Joint inversion of gravity and adjoint-state first arrival tomography

Joint inversion of different geophysical data is used to identify different geological structures and estimate some physical parameters. This study is aimed to do joint inversion of gravity and adjoint-state first arrival tomography by using the Gardner relation. Gardner's relation is an equation that relates seismic velocity to density. 
In the gravity method, forward modeling aims to compute the gravity response at the surface due to a density distribution in the subsurface (Boulanger & Chouteau, 2001). RCG method introduced by Zhdanov (2015) has been used for inversion of gravity.
In the forward modeling of the seismic method, the eikonal equation is used to model the first arrival traveltime, which approximates the high-frequency wave equation. The eikonal equation is nonlinear. There are different methods to solve it, such as Raytracing, Fast Marching, Fast Sweeping, and Finite Difference. This solving method is usually based on a beam, grid, or graph. This study examines the fast sweeping method (FSM) based on rectangular grids by Zhao (2005) has presented.
Estimating model parameters from measured data generally minimizes the misfit function in geophysics. A classic way to solve a minimum problem is to determine the minimum of a series of linear problems sequentially. This formulation requires Frechet derivatives (Jacobin matrices), which need heavy and lengthy calculations. If minimization appears as a nonlinear optimization problem, only the misfit function gradient is required. In this study, the adjoint-state method is used to calculate the gradient.
In the 1970s, the adjoint-state method was developed to efficiently calculate the gradient. It is now a well-known numerical community method for calculating the gradient of a function misfit. It can be used when this function misfit depends on the model parameters through state variables.  State variables are solutions to the forward problem (Plessix, 2006).  For example, state equations can be beam equations in the tomography problem that is the subject of this study. State variables are spatial coordinates and slowness vectors that describe beam paths. The adjoint-state method is effective because only one additional linear system has to be solved.
This study performs the joint inversion of data based on the sequential inversion method using the petrophysical constraint, Gardner relation. The inversion process performs in several loops. In each loop, the individual seismic inversion is first performed in several iterations. The velocity model obtained using the Gardner relation is converted into a density model. The obtained density model is the initial model for individual inversion of gravity in the desired iterations. Finally, the obtained gravity model is converted to a velocity model using the inverse Gardner relation, and the above process is repeated to achieve the desired result. The results of the above inversion can be used to image shallow subsurface models and to limit the location of anomalies in hydrocarbon exploration in layered sediment environments such as basalt and salt dome modeling.