Details of the Specific Inversion Algorithm Used

ray happens

to penetrate

more than one block, only the block "·ith the longest ray segment is

used, and

it is

assumed that the

ray

traverses the entire

layer within

that single

block.

This

greatly

simplifies the geometrical considerations that

have

to

be made. Once ray segments are associated with an entire block,

all

ray lengths within

any

block are

approximately equal and equation

I.l

can

be simplified

to

sb

=

~dr /~trb without

perceptible alteration of

the

r r

inverse. Comparison

of the

mverse

constructed

using

this

formulation with

that produced with the

use of

I.l

("·here

in both instances

the one

block

per layer

approximation has

been

used) gives

a

difference in

the

most deviant

blocks between the

two

inverses

of

less

than

1% ,

and most

blocks are unal-

tered to within the four significant places kept in

the data

files.

-44-

procedure such that the

ray

trajectories are

recalculated

between

iterations.

This

has not b

een done. In

this chapter all rays have been traced through the simple structure shown

in Figure II.5.

Some

justification for

use of

this

con- stant structure

is

that travel time variations are second-order on the

ray

path,

and hence mislocating

a

ray slightly will

not

have

a serious effect on

the

delay.

However,

the ability to properly

locate

and

reconstruct

a

velocity

anomaly

is dependent

directly upon the ability to

locate

the

ray's

positions, and

this

can be quite sensitive to the

velocity

structure.

Partly for

this

reason

the

image resolution is

expected to

f

all off somewhat away

from the

earth's surface (where

the

stations are

located and the ray's positions are known). The

man-

tle velocity variations

are only of a few

percent,

however, and the problems associated

with

mantle

heterogeneity

are not expected to be of too much consequence.

Block sub -binning

The teleseismic ray set used in

this study includes only rays that are more

vertical than 45° from normal incidence. (If ever a

ray is

more

horizontal

than

this

it is excluded from

the data set.)

The result is

a tendency to blu

r

the

image more strongly in

the the vertical direction, and horizontal structure will

the

refore be

the more difiicult to

resolve. T

o

partly

compensate for this prob-

lem a scheme has been

emp

loyed

that

reduces the weights of those rays which traverse the

structure

in

the

directions

most commonly taken.

The procedure for accomplishing

this

is to divide

the

rays

that

hit

any

particular block into

subsets according to

ray parameter and azimuth (as in

dicated

by

the template

in Figure II.

Ga)

. T

hese

in

dividual slo\\·ness estimations a

re then averaged in

VR (

5ec/

km)

0 0~~~2~~3~~4--~5--~6-,r7--~8 __ 9~- I ~O __ I _ I_

-

0 0

0 0

N

EO

.:11! 0

- r<)

.c.

-

_0..0

Q.)o oc:::t

0 0 1.0

0 0 c.!>

0

1'-0

0 0

CX)

Figure II.5. One-dimensional P-velocity structure used to guide ray paths. This structure is a discretized representation of the structure determined for the Gulf of California by Walck (1984). This structure is used only in determining the ray paths, and because the ray paths are fairly vertical the exact structure is not critical.

-46-

counts/ bin 10

Figure Il.6. Figures showing the details of the binning scheme. The nine bins are shown in (a) in ray parameter-azimuth space. North is to the top and the numbers are in units of seconds/degree. Part (b) shows the weight given each bin (as a function of hit count) prior to averaging to obtain a whole slowness estimate. For more discussion refer to the text.

order to obtain a ·whole-block slo,vness estimation.

The

aYerage

is

performed

by weighting

each subset by the

functi

on shown

in

Figure II.6b.

This fun

ction

has the

effect of

some\\·hat increasing the weight

of those subsets

which are more frequently hit. -otice however,

that

this weight

decreases

the impor- tance of individual rays that fall within

the often

hit

subsets. \Vith

the block

size

used

(15

km square in map view and 30 km

deep) the

number

of hits per

bin

range

in value from zero to

over

a hundred, with a median value for bins

actually hit of three.

These weights have also been used to describe in a simple manner the

quality of oYerall ray coverage

experienced

by a block.

\

Vhen trying to

describe simply the quality

of

sampling, a probl

em exists

because both the

number of

rays

hitting a block and their distribution in azimuth and

in ray parameter are important. The

method we have

chosen

to display ray coverage

quality is

to show the aYerage subset

\veight just discussed.

This

is

a number

that increases

with hit

count.

but

also

increases \\·ith

the diversity in

ray COY-

erage.

Th

e

valu

e of this

number is

zero

if the

block

is not hit, and

cannot exceed,

for a well-hit block, a value of unity. Plots

of

the hit quality are

shown

in Figure

II.7. These

plots will be important to refer to later when the

various inYerses, of both synthetic and actual

data are

presented. The

coYer-

age

estimator

is

seen to decrease towards the margin

of the

inversion domain

in

general, and below the Pacific

Ocean in

particular. Also , the quality of

sampling is seen

to dec rease with depth

.

This property

is

rel

ate

d fundam

en-

tally to the width

of the seism

ic

array, since at depths

roug hly

comparable

to

the array

width angul

a

r

coverage begins to rap

idly

diminish.

~ lA ' ~~~~bci L )%:

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; }~ ! ~~~~ ~~; ;~~ : :::::::::;:~~ ~~ ~~~ = ~~~~~~~· ~

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·

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0ot l'',Po0Pl0o(' :~·:f.,'-', • .. • ~· • · ""' 1< ,p0o0t 0r o, -: • ',,o, ~.~ ~.: _, .ono0o0c'o' 1 0.• ~-oc -:~t' ; ',o~o~G~o~c.:· ... :!~ ·-~ ~~~-.... '::.. _..,. •" · on no c .... , ·, ,, • .., [·'' ,. ,,, v • .... L.++++·o,,.,++''·'··· -. 0 •r~ + -+'•· .. "\"I··-· ~ 'II' 'o : ..... oct::·.+.+.··~.; .• ·., .,; Po0/oto0• c ... :.~·-· :·.t~. -~ ·BOG-I :+: "·Co.·;.;: >,to poe ,a Oar' ~.; :·~ 1·~+:·:; ~::.~ ~:

-o-- -tO(). -zoo- -SilO- D D' ;t--.;;;t.' ··: • '.,,' :·: ~ '~ ;,:•:•:"~t :~~0o:o~')~,r ::co~~ ~· ... ' • • " l o; 0 0 I) t. • e tl (. (I • I' 0 , + ... • Jno•••••n•uoo. ~.-· ;f:* '• oo,.~;.•.•.•.•.•.•·.tt.•,ou•oo,., ito ~++.f.+,.: .. ~;;; ot. ~Ooo:•:•:•:•:•:•~·~•=v~Oof)< o' 0 • ~++++{ • ': Pc ooJo • • • • • • • o o fJJo0o0cPc ~ .. •..:·· 1 p0~0u0•.•.•.•.•.v•••'-••l)o0ono0o0 ~ ' ~t >~o~o0 o:•;(f:•:•:•:•:•:•:.,~o~>~"'0 o' · ,. J r o0o • • • • • • • • o n f"' o01°c ~o0 • , ~o:•~•:•:•:•:•:•:•~o~ " ~?~o~ -41)(). ~t.;

('I () rt ,, •• •1 ••• '). l 0,., ·~· ,·Jo~t,~o"•••~'.••rt••ct:~,c ', .,.,, lj.~ .. o0 0 0o0o•o•••o"o0• 4 c,, ^{1 1},.; .• .·;;Jo0c>o0e~nn••,pn°o~ c-,. • ~ ' p o ::Jooo•,po•eOc.' .. ol . Jc ~1otJd1rPoeto0o~_ ~·· ·,. o' '/ J • 0 0 0 0 0 ~:o-~ ~ ) t ~ JOC ::Jt. ')00000 hO~t; o l • ,t. ~: ~·:· ~(' 10( ~ ~O~O~')~t '•~· .. :( j ~} ·~~~ {.? I~\)~~~~ "~Q~Q6l '4' ...... :. -. )o~,

-I!CJO- -eoo- -?CJO- of: .. { ~~o00< 'J~o~O ~ ;>\,~~++ .i',.: ,:~ .. 80().-I ~o' 1-.t~ 10 "c 'oco,c-.;! ,.~-r-:-t~t'!'ttt+'i';.J -o-- -too- -200- -SilO- -41)(). -1100- -eoo- -?CJO- -&oo- E E' ~1'+1'+1'+1'+++1'+1'.( u,., ·/++~· J ,.,, ··4 'e~••CJ••••o·.._,++<t· ~!+!+!+!:!+~}+'~"+ .~ ~~~~c)~ ·~o:•:•:•~r·, t•!~ ~!•!+!:r \!<";f '<!::;.,':OJc=t·~ ~o~e~o~o:tt:c· ' ·t~ +++++; ~)o++.f' '*:>0-:;,el'oGolo'-, oee!tOO~OOOC • } :+++++; :~+-4-o ..0 ::r: ',o,. . .,ooo, -'ooooooooc'~, . ,.,.. ~+++,.·:"">+of:,' ;-.,.:: "'"!' 'oci o0o0rPo0 l ,+++;:+'+++~ .'<-o~-~. ,f oc ·~o~o~o~o~r ,o~p ~ + + +,.. ·~ . .. ~ 0 0 0 0 0 + •.. + + -t :-; , ~ 0 ) .t-.• 0o0o0o0o0oc.. ,o~c. ~ . ., · .. +!-t~:.~: J0... oc. /"'o0o0o0ego0o0o') o0r< ~·::~++~·!·::l·· >0o~o0 o~ 0~~o~o~•:o:o~o~o~o~<"o .... ~·<·' ····:·~ p o0o0o0o.•0$0o0o0o o0c..r,o( _, ;-~~. ~ro)o0o0o.e.f'J0o0o0o~Ot" ">,. ' 't J. <

e ~,. \Jo~o~o~o~•~o~o~o~o0o':p~' ~ -~o0o0C'c ro,..,~o·'·Pr' JC.vJO~t < ,.:::

,.~o~., 0t '. ~o1'o~ "~ 'l,, 0 \) ( ,t I Q .~ 'o .0.' ,c o0o~ -l't o0• ,0~ . : ,...<" ,o ,, 0d ' ·~ ' ~ • I ~c ·o.~J ' ~!~~. Figure 11.7. Displays of the hit quality factor discussed in the text. This is a measure of the variety and number of hits a block experiences, and is determined by tracing rays back from the stations shown in Figure II.3 through the structure shown in Figure IJ.5 according to the ray information given in Figure 11.1. An ideally hit block will have the maximal hit quality factor of one, while an unhit block will receive a value of zero. Practically speaking, a hit quality factor of over O.J 8 seems to indicate sufficient ray coverage to resolve most struc- ture.

~

.-. ·~

···:· .· ..

.:·:·:· 4 0 ar:~· )~060~~ hooooo,

il ::

~n

l81ill

00 1

~·~· ·

... ... • ... + +.+_..._+

I ,.t:... 00 I

<

0 0

-

^I

0 0

-

^I

Q

0

c:)

<

I

0 0 N I

0 0

~

I

0 0

~

I

0 0

~ 0 0

r

0 '

0 r

I

0 0 l(l

I

0 0

"."

0 0

c I

0 0

"i' 0 0

r--1

0 0

r;-

w

Q 0

cc

<

0 0 a) I

'

0 0 a)

I

u

(.)

0 ' _I

I

0 '

:

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-

^'

0 0

-

^'

0 0

-

^'

'

0 N 0 I

0 0

~

0 0

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'

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~

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^I

0 0

"r

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"'

^I

0 0

"."

0 0 l(l

'

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"."

0 0

"'

^I

0 0

"i'

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"i' '

0 0 r;-

0 '

0

!'-

'

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r;- 0 0

"i'

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0

"i'

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^'"'0 (])

c

::J

- ^c

₀

...

()

- .

-50-

Figure 11.7 (continued}

Figure 11.7 (continued)

-52-

Figure 11.7 (continued)

0-30 km

/""/

/ / / / /Y

/ / _;<

\

-53-

270-300 km

120-150 km 4 50-480 km

~ • ~o~•~•

0.10 0.03 0.01 0.0001

a)

Figure II.8. An average reconstruction of an anomalous block of unit magnitude to a single back projection. A delay set has been constructed by projecting the actual ray set through the assumed structure, and nine individual responses have been determined and averaged.

In (a) the reconstruction is formed by straight back projection, while in (b) the binning described in the text has been applied. Notice the improved ability to attenuate the prom- inent streaks with the use of binning. The response shown in panel (b) has been windowed and used as the point response in the deconvolution step. Another point to be made is that, because the single block response is similar to that block's averaging kernel, the response can be used to estimate the resolution capabilities of the back projection.

-54-

0 0 0

c!,

0 0

0 0

'

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cD

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0 0

cD l'-

'

0 0

10

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0

•

^I

c!,

0

C')

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N I

...

_{I I}

I I I

0

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...

_{I I}

f:Q

u

0

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...

_I

<

'

<

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0 0 C') I

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0

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•

^I 10 ^{I I}

'<"-

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,~' '\.

~"""

0 0

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l'- cD

I I

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(1)

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c:

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r- cc I

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Q) ::J c

·- - ^c

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() ~

co .

-56-

•

0-30 km 2 70-300 km

120-150 km 4 50-480 km

Figure 11.8b

~

6 6 6

6

^{I I 0}

-

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^"'

^{0 I}

~

6 6

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-57-

r.q

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< L_ __________________ ~

-58-

0 ^{I I I} 0 I

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Deconvolution

Deconvolution is accomplished in the ·wavenumber domain through divi- sion ·with an empirically determined point spread function (Green's function).

In practice, two steps have been taken to insure stability. First, the spread function has been windowed so as to include only 7 blocks in the vertical direction, 9 blocks in the two horizontal directions. To keep windowing from producing ringing problems the window boundaries have been tapered. \rin- dowing effectively removes the longest wavelength components from the Green's function. The second step is to clamp the "·avenumber domain representation of the spread function so that it never falls below some specified level, which in effect eliminates the the high \\·avenumber components from the deconvolution. This level has been chosen to be the R\1S value of the response function.

\Vhen the ray set is not homogeneous, the response functions produced from differing locations are themselns different. To minimize this problem, the spread function used in deconvolution is constructed by taking the average of nine point spread functions sampled from regularly spaced locations within the inversion domain. Such averaging is justified if the response function does not vary much ·within the windowed region, and in fact it does not. The pri- mary purpose of the windowing is to remove the distal portion of the spread function since this is its least constrained portion. The average response func- tion, prior to \\·indowing, is shown in Figure II.8. The resulting pattern is a map of the ray paths hitting the anomalous block. This is easily seen, for instance. by comparing Figure ILl b with the lower right panel of Figure II.8a

-60-

(map view). Figure Il.8a shows the spread function produced without the use of binning, while Figure II.8b has binning included. l'\otice the degree of suc- cess binning has in attenuating the strong streaks.

Space Averaging

One final step is included, ·which is the application of a moving average window after each iteration. The inversion space is spanned by 51 blocks in the E-\V direction, 37 blocks in the N-S direction, and 25 blocks in depth, where each block is 15 km on a side in the horizontal directions and 30 km deep. In most parts of the inversion space the ray coverage is not adequate to warrant the use of blocks this small. The averaging window is therefore designed to vary in size in inYerse proportion to the hit quality estimator (Fig- ure II.7). \Veil hit blocks (hit quality factor greater than 0.40) are averaged only ·with that block's four nearest horizontal neighbors, with the sum of these blocks given a weight equal to that of the central block. Blocks of intermedi- ate hit quality factor (factor between 0.18 and 0.40) are further averaged "·ith the eight next nearest horizontally located blocks, the sum of which is also given a weight equal to the central block. For poorly hit blocks (hit quality factor between 0.05 and 0.18) the eight next nearest horizontally located blocks are also included, again with an amount of "·eight equal to the central block. If a block is more poorly hit than hit quality factor 0.05, no inYerse is determined for that block. In all cases, the anraging also takes into account the hit quality factor of each participating block by weighting that block in proportion to its particular hit quality factor. The use of such an average does not distort the inverse. In fact, with the weighting scheme used, for each

iteration the combination of several smaller blocks into a fe''' larger blocks giYes the same result as "·ould be obtained

if

the inYersion \vere run originally with the larger blocks. This point is well illustrated by comparing the inver- sion just discussed with that discussed in Humphreys et a!. (198-l), where the individual block volume is about 8 times as large. Both this and the more detailed inYersion are sho,vn below.