Towards Reducing Structural Interpretation Uncertainties Using Seismic Data

Université de Lorraine

Abstract

Subsurface structural models are routinely used for resource estimation, numerical simulations, and risk management; it is therefore important that subsurface models represent the geometry of geological objects accurately. The first step of building a subsurface model is usually interpreting structural features, such as faults and horizons, from a seismic image; the identified structural features are then used to build a subsurface model using interpolation methods. Subsurface models built this way therefore inherit interpretation uncertainties since a single seismic image often supports multiple structural interpretations. In this manuscript, I study the problem of reducing interpretation uncertainties using seismic data. In particular, I study the problem of using seismic data to determine which structural models are more likely than others among an ensemble of geologically plausible structural models. I refer to this problem as appraising structural models using seismic data. The first Part of the manuscript is devoted to seismic imaging. I first propose to use reverse-time migration (RTM) as a preconditioner for waveform inversion. Numerical experiments show that the proposed preconditioner accelerates both linearized waveform inversion (least squares reverse time migration) and nonlinear waveform inversion (full waveform inversion) by at least an order of magnitude. I justify the positive numerical performance of the proposed preconditioner by showing algebraically that a low pass filter of the RTM image can approximate the diagonal elements of the Hessian matrix of the objective function under appropriate assumptions. However, I am still unable to propose a physical meaning of the low pass filtering and how it relates the RTM image to the elements of the diagonal of the Hessian matrix. Then, I propose a generalized extended Kirchhoff imaging operator for velocity modeling; the operator is generalized in the sense that it describes multiple data-domain extensions (e.g. shot, offset, and angle extensions) and image-domain extensions (e.g. time-lag and space-lag extensions) simultaneously. The advantages of the proposed generalized extended operator are twofold: firstly, it allows a unified implementation for multiple extensions (i.e. a single implementation that is valid for multiple extensions); secondly, the operator leads to a unified gradient-based migration velocity analysis (MVA) scheme. I confirm the ability of the proposed generalized extended operator to capture image distortion due to inaccurate velocity by applying it to a ray-based MVA experiment. The second Part of the manuscript is devoted to structural modeling, particularly implicit structural interpolation. I introduce Finite Difference Structural Implicit Modeling (FDSIM). The advantages of FDSIM are twofold: firstly, it is relatively easy to implement and to optimize since it is based on finite differences on regular grids; secondly, because it handles discontinuities by rasterization, FDSIM has shown to easily handle very complex fault networks. The main disadvantage of the method is that it may require a very fine resolution depending on the complexity of the fault network, sometimes leading to memory limits. I also propose new regularization operators; the particularity of these operators is that they do not need to be implemented on boundary nodes, a property which is very appealing in implicit modeling where boundary conditions are usually unknown. I then introduce Finite Element Structural Implicit Modeling (FESIM). FESIM is based on a finite element implementation of the newly proposed regularization operators. I show that the conventional finite element familiar for solving boundary value problems has to be slightly modified for implicit modeling where boundary conditions are usually unknown. The third Part of the manuscript is devoted to appraising structural models/interpretations using seismic data. I introduce and formalize the problem of appraising structural interpretations using seismic data. I propose to solve the problem by generating synthetic data for each structural interpretation and then compute misfit values for each interpretation; this allows us to rank the different structural interpretations. The main challenge of appraising structural models using seismic data is to propose appropriate data misfit functions. I derive a set of conditions that have to be satisfied by the data misfit function for a successful appraisal of structural models. I argue that since it is not possible to satisfy these conditions using vertical seismic profile (VSP) data, it is not possible to appraise structural interpretations using VSP data in the most general case. The conditions imposed on the data misfit function can in principle be satisfied for surface seismic data. In practice however, it remains a challenge to propose and compute data misfit functions that satisfy those conditions. I conclude the manuscript by highlighting practical issues of appraising structural interpretations using surface seismic data. I propose a general data misfit function that is made of two main components: (1) a residual operator that computes data residuals, and (2) a projection operator that projects the data residuals from the data-space into the image-domain. This misfit function is therefore localized in space as it outputs data misfit values in the image-domain. However, I am still unable to propose a practical implementation of this misfit function that satisfies the conditions imposed for a successful appraisal of structural interpretations; this is a subject for further research.

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BibTeX Reference

@PHDTHESIS{,
    author = { Irakarama, Modeste },
     title = { Towards Reducing Structural Interpretation Uncertainties Using Seismic Data },
     month = { "apr" },
      year = { 2019 },
    school = { Université de Lorraine },
  abstract = { Subsurface structural models are routinely used for resource estimation, numerical simulations, and risk management; it is therefore important that subsurface models represent the geometry of geological objects accurately. The first step of building a subsurface model is usually interpreting structural features, such as faults and horizons, from a seismic image; the identified structural features are then used to build a subsurface model using interpolation methods. Subsurface models built this way therefore inherit interpretation uncertainties since a single seismic image often supports multiple structural interpretations. In this manuscript, I study the problem of reducing interpretation uncertainties using seismic data. In particular, I study the problem of using seismic data to determine which structural models are more likely than others among an ensemble of geologically plausible structural models. I refer to this problem as appraising structural models using seismic data.

The first Part of the manuscript is devoted to seismic imaging. I first propose to use reverse-time migration (RTM) as a preconditioner for waveform inversion. Numerical experiments show that the proposed preconditioner accelerates both linearized waveform inversion (least squares reverse time migration) and nonlinear waveform inversion (full waveform inversion) by at least an order of magnitude. I justify the positive numerical performance of the proposed preconditioner by showing algebraically that a low pass filter of the RTM image can approximate the diagonal elements of the Hessian matrix of the objective function under appropriate assumptions. However, I am still unable to propose a physical meaning of the low pass filtering and how it relates the RTM image to the elements of the diagonal of the Hessian matrix. Then, I propose a generalized extended Kirchhoff imaging operator for velocity modeling; the operator is generalized in the sense that it describes multiple data-domain extensions (e.g. shot, offset, and angle extensions) and image-domain extensions (e.g. time-lag and space-lag extensions) simultaneously. The advantages of the proposed generalized extended operator are twofold: firstly, it allows a unified implementation for multiple extensions (i.e. a single implementation that is valid for multiple extensions); secondly, the operator leads to a unified gradient-based migration velocity analysis (MVA) scheme. I confirm the ability of the proposed generalized extended operator to capture image distortion due to inaccurate velocity by applying it to a ray-based MVA experiment.

The second Part of the manuscript is devoted to structural modeling, particularly implicit structural interpolation. I introduce Finite Difference Structural Implicit Modeling (FDSIM). The advantages of FDSIM are twofold: firstly, it is relatively easy to implement and to optimize since it is based on finite differences on regular grids; secondly, because it handles discontinuities by rasterization, FDSIM has shown to easily handle very complex fault networks. The main disadvantage of the method is that it may require a very fine resolution depending on the complexity of the fault network, sometimes leading to memory limits. I also propose new regularization operators; the particularity of these operators is that they do not need to be implemented on boundary nodes, a property which is very appealing in implicit modeling where boundary conditions are usually unknown. I then introduce Finite Element Structural Implicit Modeling (FESIM). FESIM is based on a finite element implementation of the newly proposed regularization operators. I show that the conventional finite element familiar for solving boundary value problems has to be slightly modified for implicit modeling where boundary conditions are usually unknown.

The third Part of the manuscript is devoted to appraising structural models/interpretations using seismic data.  
I introduce and formalize the problem of appraising structural interpretations using seismic data. I propose to solve the problem by generating synthetic data for each structural interpretation and then compute misfit values for each interpretation; this allows us to rank the different structural interpretations. The main challenge of appraising structural models using seismic data is to propose appropriate data misfit functions. I derive a set of conditions that have to be satisfied by the data misfit function for a successful appraisal of structural models. I argue that since it is not possible to satisfy these conditions using vertical seismic profile (VSP) data, it is not possible to
appraise structural interpretations using VSP data in the most general case. The conditions imposed on the data misfit function can in principle be satisfied for surface seismic data. In practice however, it remains a challenge to propose and compute data misfit functions that satisfy those conditions. I conclude the manuscript by highlighting practical issues of appraising structural interpretations using surface seismic data. I propose a general data misfit function that is made of two main components: (1) a residual operator that computes data residuals, and (2) a projection operator that projects the data residuals from the data-space into the image-domain. This misfit function is therefore localized in space as it outputs data misfit values in the image-domain. However, I am still unable to propose a practical implementation of this misfit function that satisfies the conditions imposed for a successful appraisal of structural interpretations; this is a subject for further research. }
}