MAGNETIC PARAMETERS INVERSION METHOD WITH FULL TENSOR GRADIENT DATA

. Retrieval of magnetization parameters using magnetic tensor gradient measurements receives attention in recent years. Determination of sub- surface properties from the observed potential ﬁeld measurements is referred to as inversion. Little regularizing inversion results using full tensor magnetic gradient modeling so far has been reported in the literature. Traditional magnetic inversion is based on the total magnetic intensity (TMI) data and solving the corresponding mathematical physical model. In recent years, with the de-velopment of the advanced technology, acquisition of the full tensor gradient magnetic data becomes available. In this paper, we study invert the magnetic parameters using the full tensor magnetic gradient data. A sparse Tikhonov regularization model is established. In solving the minimization model, the conjugate gradient method is addressed. Numerical and ﬁeld data experiments are performed to show feasibility of our algorithm.


1.
Introduction. In geophysical prospecting, data measured at, above, or below the ground are obtained during field survey (a forward problem), and extraction of the physical properties of the Earth from the data is a mathematical problem (an inverse problem) which is essential for processing and interpretation. Meanwhile, the use of magnetics for geophysical exploration is widely studied.
Traditional magnetic data are total magnetic intensity (TMI). With the development of a high temperature superconducting quantum interference devices (SQUIDs) operating in liquid nitrogen, a novel rotating magnetic gradiometer system has been designed. In China, we have designed a low temperature SQUIDs system, and we perform successively the field work in 2016. This system allows to measure components of the gradient tensor. Gradient measurements also provide valuable additional information, compared to conventional total-field measurements, when the field is undersampled. Many discussions are given on the advantages of magnetic gradient tensor surveys as compared to the conventional total magnetic intensity (TMI) surveys [13,14,3,12,4].
Inversion of physical parameters, such as the magnetic susceptibility and the magnetization, are main scientific problems using magnetic field data [9,10]. A lot of research works have been done so far. Wang and Hansen [18] reformulated the gravimetric-magnetic model in wavenumber domain into coordinates invariance form, and extended the original magnetic inversion method CompuDept into threedimensional case, which allowed a large amount of airborne magnetic data being involved in inversion; Li and Oldenburg [5] recovered 3D susceptibility models by incorporating a priori information into the model objective function using one or more appropriate weighting functions; Pignatelli et al. [8] considered using dipole source to approximate the discrete gridded model of the anomaly, encompassed the depth weighting function into the discrete potential field function and employed the L-M method to solve the corresponding linear equation to get the solution with depth resolution.
According to the potential theory, magnetic inverse problems are ill-posed [17]. The main problems are the nonuniqueness and instability of the solution. Therefore it is crucial to choose proper norm of the model to restrict the solution space of the model. This is meaningful in reducing ill-posedness and enhancing numerical stability. The norm of the model should be chosen according to the a priori information of the model. Due to the fact that the magnetic data lack the resolution in depth, pure norm constraint on the model is not sufficient to reflect the medium layers. To overcome this problem, there are two ways. One is based on the Tarantola's statistical theory [15], assuming that the data and the model are both uncertain and obey the Gaussian distribution, and constructing the fitting function using the maximum a posteriori likelihood function; another is based on Tikhonov regularization theory [16]. It can be proved these two forms are equivalent under proper conditions. Retrieval of magnetization parameters using magnetic tensor gradient measurements receives attention in recent years. The direct determination of subsurface properties (e.g., position, orientation, magnetic susceptibility) from the observed potential field measurements is referred to as inversion.
Due to the difficulty of acquiring field measured magnetic gradient tensor data, most of the results in literature are based on the total magnetic intensity (TMI) data. In methodology, previous studies mainly focus on statistical regularization and transform-based filtering method to recover physical parameters. Little optimizing and regularizing inversion results using gradient tensor modeling so far has been reported in the literature. In this paper, we will report our recent results using our device. In inversion realization, we study magnetic parameters (dipole source) inversion with regularization using magnetic gradient tensor data in this paper. Our airborne magnetic field survey using the low temperature SQUID system is performed in an area consists of paramagnetic and ferromagnetic material. Our new contributions to literature in this paper are: (1) a three-dimensional Tikhonov regularization model is established for the magnetic parameters (dipole source) inversion using magnetic gradient tensor (MGT) data; (2) this is the first time to report inversion results using our new data with our device; and other than TMI data, full tensor gradient data are compared to draw the conclusion that better inversion results can be obtained with full tensor gradient data.

2.
Method. In this Section we describe the mathematical modeling of the considered problem involving not only classical approach connected with using total magnetic intensity data but also with tensor gradient data.
2.1. Mathematical modeling. The equation describing magnetic field B f ield dipole of dipole sources m is defined as , which corresponds to allocation of the triaxial sensor that measures magnetic field B f ield dipole , and point (x, y, z) of dipole source m, µ 0 is a permeability in vacuum.
Transforming B f ield dipole into following form and redefining the variables as i = x, y, z and p = (p x , p y , p z ) ≡ (x s , y s , z s ), we have following representation for components of vector B f ield dipole : Taking derivative of B i dipole with respect to spatial variable i = x, y, z and j = x, y, z = i, we have the diagonal elements and non-diagonal elements of tensor matrix B tensor : Note, that we define full tensor magnetic gradient B tensor , which unlike to magnetic induction B f ield dipole (that has only 3 components) has 9 components and can be written in the following matrix form: actually, we have only 5 different components of the tensor matrix. Thus, for the whole object, for volume V of which we want to restore the magnetic moment density M (M = M x i + M y j + M z k ), we have the following 3D Fredholm integral equations of the 1 st kind: which can be rewritten as the following system of two 3D Fredholm integral equations of the 1 st kind: Kernels K T M I and K M GT of these integral equations can be written as In this paper we consider the model included the magnetic gradient tensor data only (thus K = K M GT ).

2.2.
Tikhonov regularization. Then we will be assume that M ∈ W 2 2 (P ), B ∈ L 2 (Q), and operator A with kernel K is continuous and unique. Norms of the right-hand side of equation (2) and the solution are introduces as follows: Suppose that instead of accurately knownB and A their approximate values B δ and A h are known, such that B δ −B L2 ≤ δ, A−A h W 2 2 →L2 ≤ h. So, the inverse problem is ill-posed and it is necessary to build a regularizing algorithm for its solving. We use the algorithm based on minimization of the Tikhonov functional [16] ( For any α > 0 an unique extremal of the Tikhonov functional M α η , η = {δ, h}, which implements minimum of F α [M ], exists. To select the regularization parameter the generalized discrepancy principle can be used [2]. When we choose the parameter α = α(η) accordingly to the generalized discrepancy principle The minimal element of the Tikhonov functional for fixed α > 0 can be found by the application of the conjugate gradient method.

2.3.
Numerical aspects of the algorithm. For numerical minimization of functional (3) we used algorithms which were described in details at works [6,7], including some recommendations of its effective parallelization.
After discretization an approximate solution M , which realizes the minimum of functional (3), can be found as a solution of the system where R -finite-difference approximation of the operator R: For numerical solving of system (5) we use the conjugate gradient in the following form.
Let M (s) -minimizing sequence, p (s) , q (s) -auxiliary vectors, p (0) = 0, M (1)any arbitrary point. Then formulae of the conjugate gradient method for searching of solution M (N ) of system (5) can be rewritten as follow: It should be noted that in numerical experiments we can put α = 0, M (1) = 0 and use the iteration number s as the regularization parameter (in this case the 3. Results of calculations.  Figure 1(a). The results of calculations which used TMI, TMI+MGT and MGT models are represented of Figure 1(b,c,d). The root-mean-square error in the components of the reconstructed vector M is 0.12263 for TMI-model, 0.12262 -for TMI+MGT-model and 0.12527 for MGT-model. This means that all of the models produce "equal" results, but Figure 1(c) shows that MGT-model produce more detailed solution concerned to the small details. The main conclusion from testing calculations is that 1) MGT-model is able to produce the better reconstruction for the magnitude of the small details of the solution, 2) the MGT-model should be used alone without combining TMI-and MGT-data.

3.2.
Field data applications. Then we performed some testing calculations with real field data (there are only the tensor data with the first 5 tensor components of the MGT-data, other 3 components of the dipole field TMI-data are absent).
Our data was measured using the low-temperature SQUID. With a superconducting loop, SQUIDs can detect minute changes of flux. In our device, only changes perpendicular to the loop are detected, hence they are vector sensors. Using the new device, we can obtain abundant information of the magnetism with low noise, which may yield high resolution inversion results and give us sufficient quantitative analysis. Our airborne magnetic field survey using the low temperature SQUID system is performed in an area consists of paramagnetic and ferromagnetic material, which is a typical test area in North China. With the new type of data, we perform numerical inversion using our proposed regularization method.  For calculations in both examples we used 128 processors of the shared research facilities of HPC computing resources "Lomonosov-1" at Lomonosov Moscow State University [11]. Time of calculations ∼ 5 minutes. 4. Conclusion. Using full tensor magnetic gradient data to retrieve the interested medium parameters is an important inverse problem in geophysics. Usually it is hard to obtain the practical full tensor magnetic gradient data. In this paper, we report our recent results using data measured by the low temperature SQUIDs system designed by SIMIT, Chinese Academy of Sciences. We establish a threedimensional Tikhonov regularization model and use the conjugate gradient method to solve the large scale inverse problem. We first perform synthetic tests using the proposed regularization method to invert the physical parameters, then practical data are performed. Our inversion results reveal that the full tensor magnetic gradient data can distinguish fine structures of anomalies underground very well. Therefore, using full tensor magnetic gradient data for geophysical prospecting is a useful technique in the future.