modes—in order to infer the interrelations between fault zone structures and earthquake processes.
As far as I know, seismic guided waves trapped in a fault zone were first discovered in a three-dimensional vertical seismic profiling experiment conducted in the area surrounding a borehole drilled into the fault zone of the Oroville, California, earthquake of 1975 (13, 14). Records of borehole seismographs placed near the fault zone showed an unusually long-period wave train when the seismic source (a thumper) crossed the fault zone at the surface. The characteristic waveform was attributed to a Love wave-type mode trapped in a low-velocity, low-Q zone. Similar trapped modes were also identified in some of the borehole seismograms obtained at the San Andreas fault near Parkfield, California (15).
The Landers, California, earthquake of 1992 offered a wealth of data for studying various aspects of fault zone trapped modes. By comparing the observed waveform with the synthetic waveform for a low-velocity, low-Q zone, Li et al. (12) estimated a fault zone width around 180 m, with shear velocity of 2.0–2.2 km/sec and a Q value of ≈50.
Interestingly, a similar estimate of fault zone width was made in an entirely different study. From a detailed study of tension cracks on the surface, Johnson et al. (16) concluded that the Landers fault rupture is not a distinct slip across a fault plane but rather a belt of localized shearing spread over a width of 50–200 m. They suggested that this might be a common structure of an earthquake fault, which might have been unrecognized previously because the shearing is small, and surficial material is usually not as brittle as in the Landers area. We identify this shear zone with the low-velocity, low-Q zone found from the trapped modes because their widths are virtually the same. Since the trapped modes were observed from aftershocks with focal depths of >10 km, we conclude that the shear zone found by Johnson et al. (16) extends to the same depth.
More recently, Li et al. (17) conducted an active experiment by shooting explosives in the Landers fault zone. They successfully recorded not only the direct trapped modes but also the reflection from the bottom of the fault zone at a depth of ≈10 km.
A CALCRUST (California Consortium for Crustal Studies) seismic reflection line was shot in the area of a similar geological setting as the Landers epicentral area. The resultant CDP time section (18) clearly presents the transition from nonreflective upper crust to strongly reflective lower crust, which has been identified with the ductile part of the crust globally (19). The transition occurs at a depth of 10 km, in agreement with the bottom depth of the low-velocity, low-Q zone found from the trapped modes. Thus, combining all these observations, we have strong evidence supporting the hypothesis that the shear zone found near the surface extends to the top of the ductile part of the crust.
Furthermore, if we identify the low-velocity, low-Q zone with the breakdown zone of the specific barrier model (6), the source of controlled fmax will be of the order of rupture velocity divided by the zone width (≈10 Hz), in agreement with our observations.
In any case, the shear zone with a width of 200 m can serve only as the micromechanism of earthquakes with a fault length much longer than, say, 1 km. Thus, earthquakes associated with this fault zone must be scale dependent, and we should observe a departure from self-similarity.
Let us now turn to recent results from coda Q−1 studies relevant to the subject of the present paper. Since coda waves are not explained in any existing seismology textbook, I shall first briefly describe what they are.
When an earthquake occurs in the earth, seismic waves are propagated away from the source. After P waves, S waves, and various surface waves are gone, the area around the seismic source is still vibrating. The amplitude of vibration is uniform in space, except for the local site effect, which tends to amplify the motion at soft soil sites compared to hard rock sites. These residual vibrations are called seismic coda waves and they decay very slowly with time. The rate of decay is roughly the same independent of the locations of seismic source and recording station, as long as they are located in a given region.
The closest phenomenon to this coda wave is the residual sound in a room, first studied by Sabine (20). If someone shoots a gun in a room, the sound energy remains for a long time because of incoherent multiple reflections. This residual sound has a very stable, robust nature similar to seismic coda waves, independent of the locations where the gun was shot or where the sound in the room was recorded. The residual sound remains in the room because of multiple reflections at the rigid wall, ceiling, and floor of the room. Since we cannot hypothesize any room-like structure in the earth, we attribute seismic coda waves to backscattering from numerous heterogeneities in the earth. We may consider seismic coda as waves trapped in a random medium.
The seismic coda waves from a local earthquake can be best described by the time-dependent power spectrum P(ω|t), where ω is the angular frequency and t is the time measured from the origin time of the earthquake. P(ω|t) can be measured from the squared output of a bandpass filter centered at frequency ω or from the squared Fourier amplitude obtained from a time window centered at t. The most extraordinary property of P(ω|t) is the simple separability of the effects of seismic source, propagation path, and recording site response expressed by the following equation. The coda power spectrum Pij(ω|t) observed at the ith station due to the jth earthquake can be written as
Pij(ω|t)=Sj(ω) Ri(ω) C(ω|t), 
for t greater than about twice the travel time of S waves from they jth earthquake to the ith station. Eq. 1 means that Pij(ω|t) can be written as a product of a term that depends only on the earthquake source, a term that depends only on the recording site, and a term common to all the earthquakes and recording sites in a given region.
The above property of coda waves expressed by Eq. 1 was first recognized by Aki (21) for aftershocks of the Parkfield, California, earthquake of 1966. The condition that Eq. 1 holds for t greater than about twice the travel time of S waves was found by the extensive study of coda waves in central Asia by Rautian and Khalturin (22). Numerous investigators demonstrated the validity of Eq. 1 for earthquakes around the world, as summarized in a review article by Herraiz and Espinosa (23). In general, Eq. 1 holds more accurately for a greater lapse time t and for higher frequencies (e.g., see ref. 24). Coda waves are a powerful tool for seismologists because Eq. 1 offers a simple means to separate the effects of source, path, and recording site. The equation has been used for a variety of practical applications, including mapping of the frequency-dependent site amplification factor (e.g., see ref. 25), discrimination of quarry blasts from earthquakes (24), the single station method for determining frequency-dependent attenuation coefficients (26), and normalizing the regional seismic network data to a common source and recording site condition (27).
In the following, I focus on the common decay function C(ω|t) on the right side of Eq. 1. I first introduce coda Q to characterize C(ω|t) in the framework of single-scattering theory and then summarize the current results on what the