficiency of our convolutional equivalent-layer method for processing gravity-gradient data. Note that the number of flops required to estimate the mass distribution by our method decreased by approximately 8 orders of magnitude compared with the classical approach.

Figure 3 shows the runtime to solve the linear system with the maximum value of N equal to 10,000 points and L = 5 components, i.e., $N_t$ 50,000 gravity-gradient data using the classical (black line), standard CLGS (red line), and our method (blue line). The quantity of data for runtime comparison is limited by using the classical equivalent layer technique on a processor Intel(R) Core(TM) i9-9880H CPU @ 2.30 GHz with 16 GB of RAM memory.

We emphasize that the values of $it$ (in Figure 2) and the maximum number of observations N (in Figure 3) were selected arbitrarily and serve solely illustrative purposes. These choices aim to highlight the scalability behavior of the methods. Specifically, (i) the computational cost of our method (blue line in Figure 2) is approximately eight orders of magnitude lower than that of the classical approach (black line) and (ii) the runtime of both the classical (black line) and standard CGLS (red line) methods in Figure 3 increases rapidly with the number of observations, while the runtime of our method (blue line) remains negligible.

2.5. Criterion for Selecting Layer Depth

There is no analytical criterion in the literature for choosing the depth of the equivalent layer. In our work, the criterion for selecting the layer depth was based on [18]. This criterion asserts that the depth of the layer in relation to the observed data should fall within a range of 2.5 to 6 times the data spacing. Specifically, we first compute the average elevation of the observation points using the z values. The equivalent layer is then positioned at a depth offset determined by the maximum data spacing, scaled by a factor within the recommended range.

2.6. Interpolating Data

To reduce the effect of over sampling along flight and tight lines, we applied a block-averaging approach similar to that proposed by [31] in the data system (Figure 1). We have first divided the survey area into a regular grid of adjacent horizontal blocks along the x and y directions, respectively. To the horizontal center of each block, we assign the averaged gravity-gradient data and vertical coordinate (z-coordinates) of the observation points within the corresponding block. The result is a new dataset on a regular grid of points, with block-averaged gravity-gradient data and vertical coordinates.

3. Synthetic Data Application

We simulated Falcon AGG and FTG data to demonstrate the efficiency of our method for jointly processing gravity-gradient tensor components. These datasets were produced by multiple simulated geologic sources generating strong-interfering anomalies more closely to a real-world scenario. Specifically, the model consists of 302 rectangular prisms, divided into two groups. The first group represents the main geologic bodies and consists of two prisms, each with a density contrast of 1.0 g/${\mathrm{cm}}^3$, a top depth of 100 m, and dimensions of $2000\times 1000\times 300$ m (the smaller body) and $5000\times 2000\times 300$ m (the larger body) along the x-, y-, and z-axes, respectively. The second group, representing geologic noise, consists of 300 small, shallow cubes with side lengths and top depths uniformly distributed between 100 and 200 m, and between 0 and 10 m, respectively. These sources produce anomalies with predominantly short-wavelength spectral content.

Both data simulations followed the flight pattern of a real airborne gravity-gradient survey over the Kauring test site, Australia [40]. To each gravity-gradient component, we introduced additive Gaussian-distributed noise, with zero mean and a standard deviation equal to 7% of the maximum absolute value of the corresponding noise-free gravity-gradient data. Specifically, the standard deviations of the noise (in eotvos) for each component were the following: $xx$ = 2.19, $xy$ = 1.94, $xz$ = 4.47, $yy$ = 3.83, $yz$ = 4.59, $zz$ = 5.46, and $uv$ = 1.48.

The simulated synthetic data consisted of 114 flight lines each with 1275 points, totaling 145,350 observations, at vertical coordinates z (Figure 4) ranging from −296 to −440 m and a mean and standard deviation of −353.34 m and 23.56 m, respectively.

We applied our method to simulated Falcon and FTG data using a single equivalent layer with one source beneath each observation point at a constant depth of $z_c$ = 23.54 m. Transformations between the north-east and data systems (Figure 1 and steps 1 and 5 of Algorithm 1) were computed with $\psi =-30°$.

Here, we illustrate the transformation by predicting the ${\mathbf{g}}_{zz}$ component and a quantity we refer to as the gradient magnitude of ${\mathbf{g}}_z$ (i.e., $∥\nabla {\mathbf{g}}_{\mathbf{z}}∥$), defined as the square root of the sum of the squares of the predicted ${\mathbf{g}}_{xz}$-, ${\mathbf{g}}_{yz}$-, and ${\mathbf{g}}_{zz}$-components.

3.1. Falcon AGG Data

The simulated Falcon AGG components ${\mathbf{g}}_{xy}$ and ${\mathbf{g}}_{uv}$ are shown in Figure 5. Each component has N = 145,350 observation points, totaling $N_t$ = 290,700 simulated data. After transforming the simulated data from the north-east system to the data system (step 1 of Algorithm 1), we estimated the mass distribution at 39.02 s and computed the predicted data in the data system (steps 3 and 4 of Algorithm 1). Refer to the Supplementary Material for the Falcon AGG data transformed into the data system (see Figure S1).

Figure 6a,b show, in the north–east system, the residuals between the simulated data (Figure 5) and the predicted data for the two components of the Falcon AGG system. The insets display histograms of the residuals, with means of 0.16 and 0.18 eotvos, and standard deviations of 2.04 and 1.70 eotvos, for the ${\mathbf{g}}_{xy}$ and ${\mathbf{g}}_{uv}$ components, respectively. The bell-shaped histograms and the estimated standard deviations are close to those of the added Gaussian noise (1.94 and 1.48 eotvos), which suggests an excellent fit to the data. Figure 6c shows the root mean square error (RMSE) as a function of iterations. The RMSE curve flattens just below 2 eotvos, consistent with the noise levels, further supporting that our method has converged properly. Taken together, the residual distribution and the RMSE convergence indicate that the solution fits the data within the expected noise level, without evidence of overfitting.

Figure 7 shows two transformation results predicted by the estimated equivalent layer using our method. Despite the strong presence of geologic noise, simulated by 300 small sources, both the predicted ${\mathbf{g}}_{zz}$ (Figure 7a) and the predicted $∥\nabla {\mathbf{g}}_{\mathbf{z}}∥$ (Figure 7b) provide a clear definition of the boundaries of the two main geologic bodies. The geologic noise is visible as short-wavelength components in these transformation results.

3.2. FTG Data

Figure 8 shows the simulated FTG components ${\mathbf{g}}_{xx}$, ${\mathbf{g}}_{xy}$, ${\mathbf{g}}_{xz}$, ${\mathbf{g}}_{yy}$, and ${\mathbf{g}}_{yz}$. Each component has N = 145,350 observation points. After transforming the simulated data from the north-east system to the data system (step 1 of Algorithm 1), we estimated the mass distribution at 98.20 s using a total of $N_t$ = 726,750 simulated data and computed the predicted data by applying steps 3 and 4 of Algorithm 1. The simulated FTG data, transformed into the data system, is provided in the Supplementary Material (see Figure S2).

The estimated mass distribution produces the predicted data that acceptably fit the simulated FTG components (Figure 8a–e). This is corroborated by Figure 9a–e, which shows the residuals between the simulated (Figure 8a–e) and predicted FTG data. The bell-shaped histograms of the residuals (shown in Figure 9 as insets) also confirm the acceptability of the data fit with means varying between −0.77 and 0.32 eotvos, as well as the standard deviations varying between 2.24 and 4.52 eotvos. The RMSE convergence curve in Figure 8f stops decreasing below 4 eotvos, which matches the level of random noise added to the data. This indicates that the predicted gravity-gradient components have minimized the random errors, and further improvement is limited by the noise floor.

Similar to the previous transformation results, Figure 10a,b also defines the boundaries of the two main geologic bodies, despite the presence of short-wavelength components. These components are not caused by random noise, but are primarily due to geologic noise simulated by 300 small, shallow cubes.

A quantitative comparison between our proposed method and the conjugate gradient for least squares (CGLSs) algorithm is provided in the Supplementary Material. Using reduced versions of the synthetic Falcon AGG and FTG datasets, we assessed the accuracy and performance of both methods through residual maps, histograms, and RMSE convergence curves. The results demonstrate that BTTB achieves comparable accuracy to CGLS while offering significantly lower computational cost (Figures S4–S11, in the Supplementary Materials).

To evaluate the robustness of our inversion method, we conducted a sensitivity analysis using synthetic Falcon AGG datasets contaminated with Gaussian noise at three levels, 5%, 7%, and 9%, of the maximum signal amplitude. For each level, six independent noise realizations were generated and inverted. Under all noise conditions examined, our method demonstrated stable convergence behavior, with residuals closely matching the noise characteristics: unbiased, Gaussian-distributed, and spatially uncorrelated. The final RMSE values converged to levels consistent with the added noise, and the transformation results consistently delineated the boundaries of the modeled geologic bodies. These findings confirm that the method is robust, repeatable, and maintains reliable performance under increasing noise conditions. Full details and figures are provided in the Supplementary Materials (see Figures S12–S20).

4. Real Data Applications

We applied our method to the Falcon AGG data from the Kauring gravity gradiometry test site, Australia [40] and FTG data from the Vinton salt dome, LA, USA [3,41]. In these real-world applications, whether using Falcon AGG data over Kauring gravity gradiometry or FTG data over the Vinton salt dome, we emphasize that the fitted data were computed at the original observation points, not at the reduced points. Aiming to process large gravity-gradient data, we also applied the method to more than 1,250,000 Falcon AGG data points from the Canobie Airborne Gravity Survey, Australia.

4.1. Kauring

With funding from the Western Australian government’s Exploration Incentive Scheme, a matching industry contribution, and support from the Geological Survey of Western Australia and Geoscience Australia, the Kauring test site situated near York in Western Australia, was established to support coordinated geophysical data collection [42]. The site is located in a farming region characterized by gently undulating granitic terrain, featuring occasional incised valleys, with a maximum topographic relief of about 75 m. According to [43], the gravity anomaly at the Kauring test site is attributed to iron-rich rocks underlying a small topographic rise near the center of the map. Ref. [44] interpreted that the amplitude of the complete Bouguer gravity anomaly field in the Kauring test site gradually decreases from west to east. Higher anomaly values are concentrated in the southwestern part of the study area, while lower values are found in the northeast. Although it remains uncertain whether the linear gravity ridge is produced by similar rocks beneath the soil cover, airborne geophysical surveys support this interpretation.

The observed Falcon AGG components ${\mathbf{g}}_{xy}$ and ${\mathbf{g}}_{uv}$ (Figure 11a,b) over the Kauring test site, Australia, consists of 114 northwest–southeast oriented lines with line spacing of 50 m, totaling 164,624 observed points at the z-coordinates shown in Figure 4. We have transformed the gravity-gradient data from the north-east system to the data system (step 1 of Algorithm 1) by using $\psi =-30°$ (Equation (12)). The observed Falcon AGG data, transformed to the data system, is available in the Supplementary Materials (see Figure S3). We then interpolated the dataset in the data system (step 2 of Algorithm 1) to obtain 1275 points in each of the 114 flight lines, which corresponds to 145,350 reduced points. We placed one equivalent source directly beneath each interpolated point, at a constant depth of $z_c$ = 23.54 m.

After approximately 120 iterations and just a few seconds of computation (Figure 12c), we estimated a mass distribution that fits the ${\mathbf{g}}_{xy}$ and ${\mathbf{g}}_{uv}$ components. Using this model, we calculated the predicted data and the corresponding residuals (Figure 12a,b) for the two AGG components in the north–east coordinate system. The insets in these figures show the histograms of the residuals, which have means of 0.11 and −1.22 eotvos, and standard deviations of 3.88 and 3.40 eotvos, for the ${\mathbf{g}}_{xy}$ and ${\mathbf{g}}_{uv}$ components, respectively. Combining the RMSE curve (Figure 12c), which stabilizes below 4 eotvos, with the residual distributions (insets in Figure 12a,b) supports the conclusion that the data are well-fitted within the expected noise level.

The predicted ${\mathbf{g}}_{zz}$ and $∥\nabla {\mathbf{g}}_{\mathbf{z}}∥$, shown in Figure 7b and Figure 13a, respectively, reveal strong linear anomalies in the southwestern part and a set of isolated anomalies in the northeastern part. These results are consistent with those reported in [44].

4.2. Vinton

The Vinton salt dome, located in southwestern Louisiana within the onshore Gulf of the Mexico Basin, is a geological structure situated in a region known for its long-standing oil and gas development history [45]. Geologically, the dome consists of a massive salt core overlain by a well-defined cap rock, which exhibits a vertical gradation from limestone at the top to gypsum and anhydrite at the bottom. This cap rock is embedded within a sedimentary sequence of interbedded sandstone and shale, reflecting the region’s complex depositional and diagenetic history. Analysis of lithological density ranges led [3] to conclude that the gravity-gradient response is largely attributable to the cap rock. Geophysical interpretations reveal a 3D cap rock structure with a subtly elongated shape, consistent with the strike of the main fault in the study area [3,46].

Figure 14 shows the observed components ${\mathbf{g}}_{xx}$, ${\mathbf{g}}_{xy}$, ${\mathbf{g}}_{xz}$, ${\mathbf{g}}_{yy}$ and ${\mathbf{g}}_{yz}$ over the Vinton salt dome. The selected area over the dome has flight lines, each with 9349 points, totaling 46,745 observations, at vertical coordinates z-coordinates ranging from −62.14 to −109.32 m. Because the flight lines are oriented along the north–south direction, the data and north-east systems coincide and we by-passed the step 1 of Algorithm 1. We interpolated the data to obtain 6000 points, totaling 30,000 observations. We placed one equivalent source directly beneath each interpolated point, at a constant depth of 1981.81 m.

After 500 iterations (Figure 15f) and 4.3 s, we estimated the mass distribution using the measured components, i.e., ${\mathbf{g}}_{xx}$, ${\mathbf{g}}_{xy}$, ${\mathbf{g}}_{xz}$, ${\mathbf{g}}_{yy}$, and ${\mathbf{g}}_{yz}$ components. This estimated model produces predicted data that fit the observed FTG components reasonably well (Figure 14a–e). This result is corroborated by the residuals shown in Figure 15a–e, which compare the observed data (Figure 14a–e) with the predicted values. The histograms of the residuals (insets in Figure 15a–e) exhibit approximately Gaussian distributions, with means ranging from −0.29 to 0.26 eotvos and standard deviations between 5.79 and 8.83 eotvos, further confirming the quality of the data fit. As in the synthetic tests, the RMSE curve (Figure 15f) reaches a value below 7.4 eotvos, which is consistent with the residual distributions shown in the insets of Figure 15a–e.

The transformation results, shown in Figure 16, clearly delineate the boundary of the cap rock of the Vinton salt dome. We call attention to the presence of short-wavelength components in both the predicted ${\mathbf{g}}_{zz}$ (Figure 16a) and the predicted $∥\nabla {\mathbf{g}}_{\mathbf{z}}∥$ (Figure 16b). Based on synthetic test results (Figure 7 and Figure 10), where simulated geologic noise appears as short-wavelength components in the transformation outputs, we infer that the short-wavelength features in Figure 15 may be due to shallow, small-scale geologic sources with relative high density contrasts.

4.3. Canobie

The Canobie survey area, located in northern Queensland, Australia, is being explored under contract with Geoscience Australia. It spans latitudes $19°$$10\prime$ S to $19°$$52\prime$ S and longitudes $140°$$27\prime$ E to $141°$$01\prime$ E [47]. The area lies within the Mount Isa Province, specifically within the Canobie Domain, which is part of the Eastern Fold Belt Subprovince [48]. The Mount Isa Province is a Paleoproterozoic to Mesoproterozoic terrane that forms part of the North Australian Craton. It is recognized as a major metallogenic province, hosting significant deposits of copper, gold, lead, zinc, silver, and uranium. The Canobie Domain is characterized by folded Proterozoic sedimentary and volcanic basement rocks typical of the eastern fold belt of the Mount Isa Inlier. Its definition is based primarily on geophysical characteristics, notably a distinctively high magnetic response [48,49]. The Eastern Fold Belt Subprovince, which includes the Canobie Domain, is particularly known for shear- and vein-hosted copper deposits, as well as iron oxide–copper–gold (IOCG) mineralization, often associated with mafic magmatism and structural features.

The Canobie Airborne Gravity Survey of Northern Queensland, Australia, aims to attract exploration to greenfield areas within the Mount Isa Basin. This survey measured the Falcon AGG components ${\mathbf{g}}_{xy}$ and ${\mathbf{g}}_{uv}$ using two acquisition channels, A and B. The observed ${\mathbf{g}}_{xy}$ component was computed as the average of the measurements from channels A and B, and the same procedure was applied to obtain the observed ${\mathbf{g}}_{uv}$ component. These averaged results are shown in Figure 17a and Figure 17b, respectively.

The estimated mass distribution results in predicted AGG components (Figure 18) that fit the observed data with satisfactory accuracy. This is supported by the residual maps shown in Figure 19 and by the corresponding histograms, which show near-Gaussian distributions centered around zero eotvos. The standard deviations of the residuals are 4.06 eotvos for the ${\mathbf{g}}_{xy}$ component and 3.94 eotvos for the ${\mathbf{g}}_{uv}$ component. Additionally, the observed AGG data range from approximately −28.8 to 38.0 eotvos, giving a total signal amplitude of about 66.8 eotvos. In this context, an RMSE of around 4 eotvos corresponds to a signal-to-noise ratio of roughly 15:1, indicating a robust inversion result. Furthermore, the RMSE convergence curve (Figure 19c) flattens after approximately 300 iterations, converging to a value just below 4 eotvos. The agreement between the residual statistics and the final RMSE, together with the absence of large coherent patterns in the residuals, suggests that the inversion has successfully recovered the dominant geologic signal, with the remaining misfit primarily reflecting random measurement noise.

Most of the survey area is characterized by strong, linear north–south and northeast-trending anomalies, except in the northeastern part, which displays a set of more isolated anomalies. This pattern is evident in both the predicted ${\mathbf{g}}_{xy}$ and ${\mathbf{g}}_{uv}$ components (Figure 18), as well as in the transformed results, the predicted ${\mathbf{g}}_{zz}$ component and $∥\nabla {\mathbf{g}}_{\mathbf{z}}∥$, shown in (Figure 20).

5. Conclusions

We present a cost-effective equivalent-layer technique for jointly processing gravity-gradient components. The method’s computational efficiency stems from the Block-Toeplitz Toeplitz-Block structure of sensitivity matrices for regular grids, enabling efficient vector–matrix product evaluations via 2D FFT. This approach estimates a mass distribution over a single equivalent layer using all gravity-gradient components, while requiring significantly fewer floating-point operations than the classical equivalent-layer technique or the CGLS method. On typical laptop hardware, the method handled over 290,000 AGG data points in just tens of seconds and is scaled to process between 726,000 FTG and 1,250,000 AGG data points in under a few minutes, demonstrating strong computational performance on large-scale data.

Tests on synthetic data confirm the method’s ability to handle irregularly spaced points, non-horizontal surfaces, and flight lines with arbitrary orientations, as well as data from different acquisition systems (e.g., AGG and FTG). The synthetic tests incorporated both random Gaussian noise and complex geologic noise from multiple small sources, simulating realistic survey conditions. The estimated mass distribution produced predicted data that fit noisy observations within expected error margins. The transformation results, specifically the vertical gradient component and the magnitude of the vertical gravity-gradient vector, effectively delineated the boundaries of the main geological bodies, despite the presence of strong short-wavelength features caused by geological noise.

Applications to real datasets, including Falcon AGG data from the Kauring test site and Canobie survey in Australia, and FTG data over the Vinton salt dome in the USA, demonstrate the method’s robustness across diverse geological settings, survey geometries, and acquisition systems. In all cases, the estimated mass distributions produced predicted gravity-gradient components that matched observations within expected noise levels. The residuals were near-Gaussian and uncorrelated, and convergence was achieved rapidly. The transformation results, particularly the vertical gradient and total gradient magnitude, successfully delineated key geological features such as the linear anomalies in Kauring and Canobie, and the cap rock boundary at Vinton, even in the presence of complex short-wavelength geological signals.

Sensitivity analyses across multiple noise levels demonstrated stable convergence and robustness, with residuals matching noise characteristics and consistent results across repeated runs. Moreover, comparative evaluations against the classical equivalent-layer technique and the conjugate gradient for least squares (CGLSs) method showed that our method achieves comparable accuracy while requiring significantly fewer floating-point operations and lower computational cost. These findings highlight the method’s reliability, efficiency, and practicality for processing large-scale airborne gravity-gradient datasets.

Extending the method to jointly process AGG and FTG data from the same area presents no methodological obstacles, requiring only that data from both datasets be acquired over the same region.