Three‐dimensional potential field data inversion with L0 quasinorm sparse constraints

The quantitative explanation of the potential field data of three‐dimensional geological structures remains one of the most challenging issues in modern geophysical inversion. Obtaining a stable solution that can simultaneously resolve complicated geological structures is a critical inverse problem in the geophysics field. I have developed a new method for determining a three‐dimensional petrophysical property distribution, which produces a corresponding potential field anomaly. In contrast with the tradition inverse algorithm, my inversion method proposes a new model norm, which incorporates two important weighting functions. One is the L0 quasi norm (enforcing sparse constraints), and the other is depth‐weighting that counteracts the influence of source depth on the resulting potential field data of the solution. Sparseness constraints are imposed by using the L0 quasinorm on model parameters. To solve the representation problem, an L0 quasinorm minimisation model with different smooth approximations is proposed. Hence, the data space (N) method, which is much smaller than model space (M), combined with the gradient‐projected method, and the model space, combined with the modified Newton method for L0 quasinorm sparse constraints, leads to a computationally efficient method by using an N × N system versus an M × M one because N ? M. Tests on synthetic data and real datasets demonstrate the stability and validity of the L0 quasinorm spare norms inversion method. With the aim of obtaining the blocky results, the inversion method with the L0 quasinorm sparse constraints method performs better than the traditional L2 norm (standard Tikhonov regularisation). It can obtain the focus and sparse results easily. Then, the Bouguer anomaly survey data of the salt dome, offshore Louisiana, is considered as a real case study. The real inversion result shows that the inclusion the L0 quasinorm sparse constraints leads to a simpler and better resolved solution, and the density distribution is obtained in this area to reveal its geological structure. These results confirm the validity of the L0 quasinorm sparse constraints method and indicate its application for other potential field data inversions and the exploration of geological structures.     

Inversion of controlled‐source electromagnetic data using a model‐based approach

A two‐and‐half dimensional model‐based inversion algorithm for the reconstruction of geometry and conductivity of unknown regions using marine controlled‐source electromagnetic (CSEM) data is presented. In the model‐based inversion, the inversion domain is described by the so‐called regional conductivity model and both geometry and material parameters associated with this model are reconstructed in the inversion process. This method has the advantage of using a priori information such as the background conductivity distribution, structural information extracted from seismic and/or gravity measurements, and/or inversion results a priori derived from a pixel‐based inversion method. By incorporating this a priori information, the number of unknown parameters to be retrieved becomes significantly reduced. The inversion method is the regularized Gauss‐Newton minimization scheme. The robustness of the inversion is enhanced by adopting nonlinear constraints and applying a quadratic line search algorithm to the optimization process. We also introduce the adjoint formulation to calculate the Jacobian matrix with respect to the geometrical parameters. The model‐based inversion method is validated by using several numerical examples including the inversion of the Troll field data. These results show that the model‐based inversion method can quantitatively reconstruct the shapes and conductivities of reservoirs.     

Multi‐method virtual electromagnetic experiments for developing suitable monitoring designs: A fictitious CO2 sequestration scenario in Northern Germany

We present a numerical study for 3D time‐lapse electromagnetic monitoring of a fictitious CO₂ sequestration using the geometry of a real geological site and a suite of suitable electromagnetic methods with different source/receiver configurations and different sensitivity patterns. All available geological information is processed and directly implemented into the computational domain, which is discretized by unstructured tetrahedral grids. We thus demonstrate the performance capability of our numerical simulation techniques. The scenario considers a CO₂ injection in approximately 1100 m depth. The expected changes in conductivity were inferred from preceding laboratory measurements. A resistive anomaly is caused within the conductive brines of the undisturbed reservoir horizon. The resistive nature of the anomaly is enhanced by the CO₂ dissolution regime, which prevails in the high‐salinity environment. Due to the physicochemical properties of CO₂, the affected portion of the subsurface is laterally widespread but very thin. We combine controlled‐source electromagnetics, borehole transient electromagnetics, and the direct‐current resistivity method to perform a virtual experiment with the aim of scrutinizing a set of source/receiver configurations with respect to coverage, resolution, and detectability of the anomalous CO₂ plume prior to the field survey. Our simulation studies are carried out using the 3D codes developed in our working group. They are all based on linear and higher order Lagrange and Nédélec finite‐element formulations on unstructured grids, providing the necessary flexibility with respect to the complex real‐world geometry. We provide different strategies for addressing the accuracy of numerical simulations in the case of arbitrary structures. The presented computations demonstrate the expected great advantage of positioning transmitters or receivers close to the target. For direct‐current geoelectrics, 50% change in electric potential may be detected even at the Earth's surface. Monitoring with inductive methods is also promising. For a well‐positioned surface transmitter, more than 10% difference in the vertical electric field is predicted for a receiver located 200 m above the target. Our borehole transient electromagnetics results demonstrate that traditional transient electromagnetics with a vertical magnetic dipole source is not well suited for monitoring a thin horizontal resistive target. This is due to the mainly horizontal current system, which is induced by a vertical magnetic dipole.     

Inversion of elongated magnetic anomalies using magnitude transforms

The calculable magnitudes of the anomalous magnetic field from simple 2D sources and their gradients and Laplacians appear as ratios that can be synthesized in functional forms, corresponding to the different source shapes. Field components and first‐order derivatives are involved in the inversion procedures presented. The structural index and source depth are estimated independently of each other. The applied functions allow magnetic profiles and magnetic maps to be shape‐ and depth‐converted with immediate imaging of the inversion results. The contours of these functions outline elongated loops around the 2.5D anomaly axis on magnetic maps. The width of the loops reflects the depth and structural index N of the source in the scale units of the inverted map. Model and field tests illustrate the effectiveness of this approach for fast automatic inversion of large sets of magnetic data for depth, shape, length and location of simple sources.     

Controlled‐source electromagnetic monitoring of reservoir oil saturation using a novel borehole‐to‐surface configuration

To advance and optimize secondary and tertiary oil recovery techniques, it is essential to know the areal propagation and distribution of the injected fluids in the subsurface. We investigate the applicability of controlled‐source electromagnetic methods to monitor fluid movements in a German oilfield (Bockstedt, onshore Northwest Germany) as injected brines (highly saline formation water) have much lower electrical resistivity than the oil within the reservoir. The main focus of this study is on controlled‐source electromagnetic simulations to test the sensitivity of various source–receiver configurations. The background model for the simulations is based on two‐dimensional inversion of magnetotelluric data gathered across the oil field and calibrated with resistivity logs. Three‐dimensional modelling results suggest that controlled‐source electromagnetic methods are sensitive to resistivity changes at reservoir depths, but the effect is difficult to resolve with surface measurements only. Resolution increases significantly if sensors or transmitters can be placed in observation wells closer to the reservoir. In particular, observation of the vertical electric field component in shallow boreholes and/or use of source configurations consisting of combinations of vertical and horizontal dipoles are promising. Preliminary results from a borehole‐to‐surface controlled‐source electromagnetic field survey carried out in spring 2014 are in good agreement with the modelling studies.     

Parsimonious wave‐equation travel‐time inversion for refraction waves

We present a parsimonious wave‐equation travel‐time inversion technique for refraction waves. A dense virtual refraction dataset can be generated from just two reciprocal shot gathers for the sources at the endpoints of the survey line, with N geophones evenly deployed along the line. These two reciprocal shots contain approximately 2N refraction travel times, which can be spawned into refraction travel times by an interferometric transformation. Then, these virtual refraction travel times are used with a source wavelet to create N virtual refraction shot gathers, which are the input data for wave‐equation travel‐time inversion. Numerical results show that the parsimonious wave‐equation travel‐time tomogram has about the same accuracy as the tomogram computed by standard wave‐equation travel‐time inversion. The most significant benefit is that a reciprocal survey is far less time consuming than the standard refraction survey where a source is excited at each geophone location.     

地面可控源频率测深三维非线性共轭梯度反演

讨论了地面可控源电磁勘探三维非线性共轭梯度反演的可行性以及反演过程中考虑场源的必要性.反演采用非线性共轭梯度反演方法.反演过程中,模型响应利用交错网格有限差分技术计算.反演数据采用与发射源平行的电场x分量E_x.利用层状导电模型作为背景,设计了两个理论模型进行数值试验:第一个模型中包含两个电阻率异常,以检验反演的有效性;第二个模型中,在测区外设置了一个低阻异常,以考察源的信息在反演中的作用.两个模型的反演分别从层状背景模型开始,迭代120次后终止.数值试验结果表明,(1)非线性共轭梯度反演所获得的电阻率分布和理论模型吻合较好;(2)非线性共轭梯度算法收敛速度较慢,需要较多的迭代次数完成反演;(3)对于可控源频率电磁勘探,必须考虑源位置信息.因此,本文采用考虑场源信息的地面可控源非线性共轭梯度反演方法能完成真正意义上的可控源频率电磁测深数据的反演.     

Inversion of potential field data using the structural index as weighting function rate decay

Nonparametric inverse methods provide a general framework for solving potential‐field problems. The use of weighted norms leads to a general regularization problem of Tikhonov form. We present an alternative procedure to estimate the source susceptibility distribution from potential field measurements exploiting inversion methods by means of a flexible depth‐weighting function in the Tikhonov formulation. Our approach improves the formulation proposed by Li and Oldenburg (1996, 1998) , differing significantly in the definition of the depth‐weighting function. In our formalism the depth weighting function is associated not to the field decay of a single block (which can be representative of just a part of the source) but to the field decay of the whole source, thus implying that the data inversion is independent on the cell shape. So, in our procedure, the depth‐weighting function is not given with a fixed exponent but with the structural index N of the source as the exponent. Differently than previous methods, our choice gives a substantial objectivity to the form of the depth‐weighting function and to the consequent solutions. The allowed values for the exponent of the depth‐weighting function depend on the range of N for sources: 0 ≤N≤ 3 (magnetic case). The analysis regarding the cases of simple sources such as dipoles, dipole lines, dykes or contacts, validate our hypothesis. The study of a complex synthetic case also proves that the depth‐weighting decay cannot be necessarily assumed as equal to 3. Moreover it should not be kept constant for multi‐source models but should instead depend on the structural indices of the different sources. In this way we are able to successfully invert the magnetic data of the Vulture area, Southern Italy. An original aspect of the proposed inversion scheme is that it brings an explicit link between two widely used types of interpretation methods, namely those assuming homogeneous fields, such as Euler deconvolution or depth from extreme points transformation and the inversion under the Tikhonov‐form including a depth‐weighting function. The availability of further constraints, from drillings or known geology, will definitely improve the quality of the solution.     

Estimating magnetic dike parameters using a non‐linear constrained inversion technique: an example from the Särna area,west central Sweden

In this paper, we describe a non‐linear constrained inversion technique for 2D interpretation of high resolution magnetic field data along flight lines using a simple dike model. We first estimate the strike direction of a quasi 2D structure based on the eigenvector corresponding to the minimum eigenvalue of the pseudogravity gradient tensor derived from gridded, low‐pass filtered magnetic field anomalies, assuming that the magnetization direction is known. Then the measured magnetic field can be transformed into the strike coordinate system and all magnetic dike parameters – horizontal position, depth to the top, dip angle, width and susceptibility contrast – can be estimated by non‐linear least squares inversion of the high resolution magnetic field data along the flight lines. We use the Levenberg‐Marquardt algorithm together with the trust‐region‐reflective method enabling users to define inequality constraints on model parameters such that the estimated parameters are always in a trust region. Assuming that the maximum of the calculated g_zz (vertical gradient of the pseudogravity field) is approximately located above the causative body, data points enclosed by a window, along the profile, centred at the maximum of g_zz are used in the inversion scheme for estimating the dike parameters. The size of the window is increased until it exceeds a predefined limit. Then the solution corresponding to the minimum data fit error is chosen as the most reliable one. Using synthetic data we study the effect of random noise and interfering sources on the estimated models and we apply our method to a new aeromagnetic data set from the Särna area, west central Sweden including constraints from laboratory measurements on rock samples from the area.     

An alternative approach in establishing relation between vertical and horizontal gradients of 2D potential field

The relation in which the vertical and horizontal gradients of potential field data measured along a profile across a two‐dimensional source are a Hilbert transform pair is re‐established using complex domain mathematics. In addition, a relation between the measured field and its vertical gradient in terms of a closed‐form formula is also established. The formula is based on hypersingular or Hadamard's finite‐part integral. To estimate the vertical gradient directly from the field data, Linz's algorithm of computing Hadamard's finite‐part integral is implemented. Numerical experiments are conducted on synthetically generated total magnetic intensity data with a mild level of noise contamination. A model of a magnetically polarised vertical thin sheet buried at a finite depth within a non‐magnetic half‐space was considered in generating the synthetic response. The results from numerical experiments on the mildly noise‐contaminated synthetic response are compared with those from using classical Fourier and robust regularised Hilbert transform‐based techniques.     

Fractal properties of medium and seismoelectric phenomena

Electrokinetic phenomena in a water-porous medium with a fractal structure above percolation threshold are theoretically investigated. Fracture zone with space-variable porosity is considered as a model of an earthquake hypocenter zone in which the electrokinetic current results from fluid filtration in a fractal pore network. A critical exponent of the streaming potential coefficient is found to depend on both the transport critical exponent and correlation length critical exponent. In this model, logarithmic dependence of electric field amplitude E on the earthquake magnitude M is derived which is compatible with the one observed by the VAN group. Without fractal properties, this form of dependence contradicts the empirical data. The electromagnetic field far from the hypocenter is calculated, which leads to the prediction of weak magnetic field variations. To explain the observed amplitude of VAN's Seismic Electric Signals (SES), the electric source must be at a distance of about 10 km from the registration point if the medium is homogeneous. Therefore, some conductive channel(s) are needed to explain the long distance selective SES transmission.     

Multi‐dimensional analyses of the SEAM controlled source electromagnetic data—the story of a blind test of interpretation workflows

Using a subset of the SEG Advanced Modeling Program Phase I controlled‐source electromagnetic data, we apply our standard controlled‐source electromagnetic interpretation workflows to delineate a simulated hydrocarbon reservoir. Experience learned from characterizing such a complicated model offers us an opportunity to refine our workflows to achieve better interpretation quality. The exercise proceeded in a blind test style, where the interpreting geophysicists did not know the true resistivity model until the end of the project. Rather, the interpreters were provided a traditional controlled‐source electromagnetic data package, including electric field measurements, interpreted seismic horizons, and well log data. Based on petrophysical analysis, a background resistivity model was established first. Then, the interpreters started with feasibility studies to establish the recoverability of the prospect and carefully stepped through 1D, 2.5D, and 3D inversions with seismic and well log data integrated at each stage. A high‐resistivity zone is identified with 1D analysis and further characterized with 2.5D inversions. Its lateral distribution is confirmed with a 3D anisotropic inversion. The importance of integrating all available geophysical and petrophysical data to derive more accurate interpretation is demonstrated.     

Transient electromagnetic surveys with unimodal transverse magnetic field: ideas and results

The theory behind transient electromagnetic surveys can be well described in terms of transverse magnetic and transverse electric modes. Soundings using transverse magnetic and transverse electric modes require different source configurations. In this study, we consider an alternating transverse magnetic field excitation by a circular electric dipole. The circular electric dipole transmitter is a horizontal analogue of the vertical electric dipole. Offshore surveys using circular electric dipole might represent an alternative to the conventional marine controlled‐source electromagnetic method at shallow sea and/or for exploring relatively small targets. Field acquisition is carried out by recording either electric or magnetic responses. Electric responses bear information on the 1D structure of a layered earth and successfully resolve high‐resistivity targets in marine surveys. Land‐based circular electric dipole soundings are affected by induced polarisation. On the contrary, magnetic responses are absent on the surface of a 1D earth, and as a result, they are very sensitive to any and even very small 3D conductivity perturbations. In addition, they are sensitive to induced polarisation or some other polarisation effects in the subsurface. At present, circular electric dipole transmitters and magnetic receivers are successfully used in on‐land mineral and petroleum exploration.     

Full waveform inversion of SH‐ and Love‐wave data in near‐surface prospecting

We develop a two‐dimensional full waveform inversion approach for the simultaneous determination of S‐wave velocity and density models from SH ‐ and Love‐wave data. We illustrate the advantages of the SH/Love full waveform inversion with a simple synthetic example and demonstrate the method's applicability to a near‐surface dataset, recorded in the village ?achtice in Northwestern Slovakia. Goal of the survey was to map remains of historical building foundations in a highly heterogeneous subsurface. The seismic survey comprises two parallel SH‐profiles with maximum offsets of 24 m and covers a frequency range from 5 Hz to 80 Hz with high signal‐to‐noise ratio well suited for full waveform inversion. Using the Wiechert–Herglotz method, we determined a one‐dimensional gradient velocity model as a starting model for full waveform inversion. The two‐dimensional waveform inversion approach uses the global correlation norm as objective function in combination with a sequential inversion of low‐pass filtered field data. This mitigates the non‐linearity of the multi‐parameter inverse problem. Test computations show that the influence of visco‐elastic effects on the waveform inversion result is rather small. Further tests using a mono‐parameter shear modulus inversion reveal that the inversion of the density model has no significant impact on the final data fit. The final full waveform inversion S‐wave velocity and density models show a prominent low‐velocity weathering layer. Below this layer, the subsurface is highly heterogeneous. Minimum anomaly sizes correspond to approximately half of the dominant Love‐wavelength. The results demonstrate the ability of two‐dimensional SH waveform inversion to image shallow small‐scale soil structure. However, they do not show any evidence of foundation walls.     

Analysing two‐dimensional effects in central loop transient electromagnetic sounding data using a semi‐synthetic tipper approach

We present a simple and feasible approach to analyse and identify two‐dimensional effects in central loop transient electromagnetic sounding data and the correspondingly derived quasi two‐dimensional conductivity models. The proposed strategy is particularly useful in minimising interpretation errors. It is based on the calculation of a semi‐synthetic transient electromagnetic tipper at each sounding and for each observational transient time point. The semi‐synthetic transient electromagnetic tipper is derived from the measured vertical component of the induced voltage and the synthetically calculated horizontal component. The approach is computationally inexpensive and involves one two‐dimensional forward calculation of an obtained quasi two‐dimensional conductivity section. Based on a synthetic example, we demonstrate that the transient electromagnetic tipper approach is applicable in identifying which transient data points and which corresponding zones in a derived quasi two‐dimensional subsurface model are affected by two‐dimensional inhomogeneities. The one‐dimensional inversion of such data leads to false models. An application of the semi‐synthetic transient electromagnetic tipper to field data from the Azraq basin in Jordan reveals that, in total, eight of 80 investigated soundings are affected by two‐dimensional structures although the field data can be fitted optimally using one‐dimensional inversion techniques. The largest semi‐synthetic tipper response occurs in a 300 m‐wide region around a strong lateral resistivity contrast. The approach is useful for analysing structural features in derived quasi two‐dimensional sections and for qualitatively investigating how these features affect the transient response. To avoid misinterpretation, these identified zones corresponding to large tipper values are excluded from the interpretation of a quasi two‐dimensional conductivity model. Based on the semi‐synthetic study, we also demonstrate that a quantitative interpretation of the horizontal voltage response (e.g. by inversion) is usually not feasible as it requires the exact sensor position to be known. Although a tipper derived purely from field data is useful as a qualitative tool for identifying two‐dimensional distortion effects, it is only feasible if the sensor setup is sufficiently accurate. Our proposed semi‐synthetic transient electromagnetic tipper approach is particularly feasible as an a posteriori approach if no horizontal components are recorded or if the sensor setup in the field is not sufficiently accurate.     

CSAMT单分量数据解释方法

可控源音频大地电磁法(CSAMT)一直沿用大地电磁法(MT)的办法,通过计算电场分量与磁场分量的比值,求取卡尼亚视电阻率.而CSAMT场源已知,电场分量和磁场分量都与地下电阻率存在一定的关系,可以单独采用CSAMT电场分量或者磁场分量提取地下介质的视电阻率.本文通过分析电场分量与磁场分量的数据特性,提出利用CSAMT电场单分量数据进行视电阻率的计算,用改进的广义逆矩阵反演方法,使初始模型中的地电层数等于频道个数,克服了以往反演计算中层数较少的问题;实现全场区电场分量视电阻率曲线的拟合反演.同时对单分量视相位计算方法进行分析,结合山西大同地区积水采空区探测及数据解释结果,论证本文提出的单分量解释方法的有效性.     

Step-on versus step-off signals in time-domain controlled source electromagnetic methods using a grounded electric dipole

The time-domain controlled source electromagnetic method is a geophysical prospecting tool applied to image the subsurface resistivity distribution on land and in the marine environment. In its most general set-up, a square-wave current is fed into a grounded horizontal electric dipole, and several electric and magnetic field receivers at defined offsets to the imposed current measure the electromagnetic response of the Earth. In the marine environment, the application often uses only inline electric field receivers that, for a 50% duty-cycle current waveform, include both step-on and step-off signals. Here, forward and inverse 1D modelling is used to demonstrate limited sensitivity towards shallow resistive layers in the step-off electric field when transmitter and receivers are surrounded by conductive seawater. This observation is explained by a masking effect of the direct current signal that flows through the seawater and primarily affects step-off data. During a step-off measurement, this direct current is orders of magnitude larger than the inductive response at early and intermediate times, limiting the step-off sensitivity towards shallow resistive layers in the seafloor. Step-on data measure the resistive layer at times preceding the arrival of the direct current signal leading to higher sensitivity compared to step-off data. Such dichotomous behaviour between step-on and step-off data is less obvious in onshore experiments due to the lack of a strong overlying conductive zone and corresponding masking effect from direct current flow. Supported by synthetic 1D inversion studies, we conclude that time-domain controlled source electromagnetic measurements on land should apply both step-on and step-off data in a combined inversion approach to maximize signal-to-noise ratios and utilize the sensitivity characteristics of each signal. In an isotropic marine environment, step-off electric fields have inferior sensitivity towards shallow resistive layers compared to step-on data, resulting in an increase of non-uniqueness when interpreting step-off data in a single or combined inversion.     

Reconstructing the gravity gradient anomaly field from surveys with wide line spacing using equivalent source processing: an error analysis

When anomalous gravity gradient signals provide a large signal‐to‐noise ratio, airborne and marine surveys can be considered with wide line spacing. In these cases, spatial resolution and sampling requirements become the limiting factors for specifying the line spacing, rather than anomaly detectability. This situation is analysed by generating known signals from a geological model and then sub‐sampling them using a simulated airborne gravity gradient survey with a line spacing much wider than the characteristic anomaly size. The data are processed using an equivalent source inversion, which is used subsequently to predict and grid the field in‐between the survey lines by means of forward calculations. Spatial and spectral error analysis is used to quantify the accuracy and resolution of the processed data and the advantages of acquiring multiple gravity gradient components are demonstrated. With measurements of the full tensor along survey lines spaced at 4 × 4 km, it is shown that the vertical gravity gradient can be reconstructed accurately over a bandwidth of 2 km with spatial root‐mean square errors less than 30%. A real airborne full‐tensor gravity gradient survey is presented to confirm the synthetic analysis in a practical situation.     

3D magnetotelluric modelling including surface topography

An edge finite‐element method has been applied to compute magnetotelluric (MT) responses to three‐dimensional (3D) earth topography. The finite‐element algorithm uses a single edge shape function at each edge of hexahedral elements, guaranteeing the continuity of the tangential electric field while conserving the continuity of magnetic flux at boundaries. We solve the resulting system of equations using the biconjugate gradient method with a Jacobian preconditioner. The solution gives electric fields parallel to the slope of a surface relief that is often encountered in MT surveys. The algorithm is successfully verified by comparison with other numerical solutions for a 3D‐2 model for comparison of modelling methods for EM induction and a ridge model. We use a 3D trapezoidal‐hill model to investigate 3D topographic effects, which are caused mainly by galvanic effects, not only in the Zxy mode but also in the Zyx mode. If a 3D topography were approximated by a two‐dimensional topography therefore errors occurring in the transverse electric mode would be more serious than those in the transverse magnetic mode.