Estimating medium statistical parameters using first arrival travel times

Several seismic and bore-hole log studies support the existence of random inhomogeneities in the Earth's crust. Therefore, besides deterministic methods of velocity estimation, the use of statistical methods may significantly improve the estimations of the velocity field and can help to quantify the textural properties of the medium which is important for petrophysical interpretations. The studies for the velocity determination of the subsurface are mainly based on the homogeneously layered medium assumption. However the Earth is a complex structure which involves both small and large scale inhomogeneities. Because of these inhomogeneities the seismic travel times fluctuates around the expected travel times. These fluctuations can be used to estimate the characteristic scale lengths of the inhomogeneities and the other statistical parameters such as velocity variance, and covariance function. The statistical parameters of inhomogeneities can be used in seismic inversion combined with geostatistical approaches in characterization of hydrocarbon reservoirs and in lithologic description of rocks. To estimate the statistical parameters of a seismic medium, a method based on ray theory (high frequency approach) is exploited. Ray theory is valid if the characteristic scale lengths of inhomogeneities are larger than the seismic wavelength and the width of the first Fresnel Zone. Under these conditions scattering and localization effects are ignored. In the proposed method first order velocity (or refractive index) perturbations and a dispersion free medium are considered. It is assumed that the inhomogeneities are larger than the dominant seismic wavelength. The method links the travel time fluctuations to medium statistical parameters (variance of the medium fluctuations and inhomogeneity scale lengths in different directions) via the variance and covariance functions of the travel time fluctuations. Both isometric and anisometric (statistically anisotropic or different correlation lengths lx, ly and lz in x, y and z directions respectively) inhomogeneities can be considered. Travel time fluctuations from a sufficient number of medium realizations are required for a good statistical sampling and for the application of the method. In previous studies the derived formulas are numerically verified by ray method considering a medium with Gaussian auto-correlation function (ACF). Some questions regarding to the assumptions of the Gaussian random medium have arisen in different occasions, since many stochastic parameter estimation studies suggest a von Kármán ACF for the Earth medium. It has been a question whether the studied medium is Gaussian or not, and whether the formulas derived by considering Gaussian random medium will give successful estimations for media having non-Gaussian ACF's, such as Exponential and von Kármán. In this thesis it is tested whether it is possible to estimate the medium fluctuations and the characteristic scale lengths of the inhomogeneities by the suggested method, even the random medium is not represented by a Gaussian ACF. Ray theory is a high frequency approximation to wave propagation, on the other hand the data recorded in seismic/seismology is finite frequency. Another motivation of this thesis is to understand the limits and the success of the method in finite frequency conditions. For this purpose, wave propagation simulations in random media are necessary. To compare the ray theoretical and wave propagation results, the ray theoretical conditions are considered and a serial acoustic finite difference wave propagation code has been used for simulation. For one random medium realization, the computation time of the wave propagation has last 95 hours, although the ray theory conditions were not totally fulfilled due to the requirement of high computation power. For ensemble averaging, 100 random media realizations are used in ray theory. When the computation time of wave propagation simulation for one random medium realization is considered, the parallel computing became necessary. Therefore the wave propagation code has been parallelized and tested. At this stage, a high performance computing system is required. Especially, when the ray theory conditions (the wavelength should be much smaller than the inhomogeneity scale length) and finite difference conditions (stability and grid dispersion) are considered, very dense computation grid is required which is very compute expensive. In this study, medium which has anisometric inhomogeneities and linearly increasing velocity with depth is considered. The random and the mean parts of the velocity field is calculated. For the ray tracing the refractive index medium is prepared, while for the acoustic modeling the velocity field is used. Considering Gaussian, Exponential and Von Kármán autocorrelation functions 100 random medium realization are performed. Using 100 different refractive index media for each autocorrelation function, the travel time fluctuations are calculated for each autocorrelation function by tracing the rays over the random media. Then using these travel time fluctuations, variances for each distances from source are calculated. The best fit between the calculated and theoretical relation of travel time variance is obtained and the the model parameters, the product of the vertical inhomogeneity scale length and the variance of the medium, and the inhomogeneity scale length ratio (ratio of vertical to horizontal scale lengths) are estimated. Next the travel time fluctuations are used for the calculation of the covariance function which will provide the estimation of the medium parameters separately. The covariance functions are calculated by windowing the travel time fluctuation data at determined distances. From the covariances, the correlation distances for each window is calculated. These correlation distances are related to the horizontal inhomogeneity scale length. Using these calculated correlation distances and the theoretical relations the horizontal inhomogeneity scale length is estimated. This information is used in formerly estimated parameters and the vertical inhomogeneity scale length, and the variance of the medium are estimated subsequently. The results for each random media are compared with the actual medium parameters. It is seen that random media having Gaussian and exponential ACFs give reliable estimates while parameter estimations obtained from von Kármán medium have relatively larger errors. To understand the limits and the success of the method in finite frequency conditions acoustic wave propagation is simulated in the same random media that are used in ray method. In this case only random media with Gaussian ACF having anisometric inhomogeneities and linearly increasing velocity with depth is considered. To simulate acoustic waves, a script written for Matlab is rewritten in Fortran by using MPI library. In this way, the simulation speed is increased about 50 fold. The acoustic wave propagation is performed by using the generated random media. Due to the linear increase of velocity, diving waves are obtained. The first arrival times are automatically picked by using Akaike Information Criterion (AIC), which is formulated for detection of first arrival P-waves. Theoretical curve for diving waves are fitted to the observed travel times to estimate the near-surface velocity and the velocity gradient. The travel time fluctuations are obtained by calculating the difference between the theoretical and observed travel times By using the travel time fluctuations of 100 random media, the travel time variance is calculated. The best fit between the calculated and theoretical relation of travel time variance is obtained and the model parameters, the product of the vertical inhomogeneity scale length and the variance of the medium, and the inhomogeneity scale length ratio are estimated. Next the travel time fluctuations are used for the calculation of the covariance function which will provide the estimation of the horizontal inhomogeneity scale length and hence the vertical inhomogeneity scale length, and the variance of the medium as described in the previous paragraph. The results obtained by ray method and acoustic modeling are compared by considering the actual parameters of the medium. It is observed that the parameters obtained from acoustic modeling traveltimes have larger errors when compared to the ray method. This discrepancy may be due to the phenomenon known as velocity shift in literature. As a future study the reason of this discrepancy can be investigated. In this thesis the formulas derived in a former study by considering ray theory and random media with Gaussian ACF is examined by considering finite frequency modeling and by using ACFs different than Gaussian ACF. It is seen that there are discrepancies in the estimated parameters between acoustic modeling and ray method. Also the estimated parameters obtained from random media generated by considering von Kármán ACF have larger errors. These results should be considered when using these relations.