N. Grobbe

Experimental validation of the electrokinetic theory and development of seismoelectric interferometry by cross-correlation

We experimentally validate a relatively recent electrokinetic formulation of the streaming potential (SP) coefficient as developed by Pride (1994). The start of our investigation focuses on the streaming potential coefficient, which gives rise to the coupling of mechanical and electromagnetic fields. It is found that the theoretical amplitude values of this dynamic SP coefficient are in good agreement with the normalized experimental results over a wide frequency range, assuming no frequency dependence of the bulk conductivity. By adopting the full set of electrokinetic equations, a full-waveform wave propagation model is formulated. We compare the model predictions, neglecting the interface response andmodeling only the coseismic fields, with laboratory measurements of a seismic wave of frequency 500 kHz that generates electromagnetic signals. Agreement is observed between measurement and electrokinetic theory regarding the coseismic electric field. The governing equations are subsequently adopted to study the applicability of seismoelectric interferometry. It is shown that seismic sources at a single boundary location are sufficient to retrieve the 1D seismoelectric responses, both for the coseismic and interface components, in a layered model.

Seismo-Electromagnetic Thin-Bed Responses : Natural Signal Enhancements?

We study if nature can help us overcome the very low signal-to-noise ratio of seismo-electromagnetic converted fields by investigating the effects of thin-bed geological structures on the seismo-electromagnetic signal. To investigate the effects of bed thinning on the seismo-electromagnetic interference patterns, we numerically simulate seismo-electromagnetic wave propagation through horizontally layered media with different amounts and thicknesses of thin beds. We distinguish two limits of bed thickness. Below the upper limit, the package of thin beds starts acting like an “effective” medium. Below the lower limit, further thinning does not affect the seismo-electromagnetic interface response signal strength anymore. We demonstrate seismo-electromagnetic sensitivity to changes in medium parameters on a spatial scale much smaller than the seismic resolution. Increasing amounts of thin beds can cause the interface response signal strength to increase or decrease. Whether constructive or destructive interference occurs seems to be dependent on the seismo-electromagnetic coupling coefficient contrasts. When the combined result of the contrast, between upper half-space and package of thin beds and the internal thin-bed contrast, is positive, constructive interference occurs. Destructive interference occurs when the combined contrast is negative. Maximum amplitude tuning occurs for thicknesses of thin-bed packages similar to the dominant pressure and shear wavelengths. Artifacts due to model periodicity are excluded by comparing periodic media with random models. By simulating moving oil/water contacts during production, where the oil layer is gradually being thinned, seismo-electromagnetic signals are proven very sensitive to oil/water contacts. An oil layer with a thickness of <1% of the dominant shear wavelength is still recognized.

Unified multi‐depth‐level field decomposition

Wavefield decomposition forms an important ingredient of various geophysical methods. An example of wavefield decomposition is the decomposition into upgoing and downgoing wavefields and simultaneous decomposition into different wave/field types. The multi-component field decomposition scheme makes use of the recordings of different field quantities (such as particle velocity and pressure). In practice, different recordings can be obscured by different sensor characteristics, requiring calibration with an unknown calibration factor. Not all field quantities required for multi-component field decomposition might be available, or they can suffer from different noise levels. The multi-depth-level decomposition approach makes use of field quantities recorded at multiple depth levels, e.g., two horizontal boreholes closely separated from each other, a combination of a single receiver array combined with free-surface boundary conditions, or acquisition geometries with a high-density of vertical boreholes. We theoretically describe the multi-depth-level decomposition approach in a unified form, showing that it can be applied to different kinds of fields in dissipative, inhomogeneous, anisotropic media, e.g., acoustic, electromagnetic, elastodynamic, poroelastic, and seismoelectric fields. We express the one-way fields at one depth level in terms of the observed fields at multiple depth levels, using extrapolation operators that are dependent on the medium parameters between the two depth levels. Lateral invariance at the depth level of decomposition allows us to carry out the multi-depth-level decomposition in the horizontal wavenumber–frequency domain. We illustrate the multi-depth-level decomposition scheme using two synthetic elastodynamic examples. The first example uses particle velocity recordings at two depth levels, whereas the second example combines recordings at one depth level with the Dirichlet free-surface boundary condition of zero traction. Comparison with multicomponent decomposed fields shows a perfect match in both amplitude and phase for both cases. The multi-depth-level decomposition scheme is fully customizable to the desired acquisition geometry. The decomposition problem is in principle an inverse problem. Notches may occur at certain frequencies, causing the multi-depth-level composition matrix to become uninvertible, requiring additional notch filters. We can add multi-depth-level free-surface boundary conditions as extra equations to the multi-component composition matrix, thereby overdetermining this inverse problem. The combined multi-component–multi-depth-level decomposition on a land data set clearly shows improvements in the decomposition results, compared with the performance of the multi-component decomposition scheme.

Turning One-sided Illumination into Two-sided Illumination by Target-enclosing Interferometric Redatuming

We present a novel method to transform seismic data with sources at the surface and receivers above and below a selected target zone in the subsurface into virtual data with sources and receivers located at the initial receiver locations. The method is based on inverting a series of multidimensional equations of the convolution- and the correlation-type. The required input data can be computed from surface seismic data with a new iterative scheme that is currently being developed. The output data contains virtual sources that illuminate the target not only from above (as in the original data), but also from below, facilitating the needs of seismic imaging and inversion in an optimal way. The method is nonlinear in the sense that all internal multiples are correctly accounted for and true amplitude in the sense that the virtual sources are forced to inherit uniform radiation patterns even though the overburden is strongly heterogeneous.

Electromagnetic and seismoelectric sensitivity analysis using resolution functions

For multi-parameter problems, such as the seismoelectric system, sensitivity analysis through resolution functions is a low-cost, fast method of determining whether measured fields are sensitive to certain subsurface parameters. We define a seismoelectric resolution function for the inversion of a bulk density perturbation. The synthetic data and Green’s functions required to construct the resolution function are computed using the seismoelectric modelling code ESSEMOD. First, we consider the purely electromagnetic problem with a conductivity perturbation at a single point in an isotropic homogenous half-space. The result is nearly identical to a published result based on analytical Green’s functions. It correctly maps the position of the scatterer. Next, we perform an electromagnetic sensitivity analysis for the case of a layered background medium. Again, the resolution function is capable of correctly mapping the scatterer when it is above as well as below a layer of increased conductivity; although in the latter case with less resolution. Finally, we generate multi-component synthetic data with our forward modelling code and compute the seismoelectric resolution function for inversion of a bulk-density perturbation. We find that the seismoelectric system is sensitive to a perturbation in bulk density and that the position of the perturbation can be correctly recovered.

Sparse Inversion for Solving the Coupled Marchenko Equations Including Free-surface Multiples

We compare the coupled Marchenko equations without free-surface multiples to the coupled Marchenko equations including free-surface multiples. When using the conventional method of iterative substitution to solve these equations, a difference in convergence behaviour is observed, suggesting that there is a fundamental difference in the underlying dynamics. Both an intuitive explanation, based on an interferometric interpretation, as well as a mathematical explanation, confirm this difference, and suggest that iterative substitution might not be the most suitable method for solving the system of equations including free-surface multiples. Therefore, an alternative method is required. We propose a sparse inversion, aimed at solving an under-determined system of equations. Results show that the sparse inversion is indeed capable of correctly solving the coupled Marchenko equations including free-surface multiples, even when the iterative scheme fails. Using sparsity promotion and additional constraints, it is expected to perform better than iterative substitution when working with incomplete data or in the presence of noise. Also, simultaneous estimation of the source wavelet is a potential possibility.

Coupled Seismo-Electromagnetic Interferometry for 2D Homogeneous SH-TE Scenarios

We here explore the application of interferometric principles to the coupled seismo-electromagnetic system. We consider 2D homogeneous space scenarios, and focus on one of the two seismo-electromagnetic propagation modes, the SH-TE mode, where horizontally polarized shear-waves are coupled to the transverse electric mode. We start by presenting the theory for retrieving seismo-electromagnetic Green’s function responses via interferometry by cross-correlation. Using explicit homogeneous space solutions, we numerically investigate the interferometric retrieval of an electric field response and a magnetic field response due to a seismic source. We first study the theoretically desirable circular source configuration, providing illumination from all sides, followed by a more realistic line source configuration, exploiting the interferometric far-field approximation. These two numerical examples prove for the selected source-receiver type combinations, that we can indeed retrieve correct dynamic seismo-electromagnetic 2D SH-TE responses using seismic boundary sources only. This is a promising result for the next step: applying 3D seismo-electromagnetic interferometry in the field.

Wavefield Decomposition of Field Data, Using a Shallow Horizontal Downhole Sensor Array and a Free-surface Constraint

SUMMARY Separation of recorded wavefields into downgoing and upgoing constituents is a technique that is used in many geophysical methods. The conventional, multi-component (MC) wavefield decomposition scheme makes use of different recorded wavefield components. In recent years, land acquisition designs have emerged that make use of shallow horizontal downhole sensor arrays. Inspired by marine acquisition designs that make use of recordings at multiple depth levels for wavefield decomposition, we have recently developed a multi-depth level (MDL) wavefield decomposition scheme for land acquisition. Exploiting the underlying theory of this scheme, we now consider conventional, multi-component (MC) decomposition as an inverse problem, which we try to constrain in a better way. We have overdetermined the inverse problem by adding an MDL equation that exploits the Dirichlet free-surface boundary condition. To investigate the successfulness of this approach, we have applied both MC and combined MC-MDL decomposition to a real land dataset acquired in Annerveen, the Netherlands. Comparison of the results of overdetermined MC-MDL decomposition with the results of MC wavefield decomposition, clearly shows improvements in the obtained one-way wavefields, especially for the downgoing fields.