Joint Inversion Algorithm Research Articles

Three-Dimensional Joint Inversion of EM and Acoustic Data Based on Contrast Source Inversion

In this article, we study a 3-D joint inversion algorithm for acoustic and electromagnetic data. This algorithm is based on the contrast source inversion method, which aims to reconstruct acoustic and electromagnetic parameters by updating two contrasts and two contrast sources alternately. The contrast is defined as the relative difference between the parameter, such as permittivity, of the object and the background, while the contrast source is expressed as the function of the contrast and the total field to partially linearize the problem. Besides, two multiplicative regularization terms are employed to alleviate the ill-posedness. Because the physical parameters in electromagnetics and acoustics share the same structure, we use the gradient of the contrast in acoustics and electromagnetics to constrain each other to achieve the enhanced structural similarity between inverted models. Then, by minimizing the predefined cost function using the conjugate gradient method, those acoustic and electromagnetic parameters, such as compressibility, attenuation, permittivity, and conductivity, can be reconstructed. Two numerical examples are given to compare the results of separate inversion and joint inversion.

Structural joint inversion on irregular meshes

SUMMARY Structural joint inversion of several data sets on an irregular mesh requires appropriate coupling operators. To date, joint inversion algorithms are primarily designed for the use on regular rectilinear grids and impose structural similarity in the direct neighbourhood of a cell only. We introduce a novel scheme for calculating cross-gradient operators based on a correlation model that allows to define the operator size by imposing physical length scales. We demonstrate that the proposed cross-gradient operators are largely decoupled from the discretization of the modelling domain, which is particularly important for irregular meshes where cell sizes vary. Our structural joint inversion algorithm is applied to a synthetic electrical resistivity tomography and ground penetrating radar 3-D cross-well experiment aiming at imaging two anomalous bodies and extracting the parameter distribution of the geostatistical background models. For both tasks, joint inversion produced superior results compared with individual inversions of the two data sets. Finally, we applied structural joint inversion to two field data sets recorded over a karstified limestone area. By including geological a priori information via the correlation-based operators into the joint inversion, we find P-wave velocity and electrical resistivity tomograms that are in accordance with the expected subsurface geology.

Two dimensional joint inversion of direct current resistivity and radiomagnetotelluric data based on unstructured mesh

Using the unstructured mesh, a new two-dimensional joint inversion algorithm has been developed for Radiomagnetotelluric and Direct current resistivity data. The unstructured mesh is generated with triangular cells, whose vertical and lateral lengths increase towards the depths. The Finite Element Method (FEM) has been used in the forward modelling part of the developed joint inversion algorithm. In the previous studies, structured grid-based joint inversion algorithms have been developed using the Finite Difference Method (FDM). In the structured grid-based algorithms, when the mesh is being generated with rectangular cells, the vertical lengths of the cells get bigger towards the depths while the lateral lengths remain constant. With the structured mesh, the undulated surface topography cannot be represented well enough. Also, because of the incompatible aspect ratio of model cell sizes in deeper model sections, the resolution of the model parameters will get smaller and cannot be resolved well with the structured grids. Imaging of surface topography and underground resistivity structures by the new algorithm requires fewer elements than those using structured grids. Therefore, the developed algorithm is faster than traditional 2D inversion algorithms. Furthermore, the resolution of the deeper model parameters has been increased by using the definition of the unstructured grid. A regularized inversion scheme with a smoothness-constrained stabilizer has been employed to invert the data. First, we have tested the developed joint inversion algorithm using synthetic data simplified from archaeological and mine site scenario and the results have been compared with the conventional algorithms using structured grids. We have also tested our algorithm with the real data which were collected from mineral investigation site at approximately 10 km east of the Elbistan district of Kahramanmaraş province, in the west of the Taurus Mountains, Turkey. The results show that the developed joint inversion algorithm is a powerful tool to detect both resistive and conductive targets.

Nonlinear creep model and parameter identification of mudstone based on a modified fractional viscous body

To study the evident nonlinear creep characteristics of mudstone, the creep process of rock is regarded as the superposition of linear and nonlinear creep processes. Using an element combination model for reference, we introduce the mechanical element (FC) with a fractional derivative based on fractional calculus theory. Compared with the Newtonian element, the FC can react more effectively to the nonlinear gradual process of rheological problems. The variable viscosity coefficient FC element with fractional calculus is connected to both Kelvin and spring elements in series instead of a Newton body element. In addition, considering the effects of load level and time damage variables, a five-element nonlinear creep damage model is proposed that can simulate the entire process of mudstone creep, particularly the accelerated creep phase of mudstone, and both constitutive and creep equations of the model can be derived. Based on the creep model’s use of multiple parameters under graded loading, this study employs a modified particle swarm optimization algorithm with a genetic crossover factor and least-square method joint inversion algorithm. Using creep experimental data of mudstone, this study identifies the parameters for the proposed five-element fractional creep damage model and derives reasonable parameter values. A parameter sensitivity analysis of the constitutive model is also conducted. The results show that the size of the stress loading level $$ \sigma $$ determines the length of the stable creep phase of the mudstone and the time to enter the accelerated creep phase. The derivative order $$ \gamma $$ becomes a direct influence factor on the rock creep rate.

Two‐dimensional data‐space joint inversion of magnetotelluric, gravity, magnetic and seismic data with cross‐gradient constraints

ABSTRACTIn recent years, joint inversion has been widely used for integrated geological interpretation. We extended a data‐space joint inversion algorithm of magnetotelluric, gravity and magnetic data to include first‐arrival seismic travel‐time and normalized cross‐gradient constraints. We describe the main features of the algorithm and apply it to synthetic data generated for hypothetical models. For the synthetic data, we find that the joint inversion with multiple parameters is superior to the joint inversion with two or even three parameters, which can reduce the multisolution of inversion results more effectively. Furthermore, data‐space joint inversion involves fewer memory requirements and better calculation speeds than traditional model‐space joint inversion. The normalized cross‐gradient constraints can better couple model parameters of different magnitudes compared with traditional unnormalized cross‐gradient constraints, resulting in higher levels of structural similarity among resistivity, density, magnetic susceptibility and velocity models.

Alternating Joint Inversion of Controlled-Source Electromagnetic and Seismic Data Using the Joint Total Variation Constraint

An alternating joint inversion method for controlled-source electromagnetic (CSEM) and seismic data is developed. The structural constraint is used for correlating and constraining the electromagnetic (EM) resistivity and seismic velocity parameters during the inversion. The structural coupling used is the joint total variation (JTV) constraint, which is incorporated into the objective function of the individual EM or seismic inversion to enforce the structural similarity between the resistivity and velocity. In this paper, the conventional cross-gradient constraint is not preferred as it can only be used for enforcing structural similarity for 2-D or 3-D case since it is always zero for 1-D joint inversion. The JTV constraint can be applied for 1-D joint inversion, as well as for the 2-D/3-D case, which is of a broader interest. The improved Gauss–Newton (GN) is used for minimizing the objective function and for reconstructing the subsurface resistivity and velocity. The alternating joint inversion algorithm is applied for integrating land CSEM data with cross-well seismic data for subsurface reservoir evaluation and water–oil identification. Numerical examples show that the developed joint inversion can improve the inversion results significantly over those from the separate EM or seismic inversion.

A novel pulse sequence and inversion algorithm of three-dimensional low field NMR technique in unconventional resources.

Compared with two-dimensional (2D) nuclear magnetic resonance (NMR) technique like correlations among the transversal relaxation time (T2), the longitudinal relaxation time (T1), and the diffusion coefficient correlation (D), three-dimensional (3D) NMR technique is superior with the complete measurement of T2, T1, and D simultaneously. It can solve the problem of overlaps in 2D correlation map and is helpful to characterize relaxation components in unconventional resources such as tight gas and oil shale. However, the existed 3D NMR technique is restricted due to the loss of short relaxation information and the inversion inaccuracy that caused by the incomplete measurement of the diffusion editing window. We developed a tri-window pulse sequence to collect the full decaying information of porous media. In the first window, the inversion-recovery pulse sequence is applied for T1 encoding. In the second window, D and T2 are encoded by an adjustable continuous pulse field gradient and echo spacing (TE). In the last window, CPMG with the shortest TE is used to acquire diffusion-free relaxation information. We then proposed a joint inversion algorithm named "composite-data-processing" to obtain the 3D correlation map. The algorithm adopts the dimension reduction technique and the truncated singular value decomposition (TSVD) to speed up the inversion process and enhance the inversion stability. Numerical simulations show that good estimations of the inversion results are obtained at different signal to noise ratios (SNRs). Our results suggest that the novel pulse sequence and inversion algorithm of 3D NMR can be effectively applied to the exploration of unconventional resources.

Study on Joint Inversion Algorithm of Acoustic and Electromagnetic Data in Biomedical Imaging

In this paper, we study a joint inversion algorithm to reconstruct acoustic and electromagnetic (EM) data in biomedical imaging. This algorithm is based on contrast source inversion algorithm. We define contrast source and contrast functions in both acoustics and EMs, respectively, and apply two reciprocal regularization operators to link acoustic and EM inversion. By minimizing the cost function using the conjugate gradient method, we can simultaneously reconstruct compressibility, attenuation, permittivity, and conductivity of different human tissues. Numerical experiments show that acoustic and EM inversion can compensate each other and achieve a better reconstruction than individual inversion. Besides, the joint inversion method is particularly sensitive to compressibility and conductivity and thus may be used to monitor air and water contents in human thorax as well as other applications in biomedicine.

Joint Inversion of Seismic and Audio Magnetotelluric Data with Structural Constraint for Metallic Deposit

Audio magnetotelluric (AMT) and seismic methods are widely used to detect metallic deposits. However, each geophysical method only provides partial information of the underground target. Besides, individual methods have inherent limitations and ambiguity which leads to non-uniqueness when solving the inverse problem. To obtain a more robust and consistent ore deposit model, it is best to integrate different geophysical methods and data types. Towards this effort, we propose a joint inversion algorithm using cross-gradient constraint to build a connection between seismic and AMT data, and simultaneously invert for a resistivity and P-wave velocity model. Compared with separate AMT Gauss–Newton inversion and seismic Full waveform inversion (FWI) method, we can get more detailed and robust inversion results. In addition, frequency domain FWI with the Limited-Memory-Broyden–Fletcher–Goldfarb–Shanno (L-BFGS) algorithm provides an effective way to reduce computer memory usage and improve convergence speed. This joint inversion algorithm has been tested using simple synthetic models with two cross targets. The results obtained with separate inversions were compared with those obtained with joint inversion. Then, we applied the algorithm to geophysical models of the Jinchuan sulfide deposit. The AMT results obtained with joint inversion of seismic data were better than those obtained with separate AMT inversion. The joint inversion approach appears more robust than the traditional separate FWI inversion and it is recommended that the proposed algorithm be considered in future projects of real field data.

MOPSO: a new computing algorithm for joint inversion of Rayleigh wave dispersion curve and refraction traveltimes

Adequate estimation of shear-wave (VS) and P-wave (VP) velocity profiles is one of the significant objectives in the course of seismic surveys; however, the main problem in obtaining VS and VP is the non-uniqueness of surface waves and refracted seismic inversion results. Moreover, the hidden-layer problem often exists in the case of the refraction seismic method. The main purpose of this study is to cope with the above problems and reconstruct subsurface structures by joint inversion of Rayleigh wave dispersion curve and refraction traveltimes. The proposed joint inversion is based on a multi-objective particle swarm optimisation (MOPSO) strategy as a new tool for joint inversion of seismic datasets. The Pareto front concept was applied in the proposed joint inversion scheme. Using the Pareto front, the presented inversion algorithm provided a useful tool to evaluate the results (i.e. the number of layers, thicknesses or Poisson ratio values for the estimated models). The proposed algorithm was tested on two synthetic datasets and also on an experimental dataset. Furthermore, the joint inversion results were compared with the results of individual datasets inverted using PSO inversion algorithm. To verify the applicability of the proposed method, it was applied at a sample site located in Tabriz city, north-western Iran. For a real dataset, the refraction microtremor (ReMi) was used to obtain Rayleigh wave dispersion curves. Moreover, sourced from a vertical-incident seismic source (sledgehammer), seismic refraction data were recorded via vertical component geophones. The results showed that the proposed joint inversion technique can considerably reduce uncertainties of the inverted models.

Two-dimensional joint inversion of Magnetotelluric and local earthquake data: Discussion on the contribution to the solution of deep subsurface structures

Joint inversion of data sets collected by using several geophysical exploration methods has gained importance and associated algorithms have been developed. To explore the deep subsurface structures, Magnetotelluric and local earthquake tomography algorithms are generally used individually. Due to the usage of natural resources in both methods, it is not possible to increase data quality and resolution of model parameters. For this reason, the solution of the deep structures with the individual usage of the methods cannot be fully attained. In this paper, we firstly focused on the effects of both Magnetotelluric and local earthquake data sets on the solution of deep structures and discussed the results on the basis of the resolving power of the methods. The presence of deep-focus seismic sources increase the resolution of deep structures. Moreover, conductivity distribution of relatively shallow structures can be solved with high resolution by using MT algorithm. Therefore, we developed a new joint inversion algorithm based on the cross gradient function in order to jointly invert Magnetotelluric and local earthquake data sets. In the study, we added a new regularization parameter into the second term of the parameter correction vector of Gallardo and Meju (2003). The new regularization parameter is enhancing the stability of the algorithm and controls the contribution of the cross gradient term in the solution. The results show that even in cases where resistivity and velocity boundaries are different, both methods influence each other positively. In addition, the region of common structural boundaries of the models are clearly mapped compared with original models. Furthermore, deep structures are identified satisfactorily even with using the minimum number of seismic sources. In this paper, in order to understand the future studies, we discussed joint inversion of Magnetotelluric and local earthquake data sets only in two-dimensional space. In the light of these results and by means of the acceleration on the three-dimensional modelling and inversion algorithms, it is thought that it may be easier to identify underground structures with high resolution.

Joint Inversion of Gravity and Magnetotelluric Data for the Depth-to-Basement Estimation

It is well known that both gravity and magnetolluric (MT) methods can be used for the depth-to-basement estimation due to the density and conductivity contrast between the sedimentary basin and the underlaid basement rocks. In this case, the primary targets for both methods are the interface between the basement and sedimenary rocks as well as the physical properties of the rocks (density and conductivity). The solution of this inverse problem is typically nonunique and unstable, especially for gravity inversion. In order to overcome this difficulty and provide a more robust solution, we have developed a method of joint inversion to recover both the depth to the basement and the physical properties of the sediments and basement using gravity and MT data simultaneously. The joint inversion algorithm is based on the regularized conjugate gradient method. To speed up the inversion, we use an effective forward modeling method based on the surface Cauchy-type integrals for the gravity field and the surface integral equation representations for the MT field, respectively. We demonstrate the effectiveness of the developed method using several realistic model studies.

Joint inversion of induction and galvanic logging data in axisymmetric geological models

We have developed a computational algorithm for joint inversion of array induction and galvanic data in 2D models. It is based on a finite-difference solver and nonlinear minimization and is designed to develop realistic geoelectric models of complex fluid-saturated formations. The algorithm is tested and verified on noisy synthetic induction and galvanic data. The obtained 2D inversion results are compared with those corresponding to the traditional 1D radial approach. The developed algorithm was used to conduct joint 2D inversion of VEMKZ and BKZ logs from the Fedorovskoe and Vostochno-Surgutskoe oil fields in the E–W striking Ob’ River area.

2D joint inversion of CSAMT and magnetic data based on cross-gradient theory

A two-dimensional forward and backward algorithm for the controlled-source audio-frequency magnetotelluric (CSAMT) method is developed to invert data in the entire region (near, transition, and far) and deal with the effects of artificial sources. First, a regularization factor is introduced in the 2D magnetic inversion, and the magnetic susceptibility is updated in logarithmic form so that the inversion magnetic susceptibility is always positive. Second, the joint inversion of the CSAMT and magnetic methods is completed with the introduction of the cross gradient. By searching for the weight of the cross-gradient term in the objective function, the mutual influence between two different physical properties at different locations are avoided. Model tests show that the joint inversion based on cross-gradient theory offers better results than the single-method inversion. The 2D forward and inverse algorithm for CSAMT with source can effectively deal with artificial sources and ensures the reliability of the final joint inversion algorithm.

2D data-space cross-gradient joint inversion of MT, gravity and magnetic data

We have developed a data-space multiple cross-gradient joint inversion algorithm, and validated it through synthetic tests and applied it to magnetotelluric (MT), gravity and magnetic datasets acquired along a 95km profile in Benxi-Ji'an area of northeastern China. To begin, we discuss a generalized cross-gradient joint inversion for multiple datasets and model parameters sets, and formulate it in data space. The Lagrange multiplier required for the structural coupling in the data-space method is determined using an iterative solver to avoid calculation of the inverse matrix in solving the large system of equations. Next, using model-space and data-space methods, we inverted the synthetic data and field data. Based on our result, the joint inversion in data-space not only delineates geological bodies more clearly than the separate inversion, but also yields nearly equal results with the one in model-space while consuming much less memory.

3-D cross-gradient joint inversion of seismic refraction and DC resistivity data

We present a 3-D cross-gradient joint inversion algorithm for seismic refraction and DC resistivity data. The structural similarity between seismic slowness and resistivity models is enforced by a cross-gradient term in the objective function that also includes misfit and regularization terms. A limited memory quasi-Newton approach is used to perform the optimization of the objective function. To validate the proposed methodology and its implementation, tests were performed on a typical archaeological geophysical synthetic model. The results show that the inversion model and physical parameters estimated by our joint inversion method are more consistent with the true model than those from single inversion algorithm. Moreover, our approach appears to be more robust in conditions of noise. Finally, the 3-D cross-gradient joint inversion algorithm was applied to the field data from Lin_an ancient city site in Hangzhou of China. The 3-D cross-gradient joint inversion models are consistent with the archaeological excavation results of the ancient city wall remains. However, by single inversion, seismic slowness model does not show the anomaly of city wall remains and resistivity model does not fit well with the archaeological excavation results. Through these comparisons, we conclude that the proposed algorithm can be used to jointly invert 3-D seismic refraction and DC resistivity data to reduce the uncertainty brought by single inversion scheme.

Two dimensional joint inversion of direct current resistivity, radio-magnetotelluric and seismic refraction data: An application from Bafra Plain, Turkey

Direct current resistivity, radio-magnetotelluric and seismic refraction methods are widely used in the identification of near surface structures with collected data generally being interpreted separately. In recent decades, the use of joint inversion algorithms in geosciences has become widespread to identify near surface structures. However, there is no developed joint inversion algorithm using direct current resistivity, radio-magnetotelluric and seismic refraction methods. In this study, we developed a new two-dimensional joint inversion algorithm for direct current resistivity, radio-magnetotelluric and seismic refraction data based on a cross gradient approach. In addition, we proposed a new data weighting matrix to stabilize the convergence behavior of the joint inversion algorithms. We used synthetic data to show the advantage of the algorithm. The developed joint inversion algorithm found resistivity and velocity models that are better than the individual inversion of each data set. We also tested an algorithm with the field data collected in the Bafra Plain (Samsun, Turkey) to investigate saltwater intrusion. In comparing the field data inversion results with the sounding log, it can be seen that the developed joint inversion algorithm with the proposed data weighting matrix recovered the resistivity and velocity model better than the individual inversion and classical joint inversion of each data set. Our results showed that a more unique hydrogeological scenario might be obtained, especially in highly conductive media, with the joint usage of these methods.

Mapping soil salinity and a fresh-water intrusion in three-dimensions using a quasi-3d joint-inversion of DUALEM-421S and EM34 data

In order to understand the drivers of topsoil salinization, the distribution and movement of salt in accordance with groundwater need mapping. In this study, we described a method to map the distribution of soil salinity, as measured by the electrical conductivity of a saturated soil-paste extract (ECe), and in 3-dimensions around a water storage reservoir in an irrigated field near Bourke, New South Wales, Australia. A quasi-3d electromagnetic conductivity image (EMCI) or model of the true electrical conductivity (σ) was developed using 133 apparent electrical conductivity (ECa) measurements collected on a 50m grid and using various coil arrays of DUALEM-421S and EM34 instruments. For the DUALEM-421S we considered ECa in horizontal coplanar (i.e., 1mPcon, 2mPcon and 4mPcon) and vertical coplanar (i.e., 1mHcon, 2mHcon and 4mHcon) arrays. For the EM34, three measurements in the horizontal mode (i.e., EM34-10H, EM34-20H and EM34-40H) were considered. We estimated σ using a quasi-3d joint-inversion algorithm (EM4Soil). The best correlation (R2=0.92) between σ and measured soil ECe was identified when a forward modelling (FS), inversion algorithm (S2) and damping factor (λ=0.2) were used and using both DUALEM-421 and EM34 data; but not including the 4m coil arrays of the DUALEM-421S. A linear regression calibration model was used to predict ECe in 3-dimensions beneath the study field. The predicted ECe was consistent with previous studies and revealed the distribution of ECe and helped to infer a freshwater intrusion from a water storage reservoir at depth and as a function of its proximity to near-surface prior stream channels and buried paleochannels. It was concluded that this method can be applied elsewhere to map the soil salinity and water movement and provide guidance for improved land management.

Joint inversion of gravity and traveltime data using a level-set-based structural parameterization

We have developed a new level-set-based structural parameterization for joint inversion of gravity and traveltime data, so that density contrast and seismic slowness are simultaneously recovered in the inverse problem. Because density contrast and slowness are different model parameters of the same survey domain, we assume that they are similar in structure in terms of how each property changes and where the interface is located, so that we are able to use a level-set function to parameterize the common interface shared by these two model parameters. The level-set parameterization makes it easy to maintain the structural similarity between the two geophysical properties. The inversion of gravity and traveltime data is carried out by minimizing a joint data-fitting function with respect to density contrast, slowness, as well as the level-set function. An adjoint state method is used to compute the traveltime gradient efficiently. We have tested our algorithm on various synthetic examples, including a 2D ovoid model and the 2D SEG/EAGE salt model. The results show that the joint-inversion algorithm effectively improves recovery of subsurface features. To the best of our knowledge, this is the first time that the level-set method was used to structurally link density contrast and slowness distribution systematically.

Reservoir property mapping and monitoring from joint inversion of time-lapse seismic, electromagnetic, and production data

We have developed a deterministic multiphysics joint inversion approach integrating seismic, electromagnetic (EM), and production data to map relevant reservoir properties, such as permeability and porosity, and the time evolution of the flooding front movement, i.e., saturation changes with time. These measurements are complementary in terms of their sensitivity to individual reservoir properties and their coverage of reservoir volumes. As a consequence, integration reduces ambiguities in the interpretation. In the workflow, a reservoir model is first built based on prior information. The production data are simulated by evolving the model in time based on the known well-control strategy. Simultaneously, the temporal and spatial distribution of fluid properties, such as saturation, salt concentration, density, and pressure are also obtained from the forward modeling. These properties, together with in situ rock properties, are transformed to formation resistivity and elastic properties using prescribed petrophysical relationships, such as Archie’s law and effective medium rock-physics models. From the transformation results, synthetic EM and full-waveform seismic data can be subsequently simulated. A Gauss-Newton optimization scheme is used to iteratively update the reservoir permeability and porosity fields until the mismatch between the synthetic data and the observed data becomes less than a predefined threshold. This inverse problem is usually highly underdetermined; hence, it is necessary to bring in prior information to further constrain the inversion. Different regularization approaches are investigated to facilitate incorporation of prior information into the joint inversion algorithm.