Research On Magnetization Inversion Methods Based On Adaptive Weighted L_p-norm

Inversion of 3D physical properties using gravity or magnetic potential field data is an essential tool for quantitative interpretation and has been wildly used in deep structural studies,mineral resource exploration and engineering exploration,etc.Among them,magnetization inversion is an essential method to study the spatial distribution features of magnetic geological bodies,which is of great significance in the quantitative interpretation of related geological problems.While further analysis is still needed due to multiple solutions of the magnetization inversion and the complexity of real geological scenarios.In the magnetization inversion,the constructed objective function plays a key role in the distribution features of the inverted results,which greatly affects the understanding of the distribution features of the underground magnetic bodies.The classical magnetization inversion is carried out by using L₂-norm objective function,which is usually too smooth to clearly recover the boundaries of geological bodies.The L_p-norm inversion has been developed gradually,and the inverted result by the mostly used L₁-or L₀-norm objective function is more focused and can delineate the boundaries of geological bodies more easily.In the study of L_p-norm objective function,there are few systematic studies on the p-value selection.And the approximate formula of L_p-norm is usually derived based on the magnetization alone,without considering the constraint function acting on the magnetization.As a result,the model constraint function in different L_p-norm model constraint term of the total objective function is still weighted the same.In addition,few studies focused on how to adaptively choose p-values to invert the geological bodies with different distribution features.Finally,it is found that the introduced classical depth weighting constraint function in the magnetization inversion may cause the"tail"phenomenon while weakening the"skin"effect.To deal with the above problems,the magnetic anomaly data and magnetic anomaly amplitude（T_a）data are used to invert the magnetization amplitudein the condition of only induced magnetism exist or both induced magnetism and remanent magnetism exist.According to Lawson’s algorithm,the approximate formula of L_p-norm,the L_p-norm objective function is derived.Based on which,the systematic study of L_p-norm magnetization inversion is carried out.The main research results obtained are:（1）The weighted L_p-norm objective function is built according to the first-order derivative approximation of Lawson formulation,at which the magnetization and the model constraint function are considered as a whole part.The systematic comparative study of the L_p-norm inversion and weighted L_p-norm inversion with p between 0 and 3 is conducted by using quantitative accuracy evaluation.According to the model testing results,it is sufficient to meet the demand of magnetization inversion by taking p between 0 and 2,and the inversion is not easy to converge stably when taking p higher than 2,and the corresponding inverted result tends to be much blurry.Usually taking p between 0 and 1 helps better recover the real magnetization amplitude and the spatial distribution features of the target geological bodies,while when taking p higher than 1,the recovered magnetization amplitude is smaller than the real value and the recovered targets tend to be blurry.In addition,the accuracy of the weighted L_p-norm inversion are improved to some extent,comparing with the L_p-norm inversion.When remanent magnetization exists,the magnetic anomaly data is recommended to be used for the inversion when the remanent magnetization directions are basically the same and the total magnetization direction can be easily estimated,while the magnetic anomaly amplitude（T_a）data should be used when remanent magnetization directions are various or the total magnetization direction is difficult to be estimated.（2）According to the research of the weighted L_p-norm objective function,a smaller p-value is more favorable to delineate the boundaries of the geological bodies or to reflect the obvious variation of the magnetization.Based on the above understanding,the adaptive weighted L_p-norm objective function is established by combining the normalized vertical derivatives of the maximum-edge-recognition methods（NVDR-MERMs）,to obtain a more realistic 3D spatial distribution of magnetization that can take account of different spatial distribution features.In which,the size of p-value is determined based on the results of NVDR-MERMs,and is generally desirable between 0 and 1.According to the model testing results,the adaptive weighted L_p-norm inversion has achieved a high accuracy.（3）The depth weighting constraint function introduced in the adaptive weighted L_p-norm magnetization inversion is studied.Starting from the forward theory,the decaying feature of forward kernel function is studied,and it is believed that the essence of the introduced depth weighting function is to take column transformation of the forward kernel function,thereby weakening the rapid decay with depth of the forward kernel function.Therefore,the improved depth weighting functions based on the column transformation of the forward kernel function are proposed to eliminate the possibly existed"tailing"phenomenon by the classical depth weighting function,as well as to improve the focusing ability,vertical and horizontal resolution of the magnetization inversion,comprehensively.（4）Aiming at the study of the spatial distribution features of magneticgeological bodies in terrestrial area or in the ocean,the adaptive weighted L_p-norm magnetization inversion method is applied to delineate the metallogenic geological bodies of the Jinchuan Cu-Ni deposit and to recover the magnetic targets of the Ita Mai Tai guyots.The inverted results have confirmed the effectiveness and practicability of the proposed adaptive weighted L_p-norm magnetization inversion method.