# Forward modelling

In document Seismic inversion for earthquake location and 3-D velocity structure (Page 104-114)

## VIEWED AS AN INVERSE PROBLEM

### 6.2 Forward modelling

To solve any inverse problem we must be able to make theoretical predictions of all

observables in the dataset. The forward modelling forms one of the fundamental parts of an

inverse problem. Usually the degree of difficulty involved in solving the forward problem

governs the entire approach to the inverse problem. We mentioned above that ray tracing in

a laterally heterogeneous media is a computationally exhaustive task. Even with modem day

computers, substantial time and effort is required to determine many thousands of travel

times. This is the primary reason for the initial linearized inversion. In this case we need

only solve the forward problem in the starting model, which is usually chosen to be laterally

homogeneous. Problems may still exist in finding an adequate initial model, nevertheless the

complexity of the forward problem is substantially reduced.

6.2.1 Ray Tracing through a laterally homogeneous velocity model

For the phases observed in our dataset ray tracing through laterally homogeneous

models is relatively straightforward. The Doyle, Everingham & Hogan (1959), [DEH model

used in the hypocentre location work earlier (Fig. 2.6)] provides a useful starting model. It

consists of a simple layer over a half space and represents the simplest class of model that

provides a reasonable travel time fit across the entire region. The absence of any velocity

gradients in the model further simplify the travel time calculations. The Pg phase is modeled

as a direct ray and the Pn phase as a head wave arrival (Fig. 6.9). Travel time expressions

for rays in a laterally homogeneous media may be easily derived from the ray equations. The

required Pg and Pn times are just special cases of these, and may be found in most texts on

the subject (e.g. Lee & Stewart 1981). We merely repeat the formulae here.

For the direct Pg ray we have simply,

3-D Linear Inversion 6.7

where r a = (xa,ya,za) is the position vector of the source at point A, r b is the receiver at B,

and al is the P-wave velocity in the Crustal layer. For the Pn head wave phase the travel time is given by,

_A

OCo

2h - z„ - Zi

cos L (6.2)

2 2 1/2

where A = [ (xa - xb) + (ya - yb) ] , the epicentral distance between A and B, h is the

Crustal thickness, (Xj is the P-wave velocity in the upper Mantle, and ic is the critical angle of

the ray between the two regions as given by Snell's law,

1 a i

• • -1 I

ir = sin — a 2

The S-wave phase is modeled as a direct ray similar to the Pg only the velocity a is

replaced with the shear wave velocity ß. In the DEH model the values of these velocities are

given by,

a l = 6.03 km/s Crustal Pg velocity

a 2 = 7.86 km/s upper Mantle Pn velocity (for epi < 400 km)

= 8.16 km/s f t t f i f f t

(for epi > 400 km)

### P

= 3.16 km/s shear wave velocity (for epi < 400 km)

= 4.53 km/s t f 11 f t 11

(for epi > 400 km)

The second velocity quoted for CX2 is used to model phases which bottom deeper in the upper

Mantle. Because of the magnitude-distance ratio involved, in practice, only a very few of

these are actually observed (see Fig. 6.7)

Note that the above equations are given in a rectangular coordinate system. Since we

are dealing with regional distance scales (approx 7° x 6°) ideally some projection of

geometric to rectangular coordinates should be used. Here we have employed the

3-D Linear Inversion 6.8

length scales is found to be less than 2 %. Overall the projection forms only a minor correction.

Figure 6.9 (a) Diagram of Pg, Direct Crustal ray.

Figure 6.9 (b) Diagram of Pn, Head wave or critically refracted phase.

As with most inversion problems, approximations are introduced in order to make

the forward problem tractable. Errors introduced by the forward modelling are possibly the

most troublesome of all since it is here that an appreciation of the physical nature of the

3-D Linear Inversion 6.9

relationship is taken then little can be done to rectify the situation at a later stage. For

example, if the forward modelling errors are of a comparable size to the observed travel time

residuals then we have no way of distinguishing between real velocity structure effects and

artificially induced heterogeneities. In our case we have a very simple starting model which

allows no curvature in the raypaths. However we are aided somewhat by Fermat's principle

which tells us that for a fixed velocity field the travel time is stationary with respect to small

variations in the raypath. We do not expect therefore that the forward modelling errors are

significantly large. Furthermore the comparison between the relatively crude straight ray

approximation and the sophisticated full 3-D ray (to be used in the nonlinear inversion) will

provide useful insight into actual size of the forward modelling errors and their effect on the

inversion.

We note in passing that by differentiating the above travel time expressions with respect to the source parameters ra = (xa,ya,za), we obtain, reassuringly, a special case of

(5.9), the Frechet derivative expressions for a general ray. Since we require only the values

of the slowness at the source and the direction of the ray there, these terms are easily

calculated. The Frechet derivatives for velocity model parameters are also determined at this

stage. However these are best discussed in conjunction with the velocity parameterisation

which we have as yet left unspecified. Overall then the ray tracing in the linearized inversion

is computationally very cheap and performed without difficulty.

6 2 2 Seismic velocity field parameterisation

It has been noted by many authors including Crosson (1976), Tarantola &

Nercessian (1984) and Koch (1985a) that one of the most important aspects of an inversion

problem is the type of model parameterisation used. In the seismic case the way in which we

may parameterise the velocity structure is constrained by both the quantity and quality of the

available data. Essentially it forms the window through which we attempt to view the real

earth. The results of any inversion can be only as flexible, or as detailed, as the

3-D Linear Inversion 6.10

In the 3-D inversion, the parameterisation places a lower limit on the scale length of

heterogeneities that can be determined. If too coarse a parameterisation is used then any

information on structure at smaller scales will be lost. Ideally the nature of the

parameterisation should be governed by the wavelength of spatial resolution in the data.

Usually the only indication of this is given by the spatial distribution of the raypaths, and so

the parameterisation scale is chosen accordingly. (The term 'parameterisation scale' is taken

to mean the size of the model region that each parameter influences.)

Another important factor is the type of inversion algorithm used. Some methods will

suffer from instabilities if the model is parameterised too finely. This fact has tended to

encourage the large constant velocity block parameterisation used in the early work of Aki et. al. (1977) and others. There is, however, a certain dissatisfaction with an inversion procedure that dictates the type and size of parameterisation. Methods based on the inversion

of large scale matrices usually treat the problem as an over-determined one and hence they

assume that all model parameters are well sampled, and are resolvable, with the data. In

most problems using real data there exist local areas of under-determination (regions where

the model is poorly sampled) which subsequently induce instabilities into the procedure.

These effects are actually the fault of the inversion procedure rather than the

parameterisation. Ideally an inversion algorithm should not become unstable simply because

there are regions within the model which are poorly sampled. A uniform increase of the

scale of parameterisation (be it a block structure or otherwise) may overcome this problem,

however this will almost certainly result in the loss of detail elsewhere in the model. More

recently authors have tended to use as flexible a model parameterisation as possible in order

to resolve as much detail as the data allows (Tarantola & Nercessian 1984, Nolet 1987a).

This requires increasing the number of model parameters and hence a very large scale

problem results. The problem of local under-determination is dealt with in a different way,

usually by smoothing i.e. introducing some dependence between model parameters that

influence closely spaced regions of the model. Although applying smoothing is very similar

3-D Linear Inversion 6.11

smoothing technique is more versatile. In this case the scale length may be varied between

parts of the model, or even during the inversion itself.

The search for an optimal parameterisation is extremely difficult. Koch (1985a)

attempts a comparison of different types via a statistical analysis of inversion results, as does

Aki et al. (1977). However this type of approach can really only suggest a preferred type or scale from the few tested. Ideally, the type of model parameterisation should be determined

by the available data and the data-model relationship. In section 5 we rather loosely defined

the velocity parameterisation by using a series of orthogonal functions in space. In particular

we choose to work directly with the slowness field rather than the velocity field, because the

integrand in the travel time integral given by (5.3b) is linear in slowness but inversely

proportional to the velocity. This is a good example of how the nature of the forward

problem suggests the type of model parameter to be used. However we must still decide on

the particular form of the orthogonal functions in (5.12).

Since we are at present considering a linear inversion using relatively simple

approximations to raypaths, it seems unnecessary to concoct too elaborate a

parameterisation. Essentially the aim of the linearized inversion is to establish whether we

can image laterally heterogeneous seismic structure from the available data and to provide a

basis for a more detailed nonlinear study. In the linear inversion then we will follow many

previous authors and use a simple constant velocity cellular parameterisation, only we

include a relatively large number of cells to allow flexibility in the model. Each model

parameter affects only one cell and represents the value of the slowness in that cell. The cell

dimensions are determined by considering the variation of raypath density across the model.

Lateral dimensions of a half degree by half degree were chosen, which produces 128 cells in

a layer (Fig. 6.10). In order to examine lateral variations of average Crustal and upper

Mantle P velocities, we use one layer for the Crust and a second below for the upper Mantle.

For the S-wave structure a single layer is used. Since the DEH velocity model (above) is

g u re 6.10 A p e rs p e c ti v e view o f the veloc ity mode l p ara m ctc ris ali on fo r S E A u stra lia , incl udi ng th e S E tw ork rep re sen te d b y triangles. The p eri m ete r o f d ie 1 28 cells in each layer i s sho wn in b o ld .

3-D Linear Inversion 6.12

With the cellular structure the Frechet derivatives become particularly simple. As

shown in section 5.2 for the cellular structure the orthogonal functions are simply block

functions in space and the Frechet derivatives become the pathlengths of the rays in each cell

i.e.

These may easily calculated when solving the forward problem.

Because of the considerable variation in raypath density across the model (Fig. 6.6)

any regular cell pattem will result in large variations in model parameter constraint. Some

cells will have many more rays passing through them than others. The half degree by half

degree cell dimension is chosen as a compromise across the entire model. We note however

that within the network the parameterisation is likely to be too coarse, whereas, at the very

perimeter of the model it may be slightly ambitious. Here, again, the results of the linear

inversion will provide an indication of the size of heterogeneities present in the region and, in

doing so, be useful in determining the scale of parameterisation most suitable for the

nonlinear inversion.

Note for the linear inversion the choice of parameterisation is completely independent

of the ray tracing procedure, whereas the nonlinear inversion will require the tracing of rays

through a perturbed model. The relatively simple cellular parameterisation is really only

suited to the linear inversion and so a more sophisticated one must be designed for the

nonlinear case. The linear and nonlinear inversion studies will, therefore, be performed

using two different parameterisations. Since this may contribute towards differences in

results, we prefer to use the simplest parameterisation possible for the linear study. Overall

the model parameters lend themselves easily to physical interpretation, and are well suited to

3-D Linear Inversion 6.13

6 2 .3 Mo ho Topography

In south-east Australia a topic of recurring interest is the variation of Moho depth.

Several authors have reported Moho depths for the region, chiefly through seismic refraction

studies. We ask, then, whether any constraint on Crustal thickness can be determined from

our refraction data. In the above formulation of the model no attempt has been made to allow

variations of the Moho position, which is essentially the boundary between upper and lower

cells in the model. In the DEH model the Moho is represented by a first order discontinuity

in seismic velocity. To attempt an inversion for the Moho shape we must introduce extra

model parameters which should define the position of the Moho at all times. A very simple

way of proceeding is to let one model parameter represent the depth of the bottom face in

each cell of the upper layer (Fig. 6.10). This gives a reasonable first order impression of a

surface and fits in well with the cellular parameterisation of the velocity structure.

Having introduced extra model parameters we must find the corresponding elements

of the Frechet derivative matrix G in (5.6). In the section 6.2.1 all rays which encounter the

Moho discontinuity were designated as head waves. This gave a simple expression for the

travel times (eqn. 6.2). Upon differentiation of (6.2) with respect to h, the average Moho

depth, we obtain,

3T

### 3E"

2 cos i.

(6.3)

Since the Pn ray actually encounters the Moho at two points (C and D in Fig. 6.2), which

will most likely lie in different cells, we require the travel time derivative with respect to the

Moho depth at each point. This is simply half of the above expression. If the i th ray is of

type Pn then,

cos ic

«1

3-D Linear Inversion 6.14

= 0 otherwise (6.4)

and for a Pg or S ray,

where Cj and dj are the indices of the slowness cells in which the ray hits the Moho.

Using this Moho parameterisation we restrict our attention to vertical displacements

of the Moho only. Again a more elaborate type could be devised, perhaps resulting in a

more physically realistic Moho. However, at present it is unclear how much information on

Moho structure can be retrieved from the data. In addition we do not know what effect the

raypath approximation might have. As preparation for the nonlinear inversion study (see

section 7.3) we examine the case of a ray traversing two heterogeneous regions separated by

a planar discontinuity. We derive an expression for the perturbation in travel time resulting

from a small perturbation in position and inclination of the interface. (This is used to

determine partial derivatives, analogous to those above, for the nonlinear case where accurate

raypaths are known.) The result suggests that the angle at which the ray strikes the surface

plays an important role in determining how the travel time constrains the position of the

Moho. In the linear inversion the Pn ray is modeled as a critically refracted ray and so the

ray angle at the interface is given by Snell's law between the two media. At best this will be

only a crude approximation to the true angle. More importantly, in the initial model nearly all Pn rays are assumed to strike the Moho at the same angle i.e. sin'^cXj/c^). Where a x and

a 2 are given above. (The only exception here being the few sources at large epicentral

distances, for which (X2 takes the larger value 8.16 km s'1.) We might expect then that the Moho inversion will be hindered somewhat by the raypath approximations. Nevertheless it

seems worthwhile to include it as a variable if only to see whether allowing it to vary affects

3-D Linear Inversion 6.15

In document Seismic inversion for earthquake location and 3-D velocity structure (Page 104-114)