The ever-expanding need for energy, including those related to fossil fuels, is behind the drive to explore more complicated regions, such as salt and subsalt provinces. This exploration quest relies heavily on recorded surface seismic data to provide precise and detailed subsurface properties. However, conventional seismic processing algorithms including the state-of-the-art full-waveform inversion (FWI) fail to recover those features in many areas of salt provinces. Even the industrial solution with substantial involvement of manual human-interpretation has faced challenges in many regions. In this thesis, I attempt to replace those manual, and somewhat erroneous, steps to the velocity building in salt provinces with a mathematically robust algorithm under the FWI machinery. I, specifically, regularize FWI by penalizing the velocity drops with depth with a new more flexible function. Although promising, FWI is computationally very expensive, especially for large 3D seismic data. It updates an initial guess of the model iteratively using the gradient of the misfit function, which requires lengthy computations and large memory space/disc storage. Based on the adjoint state method, gradient computation usually requires us to store the source wavefield, or include an additional extrapolation step to propagate the source wavefield from its temporary storage at the boundary. To mitigate this computational overburden, I propose an amplitude excitation gradient calculation based on representing the source wavefield history by a single, specifically the most energetic arrival. In this thesis, I also propose a novel-multiscale scheme based on ux-corrected transport (FCT) to reduce artifacts in the gradient direction due to the noise present in seismic data. FCT comprises of two finite difference schemes: a transport and a diffusion to compute the flux at a grid point. I observe a couple of benefits in FCT-based FWI. First, it yields a smooth gradient at the earlier iterations of FWI by promoting the lower frequency content of the seismic data. Second, it is easily compatible with the existing FWI code, and with any objective function. The multiscale strategy starts with a large smoothing parameter and relaxes it progressively to zero to achieve the final inverted model from traditional FWI.

The thesis deals with the numerical modeling of nonreactive and nonlinearly sorbing solutes in groundwater and analysis of the effect of heterogeneity resulting from spatial variation of physical and chemical parameters on the transport of solutes. For this purpose, a hy­brid flux corrected transport (FCT) and central difference method based on operator-split approach is developed for advection-dispersion solute transport equation. The advective transport is solved using the FCT technique, while the dispersive transport is solved using a conventional, fully implicit, finite difference scheme. Three FCT methods are developed and extension to multidimensional cases are discussed. The FCT models developed are anlaysed using test problems possessing analyt­ical solutions for one and two dimensional cases, while analysing advection and dispersion dominated transport situations. Different initial and boundary conditions, which mimic the laboratory and field situations are analysed in order to study numerical dispersion, peak cliping and grid orientation. The developed models are tested to study their relative merits and weaknesses for various grid Peclet and Courant numbers. It is observed from the one dimensional results that all the FCT models perform well for continuous solute sources under varying degrees of Courant number restriction. For sharp solute pulses FCT1 and FCT3 methods fail to simulate the fronts for advection dominated situations even for mod­erate Courant numbers. All the FCT models can be extended to multidimensions using a dimensional-split approach while FCT3 can be made fully multidimensional. It is observed that a dimensional-split approach allows use of higher Courant numbers while tracking the fronts accurately for the cases studied. The capability of the FCT2 model is demonstrated in handling situations where the flow is not aligned along the grid direction. It is observed that FCT2 method is devoid of grid orientation error, which is a common problem for many numerical methods based on Cartesian co-ordinate system. The hybrid FCT2 numerical model which is observed to perform better among the three FCT models is extended to model transport of sorbing solutes. The present study analyses the case of nonlinear sorption with a view to extend the model for any reactive transport situation wherein the chemical reactions are nonlinear in nature. A two step approach is adopted in the present study for coupling the partial differential equation gov­erning the transport and the nonlinear algebraic equation governing the equilibrium sorp­tion. The suitability of explicit-implicit (EI - form) formulation for obtaining accurate solution coupling the transport equation with the nonlinear algebraic equation solved using a Newton-Raphson method is demonstrated. The performance of the numerical model is tested for a range of Peclet numbers for modelling self-sharpening and self-smearing con­centration profiles resulting from nonlinear sorption. It is observed that FCT2 model based on this formulation simulates the fronts quite accurately for both advection and dispersion dominated situations. The delay in the solute mobility and additional dispersion are anal­ysed varying the statistical parameters characterising the heterogeneity namely, correlation scale and variance during the transport of solutes and comparisons are drawn with invariant, cases. The impact of dispersion during the heterogeneous transport is discussed.