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Misfit Function (misfit + function)

Distribution by Scientific Domains

Earth and Environmental Science	100%

Selected Abstracts

Full waveform inversion of seismic waves reflected in a stratified porous medium

GEOPHYSICAL JOURNAL INTERNATIONAL, Issue 3 2010
Louis De Barros
SUMMARY In reservoir geophysics applications, seismic imaging techniques are expected to provide as much information as possible on fluid-filled reservoir rocks. Since seismograms are, to some degree, sensitive to the mechanical parameters and fluid properties of porous media, inversion methods can be devised to directly estimate these quantities from the waveforms obtained in seismic reflection experiments. An inversion algorithm that uses a generalized least-squares, quasi-Newton approach is described to determine the porosity, permeability, interstitial fluid properties and mechanical parameters of porous media. The proposed algorithm proceeds by iteratively minimizing a misfit function between observed data and synthetic wavefields computed with the Biot theory. Simple models consisting of plane-layered, fluid-saturated and poro-elastic media are considered to demonstrate the concept and evaluate the performance of such a full waveform inversion scheme. Numerical experiments show that, when applied to synthetic data, the inversion procedure can accurately reconstruct the vertical distribution of a single model parameter, if all other parameters are perfectly known. However, the coupling between some of the model parameters does not permit the reconstruction of several model parameters at the same time. To get around this problem, we consider composite parameters defined from the original model properties and from a priori information, such as the fluid saturation rate or the lithology, to reduce the number of unknowns. Another possibility is to apply this inversion algorithm to time-lapse surveys carried out for fluid substitution problems, such as CO2 injection, since in this case only a few parameters may vary as a function of time. We define a two-step differential inversion approach which allows us to reconstruct the fluid saturation rate in reservoir layers, even though the medium properties are poorly known. [source]

Joint inversion of multiple data types with the use of multiobjective optimization: problem formulation and application to the seismic anisotropy investigations

GEOPHYSICAL JOURNAL INTERNATIONAL, Issue 2 2007
E. Kozlovskaya
SUMMARY In geophysical studies the problem of joint inversion of multiple experimental data sets obtained by different methods is conventionally considered as a scalar one. Namely, a solution is found by minimization of linear combination of functions describing the fit of the values predicted from the model to each set of data. In the present paper we demonstrate that this standard approach is not always justified and propose to consider a joint inversion problem as a multiobjective optimization problem (MOP), for which the misfit function is a vector. The method is based on analysis of two types of solutions to MOP considered in the space of misfit functions (objective space). The first one is a set of complete optimal solutions that minimize all the components of a vector misfit function simultaneously. The second one is a set of Pareto optimal solutions, or trade-off solutions, for which it is not possible to decrease any component of the vector misfit function without increasing at least one other. We investigate connection between the standard formulation of a joint inversion problem and the multiobjective formulation and demonstrate that the standard formulation is a particular case of scalarization of a multiobjective problem using a weighted sum of component misfit functions (objectives). We illustrate the multiobjective approach with a non-linear problem of the joint inversion of shear wave splitting parameters and longitudinal wave residuals. Using synthetic data and real data from three passive seismic experiments, we demonstrate that random noise in the data and inexact model parametrization destroy the complete optimal solution, which degenerates into a fairly large Pareto set. As a result, non-uniqueness of the problem of joint inversion increases. If the random noise in the data is the only source of uncertainty, the Pareto set expands around the true solution in the objective space. In this case the ,ideal point' method of scalarization of multiobjective problems can be used. If the uncertainty is due to inexact model parametrization, the Pareto set in the objective space deviates strongly from the true solution. In this case all scalarization methods fail to find the solution close to the true one and a change of model parametrization is necessary. [source]

Constraints on earthquake epicentres independent of seismic velocity models

GEOPHYSICAL JOURNAL INTERNATIONAL, Issue 3 2004
T. Nicholson
SUMMARY We investigate the constraints that may be placed on earthquake epicentres without assuming a model for seismic wave speed variation within the Earth. This allows location improvements achieved using 1-D or 3-D models to be put into perspective. A simple, arrival order misfit criterion is proposed that may be used in standard location schemes. The arrival order misfit criterion does not use a seismic velocity model but simply assumes that the traveltime curve for a particular phase is monotonic with distance. Greater robustness is achieved by including a contribution from every possible pairing of stations and the effect of timing inconsistencies reduced by smoothing. An expression is found that relates the smoothing parameter to the number of observations. A typical event is studied in detail to demonstrate the properties of the misfit function. A pathological case is shown that illustrates that, like other location methods, the arrival order misfit is susceptible to poor station distribution. 25 ground truth and 5000 other teleseismically observed events are relocated and the arrival order locations compared to those found using a least-squares approach and a 1-D earth model. The arrival order misfit is found to be surprisingly accurate when more than 50 observations are used and may be useful in obtaining a model independent epicentre estimate in regions of poorly known velocity structure or the starting point for another location scheme. [source]

A hybrid fast algorithm for first arrivals tomography

GEOPHYSICAL PROSPECTING, Issue 5 2009
Manuela Mendes
ABSTRACT A hybrid algorithm, combining Monte-Carlo optimization with simultaneous iterative reconstructive technique (SIRT) tomography, is used to invert first arrival traveltimes from seismic data for building a velocity model. Stochastic algorithms may localize a point around the global minimum of the misfit function but are not suitable for identifying the precise solution. On the other hand, a tomographic model reconstruction, based on a local linearization, will only be successful if an initial model already close to the best solution is available. To overcome these problems, in the method proposed here, a first model obtained using a classical Monte Carlo-based optimization is used as a good initial guess for starting the local search with the SIRT tomographic reconstruction. In the forward problem, the first-break times are calculated by solving the eikonal equation through a velocity model with a fast finite-difference method instead of the traditional slow ray-tracing technique. In addition, for the SIRT tomography the seismic energy from sources to receivers is propagated by applying a fast Fresnel volume approach which when combined with turning rays can handle models with both positive and negative velocity gradients. The performance of this two-step optimization scheme has been tested on synthetic and field data for building a geologically plausible velocity model. This is an efficient and fast search mechanism, which permits insertion of geophysical, geological and geodynamic a priori constraints into the grid model and ray path is completed avoided. Extension of the technique to 3D data and also to the solution of ,static correction' problems is easily feasible. [source]

A practical approach to PP seismic angle tomography

GEOPHYSICAL PROSPECTING, Issue 6 2004
S.K. Foss
ABSTRACT The use of the differential semblance misfit function on common-image-point gathers in the angle domain lends itself to an automated tomographic approach through a gradient-based search in the model space. The velocity model is described by a layer-based model with linear velocity trends and a superimposed bicubic B-spline. The interfaces of the layer-based model are computed by map migration of the PP zero-offset traveltimes of key reflectors. The common-image-point gathers are produced by a restricted inverse generalized Radon transform or amplitude-versus-angle-compensated migration. We present a complete description of all 2.5D formulae for isotropic velocity analysis of PP reflections and the results for ocean-bottom seismic data. [source]

Joint inversion of multiple data types with the use of multiobjective optimization: problem formulation and application to the seismic anisotropy investigations

Constrained tomography of realistic velocity models in microseismic monitoring using calibration shots

GEOPHYSICAL PROSPECTING, Issue 5 2010
T. Bardainne
ABSTRACT The knowledge of the velocity model in microseismic jobs is critical to achieving statistically reliable microseismic event locations. The design of microseismic networks and the limited sources for calibration do not allow for a full tomographic inversion. We propose optimizing a priori velocity models using a few active shots and a non-linear inversion, suitable to poorly constrained systems. The considered models can be described by several layers with different P- and S-wave velocities. The velocities may be constant or have 3D gradients; the layer interfaces may be simple dipping planes or more complex 3D surfaces. In this process the P- and S- wave arrival times and polarizations measured on the seismograms constitute the observed data set. They are used to estimate two misfit functions: i) one based on the measurement residuals and ii) one based on the inaccuracy of the source relocation. These two functions are minimized thanks to a simulated annealing scheme, which decreases the risk of converging to a local solution within the velocity model. The case study used to illustrate this methodology highlights the ability of this technique to constrain a velocity model with dipping layers. This was performed by jointly using sixteen perforation shots recorded during a multi-stage fracturing operation from a single string of 3C-receivers. This decreased the location inaccuracies and the residuals by a factor of six. In addition, the retrieved layer dip was consistent with the pseudo-horizontal trajectories of the wells and the background information provided by the customer. Finally, the theoretical position of each calibration shot was contained in the uncertainty domain of the relocation of each shot. In contrast, single-stage inversions provided different velocity models that were neither consistent between each other nor with the well trajectories. This example showed that it is essential to perform a multi-stage inversion to derive a better updated velocity model. [source]