# Linear inversion results: Images from the data

In document Seismic inversion for earthquake location and 3-D velocity structure (Page 136-140)

## VIEWED AS AN INVERSE PROBLEM

### 6.4 Linear inversion results: Images from the data

In this section we present and discuss the results of the linear inversion study. The

initial inversion algorithm used here is that described in the previous section. In most

studies of this kind it is inevitable that some evolutionary process will occur. Repeated

application and refinement of the basic algorithm usually results in a descendant bearing only

a partial resemblance to its early ancestor. Such is the case here. In an effort to obtain a firm

understanding of the inversion problem we have accelerated this process. It is necessary to

I

determine and assess the factors which have a major influence on the inversion results in

order to interpret those results correctly. To this end we have experimented with various

features of the problem i.e. data coverage, algorithm structure, a priori information etc., at each stage performing a new inversion to properly determine its effect. This together with

the natural evolution of the algorithm, has resulted in a large amount of material to analyze.

The results presented here are only a moderate subset of the many generated, however they

are representative of the major improvements in the algorithm and important features of the

problem (at least in the view of the author).

A common way of testing an inversion algorithm in order to assess its capabilities

and limitations, is by experimenting with artificially generated data. Spakman & Nolet

(1987) use this technique to compare two large scale equation solvers applied in tomography

problems. However for this type of approach to be really useful in our case, we should

invert artificial arrival times generated from laterally heterogeneous velocity models.

Obviously in order to calculate the data we must solve the two-point ray tracing problem in a

heterogeneous media, which is the very task we seek to avoid by performing the linear

inversion. Detailed experiments with artificial data are therefore deemed to be unfeasible

3-D Linear Inversion 6.38

long term (even if it is rather frustrating in the short term!) since we are forced to consider all

problems arising from the inadequacies of such data rather than simply ignore them in an

idealized situation. Furthermore we may encounter some unforeseen problems resulting

from the use of real data, which can only improve our understanding, and help us to develop

a more realistic approach to the inversion.

Before we can actively discuss 'solutions' to the linear inverse problem we must first

define, or at least describe, what exactly we mean by a solution in the context of the arrival

time inversion. We recall from an earlier discussion that without any mechanism for solving

the two-point ray tracing problem in heterogeneous media, we are unable to make any

rigorous examination of the merits of a proposed velocity model i.e. we cannot determine

how well it fits the observations. At best we can only estimate the data fit by relying on the

linearity assumption (described in ch. 5). This will be the case with any linearized inversion

study which neglects to carry out forward modelling on the final 'solution' model. Since we

cannot determine exactly how well a given model fits the data, we have a problem in

defining a convergence criteria. The usual way of proceeding is simply to define some value of the data misfit function, which provides an acceptable level of 'estimated' data misfit.

How useful this will be is entirely dependent on the validity of the local linearity assumption

i.e. of eqn. 5.6, since this is used to estimate the arrival times in the heterogeneous model

given those calculated in the laterally homogeneous model. We must remember then, when

comparing the results of the linear inversion, that we cannot claim any one model fits the

data significantly better than any other since we do not know accurately how well any fit the

data. Because of this uncertainty inherent within the linear inversion study we do not

attempt to make such claims. Instead we examine the general features of the seismic models

resulting from the inversion and ask other questions, ones which we have at least a

reasonable possibility of resolving. We need to know whether the details contained within

the velocity maps are actually a result of the information contained in the arrival time data or

perhaps due to some other factor e.g. an artifact of the inversion algorithm. To this end we

3-D Linear Inversion 6.39

results. Only in this way can we gain a comprehensive understanding of the nature of the

To avoid labeling an inversion as 'converged' when its estimated misfit falls below

some pre-assigned value, we simply make all comparisons between different inversions at

the same stage of the inversion i.e. after the same number of iterations of the algorithms.

This position is determined by examining the rate of reduction of the estimated misfit. In

early trials it was found that by 15 - 20 iterations the rate of decrease of the estimated misfit

was less than 5 % of its original value, and also the effect of the model perturbations were

almost negligible on the overall features of the velocity structure. We shall compare all

inversion models at 20 iterations to ensure that most of the significant features have evolved.

Although this pseudo convergence criteria is perhaps less satisfying than assigning a cutoff

I

misfit value, it has its merits. Essentially we avoid the temptation of over interpreting our

results. We prefer not to choose a convergence criteria based on the required level of fit to

the data when we cannot accurately calculate that fit. By avoiding this type of condition we

can only (and should only) regard the resulting models as 'inversion images' and not models

of the real earth. It is our intention to use the linear inversion results as a handrail in

examining the nature of the problem and not a mechanism by which we generate any one

best fitting, or preferred velocity model. The nonlinear inversion is the only place where

such questions can be addressed. Even then we should treat all models as 'possible images

of the earth' since we are restricted by the quantity and type of data available. A completely

independent dataset (e.g. a set of seismic traveltime or even waveform data, collected from

the same geographical region) may 'illuminate' the earth in a different way and, if interpreted

correctly and accurately, may well produce a different image (although it might not be

fundamentally different!). One beneficial feature of the approach adopted here is that a large

number of images are generated by different algorithms and different quantities of data, and

so an examination of all of these will give us a strong impression of the stable and consistent

3 -D Linear Inversion 6.40

In brief, we restrict ourselves to simple aims in the linear study and do not attempt to

make rash claims about the likelihood or otherwise of the features generated as being

attributable to the Crustal structure in south-east Australia. We analyse and discuss the

features of the models generated and hopefully gain from the insight offered in interpreting

the images resulting from the nonlinear inversion to be performed later. In the next section

we present some of the linear inversion images and discuss what we believe to be the more

important features. This is followed by a more formal discussion and examination of the

resolution properties of the available dataset, with special reference to the trade-off

relationship between hypocentral and velocity parameters.

6.4.1 Velocity maps and hypo centre distributions

Because the bulk of the controlled source travel time data was actually collected

several months after the earthquake data, all of the early inversions were performed without

the help of the blast data. In fact the earthquake dataset itself was revised several times to

improve the coverage of events. The details given in section 6.1 are appropriate to the final

state of the earthquake dataset, which contains approximately 10 % more events than the

original. The results have been displayed in more or less chronological order, so that one

can observe the development of the algorithm and evolution of the dataset.

6.4.1.1 The two dimensional subspace inversion

Fig. 6.12 shows the Crustal P-velocity model obtained from the original 2-D

subspace scheme. The velocities from the Crustal layer of cells have been contoured to

produce a smoothly varying function representing average Crustal velocities. Contours are

plotted at regular intervals of 0.03 km s4 between 5.84 km s4 and 6.40 km s4 . The initial

model used in all of the early inversions was the laterally homogeneous DEH model, which

has a Crustal P-velocity of 6.03 km s4 . A small interval containing this value has been left

unshaded so that we may immediately observe the relatives highs and lows compared to the

### Pg v e lo c itie s

In document Seismic inversion for earthquake location and 3-D velocity structure (Page 136-140)