mat-LRTdisp

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MATLAB package for measuring multi-mode surface wave dispersion using the Linear Radon Transform (LRT)

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# mat-LRTdisp

[![DOI](https://zenodo.org/badge/228703313.svg)](https://doi.org/10.5281/zenodo.18931030)
=========================

## A) Radon Transform for Phase Velocity Dispersion
Part a) of the package extracts phase velocity dispersion from multimode surface waves using the Linear Radon Transform (LRT), as demonstrated by Luo et al. (2015). The input is a record section with surface waves windowed in time. The output is a radon panel in the period-phase velocity (Radon) domain, showing the energy contained in the individual mode branches. The code also includes an interactive tool for picking the maximum peaks corresponding to surface-wave dispersion.

An example synthetic Love wave dataset (fundamental through 4th higher mode) is included in ./pa5_5km/ to demonstrate its application.

![](https://raw.githubusercontent.com/jbrussell/mat-LRTdisp/master/preview.png)

## B) Mode Branch Isolation in Time Domain
Part b) of the package demonstrates how the LRT can be used to isolate mode branches of interest in the time domain. It is slower because the full forward operator and Radon panel (positive & negative frequencies) must be saved. Gaussian filtering of the fundamental mode is done in the Radon domain, then transformed back to the time domain using the forward operator.

Below is an example of (left) the Gaussian window applied to the fundamental mode Love wave and (right) the resulting isolation of the fundamental mode compared to the true fundamental mode synthetic

![](https://raw.githubusercontent.com/jbrussell/mat-LRTdisp/master/preview2.png)

**A note about the LRT inversion:**
We include several different methods for solving the Radon Transform inverse problem. In general, the sparser the method (so-called "high resolution LRT" methods), the more sensitive it is to noise in the data. We have found that the most stable results come from the Weighted Conjugate Guided Gradient (CGG) of Ji (2006) (i.e., CGG_weight in our notation), but you are encouraged to try others:

1) **L2**: L2-norm ; *Schultz (2012)*
2) **L1**: L1-norm ; *Schultz (2012)*
3) **Cauchy**: Cauchy norm ; *Schultz (2012)*
4) **CGsimple**: Simple conjugate gradient ; *Ji (2006)*
5) **CG_IRLS**: Conjugate gradient with iterative reweighted least squares ; *Ji (2006)*
6) **CGG_weight**: Conjugate guided gradient with model and residual weighting **preferred** ; *Ji (2006)*
7) **CGhestenes**: Conjugate gradient with an approximation in the descent calculation that is more numerically stable ; *Ji (2006); Claerbout (1992)*

The advantage of the latter 4 methods from Ji (2006) is that regularization parameters are determined by the data, rather than needing to make an apriori guess.
___
This package includes pieces of code written by Ryan Schultz, particularly for the minimization problem: [Radon-Transform_Schultz-Gu](https://github.com/RyanJamesSchultz/Radon-Transform_Schultz-Gu)

References:

Schultz, R., Gu, Y. J., 2012. 
    Flexible, inversion-based Matlab implementation of the Radon transform
    Computers and Geosciences 52, 437-442.
    doi: 10.1016/j.cageo.2012.08.013.

An, Y., Gu, Y. J., Sacchi, M., 2007. 
    Imaging mantle discontinuities using least-squares Radon transform
    Journal of Geophysical Research 112, B10303.
    doi: 10.1029/2007JB005009.
          
    Ji, J, 2006. 
    CGG method for robust inversion and its application to velocity-stack inversion. 
    Geophysics, 71(4):R59. 
    doi:10.1190/1.2209547

Luo, Y., Yang, Y., Zhao, K., Xu, Y., Xia, J., 2015. 
    Unraveling overtone interferences in Love-wave phase velocity measurements by radon transform. 
    Geophysical Journal International, 203(1):327–333.
    doi: 10.1093/gji/ggv300.

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