The frequencyBessel (FJ) transform for estimating multimodal surface wave dispersion curves
Contents
Background
Passive dense array surface wave tomography plays a significant role in probing the earth’s interior with the development of a large number of dense seismic arrays, attracting researchers’ growing interest in imaging applications in various scales. These array methods, like beamforming and MASW (multichannel analysis of surface wave) have been widely used to estimate dispersion curves in past several decades. A new array stacking technique, the frequencyBessel (FJ) transform, was proposed about five years ago with some main advantages of extracting dispersion curves of highresolution and broad frequency band (Wang et al., 2019). More importantly, this method facilitates the estimation of multimodal dispersion curves for investimating underground shear wave velocity structure (Wu et al., 2020; Li et al., 2021). Here we introduce the basic principle of this newly proposed array stacking method.
Principles of the FJ transform
The verticalvertical (ZZ) component crosscorrelations in frequency domain $C_{ZZ}(\omega, r)$ from ambient seismic noise is approximately proportional to the imaginary part of the Green’s function $G_{ZZ}(\omega, r)$ between two receivers with distance of $r$, $$ C_{ZZ}(\omega,r) \approx A\cdot Im\{G_{ZZ}(\omega, r)\}, \tag{1} $$ in which, $\omega$ is angular frequency, and $Im\{\cdot\}$ represents imaginary part. Wang et al (2019) defined the spectrum $I(\omega, k)$ of the FJ transform as $$ I(\omega, k)=\int_0^{+\infty}C_{ZZ}(\omega, r)J_0(kr)rdr, \tag{2} $$ where, $J_0$ is the zerothorder Bessel function of the first kind. For a flat multilayered medium, $G_{ZZ}(\omega, r)$ due to an isotropic source is expressed as $$ G_{ZZ}(\omega, r)=\int_0^{+\infty}g_z(\omega, \kappa)J_0(\kappa r)\kappa d\kappa, \tag{3} $$ where, $g_z(\omega, k)$ is the kernel function of the medium, and it is inversely proportional to the surface wave dispersion equation, which means that $g_{z}(\omega, \kappa)$ reaches infinity at roots (dispersion curve) of dispersion equation. Now we use $(1)$, $(2)$ and $(3)$ to obtain $$ I(\omega, k)=\int_0^{+\infty}Im\{\int_0^{+\infty}g_z(\omega, \kappa)J_0(\kappa r) \kappa d\kappa\}J_0(kr)rdr. \tag{4} $$ Note the orthogonality of $J_0$, $\int_0^{+\infty}J_0(kr)rdr\int_0^{+\infty}J_0(\kappa r)\kappa d\kappa=\frac{\delta(k\kappa)}{k}$, and thus we have $$ I(\omega, k) \approx A \cdot Im\{g_z(\omega, k)\}. \tag{5} $$ $(5)$ shows that $I(\omega, k)$ is infinity at the dispersion point. This is the basic idea that we can estimate dispersion curves utilizing the FJ transform.
Utilizing the CCFJpy package to estimate dispersion curves
There is a convenient Python package named CCFJpy for extracting dispersion curves developed by members from Xiaofei Chen’s team. We can find this package on this website (https://github.com/ColinLii/CCFJpy). Here we show the dispersion diagram result computed using this package. Crosscorrrelation functions are from synthetic ambient noise based on the following sourcereceiver configuration.
We utilize 100 receivers and 2000 sources to synthesize seismic event. Synthetic ambient noise is generated according to steps described in this blog. Here we show the dispersion diagram using CCFJpy.


White dots in the above figure are theoretical dispersion curves from the fundamental mode to the third higher mode.
Author Geophydog
LastMod 20240129