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The structure-coupled joint inversion method of gravity and magnetic data is a powerful tool for developing improved physical property models with high resolution and compatible features; however, the conventional procedure is inefficient due to the truncated singular values decomposition (SVD) process at each iteration. To improve the algorithm, a technique using damped leastsquares is adopted to calculate the structural term of model updates, instead of the truncated SVD. This produces structural coupled density and magnetization images with high efficiency. A so-called coupling factor is introduced to regulate the tuning of the desired final structural similarity level. Synthetic examples show that the joint inversion results are internally consistent and achieve higher resolution than separated. The acceptable runtime performance of the damped least squares technique used in joint inversion indicates that it is more suitable for practical use than the truncated SVD method.

The joint inversion technique (i.e., imaging) is essential in interpreting collocated gravity and magnetic data. However, handling the combination of geophysical data types, each of which is sensitive to different material properties, poses a major challenge. In the early stage, Menichetti and Guillen [

In this study we present a joint inversion framework for gravity and magnetic data using a structural similarity criterion, which is proposed as an improved version of the Fregoso and Gallardo [

The principle of the structural similarity criterion is that collocated multiple physical property models that reflect the same geological structure are considered implicitly to have common boundaries. Therefore the structural similarity criterion is required to quantitatively evaluate the structural resemblance of one or two physical property models. Haber and Oldenburg [

where

which express the structural similarity of the two models on their orthometric vertical projection surfaces. When the model gradients have the same or reverse direction, that is, their gradients are parallel, these three components tend to have very small values; otherwise they would be non-zero quantities. So, for the whole model vo-

lume beneath the study area, a built-up array

throughout the whole space. When the structures are consistent, the array

As described by Gallardo and Meju [

where subscripts 1 and 2 represent the gravity and magnetic methods; subscript 0 represents reference vector information; d is the observational data vector; p is the transformed model vector;

Following the robust statistical optimization algorithm framework developed by Fregoso and Gallardo [

where

where

The final models

The inverse of matrix

where U and V are orthonormal matrices, and

where c is the truncated index of the non-null singular sequence

where I is the identity matrix, and b is the denominator of the damping factor. The second term added in brackets has the effect of making the total matrix well-posed. Although some structural perturbation information is lost in such a scheme, the computation stability is enhanced. This linear system is easy to compute with an iterative conjugate gradient solver, which accelerates the computing rates compared to truncated SVD. Then model updates of the equation at the kth iteration is solved using:

which is the equation adopted in this paper. In this scheme, b is termed a coupling factor, and has an impact on the final achieved structural resemblance such that large values of b indicate greater model similarity. In other words, the structural coupling level may be readily controlled by tuning b by trial and error. A synthetic test below outlines the principle of choosing an optimal value of b.

We first describe a synthetic example that illustrates the basic features of the proposed joint technique, using a typical synthetic model similar to that of Frogoso and Gallardo [^{3} and magnetization amplitude was 1.0 A/m.

To illustrate the use of the coupling factor b, we assigned different values to compute the cross-gradients L-2 norm at each iteration as shown in

structures are consistent with each other, while a large value accounts for the incompatible feature. Clearly, a large b value achieved better convergence than a small value; however, if too big, the stability of the computation may be compromised. The choice often entails a trial-and-error process. A relatively large number may be chosen provided it achieves satisfactory convergence of the cross-gradients norm and that the numerical computation is stable.

We then tested and compared the efficiency of the proposed damped least-squares technique and the conventional truncated SVD. Two different mesh discretization scales were adopted with various model dimensions to demonstrate runtime performance on the same computer platform. ^{−8}) was slightly faster than SVD (final cross-gradients norm 3.9 × 10^{−8}), but for the larger model scale, the runtime of the truncated SVD (final cross-gradients norm 6.0 × 10^{−8}) dramatically increased at each iteration except the initial computation, compared to the damped method (final cross-gradients norm 2.4 × 10^{−8}). It was seen that the truncated SVD is not suitable for large-scale problems. The damped least-squares method obtains the same results but with better runtime performance.

Efficient, simultaneous 3-D structure-coupled joint inversion of collocated gravity and magnetic data using a damped least-squares technique is presented. It is consistent with the previous view that the cross-gradients constraints have a significant capacity to obtain more compatible models with higher structural resemblance than separated inversion.

The damped least-squares technique is introduced to perform efficient computation of the structural perturbation term in the linear system at each iteration. A coupling factor is proposed to regulate the final structural similarity level achieved. A relatively large value of this factor, chosen by trial and error, is suggested to achieve desired cross-gradients norm convergence on condition that the numerical computation is stable.

The synthetic experiments show that the joint inverted density and magnetization results are consistent, which reveals the geological setting. Furthermore, the runtime performance is more acceptable than that of truncated SVD, indicating its suitability for practical use.

This work was funded by National Natural Science Foundation of China, Beijing Higher Education Young Elite Teacher Project (YETP0650), The National High Technology Research and Development program (“863”

Model scale | Iteration | Runtime performance | |
---|---|---|---|

Damped least squares (s) | Truncated SVD (s) | ||

5 × 10 × 10 | 1 | 0.75 | 0.76 |

2 | 0.76 | 1.35 | |

3 | 0.79 | 1.29 | |

4 | 0.79 | 1.28 | |

5 | 0.85 | 1.33 | |

6 | 0.80 | 1.33 | |

5 × 20 × 20 | 1 | 3.03 | 3.13 |

2 | 6.18 | 102.71 | |

3 | 6.67 | 101.36 | |

4 | 6.04 | 99.52 | |

5 | 6.04 | 101.84 | |

6 | 5.92 | 101.36 |

Program) of China (No. 2013AA063901-4 and 2013AA063905-4), R&D of Key Instruments and Technologies for Deep Resources Prospecting (The National R&D Projects for Key Scientific Instruments) (No. ZDYZ2012- 1-02-04), Constrained multi-parameter inversion of geophysical technology and software systems (No. 2014AA06A613), and The Fundamental Research Funds for the Central Universities.