Adjoint-state differential arrival time tomography

The recently developed adjoint-state traveltime tomography (ATT) method offers an alternative approach to conduct traveltime tomography without the need for ray tracing or waveform modelling. Instead, it utilizes the eikonal equation to depict the minimal traveltime field from an earthquake location to any position in the computational domain. The process of tomographic inversion is formulated as an optimization problem with the goal of minimizing the difference between observed and theoretical first arrival times, which is subsequently solved using the efficient adjoint method. One advantage of differential arrival time data is that it cancels or reduces the influence of common factors, making it more sensitive to a specific subset of model parameters compared to first arrival times. To take advantage of this property, two variants of the ATT method are derived to determine velocity structure and earthquake locations in this study. The first variant, adjoint-state common-source differential arrival time tomography (ATT-CS), uses common-source differential arrival times; while the second variant, adjoint-state common-receiver differential arrival time tomography (ATT-CR), inverts common-receiver differential arrival times. Numerical examples demonstrate that the ATT-CS method is a valuable tool for imaging receiver-side fine-scale velocity structures. Conversely, the ATT-CR method is well suited for resolving source-side velocity structures. Differential arrival times also place constraints on earthquake locations. Compared to common-source differential arrival times, common-receiver differential arrival times are less sensitive to velocity errors and suitable for earthquake location determination. Both common-source and common-receiver differential arrival times are considered first-order differential arrival times. To demonstrate the ease with which the ATT method can be extended to higher-order differential arrival times, we also derive the adjoint-state second-order differential arrival time tomography method. Finally, we discuss how the adjoint-state tomography methods address multipathing.