4.2 Previous work

Historically the first significant work done on examining lateral heterogeneities via

travel time inversion was by Aki & Lee (1976) who used local earthquake as sources. This

was followed shortly after by Aki, Christoffersson & Husebye (1977) using teleseismic

travel times of first arriving P phases (the ACH method). The ’art' of the inversion process

has been essentially to determine the perturbation of a reference velocity model required to

bring the predicted and observed travel times closer. Both of these studies divided the earth

region of interest into blocks and assumed the velocity within each block to be a constant In

this way they were able to describe models with laterally varying velocities. By considering

3-D Inversion 4.5

the data outnumber the unknowns, each of which is then assumed to be adequately

constrained by the data. The ACH method became the standard technique for studying

lateral heterogeneities under a dense array of recorders using teleseismic travel times. It has

been applied to several arrays around the world including those in Norway (NORSAR) (Aki

et al. 1977), central California (Husebye et al. 1976), Hawaii (Ellsworth & Koyanogi 1977) and Montana U.S.A. (LASA) (Aki, Christoffersson & Husebye 1976).

In recent times, authors have devised more sophisticated methods to determine lateral

heterogeneities from travel time data. Chou & Booker (1979) applied the techniques of

Backus & Gilbert (1968 & 1970) to the 3-D inversion problem using synthetic data. They

considered only a purely linear problem, and represented the earth structure under study by

'ideal averaging' volumes. This resulted in a smoothly varying 3-D velocity field, a feature

also favoured in the inversion of teleseismic travel times by Thomson & Gubbins (1982),

however they achieved it by using a cubic spline interpolation over a finite number of grid

points. Both of these studies attempted to invert for velocity structure alone and avoided any

relocation of sources. They assumed that all information on velocity structure contained in

the data was confined to a region immediately beneath the receivers, hence only a small

fraction of the ray paths between source and receiver had to be considered and a much

simpler source/receiver geometry results. This assumption may be reasonable when using

teleseismic data or in cases where one is confident that the structure beneath the receivers is

sufficiently anomalous to dominate the travel time residuals. However there are still

problems in determining the depth to which the model region should be extended. A

drawback of this technique is the difficulty of depth resolution and that one is faced with the

possibility of having mapped distant anomalies into local velocity structure, especially when

azimuthal raypath coverage is limited.

Tarantola & Nercessian (1984) favour a continuous inversion approach which also

avoids the a priori discretization of the velocity model into blocks. Instead they leave the form of the velocity model unspecified and simply define it to be a continuous 3-D function.

3-D Inversion 4.6

priori information on the seismic structure may be easily incorporated into the inversion. Nercessian et al. (1984) applied this approach to determine the 3-D structure under the Mont Dore volcano. Artificial sources were used so the need for source relocation was avoided.

The introduction of hypocentral coordinates in the inversion has been considered by

Spencer & Gubbins (1980) and Pavlis & Booker (1980). Spencer (1985) presents a review

of these procedures. Both pairs of authors propose a technique that separates the

hypocentral unknowns from the velocity field unknowns. The so called 'Simultaneous' or

'Joint' hypocentre and velocity structure inversion problem, as it is sometimes known, is an

example of a multi-parameter type inversion i.e. one attempts to solve for many unknowns

of different physical dimension or type. The errors contained in the solution of one

parameter type may adversely influence those of the other. The authors deal with this

problem by a process of data projection. When solving for one parameter type the data is

projected onto the 'null space' of the other, and the algorithm updates each parameter type

sequentially after correcting the data residuals from the previous update. An advantage of

the method is that the parameters which are updated first are unbiased by the errors in the

other parameter type. It is also useful in reducing the overall computation required.

However the success of this 'unbiasing' of parameter types is crucially dependent on the

problem being linear. Which is not the case for the joint inversion problem, since the travel

times of seismic waves are nonlinearly dependent on both the position of the source and the

velocity structure through which the waves travel. For nonlinear problems, then, the initial

estimates of the unknowns must be close enough to the 'true' values for the problem to be

effectively globally linear, otherwise the projection process itself introduces errors into the

inversion (Williamson 1986).

Most inversion studies for 3-D velocity structure differ in a few major areas; the use

of teleseismic or local travel time data; the type of velocity structure parameterisation i.e.

smoothly varying or blocks; and, whether a linear or nonlinear inversion has been adopted.

A nonlinear inversion for 3-D structure has been attempted by only a few authors, notably

3-D Inversion 4.7

nonlinear approach one must trace rays through laterally heterogeneous velocity models,

which is an arduous task even when one has a suitable algorithm. In these studies the

problem is simplified somewhat by considering only teleseismic data recorded by a dense

array of receivers. In this case the raypath geometry is relatively simple in the region of the

model i.e. usually from the base of the model to the surface. However the near vertical

nature of all raypaths does have the undesirable effect of reducing the vertical resolution.

Local travel time data has only rarely been used in studies of lateral heterogeneities.

Few major developments have occurred since the work of Aki & Lee (1976). Thurber

(1983) used Pavlis & Booker's method applied to the inversion of local data for the structure

under the Coyote Lake area in central California. Interestingly enough he constructed a

smoothly varying velocity structure by interpolating between a finite number of grid points,

and an approximate 3-D ray-tracing technique to calculate the travel times. His scheme does

make an effort to account for the nonlinearity of the problem, however approximate ray

tracing techniques depend heavily on the particular type of velocity structure for which they

are designed and so are usually non-transportable to other situations. Furthermore they may

introduce some systematic bias by producing over estimates of travel times. Koch (1985b)

performed a nonlinear inversion on crustal and intermediate depth events (>100 km) using

constant velocity block models with differing sizes. However exact analytical ray tracing is

performed through the cellular structure, whilst the inversion comprises basically of a large

linear system solver. This type of procedure can become very expensive as the number of

model parameters increases. Consequently one is restricted to a small number of model

parameters and hence a relatively crude parameterisation. Hearn & Clayton (1986 a & b)

adapted the time term method of Scheidegger & Willmore (1957) and used a backprojection

technique to invert for local heterogeneous structure. However their scheme seems to

depend quite heavily on the linearized equations used, which do not include any source re­

location. The results of their work probably owe much to the excellent data coverage, which

is made possible by the extremely high density of both sources and receivers available in

3-D Inversion 4.8

To date a nonlinear inversion of local travel times has largely been avoided in favour

of the more approximate methods discussed above. In most regional seismic networks one

encounters a rather complex pattern of raypaths due to the complicated source/receiver

geometry. This fact alone suggests that inversion of local travel time data has much greater

potential for resolving lateral heterogeneities than teleseismic data, since in the local case the

raypaths sample the region of interest for their entire path length and from a wide range of

azimuths and depths. It also indicates that the local problem is intrinsically more difficult

than the teleseismic one since we cannot restrict attention to a small region of the earth where

raypaths have a simple geometry. It is therefore important to understand the extent to which

we may constrain 3-D seismic structure using local travel time data and more importantly to

develop methods for examining lateral heterogeneities from both natural and artificial sources

distributed from local to regional distance scales. This will provide encouragement to extend

the techniques to other regions of economic or tectonic importance where natural seismic

sources may be unavailable and we must rely on expensive controlled source surveys. The

use of local earthquake data in the mapping of 3-D seismic structure may also provide a more

detailed understanding and interpretation of local geological structure.

{Note: From a purely theoretical viewpoint the determination of both seismic velocity structure and earthquake hypocentres simultaneously is closely linked to the hypocentral location problem encountered earlier. Although the current problem may be formulated in similar terms, the previously used 'Direct Search' type methods are entirely inappropriate

when the number o f unknowns increases significantly. However the philosophy o f treating unknowns o f different physical dimension or type separately seems to be a reasonable one and hence will be retained.)





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