Introduction

Triangle smoothing is a widely used and efficient filtering operation that finds numerous applications in regularizing seismic inverse problems and computing local attributes (Fomel, 2007a,b). Non-stationary triangle smoothing uses a variable smoothing radius (i.e. variable strength of smoothing) along the dimensions of the input dataset. Greer and Fomel (2018) developed an iterative method to estimate the smoothing radius for non-stationary smoothing for matching two seismic datasets. The method is based on the local frequency attribute and has been applied successfully for approximating the inverse Hessian operator in least-squares migration (Greer et al., 2018).

Chen and Fomel (2021) proposed a non-stationary local signal-and-noise orthogonalization method as an alternative to the local signal-and-noise orthogonalization method (Chen and Fomel, 2015). In this approach, the stationary smoothing constraint used to obtain the local orthogonalization weights becomes non-stationary. For highly non-stationary data, the smoothing radius is small where the signal is dominant and it is large where the noise is dominant; thus, the radius adapts to achieve the optimal stability and accuracy. Wang et al. (2021) proposed a non-stationary local slope estimation method that balances both the stability and the resolution of slope perturbations by controlling the strength of triangle smoothing in the shaping regularization framework within the plane-wave destruction algorithm (Fomel, 2002). Chen (2021) introduced a multi-dimensional non-stationary triangle smoothing operator in local time-frequency transformation (Liu and Fomel, 2013). This transformation was proven to be effective in addressing the non-stationary nature of the input seismic data and thus useful in several practical applications of time-frequency analysis.

Non-stationary smoothing applications improve resolution and accuracy, but they require an additional computational cost due to the necessary radius estimation step. In a field data example performed by Chen and Fomel (2021), the radius estimation step in non-stationary local signal-and-noise orthogonalization increased computational time by a factor of 15. While the method of Greer and Fomel (2018) is robust and effective, it does not provide an optimally fast convergence. We propose an alternative method based on Gauss-Newton iteration to estimate the triangle smoothing radius for matching seismic datasets.We derive and implement the derivative of the triangle smoothing operator to guide better guesses for the radius in regularized iterative least-squares inversion.