Fast resistivity/IP inversion using a low‐contrast approximation

Geophysics ◽  
1996 ◽  
Vol 61 (1) ◽  
pp. 169-179 ◽  
Author(s):  
Les P. Beard ◽  
Gerald W. Hohmann ◽  
Alan C. Tripp
Keyword(s):  
Inversion Algorithm ◽  
Forward Solution ◽  
Near Surface ◽  
Low Contrast ◽  
Volume Integral Equation ◽  
Strike Direction ◽  
Resistivity Model ◽  
Resistivity Inversion ◽  
Full Solution ◽  
End Corrections

By computing only the diagonal terms of the volume integral equation forward solution of the 3-D DC resistivity problem, we have achieved a fast forward solution accurate at low to moderate resistivity contrasts. The speed and accuracy of the solution make it practical for use in 2-D or 3-D inversion algorithms. The low‐contrast approximation is particularly well‐suited to the smooth nature of minimum structure inversion, since complete forward solutions may be computationally expensive. By using this approximate 3-D solution as the forward model in an inversion algorithm, and by constraining the resistivities and polarizabilities along any row of cells in the strike direction to be held constant, we effect a fast 2-D resistivity inversion that contains end corrections. Because the low‐contrast solution is inaccurate for cells near the electrodes, we employ a full solution to compute the response of the near‐surface when the near‐surface environment is substantially different from the host rock. This response is stored and used in the iterative resistivity inversion in conjunction with the approximate solution. Once an adequate estimated resistivity model has been found, derivatives from this model are used with Seigel’s formula to compute the inverse solution to the linear polarizability problem in a single iteration.

2-Dimensional soil and vadose-zone representation using an EM38 and EM34 and a laterally constrained inversion model

Soil Research ◽  
2009 ◽  
Vol 47 (8) ◽  
pp. 809 ◽  
Author(s):  
J. Triantafilis ◽  
F. A. Monteiro Santos
Keyword(s):  
Electrical Conductivity ◽  
Vadose Zone ◽  
Root Zone ◽  
Clay Content ◽  
Inversion Algorithm ◽  
Near Surface ◽  
Particle Size Fractions ◽  
Murray Darling Basin ◽  
Full Solution ◽  
Soil Particle Size Fractions

The network of prior streams and palaeochannels common across the Riverine Plains of the Murray–Darling Basin act as conduits for the redistribution of water and soluble salts beneath the root-zone. To improve scientific understanding of these hydrological processes there is the need to better represent and map the connectivity and spatial extent of these physiographic and stratigraphic features. Groundbased electromagnetic (EM) instruments, which measure bulk soil electrical conductivity (σa), have been used widely to map their areal distribution across the landscape. However, methods to resolve their location with depth have rarely been attempted. In this paper we employ a 1-D inversion algorithm with 2-D smoothness constraints to predict the true electrical conductivity (σ) at discrete depth increments using EM data. The EM data we use include the root-zone measuring EM38 and the deeper sensing EM34. We collected EM38 data in the vertical (EM38v) and horizontal (EM38h) dipole modes and EM34 data in the horizontal mode and coil spacing of 10, 20, and 40 m (respectively, EM34-10, EM34-20, and EM34-40). In order to compare and contrast the value of the various EM data we carried out multiple inversions using different combinations, which include: independent inversions of (i) EM38 (root-zone) and (ii) EM34 data (vadose-zone), and in combination using (iii) EM38v, EM38h, and EM34-10 (near-surface), and (iv) all 5 EM datasets (regolith) available. The general patterns of σ are shown to compare favourably with the known pedoderms, physiographic, and stratigraphic features and soil particle size fractions collected from calibration cores drilled across the lower Macquarie Valley study area. In general we find that the EM38 assists in resolving root-zone variability, specifically duplex soil profiles and physiographic features such as prior streams, while the use of the EM34 assists in resolving the stratigraphic nature of the vadose-zone and specifically the likely location of palaeochannels and subsurface anomalies that may indicate the location of good quality groundwater and/or clay aquitards. In this case, our potential to use σ to predict clay content is limited by the non-linearity of the cumulative functions. In order to improve on the non-linearity of our inversion we need to develop a full solution of the forward problem.

RESINVM3D: A 3D resistivity inversion package

Geophysics ◽  
2007 ◽  
Vol 72 (2) ◽  
pp. H1-H10 ◽  
Author(s):  
Adam Pidlisecky ◽  
Eldad Haber ◽  
Rosemary Knight
Keyword(s):  
Model Space ◽  
Gradient Algorithm ◽  
Computational Time ◽  
Preconditioned Conjugate Gradient ◽  
Inversion Algorithm ◽  
Forward Solution ◽  
Homogeneous Half Space ◽  
Finite Volume Discretization ◽  
Model Update ◽  
Resistivity Inversion

We have developed an open source 3D, MATLAB based, resistivity inversion package. The forward solution to the governing partial differential equation is efficiently computed using a second-order finite volume discretization coupled with a preconditioned, biconjugate, stabilized gradient algorithm. Using the analytical solution to a potential field in a homogeneous half space, we evaluate the accuracy of our numerical forward solution and, subsequently, develop a source correction factor that reduces forward modeling errors associated with boundary effects and source electrode singularities. For the inversion algorithm we have implemented an inexact Gauss-Newton solver, with the model update being calculated using a preconditioned conjugate gradient algorithm. The inversion uses a combination of zero and first order Tikhonov regularization. Two synthetic examples demonstrate the usefulness of this code. The first example considers a surface resistivity survey with 3813 measurements. The discretized model space contains 19,040 cells. For this example, the inversion package converges in approximately [Formula: see text] on a [Formula: see text] Pentium 4, with [Formula: see text] of RAM. The second example considers the case of borehole based data acquisition. For this example there were 4704 measurements and 13,200 model cells. The inversion for this example requires [Formula: see text] of computational time.

A hybrid regularization scheme for the inversion of magnetotelluric data from natural and controlled sources to layer and distortion parameters

Geophysics ◽  
2012 ◽  
Vol 77 (4) ◽  
pp. E301-E315 ◽  
Author(s):  
Thomas Kalscheuer ◽  
Juliane Hübert ◽  
Alexey Kuvshinov ◽  
Tobias Lochbühler ◽  
Laust B. Pedersen
Keyword(s):  
Transfer Functions ◽  
Homogeneous Distribution ◽  
Stable Configuration ◽  
Inversion Algorithm ◽  
Magnetotelluric Data ◽  
Data Set ◽  
Near Surface ◽  
Minimum Solution ◽  
Magnetic Transfer ◽  
Distortion Parameters

Magnetotelluric (MT), radiomagnetotelluric (RMT), and, in particular, controlled-source audiomagnetotelluric (CSAMT) data are often heavily distorted by near-surface inhomogeneities. We developed a novel scheme to invert MT, RMT, and CSAMT data in the form of scalar or tensorial impedances and vertical magnetic transfer functions simultaneously for layer resistivities and electric and magnetic galvanic distortion parameters. The inversion scheme uses smoothness constraints to regularize layer resistivities and either Marquardt-Levenberg damping or the minimum-solution length criterion to regularize distortion parameters. A depth of investigation range is estimated by comparing layered model sections derived from first- and second-order smoothness constraints. Synthetic examples demonstrate that earth models are reconstructed properly for distorted and undistorted tensorial CSAMT data. In the inversion of scalar CSAMT data, such as the determinant impedance or individual tensor elements, the reduced number of transfer functions inevitably leads to increased ambiguity for distortion parameters. As a consequence of this ambiguity for scalar data, distortion parameters often grow over the iterations to unrealistic absolute values when regularized with the Marquardt-Levenberg scheme. Essentially, compensating relationships between terms containing electric and/or magnetic distortion are used in this growth. In a regularization with the minimum solution length criterion, the distortion parameters converge into a stable configuration after several iterations and attain reasonable values. The inversion algorithm was applied to a CSAMT field data set collected along a profile over a tunnel construction site at Hallandsåsen, Sweden. To avoid erroneous inverse models from strong anthropogenic effects on the data, two scalar transfer functions (one scalar impedance and one scalar vertical magnetic transfer function) were selected for inversion. Compared with a regularization of distortion parameters with the Marquardt-Levenberg method, the minimum-solution length criterion yielded smaller absolute values of distortion parameters and a horizontally more homogeneous distribution of electrical conductivity.

Computer Application in Geosciences: Imaging of the Subsurface by Structural Coupled Inversion of Potential Field Data

2014 ◽  
Vol 644-650 ◽  
pp. 2670-2673
Author(s):  
Jun Wang ◽  
Xiao Hong Meng ◽  
Fang Li ◽  
Jun Jie Zhou
Keyword(s):  
Potential Field ◽  
Field Data ◽  
Gravity Data ◽  
Computer Application ◽  
Magnetic Data ◽  
Imaging Method ◽  
Inversion Algorithm ◽  
Constant Property ◽  
Near Surface ◽  
Potential Field Data

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.

2-D and 3-D resistivity image reconstruction using crosshole data

Geophysics ◽  
1992 ◽  
Vol 57 (10) ◽  
pp. 1270-1281 ◽  
Author(s):  
Hiromasa Shima
Keyword(s):  
Field Test ◽  
Numerical Models ◽  
Electrical Potential ◽  
Nonlinear Least Squares ◽  
Three Dimensions ◽  
Inversion Method ◽  
Good Resolution ◽  
Inversion Algorithm ◽  
Linear Inversion ◽  
Resistivity Model

Theoretical changes in the distribution of electrical potential near subsurface resistivity anomalies have been studied using two resistivity models. The results suggest that the greatest response from such anomalies can be observed with buried electrodes, and that the resistivity model of a volume between boreholes can be accurately reconstructed by using crosshole data. The distributive properties of crosshole electrical potential data obtained by the pole‐pole array method have also been examined using the calculated partial derivative of the observed apparent resistivity with respect to a small cell within a given volume. The results show that for optimum two‐dimensional (2-D) and three‐dimensional (3-D) target imaging, in‐line data and crossline data should be combined, and an area outside the zone of exploration should be included in the analysis. In this paper, the 2-D and 3-D resistivity images presented are reconstructed from crosshole data by the combination of two inversion algorithms. The first algorithm uses the alpha center method for forward modeling and reconstructs a resistivity model by a nonlinear least‐squares inversion. Alpha centers express a continuously varying resistivity model, and the distribution of the electrical potential from the model can be calculated quickly. An initial general model is determined by the resistivity backprojection technique (RBPT) prior to the first inversion step. The second process uses finite elements and a linear inversion algorithm to improve the resolution of the resistivity model created by the first step. Simple 2-D and 3-D numerical models are discussed to illustrate the inversion method used in processing. Data from several field studies are also presented to demonstrate the capabilities of using crosshole resistivity exploration techniques. The numerical experiments show that by using the combined reconstruction algorithm, thin conductive layers can be imaged with good resolution for 2-D and 3-D cases. The integration of finite‐element computations is shown to improve the image obtained by the alpha center inversion process for 3-D applications. The first field test uses horizontal galleries to evaluate complex 2-D features of a zinc mine. The second field test illustrates the use of three boreholes at a dam site to investigate base rock features and define the distribution of an altered zone in three dimensions.

Turning‐ray tomography

Geophysics ◽  
1995 ◽  
Vol 60 (6) ◽  
pp. 1917-1929 ◽  
Author(s):  
Joseph P. Stefani
Keyword(s):  
Velocity Structure ◽  
Initial Point ◽  
Real Data ◽  
Point Sources ◽  
Surface Velocity ◽  
Inversion Algorithm ◽  
Low Velocity ◽  
Linear Inversion ◽  
Near Surface ◽  
Velocity Inversion

Turning‐ray tomography is useful for estimating near‐surface velocity structure in areas where conventional refraction statics techniques fail because of poor data or lack of smooth refractor/velocity structure. This paper explores the accuracy and inherent smoothing of turning‐ray tomography in its capacity to estimate absolute near‐surface velocity and the statics times derived from these velocities, and the fidelity with which wavefields collapse to point diffractors when migrated through these estimated velocities. The method comprises nonlinear iterations of forward ray tracing through triangular cells linear in slowness squared, coupled with the LSQR linear inversion algorithm. It is applied to two synthetic finite‐ difference data sets of types that usually foil conventional refraction statics techniques. These models represent a complex hard‐rock overthrust structure with a low‐velocity zone and pinchouts, and a contemporaneous near‐shore marine trench filled with low‐ velocity unconsolidated deposits exhibiting no seismically apparent internal structure. In both cases velocities are estimated accurately to a depth of one‐ fifth the maximum offset, as are the associated statics times. Of equal importance, the velocities are sufficiently accurate to correctly focus synthetic wavefields back to their initial point sources, so migration/datuming applications can also use these velocities. The method is applied to a real data example from the Timbalier Trench in the Gulf of Mexico, which exhibits the same essential features as the marine trench synthetic model. The Timbalier velocity inversion is geologically reasonable and yields long and short wavelength statics that improve the CMP gathers and stack and that correctly align reflections to known well markers. Turning‐ray tomography estimates near‐surface velocities accurately enough for the three purposes of lithology interpretation, statics calculations, and wavefield focusing for shallow migration and datuming.

Tropospheric NO2 and HCHO derived from dual-scan MAX-DOAS measurements in Uccle (Belgium) and application to S5P/TROPOMI validation

2020 ◽  
Author(s):  
Ermioni Dimitropoulou ◽  
Francois Hendrick ◽  
Martine M. Friedrich ◽  
Gaia Pinardi ◽  
Frederik Tack ◽  
...  
Keyword(s):  
Vertical Profile ◽  
A Priori ◽  
Elevation Angle ◽  
Optical Absorption Spectroscopy ◽  
Azimuthal Direction ◽  
Inversion Algorithm ◽  
Vertical Column ◽  
Near Surface ◽  
Mixing Ratios ◽  
Horizontal Variation

<p>Ground-based Multi-Axis Differential Optical Absorption Spectroscopy (MAX-DOAS) measurements of aerosols, tropospheric nitrogen dioxide (NO<sub>2</sub>) and formaldehyde (HCHO) have been carried out in Uccle, Brussels, during two years (March 2018 – March 2020). The MAX-DOAS instrument has been operating in both UV and visible (Vis) wavelength ranges in a dual-scan configuration consisting of two sub-modes: (1) an elevation scan in a fixed viewing azimuthal direction (the so-called main azimuthal direction) pointing and (2) an azimuthal scan in a fixed low elevation angle (2<sup>o</sup>). By applying a vertical profile inversion algorithm in the main azimuthal direction and an adapted version of the parameterization technique proposed by Sinreich et al. (2013) in the other azimuthal directions, near-surface  concentrations (VMRs) and vertical column densities (VCDs) are retrieved in ten different azimuthal directions.</p><p>The present work focuses on the seasonal horizontal variation of NO<sub>2 </sub>and HCHO around the measurement site. The observations show a clear seasonal cycle of these trace gases. An important application of the dual-scan MAX-DOAS measurements is the validation of satellite missions with high spatial resolution, such as TROPOMI/S5P. Measuring the tropospheric  VCDs in different azimuthal directions is shown to improve the spatial colocation with satellite measurements leading to a better agreement between both datasets. By using  vertical profile information derived from the MAX-DOAS measurements, we show that a persistent systematic underestimation of the TROPOMI  data can be explained by uncertainties in the a-priori NO<sub>2</sub> profile shape in the satellite retrieval. A similar validation study for TROPOMI HCHO is currently under progress and preliminary results will be presented.</p><p><strong>References:</strong></p><p>Sinreich, R., Merten, A., Molina, L., and Volkamer, R.: Parameterizing radiative transfer to convert MAX-DOAS dSCDs into near-surface box-averaged mixing ratios, Atmos. Meas. Tech., 6, 1521–1532, https://doi.org/10.5194/amt-6-1521-2013, 2013.</p>

Linearized tomographic inversion of first‐arrival times

Geophysics ◽  
1992 ◽  
Vol 57 (11) ◽  
pp. 1482-1492 ◽  
Author(s):  
James L. Simmons ◽  
Milo M. Backus
Keyword(s):  
Reference Model ◽  
Depth Range ◽  
Inversion Algorithm ◽  
Data Set ◽  
Tomographic Inversion ◽  
Seismic Reflection Data ◽  
Near Surface ◽  
Geometric Patterns ◽  
First Arrival ◽  
Thick Layers

A linearized tomographic‐inversion algorithm estimates the near‐surface slowness anomalies present in a conventional, shallow‐marine seismic reflection data set. First‐arrival time residuals are the data to be inverted. The anomalies are treated as perturbations relative to a known, laterally‐invariant reference velocity model. Below the sea floor the reference model varies smoothly with depth; consequently the first arrivals are considered to be diving waves. In the offset‐midpoint domain the geometric patterns of traveltime perturbations produced by the anomalies resemble hyperbolas. Based on simple ray theory, these geometric patterns are predictable and can be used to relate the unknown model to the data. The assumption of a laterally‐invariant reference model permits an efficient solution in the offset‐wavenumber domain which is obtained in a single step using conventional least squares. The tomographic image shows the vertical‐traveltime perturbations associated with the anomalies as a function of midpoint at a number of depths. As implemented, the inverse problem is inherently stable. The first arrivals sample the subsurface to a maximum depth of roughly 500 m (≈ one‐fifth of the spread length). The model is parameterized to consist of fifteen 20-m thick layers spanning a depth range of 80–380 m. One‐way vertical‐traveltime delays as large as 10 ms are estimated. Assuming that these time delays are distributed over the entire 20-m thick layers, velocities much slower than water velocity are implied for the anomalies. Maps of the tomographic images show the spatial location and orientation of the anomalies throughout the prospect for the upper 400 m. Each line is processed independently, and the results are corroborated to a high degree at the line intersections.

THEORETICAL MAGNETOTELLURIC AND TURAM RESPONSE FROM TWO‐DIMENSIONAL INHOMOGENEITIES

Geophysics ◽  
1971 ◽  
Vol 36 (1) ◽  
pp. 38-52 ◽  
Author(s):  
Charles M. Swift
Keyword(s):  
Numerical Solutions ◽  
Deep Structure ◽  
Solution Technique ◽  
Two Dimensional ◽  
Near Surface ◽  
Anomalous Field ◽  
Strike Direction ◽  
Conductivity Structure ◽  
Representative Model ◽  
Infinite Line

The “network solution” technique for obtaining numerical solutions to Maxwell’s equations in two‐dimensional inhomogeneous media, originally developed by Madden, is presented in detail in this paper. By using plane‐wave and infinite line‐current sources respectively, theoretical magnetotelluric and Turam responses can be calculated above and within an arbitrarily complex two‐dimensional earth conductivity structure. Analysis of a few representative model results emphasizes the importance of considering all the electromagnetic field components, particularly those of the anomalous field. In magnetotellurics, the vertical magnetic field is useful in determining strike direction, and only the E parallel apparent resistivities are representative of deep structure in the presence of near‐surface conductivity inhomogeneities. In Turam, a finite resistivity background causes buried conductors to appear deeper and more conductive than they would appear if the background had infinite resistivity; and strong conductors can affect the total field sufficiently to cause screening.

3-D resistivity inversion using the finite‐element method

Geophysics ◽  
1994 ◽  
Vol 59 (12) ◽  
pp. 1839-1848 ◽  
Author(s):  
Yutaka Sasaki
Keyword(s):  
Finite Element ◽  
Approximate Method ◽  
Synthetic Data ◽  
Computation Time ◽  
Inversion Method ◽  
Data Sets ◽  
Near Surface ◽  
Partial Derivatives ◽  
Homogeneous Half Space ◽  
Resistivity Inversion

With the increased availability of faster computers, it is now practical to employ numerical modeling techniques to invert resistivity data for 3-D structure. Full and approximate 3-D inversion methods using the finite‐element solution for the forward problem have been developed. Both methods use reciprocity for efficient evaluations of the partial derivatives of apparent resistivity with respect to model resistivities. In the approximate method, the partial derivatives are approximated by those for a homogeneous half‐space, and thus the computation time and memory requirement are further reduced. The methods are applied to synthetic data sets from 3-D models to illustrate their effectiveness. They give a good approximation of the actual 3-D structure after several iterations in practical situations where the effects of model inadequacy and topography exist. Comparisons of numerical examples show that the full inversion method gives a better resolution, particularly for the near‐surface features, than does the approximate method. Since the full derivatives are more sensitive to local features of resistivity variations than are the approximate derivatives, the resolution of the full method may be further improved when the finite‐element solutions are performed more accurately and more efficiently.

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