Efficient One-Step Surface-Wave Tomography through Implicit Differentiation and Jacobian Factorization

Abstract

Surface-wave tomography provides important constraints on the crust and upper mantle, but directly inverting millions of measurements for three-dimensional structure remains computationally demanding. We develop an efficient Gauss--Newton framework for one-step surface-wave tomography that jointly inverts multiperiod interstation traveltimes for shear-wave velocity, accounting for both lateral wave propagation and depth sensitivity. Implicit differentiation of the converged discrete Eikonal equations provides individual traveltime sensitivity kernels, which are constructed efficiently by reusing the Eikonal dependency structure across receivers. A factorized Jacobian retains these kernels separately from shared model and dispersion mappings, reducing storage and repeated-product costs within each linearized solve. Synthetic experiments show that the framework recovers three-dimensional velocity anomalies with fewer model updates and lower model errors than the tested comparison workflows at similar data fits. Computational benchmarks demonstrate substantial savings in operator storage and the cost of linearized model updates compared with using explicitly assembled Jacobians. We apply the framework to 6.34 million Rayleigh-wave phase traveltimes across the contiguous United States. The recovered model captures major crustal and uppermost mantle velocity variations, including the broad contrast between the tectonically active western United States and the continental interior, consistent with previous surface-wave studies. These results demonstrate a practical Gauss--Newton approach to imaging the crust and upper mantle using large surface-wave data sets.