Rayleigh Wave Dispersion Spectrum Inversion Across Scales

Abstract

Traditional approaches of using dispersion curves for S-wave velocity reconstruction have limitations, principally, the 1D-layered model assumption and the automatic/manual picking of dispersion curves. At the same time, conventional full-waveform inversion (FWI) can easily converge to a non-global minimum when applied directly to complicated surface waves. Alternatively, the recently introduced wave equation dispersion spectrum inversion method can avoid these limitations, by applying the adjoint state method on the dispersion spectra of the observed and predicted data and utilizing the local similarity objective function to depress cycle skipping. We apply the wave equation dispersion spectrum inversion to three real datasets of different scales: tens of meters scale active-source data for estimating shallow targets, tens of kilometers scale ambient noise data for reservoir characterization and a continental-scale seismic array data for imaging the crust and uppermost mantle. We use these three open datasets from exploration to crustal scale seismology to demonstrate the effectiveness of the inversion method. The dispersion spectrum inversion method adapts well to the different-scale data without any special tuning. The main benefits of the proposed method over traditional methods are that (1) it can handle lateral variations; (2) it avoids direct picking dispersion curves; (3) it utilizes both the fundamental and higher modes of Rayleigh waves, and (4) the inversion can be solved using gradient-based local optimizations. Compared to the conventional 1D inversion, the dispersion spectrum inversion requires more computational cost since it requires solving the 2D/3D elastic wave equation in each iteration. A good match between the observed and predicted dispersion spectra also leads to a reasonably good match between the observed and predicted waveforms, though the inversion does not aim to match the waveforms.

Appendix A: The wave equation dispersion spectrum inversion

In full-waveform inversion, objective functions intend to measure the mismatch between the predicted and observed data. One of the most intuitive measurements is the $L_2$ norm distance, which is given by

$$\begin{aligned} \phi (m) = \frac{1}{2}\int _{s}\int _{r}\int _{t} (d^{p}(s,r,t,m) - d^{o}(s,r,t) )^2, \end{aligned}$$

(A.1)

where $d^{p}(s,r,t,m)$ and $d^{o}(s,r,t)$ are predicted and observed data, respectively. m denotes the model parameter, which is, in this paper, shear wave velocity. $\phi (m)$ measures the waveform difference. s, r and t are indexes of time-domain common-shot-gathers, which are for source, receiver and time, respectively.

The widely used $L_2$ norm objective function suffers from the known cycle-skipping problem, which is even worse for dispersive surface waves. Inspired by the success of surface wave dispersion curve inversions (e.g., MASW), we found that the dispersion spectrum (e.g., f-v spectrum) is less influenced by the cycle-skipping than waveform. The shape of the dispersion spectrum retains the kinematic information needed for S-wave recovery in the near-surface, while the amplitude is influenced by many factors, making it difficult to predict by numerical solutions. As a result, we propose to measure the curvature difference between observed and predicted dispersion spectra. In particular, we use the local crosscorrelation to quantify the difference between the predicted and observed dispersion spectra. The objective function is given by Zhang and Alkhalifah (2019b)

$$\begin{aligned} \phi (m) = \frac{1}{2}\int _{s}\int _{f}\int _{v}\int _{f'}W(f') S^2(s,f,v,f',m)\,df' dv df ds, \end{aligned}$$

(A.2)

with

$$\begin{aligned} S(s,f,v,f',m) = \frac{\int _{F-\frac{1}{2}w}^{F+\frac{1}{2}w} |C^p(s,f,v,m)||C^o(s,f,v,f')| df}{\sqrt{\int _{F-\frac{1}{2}w}^{F+\frac{1}{2}w} |C^p(s,f,v,m)|^2 df}\sqrt{\int _{F-\frac{1}{2}w}^{F+\frac{1}{2}w}|C^o(s,f,v,f')|^2 df}}, \end{aligned}$$

(A.3)

where $S(s,f,v,f',m)$ is a local crosscorrelation function, which quantifies the local similarity (range 0 and 1) between the predicted and observed f-v spectra. Note that, the predicted and observed data are dispersion spectra in frequency-velocity domain. f and v are the available frequency and selected phase-velocity. $|C^p(s,f,v,m)|$ and $|C^o(s,f,v,f')|$ are the predicted and observed Rayleigh wave dispersion spectra (the amplitude spectrum), respectively. The calculation of such spectra will be explained later. $f'$ denotes the frequency extension of the observed spectrum. Here, we use trace stretching to calculate the phase shift (Mikesell et al. 2015). A typical extension range is $f'=0.8-1.2$. F and w denote the central point and the length of the sliding window, respectively. In practice, we use the shaping regularization introduced by Fomel (2007) to perform the crosscorrelation of the predicted and observed dispersion spectra locally. Unlike the global correlation, the local measurements can suppress cross talk between neighboring modes (the fundamental and higher modes). The penalty function, $W(f')$, forces the predicted spectrum to be similar to the observed one. Instead of the widely used linear and Gaussian penalty functions, we use a polynomial function to satisfy the following boundary conditions: $W|_{ f'_{\min }} = 0; W|_{ f'_{\max }} = 0; W'|_{ f'_{\min }} = 0; W'|_{ f'_{\max }} = 0 ; W|_1 = 1; W'|_1=0$. The proposed penalty function is decided by the selected extension range ($f'_{\min }$, $f'_{\max }$), thus there are fewer parameters to manipulate. More importantly, it reduces to zero smoothly at the extension boundaries and thus eliminating discontinuities from the objective function.

The f-v spectrum is calculated using a linear Radon transform (Luo et al. 2008). After a temporal Fourier transform of the shot gather, the linear Radon transform can be calculated for each temporal frequency component f as:

$$\begin{aligned} C(f,v) = \int _{x_{\min }}^{x_{\max }} D(f,x)e^{-\frac{i2\pi fx}{v}} dx, \end{aligned}$$

(A.4)

and its adjoint form is given by

$$\begin{aligned} D(f,x) = \int _{v_{\min }}^{v_{\max }} C(f,v)e^{\frac{i2\pi fx}{v}} dv. \end{aligned}$$

(A.5)

D(f, x) denotes the Fourier transform of the common-shot-gathers in the time axis. C(f, v) is the complex f-v spectrum. Notice that, we use the amplitude spectrum (|C(f, v)|) in the objective function [Eq. (A.3)] and the corresponding adjoint source [Eq. (A.7)]. $x_{\min }$ and $x_{\max }$ are the minimum and the maximum offsets used for the integration, respectively. $v_{min}$ and $v_{max}$ are the boundaries of the selected phase velocities for integral, which control the stacking slopes of Rayleigh waves at different frequencies. We can set a physical range for them that is dependent on the target area.

The predicted seismic data are calculated by solving a first order elastic wave equation, which is given by

$$\begin{aligned} \left( \begin{array}{cc} \rho \mathbf{I} _3 &{} 0 \\ 0 &{} \mathbf{C} ^{-1}\\ \end{array}\right) \frac{\partial \Psi }{\partial t}- \left( \begin{array}{cc} 0 &{} E^T \\ E &{} 0 \\ \end{array}\right) \Psi - \mathbf{s} = 0, \end{aligned}$$

(A.6)

where $\Psi = (v_1, v_2, v_3, \sigma _1, \sigma _2, \sigma _3, \sigma _4, \sigma _5, \sigma _6)^T$ is a vector containing three particle velocities and six stresses components, E denotes the spatial-differentiation operator, $\mathbf{I} _3$ is a 3 by 3 identity matrix. $\mathbf{C}$ represents the stiffness matrix and $\mathbf{s}$ denotes the point source used for modeling.

The gradient for S-wave velocity updating with respect to the objective function can be calculated by the chain rule, which is $\frac{\partial \phi (m)}{\partial m}=\frac{\partial \phi (m)}{\partial d^p(m)}\frac{\partial d^p(m)}{\partial m}$, where $\frac{\partial \phi (m)}{\partial d^p(m)}$ is the adjoint source and $\frac{\partial d^p(m)}{\partial m}$ is the Fréchet derivative (Plessix 2006). The adjoint source of the proposed objective function is given by Zhang and Alkhalifah (2019b)

$$\begin{aligned} \frac{\partial \phi (m)}{\partial d^p(m)} =\mathfrak {R}\left( iFFT\left( AdjRadon \left( \frac{\partial \phi (m)}{\partial |C^p|}\frac{\partial |C^p|}{\partial C^p}\right) \right) \right) \end{aligned}$$

(A.7)

and

$$\begin{aligned}&\frac{\partial \phi (m)}{\partial |C^p|} \nonumber =\int _{f'} W(f') S(f,v,f',m) \nonumber \\\qquad & \left( \frac{|C^o(f,v,f')|}{\sqrt{\int _{F-\frac{1}{2}w}^{F+\frac{1}{2}w} |C^p(f,v,m)|^2 df}\sqrt{\int _{F-\frac{1}{2}w}^{F+\frac{1}{2}w} |C^o(f,v,f')|^2 df}} - \frac{|C^p(f,v,m)| \int _{F-\frac{1}{2}w}^{F+\frac{1}{2}w} |C^p(f,v,m)||C^o(f,v,f')| df}{ \sqrt{\int _{F-\frac{1}{2}w}^{F+\frac{1}{2}w} |C^p(f,v,m)|^2 dt}^3\sqrt{\int _{F-\frac{1}{2}w}^{F+\frac{1}{2}w} |C^o(f,v,f')| df}}\right) df'. \end{aligned}$$

(A.8)

By crosscorrelating of the forward-propagated source wavefield and the backward-propagated adjoint-source [Eqs. (A.7) and (A.8)] wavefield, we can calculate the gradient of the objective function. S-wave velocity is iteratively updated using the Limited-memory Broyden–Fletcher–Goldfarb–Shanno (l-BFGS) optimization scheme (Liu and Nocedal 1989), which can be written as

$$\begin{aligned} m = m_0 - \alpha H^{-1} g, \end{aligned}$$

(A.9)

where $H^{-1}$ denotes the inverse Hessian which is approximated by the l-BFGS and $\alpha$ is the step length used for S-wave velocity updates, which is determined by Wolfe conditions (Wolfe 1969):

$$\begin{aligned}&\phi (m_{k+1}) \le \phi (m_k) + c_1 \alpha _k p_k^Tg_k, \end{aligned}$$

(A.10)

$$\begin{aligned}&p_k^Tg_{k+1} \ge c_2 p_k^Tg_k, \end{aligned}$$

(A.11)

with $p_k =-g_k$ for gradient descent and $0<c_1<c_2<1$.

The Wolfe conditions have been successfully used in solving waveform inversion problems (Métivier and Brossier 2016; Liu et al. 2019). An acceptable step length should decrease the data misfit (the Armijo rule) and the slope (the curvature condition) sufficiently. Nocedal and Wright (2006) suggested $c_1=10^{-4}$ and $c_2=0.9$ to obtain sufficient reductions for quasi-Newton methods. The iteration is terminated when there are ten fails in searching for $\alpha$.