Coda Waves and Diffuse Fields in Geophysics

Coda waves and their analysis originated in the Geophysics community where the term

“coda” wave was coined by Aki [Aki 1969, Aki & Chouet 1975]. Aki borrowed the term from music, where a coda is the last portion of a piece of music, and used it to refer to the late arriving energy in seismograms. The goal of the early work involving coda waves was to examine the heterogeneities in the Earth’s crust by extracting parameters about the source and the propagation medium (e.g. attenuation and seismic moment). This early work has been reviewed by Herraiz &

Espinosa (1987).

Work on coda waves progressed from extracting parameters from a single wave to comparative measurements in order to determine the time lapse properties of the crust. The first comparative measurement was used to determine the temporal variation of S wave velocities in the crust [Poupinet et al. 1984]. They developed the first algorithm to extract the relative velocity change between two coda waveforms. Since the original paper use earthquake doublets, the algorithm is often referred to as the Doublet Technique. A second name for the algorithm is the Cross-Spectral Moving-Window (CSMW) Technique since the algorithm utilizes the cross-spectrum of multiple time windows. The algorithm was subsequently improved and extended to include attenuation measurements [Roberts et al. 1992].

Nearly a decade later, an independent derivation of coda waves as multiply scattered waves yielded a temporal version of the CSMW Technique referred to as Coda Wave Interferometry (CWI) [Snieder et al. 2002]. Although CWI and CSMW are equivalent expressions in the time and frequency domains respectively, CWI has a stronger physics base and became the method of choice. CWI makes no assumptions as to the linearity of the relative velocity change with respect to time. However, if the relative velocity change is linear with time (as is the case for homogeneous

and thermal loads), then the CWI algorithm can be reduced to simple dilation of the second waveform until maximum correlation is achieved [Sens-Schöfelder & Wegler 2006]. Although this drastically simplifies the algorithm, the Stretching Technique becomes a grid search method that requires interpolation and thus is computationally expensive. A comparison of the CWI/CSMW and Stretching methods was presented by Hadziioannou et al. (2009) with the conclusion that the stretching method was more stable and less sensitive to noise.

Currently, the extraction of the coda wave relative velocity change has fallen out of favor in the Geophysics community [Weaver & Lobkis 2006]. The preference of many geophysics researchers now lies with the Green’s function extraction method developed by Lobkis & Weaver (2001) for NDE. This method allows for the passive (external source) reconstruction of the ballistic wave propagating between two receivers. Thus received coda waves can be transformed into traditional wave modes and traditional wave analyses (e.g. time of flight or amplitude reduction) apply.

2.6.1 Contributions to Theory

The first analytical treatment of coda waves was presented by Aki (1969). In this work, Aki proposed that the coda waves consisted of singly scattered surface skimming waves. In a later publication, Aki compared his original single scatter model to a diffusion model and noted that neither extreme accurately described the observed coda [Aki & Chouet 1975]. The early work on seismic codas is reviewed by Herraiz & Espinosa (1987). Herraiz & Espinosa divide the evolution of the coda wave modeling efforts into four main groups:

1. Transition from surface wave driven models to S wave driven models. Initiated in 1969.

2. Abandoning the diffusion model. Initiated in 1977.

3. Transition from single scattering dominated to multiple scattering dominated models.

Initiated in 1977.

4. Multiple scattering models with more realistic assumptions. Initiated in 1982.

It is concluded in this review that none of the models presented adequately describe the physical mechanisms behind the formation of the coda wave.

Several decades later, Snider et al. introduced a new method for analyzing coda waves and extracting temporal variations by comparing two coda waves [Snieder et al. 2002]. They approach the coda wave from a multiple scattering perspective and examine the statistical perturbations of the positions of scatterers. Estimates of the perturbations are derived from the cross-correlation of the coda waves. For a homogeneous load, the average perturbation is linear with time and is equivalent to a relative velocity change in the material. A review of the theory and several experimental applications is given by Snieder (2006).

2.6.2 Greens Function Extraction

Although originally derived for NDE, the method of recreating Green’s function from the cross correlation of received coda waves has seen much greater acceptance and application within the Geophysics community. The body of work concerning the extraction of Green’s function from the cross-correlation of coda waves is extensive and an early review is provided by Weaver &

Lobkis (2006). Only a brief overview is presented here as Green’s function extraction is not the focus of this thesis. The first derivations and experimental demonstrations of Green’s function recreation are given by Weaver & Lobkis (2001⁵, 2001, 2002). These early derivations were based on normal mode expansion for finite, lossless cavities, but other derivations were soon presented

5 See Lobkis & Weaver 2001

that did not require the assumption of a finite, lossless cavity. Four distinct derivations have been presented, demonstrating that the Green’s function is present in the correlation of coda waves: 1) derivation based on normal mode expansion [Lobkis & Weaver 2001], 2) derivation based on stationary phase [Snieder 2004], 3) derivation based on energy principles [Snieder et al. 2007], and 4) derivation based on ward identities [Weaver 2008].

After the initial introduction of Green’s function extraction, proof of the applicability of this method to open, heterogeneous, scattering media was provided by many authors within a few years [Derode et al. (2003a, 2003b), Weaver & Lobkis 2004, Wapenaar et al. 2005, Snieder 2007, Weaver et al. 2009]. The early work has been reviewed by Weaver & Lobkis (2006) and Larose et al. (2006). Several authors noted that the resolution of the reconstructed Green’s function could be improved if the correlation of the Green’s functions is taken across an array of receivers [Stehly et al. 2008, Froment et al. 2011]. The resulting conclusions from this multitude of work are: 1) the ballistic waveform traveling between two points can be recreated from the correlation of the received coda waves generated from a tertiary source, 2) the recreation can be improved by ensemble averaging the results from multiple sources, 3) the results can be improved by spatially distributing the sources, and 3) the method theoretically works for any material and geometry for which coda waves exist.