Harmonic and biharmonic biases in potential field inversion

Geophysics ◽  
2014 ◽  
Vol 79 (1) ◽  
pp. G15-G25 ◽  
Author(s):  
João B. C. Silva ◽  
Darcicléa F. Santos ◽  
German Garabito

We found that minimum [Formula: see text]-norm and smoothness-constrained continuous solutions of the linear inverse problem of potential field data are harmonic and biharmonic, respectively. In the case of a discrete distribution, the minimum [Formula: see text]-norm and smoothness-constrained solutions become biased toward being harmonic or biharmonic, respectively. As a result, the estimated discrete distribution of density or magnetization contrast tends to be smooth and to satisfy the maximum principle, which forces the solution maxima and minima to lie on any boundary of the discretized region. The above findings were illustrated with 2D numerical examples. The harmonic or biharmonic bias is brought forth when the strengths of the minimum [Formula: see text]-norm or the smoothness constraint are enhanced (relative to all other constraints) by approximating the continuous case (a large number of discretizing cells relative to the number of independent observations) and/or by using a regularizing parameter. We discovered that, by inspecting the rearranged normal equations, it is possible to qualify three different possibilities of designing nonharmonic estimators. Then we found that all three possibilities have in fact already been implemented in the literature, reinforcing, in this way, the validity of the theoretical demonstration.

2021 ◽  
Vol 14 (1) ◽  
Author(s):  
Luan Thanh Pham ◽  
Ozkan Kafadar ◽  
Erdinc Oksum ◽  
Ahmed M. Eldosouky

Geophysics ◽  
2014 ◽  
Vol 79 (1) ◽  
pp. IM1-IM9 ◽  
Author(s):  
Nathan Leon Foks ◽  
Richard Krahenbuhl ◽  
Yaoguo Li

Compressive inversion uses computational algorithms that decrease the time and storage needs of a traditional inverse problem. Most compression approaches focus on the model domain, and very few, other than traditional downsampling focus on the data domain for potential-field applications. To further the compression in the data domain, a direct and practical approach to the adaptive downsampling of potential-field data for large inversion problems has been developed. The approach is formulated to significantly reduce the quantity of data in relatively smooth or quiet regions of the data set, while preserving the signal anomalies that contain the relevant target information. Two major benefits arise from this form of compressive inversion. First, because the approach compresses the problem in the data domain, it can be applied immediately without the addition of, or modification to, existing inversion software. Second, as most industry software use some form of model or sensitivity compression, the addition of this adaptive data sampling creates a complete compressive inversion methodology whereby the reduction of computational cost is achieved simultaneously in the model and data domains. We applied the method to a synthetic magnetic data set and two large field magnetic data sets; however, the method is also applicable to other data types. Our results showed that the relevant model information is maintained after inversion despite using 1%–5% of the data.


2010 ◽  
Author(s):  
M. Shyeh Sahibul Karamah ◽  
M. N. Khairul Arifin ◽  
Mohd N. Nawawi ◽  
A. K. Yahya ◽  
Shah Alam

2014 ◽  
Vol 644-650 ◽  
pp. 2670-2673
Author(s):  
Jun Wang ◽  
Xiao Hong Meng ◽  
Fang Li ◽  
Jun Jie Zhou

With the continuing growth in influence of near surface geophysics, the research of the subsurface structure is of great significance. Geophysical imaging is one of the efficient computer tools that can be applied. This paper utilize the inversion of potential field data to do the subsurface imaging. Here, gravity data and magnetic data are inverted together with structural coupled inversion algorithm. The subspace (model space) is divided into a set of rectangular cells by an orthogonal 2D mesh and assume a constant property (density and magnetic susceptibility) value within each cell. The inversion matrix equation is solved as an unconstrained optimization problem with conjugate gradient method (CG). This imaging method is applied to synthetic data for typical models of gravity and magnetic anomalies and is tested on field data.


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