determination of the three-dimensional seismic structure ... · and experimental works are needed...

VOL. 82, NO. 2 JOURNAL OF GEOPHYSICAL RESEARCH JANUARY 10, 1977

DETERMINATION OF THE THREE-DIMENSIONAL SEISMIC STRUCTURE

OF THE LITHOSPHERE

Keiiti Aki

Department of Earth and Planetary Sciences, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139

1 Anders Christoffersson

Department of Statistics, University of Uppsala, Uppsala, Sweden

Eystein S. Husebye

Norwegian Seismic Array, Kjeller, Norway

Abstract. A new three-dimensional earth model- the east. The latter feature may be attributed ing is proposed as a framework to obtain more detailed and accurate information about the earth's

interior. We start with a layered medium of classic seismology but divide each layer into many blocks and assign a parameter to each block which describe• the velocity fluctuation from the average for the layer. Our data are the teleseismic P travel time residuals observed at an

array of seismographs distributed on the surface above the earth's volume we are modeling. By isolating various sources of errors and biases we arrive at a system of equations to determine the model parameters. The solution was obtained by the use of generalized inverse and stochastic inverse methods. Our method also gives a lower limit of the true rms slowness fluctuation in

the earth under the assumption of ray theory. Using P wave residual data from the Norwegian Seismid Array (Norsar), we have obtained the map of velocity anomalies at various depths up to a depth of 126 km. The'rms slowness fluctuation was found to be at least 3.1%. This is in agreement with estimates obtained from statistical

analysis of P time fluctuations based on the Chernov theory. The three-dimensional velocity anomalies are presented both by the generalized inverse and by the stochastic inverse solutions. We prefer the dual presentation because it gives the reader greater freedom in judging the results than a single '9ptimal' solution. Both methods gave essentially •he same results. The discrepancies, when they existed, were always explainable in terms of differences in the smoothing procedure which is explicitly given in the resolution matrix. The dominant features in the

obtained three-dimensional velocity image of the lithosphere beneath the Norsar array are low velocities to the west and high velocities to

Ordering of authors is made alphabetically

1 Visiting staff member at Lincoln Laboratory, Applied Seismology Group, Massachusetts Institute of Technology, Cambridge, Massachusetts 02142

to rocks of the Baltic shield which are undis-

turbed by the Caledonian orogeny or by Permian volcanism. Our result conclusively demonstrates the existence of strong small-scale inhomogeneities to the bottom of the lithosphere. More theoretical and experimental works are needed to relate these velocity anomalies with the magma ascent mechan- ism which caused the Oslo graben.

Introduction

Recent developments in solid earth geophysics, such as plate tectonics, earthquake prediction, and exploration for geothermal energy, demand in- creasingly detailed information on the three- dimensional structure of the earth's interior.

Experience from large-aperture arrays like Norsar, Lasa, and similar types of seismograph networks has shown that conventional layered medium models used for interpreting seismic refraction and wide- •ngle reflection data cannot satisfactorily explain the observed P wave travel time and amplitude anomalies.

The antipode to the conventional models is to assume that the seismic structure is so complex that it behaves as a medium with random inhomo-

geneities. Recently, this kind of model has suc- cessfully been introduced in the interpretation of observed P wave travel time and amplitude anomalies by using the Chernov [1960] theory for acoustic wave propagation in random media. The Chernov theory is based on a statistical description of the medium in terms of the mean square of fractional fluctuation of velocity, the correlation distance (the distance between two points in the medium at which velocity fluctuation becomes uncorrelated), and the linear extent of the inhomogeneous region.

The crust and upper mantle beneath the Lasa and Norsar arrays have been interpreted in terms of Chernov's random media parameters by Aki [1973], Capon [1974], Christoffersson [1975], an•-•ahle et al. [1975]. Moreover, Berteussen et al.' [1975] were able to explain about 60% of the variance of travel time residuals and amplitude anomalies observed at the Norsar array. However, the drawback with the random media description of the crust

Copyright 1977 by the American Geophysical Union. and upper mantle is that only the average proper- .

Paper number 6B0578. . 277

278 Aki et al.' Three-Dimensional Structure of the Lithosphere

ties of the scattering sources are considered. In this paper we shall introduce a new earth-

modeling approach with velocity fluctuation similar to that of the Chernov medium but defined

deterministically within the volume of crust and upper mantle structures. Our new modeling is simple but flexible and can accommodate complex geological structures in contrast to conventional layered medium models with uniform material properties within each layer. The starting point is the classical earth model which consists of homo-

geneous layers with uniform thickness and fixed average velocity. Each layer is divided into many blocks, and then each block is assigned an inde-

data observed at the array for many teleseismic events.

Ideally, we would like to divide the whole earth into blocks of size comparable to wavelength. The finite amount of available data, however, imposes limits to the block size and volume of the earth to be considered. Since we are using ray theory for computing the travel time, it is implicit that the inhomogeneities are smooth within a wavelength. On the other hand, anomalous structures with scale length much shorter than the wavelength will behave as an equivalent homogeneous body with some average properties. It is therefore meaningless to make the block size

pendent parameter that describes slowness fluctu- much smaller than the shortest wavelength of the ation from the average value for the layer. By incoming signals. inverting the teleseismic P wave travel time data To formulate our model, we start with con- from an array of sensors placed on the earth's s•ering an arbitrary ray passing through the surface, the slowness fluctuation for each block s layer. The • denote the average velocity is calculated, and the result is examined by reso- in the layer, an• • the distance traveled lution and error analysis. through the layer h•d there been no velocity per-

We shall apply our method to the Norsar data turbations. Further, let d denote the actual and compare the resultant three-dimensional veloc- distance traveled in the l•yer, and v the actual ity anomalies with available geological and other velocity. The travel time for the actual ray in geophysical data pertinent to the region.

Formulation of the Model

the layer is then

In this section we define our model, formulate the problem of inverting the travel time data, and systematically consider the sources of errors and biases possibly affecting the final solutions. The initial model consists of homogeneous layers with constant thickness and fixed average velocity as shown in Figure 1. Then we define somewhat arbitrarily a volume of the earth under the array which contains the observed ray paths. The earth's structure outside this volume is called 'standard

earth' and is assumed known. Each layer is divided with d = • + Ad into many blocks, and the slowness fluctuation in s s s' each block will be determined from the travel time

•/Sensorsx•

= d /v (•) ts s s

We shall define the fractional slowness fluctuation m by

s

--

1/Vs = 1/Vs(1 + ms) (2)

STANDARD EARTH

Fig. 1. The initial layered model with constant thickness and average velocity v . is divided into many blocks. Our prob-

O1

lem is to estimate the velocity perturbation m for each block using the P travel time data for many teleseismic events observed at the receivers •above the model. The velocity perturbation m for a particular block in a particular layer represents an average over all the rays having most of the unperturbed path in the layer within the block.

Then, rewriting (1), we get

(d +Ad)

_ s s (1 +m s) t s - _ v

s

(3)

Consider now the arrival time t at the sur-

face for a ray going through all the layers of our model. This can be written as

NL

= + ß + c (4) t I t s s=l

where NL is the number of layers, T is the arrival time at the bottom layer, and e is a residual term containing measurement errors. Further, letting Y denote the arrival time at the bottom layer for the unperturbed ray and assuming AT = T - Y, we obtain

--

NL d

t = • + • --s(1 + m ) + AT + -- S s=l V

s

NL Ads( + • •--- 1 + m s ) + z (5) s=l v

s

Now, according to Fermat's principle the travel time is stationary with respect to path perturbations around the geometrical ray path (see the appendix). In our case this principle gives that

NL Ad

Az + • --s(1 + m ) (6) -- S s=l V

s

is of higher order than the other terms in (5) and thus may be included in the residual term e.

Aki et al.' Three-Dimensional Structure of the Lithosphere 279

Equation (5) is now written as

NL d

t = • + •. --s(! + m ) + s - S S=i V

S

= • + •. gs (1 + m s ) + s s=l

(7)

..

where g = d /v . We divide each layer into s s

blocks, and within each layer we assign to each ray only the block that contains most of the unperturbed ray path. To account for possible con- tributions from neighboring blocks when parts of the ray pass throu• them, the slowness fluctuations within the i-" block of the s •" layer are decomposed as

--

= + Am. (8) msi msi - th

where m . is the slowness anomaly of the s S1

layer averag• over rays having most of the path within the i-" block, Again rewriting, we get

culated from the initial model, standard earth, and focal parameters, all of which_ are assumed to be known. The slowness anomalies m are unknown and thus subject to estimation. By also treating m as an unknown parameter, those errors in Yøthat are common to all instruments for a given event, such as errors in origin time, and effects of anisotropy if the anisotropy is uniform throughout our block model, will be absorbed. Other types of errors in Y may affect our estimates of slowness anomalies.

Let us examine the effects of small errors in

the •tandard earth, initial model, and source parameters for the events considered. Suppose ß _ and G are unknown and thus hAave to^be replaced by their respective estimates • and g. We shall write

A • = • - a(q) (•)

and

• = G - A(•) (12) NL NL NB A

- - where A(•) is caused by errors in epicenter loca- s=l s=l i 1 sz S• tion and--standard earth model and A(G), in addi-

tion, by errors in the initial model of crust- where NB is the number of blocks in each layer, Fsi is an indicator which is 1 if the unperturbed •g in the s-" layer has most of its path in the block and is zero otherwise. The residual s*

is now the sum of s in (7! and Amsi defined in (8).

The next step is to ..express the observed arrival times at all insgrument•. We introduce the following notations:

total number of instruments, total number of blocks in the model (NL x NB)•

mantle structure under the seismograph array. . A

Using • and •_ instead of G and • in (10) gives . A

t: mc•_• + Z + A(•A) + •n + A(•)•n + s (13) If the error A(•) is of first order, the term A(•)• .is of second order and can be neglected. For •(•) the situation is different because this term is of the same order as errors in the stan-

dard earth model and errors in epicenter location, the former effect being more serious.

The effect of near-source small-scale inhomo- t

calculated fro• a standard earth, equal to (71 , ... , 7N)--;

m tr•vel time õpent in the mod• for the un- o perturbed ray, equal to •. . g • =l $

i vector of which all the •ements are unzty, equal to (1, 1, ... , 1) •

m vector of slown_ess anomalies defined by (8),

-- equal to (•igl• m12 , ... , •NL N ")T; • vector of oraer and erro• •erms, equal -- to (Sl*, , SN, )T' o ,,, . '

•11 ''' g. lM] where the elements of

vector of arrival times a t the_surface of the geneities is considered negligible bec.4use the earth, equal to (t 1, ... , t. ) '•'• long transmission path through the relatively vector of arrival times at t•e bottom layer homogeneous lower mantle smooths out such effects

on the wave front before reaching the instruments. The larger the scale length of the inhomogeneity is and the closer to the station it is, the greater will be the effect on our solution. The choice of events with good azimuthal and distance coverage will minimize the effect of the medium sized inhomogeneities at medium distances. There- fore inhomogeneities near the station but outside our block model are assumed to be small.

^To summarize, we have to assume that the errors A(k__') are of higher order than other terms in the model or that the sampling of events is such that

gt• = g if the ray to the expectation of the errors in relative arrival itnSinstrument passes times (those common to all the instruments were through the j th block, absorbed by m ) at the bottom of our model is which is in the s th layer zero. In practice, this means that the events and zero otherwise• used in the analysis should have good coverage

in azimuth and distance. Equation (9) will now become

(lO) -- --

t = • +mi+ Gm+ s

The term m represents the travel time for rays passing through the initial model_which is over-

Estimation of Model Parameters

Equation (10) contains the parameter m common to all stations and the slowness anomaly o•. The

lying the standard earth, while Gm is the travel least squares solutions for these two parameter time anom_alies due to perturbations in the blocks. sets can be determined separately if we rewrite The term ß represents the travel times through the (10) as standard earth from the focus to the bottom of

our initial model. G, •, and m can all be cal- t - • = (G - G)m + m i + Gm + ½ (14)


where the elements of G are the average of G over

all stationsß Using gij introduced earlier, we can write N N

gll- •' gil/N ..... glM - • giM/N i=l i?l

G_G = ß . ' N ' N

- •' gil/N , ... , gNM - =•lgiM/N gN1 i=l i Since Gm is common to all stations, it can be lumped together with the m . We shall write for

o

this so-called nuisaqce parameter

ci = m i + Gm (16)

The above modification of the G matrix eliminates

the necessity of somewhat arbitrarily choosing one particular station for reference, as is fre- quently done in studies of array station residuals. This point is of some importance, as observational errors in reference station residuals

may bias the final results. For a sample of n events, (14) can be written

as

travel time residual t. over all the stationsß

The estimate of the slowness anomaly m is then obtained by applying the least squares method to

t* = • - % • = • m + z - z (23)

15) where the elements of t* are the travel time residuals minus their avelage over all the stations for each event and • denotes the averaged error.

The sum of the columns in G corresponding to any one layer is identically zero because no uniform slowness perturbation i• a layer would affect t*. Thus the rank of G is less than or

equal to M - NL, the total number of blocks minus the number of layers in the model. This in turn means the slowness perturbation in any layer at best can be determined except for an additive constant. The elements of the corresponding best possible resolution matrix are

for i = j• NB denotes number of blocks in the layer to which block j belongs.

•.. = -•/N•

for i , j blocks in the same layer as block j.

i = 1 .... , n (17)

Combining all events, we write

i_ 0 ... 0 G 1 - G 1

0 i ... 0 G 2 - G 2 ß • •

ß , ,

00... iG -G -- n n

or

with

c

m

(18)

(19)

otherwise. This resolution matrix shows that only certain linear combinations of the medium parameter m have linear unbiased estimators. For

example, we can compare slowness differences between blocks in the same layer but not between two blocks in different layers. An important con- sequence here is that the result of a vertical smoothing operator cannot legitimately be compared even within the same layer if the averaging kernels are different for different blocks. To re-

solve this problem, we must know the absoiute values of the average layer slownesses with very high precision, say, better than the standard error for the estimates of the slowness

perturbation. The least squares estimate for m is obtained by solving

iO ... 0

0i ... 0

ee, i

G 1 - G 1 G 2 - G 2

(20)

and

•zf T T T

z__ = (e I , z__2 , ... , in ) T

c = (c 1, c 2, ... , c ) (21) n

The columns of I are ort•hogonal to each other and also to the columns of G. Consequently, the least squares estimates of c and m can be obtained separately. First, we h. ave

A c = • I T (22)

i.e., the i th element of c A is the average of i

•• = •t* (24) -. __

If the station density is appropriate for the chosen block specifications and the earthquake data have good coverage in azimuth and distance, the solution to this system will be unique except for the additive constant just mentioned. In practice, however, the number of rays passing through different blocks will vary considerably. Some blocks will have no ray passing through or only a very small number. We have to exclude from the analysis those blocks that have not been sampled. Even so, there will in general be some nonuniqueness in the solution. For example, if two neighboring blocks are always jointly sampled by the same rays, we cannot resolve the slowness perturbations for these blocks. To improve the uniqueness, it is important to increase the station density and the number of observations so that each individual block is sampled by several

rays comicfrom various directions. Since G G is singular, a unique solution of


(24) does not exist. Lanczos [1961] gives a 'natural' solution to this problem, which is the minimum length solution formed by excluding eigenvectors with zero eigenvalues. This solution is sometimes called 'generalized inverse' and we shall write it with the subscript g as

= -g --

where

(•)-g = VA -1vT (26) P

A is a diagonal matrix containing the nonzero e•genvalues of •T•, V is the matrix containing the corresponding eigenvectors, and

GTG = VA V T (27) P

Inserting (23) in (25) and noting that •C• = 0, we get

--g --

or

--g --

If the residual s has expectation zero, we obtain

giving the resolution matrix VV T introduced by Backus and Oilbert [ 1968].

Consœder 'now the pro•ection oœ • on •T•. This pro•ection, m , can be written as

m__p = (VApV T) (VAp-2V T) (VApV T)m__- (31 ) m : vvT•

showing that E(• ) = • . In other words, the method of gene•a-•ized-•nverse gives an estimate of the slowness perturbations _whose e•ectation is equal to the projection of m on G•G. Using this projection, we can write

--

m = m + 6 (32) -- --p --

where 6 is orthogonal to m and also to -- p •_g - Turning to the standar• errors D( ) for •g,

we have

D2(•__g) : (G•T•)-s•TE(_c•_c)•(•T•)-g (33) If we can assume that the covariance for • can be

written as o2I, we get

D 2 (•__g) = 02 (VAp-1V T) (VApV T) (VAp-1V T) : VA -1vT (34)

P

When the number of events increaseg the nonzero eigenvalues of •T• increase, and D=(•) will tend to zero as n tends to infinity, provided (1/n)•T• is well behaved in the limit. The above results can be used to obtain a lower limit for

the rms of the true slowness perturbations, as (29) gives

+ t r (•T•) -g•_e•e_ • (•7•) -g (35)

where tr represents the trace of a matrix. Then if g(_e) = q• we get

E( ) = m m + trD2( ) (36) --p -p

Letting •2(- ) deonte the tional es of D2(%), • convert timator consider

- -T- •_g •2 = m m - tr[•2( )] (37)

-- T-

Then E(• 2) =_mp _mp. From (32) it then follows that

-T- - T- 6T +__a (38) Thus

-T- E(• 2) _< m m (39)

Letting NB denote theenumber of elements in m, we can use [ (1/NB)•2] • as an estimate of the-- lower limit for the rms slowness perturbations in the actual earth.

Due to a limited amount of available data the

generalized inverse estimator may show large fluctuations in m . Therefore it may be desirable to consider other-g inversion techniques which have poorer resolution but a more stable solution. The standard techniques for inversion applied to (24) all consist of finding a matrix A of some kind and then estimating m by

m = A•Ct * (40) -est --

Because the generalized inverse solution satis- fies (24), we can write •

t* = Gm + e (41) -- ---g --

where •C% = 0. Inserting this in (40) we get __ '

• = AG •Tt* = A•C•mm + A•C% = AG•T• (42) --est -- --g -- --g

On the other hand, from (23) we get

• = A•Ct * = m•mm + A•(•-•) (43) --est .... --

showing that m . can be obtained b• a smoothing of m with the tesolution matrix AG G. These smoothed estimates will in general not fulfill the original system (24).

We shall consider an inverse solution cor-

responding to the following choice of A:

A : (•T• + 02i)-1 (44) This is a special case of the stocahstic inverse solution [Franklin, 1970], in which both data t* and solution m are considered as stochastic processes and the best (in some sense of least squares residuals in the model space• instead of the usual data space) choide of 02 is equal to the ratio of data variance to solution variance. In this paper we shall refer to this solution as the 'stochastic inverse.'

The resolution matrix can in this case be


written as

A• = VAp (Ap + 02 Ip)-ivT (45)

The standard_errors for the stochastic inverse

are, if E(z_•_ r) = o2I,

D2(_•es t) = o2[VAp(A + 02Ip)-2V T] (46) The resolution given in (45) is poorer than that for the generalized inverse, but from (46) we see that the standard errors are smaller. This solu-

tion (equations (40) and (44)) is called !damped least squares' by Levenberg [1944] and can be ob-

tained by minimizing2 I t• ---G•n i 2 + 02 I• 12 instead of It__*- •_] .--Similar solutions were considered by Aitken [1945] and Chipman [1964]. In applied mathematics and statistics the stochas•tic inverse goes under the name 'ridge regression' [e.g., Hoerl and Kennard, 1970]. One may consider this alJo a s an approximate generalized inverse, in which the effect of eigenvalues sma$1er than 02 is smoothed out. The drawback with the stochastic inverse approach when it is applied to our model is that vertical smoothing is introduced between blocks in different layers. As was pointed out previously, to use different averaging•kernels for blocks within a particular layer is not legitimate. Another alternative here is to eliminate the smalles• nonzero eigenvalues, as was done by Wiggins [1972]. This will also introduce vertical smoothing with different kernels for different blocks. A more desirable way of obtaining a smooth solution, if this is re- quired, is to first obtain the general inverse solution and then apply a predefined smoothing operator.

Model Specifications and Numerical Aspects of Estimating Slowness Perturbations

In the foregoing section we presented two approaches for estimating the slowness perturbations in our physical earth model, namely, the methods of generalized inverse and stochastic inverse. Both approaches have been utilized in analysis of Norsar data, and in the following the corresponding computational techniques will be briefly discussed. We shall also comment on the relative importance of the initial model parameters such as number of layers, size of blocks, layer thicknesses, and layer velocities.

In order to solve the system of linear equations in (24) the method of generalized inverse leads to a solution having minimum l•th [e.g., Lanczos, 1961]. If the rank of the G'G matrix equals the number of blocks sampled, K, minus the number of layers, NL, the solution for each layer

will be unique except for an additive c•nstant. However, in practice, the rank of the G G matrix will be somewhat smaller, implying that some of the blocks cannot be resolved properly. This information is actually contained in the resolution matrix and can easily be identified by checking which elements do not exhibit the best possible resolution.

We tried both the stochastic inverse (44) and the generalized inverse. Both methods gave essentially the same results. The discrepancies, when they existed, were always explainable in terms of differences in the smoothing procedure,

which is explicitly given in the resolution matrix.

In addition to the smoothing procedure our solution depends on the configuration of the blocks and their size and thickness. Such effects

have been tested empirically by varying these parameters. It was found that the effect of these model parameters was of minor importance. The essential features of the slowness perturbation pattern are repeated, except for obvious results such as that'the increasing block size wipes out small-scale inhomogeneities.

In most practical cases the lateral block size is determined by the areal density of stations of the seismic array, rather than limited by the wavelength considered earlier. Once the lateral size is fixed, the vertical size is determined from the considerations on resolution. For a

given data set, if the vertical size gets smaller, the interlayer resolution becomes poorer. Since the number of blocks in a layer is determined by the extent of the station network, the most important parameter at our disposal is the number of layers to be included in our model. The choice will depend on the following three factors: (1) effects that may be projected upon our solution due to inhomogeneity at depths below the bottom layer, (2) the significance of improving the fit to the observations by an additional layer, and (3) the limitations of currently available com- puters. The last point is not insi•nificant because the order of the matrix (•'I•) is 315 for the data used in the present study of structure under Norsar, while the total number of blocks was 5-9-9 = 405.

As the computation of the generalized inverse solution requires a computer core storage about twice as large as that for the stochastic inverse and also considerably longer processing time, the latter method may be useful in the first stage of analysis for experimenting with different block structures and initial model parameters. In addition, when there are small nonzero eigenvalues, the generalized inverse may require higher precision in order to avoid roundoff errors.

As to the second point, it is important to have some statistical measure on the significant number of layers to be included in our physical model. A convenient and straightforward test sta- tistic is the ratio of variance VR of the travel time residuals calculated for the solution to that observed. This is defined as

(47)

A more direct approach is to test whether an additional layer represents a significant variance reduction improvement. Noting the similarity of this problem to that of constructing an F test for significance of regression coefficients [Draper and Smith, 1966], we proceed by defining the following:

S T variance of the observations, equal to t*-t*;

e I •es•dual variance Sot a model with NL layers, equal to S T - (•m) •(•m)NL•

e 2 residual varianc• for--a model with NL + 1 layers, e ual to S - (•) •(•mm) •

r 1 rank of G_•G for mo•el with NL layer• r 2 rank of G • for model with NL + 1 layers.

Aki et al.: Three-Dimensional Structure of the Lithosphere 283

Then, provided that the travel time residuals have a normal distribution and are independent

F -- (e I - e2)/(r 2 - rl)

e2/[n ß N- n- r 2]

is distributed as F[r - r , n ß N- n- r 2] under the hypothesis •hat •he slowness fluctuations in the (NL + 1) layer are zero.

Application to the Norsar Data

The Norsar array in southeastern Norway consists of 22 subarrays, as shown in Figure 2, each containing six short-period seismometers. Details of the instrumentation and array opera- tion can be found in the article by Bungum et al. [1971]. The P wave travel time residuals at subarray centers for a large number of teleseismic events have been tabulated by Berteussen [1974]. The residuals were measured with respect to the

In our analysis the residual data are averaged over stations for each event, and this average is removed from the data before the inversion. The standard deviation of this residual was 0.22 s.

The parameters of the initial model for the crust and upper mantle under Norsar, given in Table 2, are determined from the results of refraction works across the southern part of Scandi- navia. (For summary, see, for example, Der and Landisman [1972] and Mass• and Alexander [1974].) Each of the five layers of our initial model is divided into 9 x 9 square blocks, with the side length 20 km. We chose five layers because the reduction of residual variance by adding the fifth layer was marginally significant.

The choice of block configuration should not affect our final solution. But we found minor differences between solutions for different con-

figurations. The main reason for this is that a number of the peripheral blocks are poorly sampled, thus resulting in exceptionally large anomalies in

Jeffreys-Bullen travel time table and source para- many of these blocks in the generalized inverse meters such as origin time, epicenter, and focal depth published by the U.S. Geological Survey. The reading error is estimated to be less than 0.1 s. These residuals have been used as station

corrections for beam steering in the routine processing at Norsar. We used 1496 readings of residuals measured for 93 events at 22 subarrays. Examples of observed residuals selected to cover various azimuths are listed in Table 1. These

events have an acceptable azimuthal and distance coverage for our inversion experiment, as shown in Figure 3.

NO RSAR N

E O•CA IVI•B RIA N R . •, LAKE

ß •e MJ½SA ß

ß ,.- . ß / ( s

0 50KM

Fig. 2. Map of Norsar subarray centers. The geology of the area is outlined. The dashed lines mark the profiles along which vertical cross sections of velocity anomalies are shown in Figure 14.

solution and poor resolution for the corresponding blocks in the stochastic inverse solution. One way of removing this effect is to average the solutions for different block configurations. Since the solutions are based on the same data set, this averaging does not necessarily decrease the standard error. We adopted this averaging in computing the stochastic inverse solution but avoided it in com-

puting the generalized inverse in which we wanted the best possible resolution.

Whereas the advantage of generalized inverse is in its high resolution, the stochastic inverse has smaller standard error. We shall present both the stochastic and generalized inverse solutions. We prefer this dual presentation of our results to a single optimal solution based on the trade- off between resolution and standard error, such as that discussed by Backus and Gilbert [1968] and Wiggins [1972]. This dual presentation will give the reader more freedom in judging our results than a single solution.

The generalized ingerse solution is depicted in Figures 4a, 5a, 6a, 7a, and 8a and also tabulated together with the estimated standard errors in Figures 4b, 5b, 6b, 7b, and 8b. The slowness perturbations are expressed in percentage of average layer slowness given in Table 2. The negative and positive slowness perturbations correspond to relatively high (H) and low (L) velocities, respectively. Significantly high and low velocity areas in the respective layers, with the magnitude of slowness perturbations larger than twice the corresponding standard errors, are shaded in Figures 4a, 5a, 6a, 7a, and 8a. All but five of the sampled blocks are perfectly resolved in the generalized inverse solution, except for the uncertainty about the additive constant for each layer as discussed in a previous section. This loss of resolution corresponds to the total rank reduction of 8. The three extra zero eigenvalues caused the loss of resolution for the

five blocks, which are marked by circles in Figures 4a, 5a, 6a, 7a, and 8a. For all other blocks the elements of the calculated resolution

matrix are in accordance with (24), demonstrating that the resolution is the best possible. The numerical values of the diagonal elements were 0.97-0.99, and the off-diagonal elements for the blocks of the same layer as the diagonal ele-

284 Aki et al.' Three.Dimensional Structure of the Lithosphere

o

I

o o o o o o o o o o

ment were -(0.01-0.03). Elements corresponding to blocks in different layers were zero. The part of the resolution matrix corresponding to the five blocks. with poor resolution is listed below.

0.14 0.24 0.24 ... ......

0.48 0.42

6'•4 ...... 61J2 0.24 ...... 0.42 0.42

The overall fit of the generalized inverse solution to the observational data is excellent. The

standard deviation of the residual is 0.09 s, which is comparable to the measurement error. The estimate of the lower limit of the root mean

square of the true velocity perturbations in the crust and upper mantle under Norsar is 3.1% using (37).

The stochastic inverse was calculated, with the damping factor e 2 = 0.02 (s per %)2 for two sets of block configurations, which were identical except for the shift in the NE-SW directions by a half-block size. This value of e2 i• 90 times greater than'the smallest eigenvalue 0.00022. Out of the total 307 nonzero eigenvalues, 129 are smaller than •2. This gives a rough idea about the resolution and standard errors through (45) and (46). The final solution was obtained by averaging over four partly overlapping blocks, i.e., two from one solution and the other two

ß

from the other solution. The averaged value is plotted at the center of gravity of the four blocks, and the contours at 1% intervals are drawn interpolating these values, as shown in Figures 9a, 10a, 11a, 12a, and 13a. The stochastic inverse solution for the center configuration together with the corresponding diagonal elements of the resolution matrix are tabulated as shown

in Figures 9b, 10b, 11b, 12b, and 13b. The diagonal elements show nearly uniform values in each layer except in peripheral zones, where they show values less than 0.5.

The standard errors were computed for the un- shifted block configuration by using (46), where the variance o 2 of the errors in the data was estimated by the sum of squared residuals It* - G• .]2 divided by the number of degrees of freedom. Since in the damped least squares,

It. _ El2 + •21•i2 is minimized instead of •* G•I 2, we obtain a conservative estimate of the true variance of errors in the data.

The computed standard errors of the stochastic inverse solutions are not tabulated because they are very uniform. The maximum is 0.39%, minimum 0.12%, and most of them lie betwee• 0.30 and 0.35%. They are quite small as compared to those of generalized inverse solutions tabulated in Figures 9b, 10b, 11b, 12b, and 13b.

Interpretation of Three-Dimensional Seismic Velocity Anomalies under Norsar

The seismic network of Norsar overlies an area of considerable geological interest and complexity [Holtedahl, 1960]. The Precambrian rocks under the array outlined in Figure 2 are part of the westward extension of the Baltic shield. Mountains formed during the Caledonian orogeny, which ends in early Devonian, lie to the west of the array. The late Precambrian to

Aki et al.' Three-Dimensional Structure of the Lithosphere ,

285

Fig. 3. Epicenter locations of earthquakes used in the analysis. early Devonian deposits in the central part of the array were partially thrusted out of their deep primary depositional basins to the north during the Caledonian orogeny. The southern part of the array straddles the Oslo graben [Oftedahl, 1960], a graben structure with its maximum tec- tonic and magmatic activity in early Permian time. Large amounts of latitic and basaltic lava flows were formed within the graben with minor amounts of felsic flows. The total volume of

lava was estimated to be of the order of

10,000 km 3 (I. Ramberg, personal communication, 1975). The eruptions took place in three stages,

,

the first and third stage as fissure eruptions and the second from several volcanic centers.

The basalts are generally alkaline basalts, and experimental petrology tells us that they derive from the upper mantle at depths around 60 km [Wyllie, 1971]. In Figure 2 the Oslo graben contours follow the boundary of Permian eruptions, but the faulting which cuts even the youngest volcanic rocks extends to'a wider area. The

northernmost branch of the fault, which defines the northwestern boundary of the graben, reaches to the center subarray O1A.

The gravity observations for the area imply

a significant positive anomaly over the Oslo graben, although it size depends somewhat on the regional correction adopted. Since the surface rocks within the graben are of slightly lower density in comparison to the surrounding Archean complex Ramberg and Smithson [1971] and Ramberg [1973] concluded that the positive gravity anomaly must be attributed to high-density material in the lower crust. They suggested that the high- density material may be intrusions from the mantle.

Norsar scientists have used two entirely dif-

TABLE 2. Initial Model Parameters for Crust

and Upper Mantle Beneath Norsar

Thickness, P Slowness, Block Size, Layer km s/km km

1 17 1/6.1 20 2 19 1/6.9 20 3 3o 1/8.2 20 4 30 1/8.2 20 5 30 1/8.2 20

286 Aki et al.- Three-Dimensional Structure of the Lithosphere

NORSAR I

I 0 50krn I I I I I J

LAYER 1 (0-17kin) j

Fig. 4a. Generalized inverse solution for layer 1 (plan view). The numbers show the fractional velocity perturbation in percent of the average layer velocity given in Table 2. The shaded areas correspond to the magnitude of solution greater than 1 time the standard error which is also listed in Figures 4b, 5b, 6b, 7b, and 8b. The letters L and H refer to low- and high-velocity anomaly, respectively. The central subarray is marked by a larger solid circle.

NW-corner

ferent approaches in studies of teleseismic P time residual s and amplitude anomalies observed under the array. In the first case, Berteussen [1974] used a least squares modeling technique for estimating Moho undulations in terms of first- to third-order polynomials. The conclusion was that the large variations in time residuals could not satisfactorily be explained by any kind of relatively simple geometrical Moho interface. The second approach, based on random medium models of the Chernov type [Dahle et al., 1975• Berteussen et al., 1975] were more successful in reducing the residual variance of P wave time observations than the Moho undulation model, more than 60% reduction for the Chernov model as com-

pared to 20% for the other. The scale length of the velocity anomaly (called 'correlation distance' in the Chernov theory) was found to be around 30-60 km. The average magnitude of the velocity anomaly was found to be about 3.3% when the vertical extent of the random medium was as-

sumed to be about 100 km. This estimate of magnitude of fluctuation is very close to our estimate of its lower limit of 3.1% using (37). The scale length of inhomogeneity also seems to be comparable between the two independent estimates shown in the contour maps of slowness fluctuation.

The contour maps of seismic velocity anomaly for layer 1 •upper crust, depth range 0-17 km) are shown in Figures 4a and 9a for the generalized and stochastic inverse solutions, respectively. In the latter map the boundary of volcanic rocks of the Oslo graben is contoured by a dashed line. However, the faults known to have occurred after the volcanic series extend to a wider area. As

was mentioned previously, a northernmost branch of the fault, which forms the western boundary of the graben, reaches to the center subarray 01A marked by a solid circle in the figure.

In both solutions we find a zone of low-

NE-corner

-0.64 0.07 5.77 6.68

-- (1.45) (1.27) (2.08) (1.75) -- --

(g• 0.82 -2.98 0.19 3.31 4.25 0.16 (1 90) (1.58) (1.36) (1.55) (1.44) (2.44)

4.58 1.39 -1.08 -0.63 0.74 3.56 2.63

(2.09) (1.99) (0.90) (0.79) (0.89) (1.49) (1.43)

(4.06) (1.51) (0.98) (0• .72)J (1.03) (0.94) __ 1.38 0.90 -3.51 -1.91 -3.78

(1.27) (1.05) (0.90) (0.83) (1.47)

-3.23 -2.28 -0.24 -0.92 0.31

(1.81) (1.36) (1.59) (1.14) (1.52)

-9.04

(3.67)

SW-corner SE-corner

Fig. 4b. The generalized inverse solution of slowness fluctuation in percent for layer 1 with the standard error. Negative and positive values correspond to relatively high and low velocities, respectively. The upper left block corresponds to the NW corner of the grid, while the center subarray block is squared. Un- resolved blocks are encircled. Initial model parameters are listed in Table 2.


NORSAR I N

I o s,om I I I I I I LAYER 2 (17-36km) I ' I

ß 1.7

ß 0.9 e- 0.2

e-0.8 ß tO

ß 1.6 ß 0.4 ß -0.1 ß-0.6

e-0.3

ß 1.0 ß0.9

e0.8

ß 0.2 ß0.6

ß -1.0 ß0.8 ß13

Fig. 5a. Generalized inverse solution for layer 2 (plan view). See Figure 4a for explanation of symbols.

velocity anomaly trending in the NNW direction generalized inverse solution. In both solutions in the western part of the area. Another signifi- a high-velocity anomaly occurs in the central cant low-velocity anomaly is located in the north- area and appears to extend to the southwest eastern corner, which is more pronounced in the toward the Oslo graben as well as to the NNW

NW-corner

1.74 -4.36

(3.49) (2.53)

3.67 0.94 -0.17

(2.89) (1.50) (1.47)

-0.77 1.03 4.10 1.19 (2.73) (1.91) (1.48) (1.09)

1.63 3.09 0.37 -0.08 (2.59) (1.65) (1.00) (0.88)

NE-corner

-1.78 -4.27

(1.51) (1.90)

-1.07 -3.97 -5.36

(1.11) (1.43) (1.60)

-0.60 -2.34 -3.22 -16.26

(1.00) (1.09) (1.57) (3.61)

-4.17 -1.85-0.27 I•.29 ! 0.83 -1.58 -4.98 (2.16) (1.35) (0.96) (0• .88)• (0.88) (1.05) (2.90) 5.74 2.53 0.98 0.88

(2.31) (1.20) (0.99) (0.92)

7.09 2.54 -1.11

(1.67) (1.23) (1.18)

9.98 4.11 -0.95

(3.16) (2.01) (2.06)

21.54

(4.11)

0.18

(0.90) (1 21)

-1.36 -1.50

(1.1•) (1.51) (2.33)

0.82 1.69

(3.23) (2.58)

SW-corner SE-corner

Fig. 5b. The generalized inverse solution for layer 2. For notations see Figure 4b.


I N NORSAR

I o I'''' 5pkm .•_ LAYER 3 (36-66km)

e-0.4 e0.8

eO.7 e-0.4 e-0.6 e-0.1

e-0.1 eO.O

el.7 eO.5 e-0.1

e1.1 eO.7 e-0.2 e-0.6


toward the nQrthern part of Lake Mj•sa (Figure 2) and the Gu•brandsdalen Valley. The high-velocity anomaly under the Oslo graben is consistent with the •bserved positive gravi.ty anomaly over the graben area. '

NW-corner

(•• 0.82 -0.42 0.76 (3 88) (1.17) (1.06)

2.54 -1.57 -1.52 -2.08

(2.06) (1.24) (0.90) (0.90)

0.65 1.31 -0.41 -0.61

(1.31) (0.90) (0.72) (0.65)

-1.44

(1.54)

2.80

(2.37)

-6.01

(2.75)

The velocity anomaly maps for the lower crust (layer 2, depth range 17-36 kin) are shown in Figures 5a and 10a. We find, under the graben, a low-velocity zone trending in the NNW direction, obliquely crossing the graben. Although

-2.03 -0.47

(1.12) (1.41)

-2.74 -2.44

(0.91) (1.06)

-0.05 -1.07

(0.65) (0.78)

2.39 0.80 -0.06 0.03 -1.16 0.12

(1.24) (0.77) (0.63) (0.57) (0.58) (0.68)

2.73 2.17 -0.21 ! 1'32 I -1.25 -1.11 (1.38) (0.75) (0.57) (0•.52)• (0.53) (0.67) -1.39 1.81 1.01 1.68 0.54 -0.09

(1.41) (0.88) (0.63) (0.58) (0.61) (0.75)

1.12 0.71 1.66 1.07 -0.17 -0.55

(1.45) •[!.03) (0.81) (0.71) (0.83) (0.99)

6.30 2.42 2.66 2.26 -1.32 -5.98

(1.77) (1.50) (1.57) (1.11) (1.26) (1.64)

5.57 4.43 4.75 4.00 5.57 (1.59) (1.32) (1.39) (1.36) (1.52)

NE-corner

-13.33 -10.20

(2.27) (2.42)

-0.33 -5.63

(1.21) (1.92)

-0.73 2.85

(0.98) (1.49) ,

-2.69 1.94

(I.00) (1.23)

-1.02 0.77

(0.91) (1.96)

-2.02 0.56

(1.16) (2.43)

1.73

(1.42)

-4.11

(3.31)

SW-corner SE-corner



NORSAR

LAYER 4 (66-96km) ,

? 50km • I

e0.5 el.0 el.0

e0.5 e-0.? e0.0 ß 06

e-0.1 e0.?

ß 1.2 ß1.1

e-10 ß 1.5 e0.3 e03 e-0.9 eQ5

Fig. 7a. Generalized inverse solution fo• i•yer 4 (plan view). See Figure 4a for explanation of symbols.

Ramberg [1973] hypothesized a high-density in- general anomaly pattern of this layer is low trusive body in the lower crust to explain the velocitieJ to the west and high velbcities to the gravity high, we find no clear counterpart of east. This pattern repeats throughout layers 3, Seismic velocity anomaly in the lower crust. The 4, and 5.

NW-corner NE-corner

21• 0.46 2.80 4.10 0.96 1.00 -1.38 -0.95 -2.02 (3.41) (1.50) (1.16) (1.03) (i.05) (1.20) (1.32) (1,68) ,

-2.40 1.54 1.90 1.41 0.45 -0.65 -1.53 0.00 -0.64

(2.73) (1,46) (1.18) (0.96) (0.89) (0.86) (0.92) (1.14) (1.31)

-0.58 2.22 1.87 1.66 0.03 -1.25 -1.29 -2,09 1.20

(1.76) (1.26) (0.88) (0.81) (0.73) (0.73) (0.83) (0.96) (1.19)

-3.29 3.76 -0.07 0.66 -1.19 -0.38 -1.70 -3.99 -0.24 (1.78) (1.28) (0.91) (0.69) (0.65) (0.66) (0.74) (0.90) (1.16)

3.16 3.92 1.40 0.51 I 1.54 ! 0.01 -1.85 -2.49 0.10 (1.56) (1.09) (0.88) (0.66) [(0.64) (0,67) (0.72) (0.95) (1.14) 2.70 2.71 1.39 -0.13 1.84 -0.11 -2.33 -1.62 -1.33

(1.59) (1.10) (0.89) (0.81) (0.68) (0.72) (0.85) (1.01) (1.19)

1.07 1.63 1.40 1.32 1.88 -1.15 -i.88 -2.65 -3.29 (1.63) (1.24) (1.09) (0.94) (0.86) (0.90) (1.11) (1.36) (1.52)

1.17 1.14 1,72 0.91 1.49 1.55 -1.93 -2.21 -5.57 (2.25) (1.70) (1.57) (1.27) (1.47) (1.67) (1.48) (1.97) (3.64)

-1.04 1.52 0.27 0.30 -0.94 0.47 -4.04 -5.32

(1.93) (1.71) (1.71) (1.65) (1.81) (1.81) (2.42) (3.95)

SW-corner SE-corner



NORSAR I 0 50km N I ,, , , , , : LAYER 5 (96-126km) I

e-0B

e-0.4 eO. 7 eO.3

el.O

eO.8 eO.5 e-0.3 eO.4 eO.6

eO.4 ß -1.0 ß-0.8

ß i.5

.


The velocity anomaly maps for the top layer anomaly pattern between layers 2 and 3 suggests of the mantle (layer 3, depth range 36-66 km) are that the anomaly may be attributed to undulation shown in Figures 6a and 11a. They show an exten- of the Moho discontinuity, which is the interface sive area of low velocity to the west and one of of the two layers. If this is the case, the crust high velocity to the east. The similarity in the is thinner to the east and thicker to the west.

NW-corner NE-corner

-3.90 -2.38 -1.55 -0.77 1.41 3.62 5.29 4.67 3.17 (2.45) (1.48) (1.2.2) (1.03) (0.93) (0.98) (1.00) (1.20) (1.31)

-0.15 -1.96 -0.35 0.66 0.27 1.39 1.26 2.45 2.32 (2.24) (1.34) (1.07) (0.90) (0.86) (0.84) (0.95) (0.99) (1.18)

-2.37 1.83 2.50 2.77 2.56 -0.58 -1.87 -2.47 -0.60 (1.34) (1.26) (0.89) (0.72) (0.68) (0.74) (0.79) (0.89) (1.11)

2.44 4.99 5.94 5.24 5.82 1.38 -4.06 -5.76 -2.06 (1.59) (0.98) (0.78) (0.65) (0.62) (0.67) (0.68) (0.85) (1.02)

4.7 i (1.21) (1.03) (0.77) (0.68) (0• .6.1)• (0.64) (0.72) (0.89) (0.99) 1.88 2.33 3.69 3.84 1.54 0.89 -1.03 -4.37 -6.35

(1.24) (1.05) (0.84) (0.70) (0.76) (0.69) (0.82) (0.97) (1.12)

-1.36 0.83 0.46 -0.34 -3.06 0.38 0.55 -2.27 -5.32 (1.35) (1.10) (0.98) (0.95) (0.84) (0.99) (0.99) (1.23) (1.32)

0.41 -0.95 -2.51 -5.06 -5.58 -0.75 2.17 -0.70 -4.26 (1.57) (1.49) (1.25) (1.08) (1.49) (1.17) (1.44) (1.47) (1.69)

-3.66 -4.07 -5.61 -5.59 -4.53 -1'.48 2.17 3.33 1.17 (1.91) (1.81) (1.59) (1.42) (1.48) (1.78) (1'.80) (1.98) (2.11)

SW-corner SE-corner



LAY E R I (0-17 KM)

-I. I -0.6 0.8

H L -0.4- I' I• •/0.5 ' 0.6 1.3 •0.8•- 0.5 - 0.5•0.6

1.7 1.6 / 0.2• -0.5 f-I.O -0.7 -0.8 _,.o

ß

-o. 5 2." /-0.,9// -0.8 \1 - 0.7 -0.6 /

\ / • OSLO I

/

•"" ! / GRABEN I I I /

/ 0 50 KM I I I I I I

Fig. 9a. Stochastic inverse solution for layer 1 (plan view). The numbers show the fractional velocity perturbation in percent of the average velocity given in Table 2. The values represent the four-point average over two block configurations, one centered at the central subarray (marked by a large solid circle) and the other shifted by a half-block length toward the southwest. The letters L and H mark areas

This is in rough agreement with the results of Berteussen [1975] and Sheppard [1976], who interpreted the Norsar residual data by the Moho undulation model.

The same pattern persists to layer 4, with the depth range 66-96 km, as shown in Figures 7a and 12a. Since the interlayer resolution for the generalized inverse is perfect, we cannot attrib- ute this pattern to the undulation of the Moho. The general increase of velocity to the east may be attributed to the presence of rocks of the Baltic shield undisturbed by the Caledonian orogeny or by Permian volcanism.

Nearly identical patterns of velocity anomalies are obtained for layer 5 (depth range 96-126 km) by the generalized and stochastic inverse solutions. This layer shows the strongest velocity variation (6% change in 50 km) among all layers for the stochastic inverse solution. Part of this anomaly may be forced from the neglected inhomogeneities below our block model. In any case, our result conclusively demonstrates the reality of strong small-scale inhomogeneities existing to great depths. The bottom of layer 5 corresponds to the bottom of the lithosphere, according to one of the models of McConnell [1975], for explaining the Fennoscandia uplift data (see also 0'Connell [1976]).

We are tempted to interpret these deep anomalies as the remains of magma ascent processes associated with the Permian volcanism of the

Oslo graben. However, we realize our ignorance on such a basic problem as whether we should associate the high-velocity or low-velocity anomaly with the remains of a magma ascent path.

of low- and high-velocity anomalies, respectively. For example, high velocity may be expected if The contours are drawn at 1% intervals. The high-density residuals precipitated from the as- corresponding diagonal elements for the resolu- cending magma have been completely crystallized tion matrix for the block configuration centered by cooling since the Permian (200 m.y. ago). at the central subarray are listed in Figures On the other hand, low velocity may be expected 9b, 10b, 11b, 12b, and 13b. if the magmatic ascent process weakened the rock

NW-corner NE-corner

0.21 -0.69 0.11 0.56

(.57) (.60) (.16) (.37)

0.08 0.43 -0.15 -0.06 0.35 -0.50

(.04) (.45) (.46) (.37) (.38) (.53)

3.02 2.10 -0.31 -0.55 -1.12 -0.06

(.49) (.43) (.60) (.66) (.69) (.51)

-0.12 4.54 -0.41 I-1.79] -1.80 -0.18 (.07) (.58) (.59) (• (.47) (.61)

2.02 1.69 -2.12 -1.34 -1.73

(.58) (.59) (.66) (.68) (.31)

0.92 0.33 0.64 -1.03 -1.30

(.42) (.63) (.38) (.64) (.58)

0.58

(.11)

-0.68

(.15)

-1.07

(.59)

-0.60

(.o4)

SW-corner SE-corner

Fig. 9b. Layer 1 tabulation of diagonal elements for the resolution matrix for the stochastic inverse solution using the 01A subarray center configuration. The upper left blocks corresponds to the NW corner of the grid, while the center subarray block is squared. Initial model parameters for layer 1 are listed in Table 2.


LAYER 2 (17-36KM)

-0.2 0.1 0.3 •- - 0-,,,,.•,•

-ø.ø--..•.• -ø.•., L øo• o.,,,_o., o.vo.

_o../o./_o., _o.. 0.4 / 1.5/ -0.1 -0.7 0•-0.2 -0.4

o., (, (o.,)_,.o _o., _o., _o., o.o _o., _o., o.o o.• •.• k-o., .-o.• -,., • -,..

o.,/,., ko.,k_o., ,.o • ,.• • o.,• -o;• -o.• -o.,

,., -,., /,.,/o.,

o., o> 0.7 O. 6 I 0.5

i / z /

/0 ( 5OKM I I I I I I

Fig. 10a. Stochastic inverse solution for layer 2 (plan view). See Figure 9a for explanation of symbols.

along its path by fracturing and heating. We also need to extend the station network for better

definition of the size and shape of these deep anomalies because the existing Norsar array covers only the northern end of the Oslo graben. A network of portable self-contained event- recording seismographs with accurate clocks will be a useful tool for such an experiment. When more refined seismic structure data become avail-

able, it will be possible to undertake detailed discussions of the seismic results in the frame-

work of current theories for magma ascent as for-

warded by Ramberg [1968, 1972], Elder [1970], and others.

Conclusion

We have presented a novel approach for three- dimensional seismic modeling of the lithosphere. In contrast to conventional methods the assumption of uniform material property within the layer is relaxed, making it possible to accommodate complex geological structures. Another important feature is that we can estimate a lower limit of the true rms velocity perturbations in the lithosphere, under the assumption of ray theory. Using Norsar P wave residuals for teleseismic events, we have estimated the three- dimensional seismic anomalies in the lithosphere to a depth of 126 km beneath the array. The fit to the observations is'very good as indicated in the final residual comparable to the measurement error. The lower limit of true rms velocity fluctuations in the lithosphere under Norsat is found to be 3.1%, in agreement with estimates obtained from a statistical analysis of P time fluctuations based on the Chernov theory. Prefer- ence was given to a dual presentation of the generalized and stochastic inverse solutions for the estimated slowness fluctuation because this gives the reader greater freedom in judging the results than a single 'optimal' solution.

The dominant features in the obtained three- dimensional velocity anomalies under Norsar are low velocities to the west and high velocities to the east. The latter feature may be attributed to rocks of the Baltic shield which are undisturbed by the Caledonian orogeny or Permian volcanism. Our result conclusively demonstrates the existence of strong small-scale inhomogeneities to the bottom of the lithosphere, which may be related to the remains of the magma ascent process responsible for the Oslo graben.

NW-corner NE-corner

0.11 -0.40

(.05) (.10)

0.49 -0.61 0.15

(.15) (.49) (.52)

0.04 1.01 1.64

(.22) (.48) (.49)

1.56 2.68 -0.61 (.34) (.48) (.57)

-0.23 1.36 -0.48 1-01.19 ! -0.68 (.31) (.48) (.57) (I .59)] (.64)

0.87 1.73 0.64 -0.41 -1.14 (.19) (.52) (.62) (.62) (.65)

1.77 1.75 -1.55 -1.98 (.41) (.57) (.59) (.58)

,.

1,04 -0.37 0.29 0,17

(.12) (.23) (.23) (.11)

-0.34 -0.09

(.43) (.39)

0.31 0.26 -0.73 -0.83

(.52) (.54) (.50) (.43)

-0.98 -0.17 -0.72 -0.59 -0.38 (.62) (.59) (.60) (.45) (.05)

-1.97 -0.59 (.56) (.17)

-0.58 -0.70

(.45) (.05)

-1.67 -0.46

(.50) (.17)

0.47

(.14)

1.13

(.o7)

SW-corner SE-corner

Fig. lob. Layer 2 tabulation of diagonal elements for the resolution matrix for the stochastic inverse solution. See Figure 9b legend for details.


LAY ER $ (36 -66KM),

0.0 -0.2 %..0.6 I.• -O.I -0.3 -0.2) 0.5 0.8 0.8 •).l. -1.5 -0.5 -0.3 -0.2 _•'O.l' - O.'l•O.O_ 0.5 O.I

_,., -0.9 0.0 /I.6 I. 0 0.7 O. I - 0.5 -0.5• k 1.7

,./o., o.,/_o., o., ,.o o., (o.,

o., ;.o o., o.,• o., ,••o.y_,.• _,.,•,., 0 I 09 05 06 O• /-I 3 --20 -I

..... •" •H'• ' 0.3 0.4 0.2 -0• .3 -2.0 -2.0 I • • I

•.O•l x O. d' -I.O ', -l.6 -1.8 -l.7 -0.1 -0.6 -0.4 -I.0 -I.2 -I.4 -I.I

-0.2 -0.8 -0.7 I -0.8 -0.9 -0.8

. ) '0.6 -0.5 -0.3 / I

0.2•-O. 4 • 0 50 KM • / i i I I I i

Fig. 11a. Stochastic inverse solution for layer 3 (plan view). See Figure 9a for explanation of symbols. ,

Appendix

Suppose that the original path OP is shifted to the ray path OP' due to a change in velocity

ß to vß + •v. in the layer i Since the from v• ß ß . horizontal component of the slowness vector is the same for OP and OP' (= sin 0/v), where v is the velocity of a half space), OP and OP' are parallel in all the layers except the perturbed layer i.

The travel time for the path OP can be ex-

NW-corner

-0.81

(.34)

0.73

(.17)

-0.62

(.14)

LAYE R 4 ( 66 - 96 KM )

- 0.7 -0.4 0.0 0.7 1.0 1,0 O. 6

- 0.9 H'•'----• -0.6 0.0 0.7 0.7 0.9 0.8 - 0.8 - I.•f•.•• - 0.8 - 0.5 0•0.5 . 1.0 -0.6 -0.8 -0.3 -0.1 -0.1 -0.2 -0.:5• 0.2 1.2

-0.5 -0.8 . 2•0. I -0.6 - 0••-•. 1 _o.• /o.•x,.o• ,.•, o.• •-o.• ?,.o••o.,

_o., o.• •.• ,.• ,o., o.• _o.• •/• ,.• ,.• ,• •., t-,.•l -•.• _,.o

o.• ,.o•,.,• ,.• ,.• /_o.•k_,., •_•.• o.• o.• o.• o.••., •.• _o.•,_,.•_•.._ •

o.• o.• _o.•o.• •.• o.o _o.•_,.• o.• o.'o • -o., k o.,; o.• •o., -o.•

,.o o., o.• o.,• -o.• -o• -q.,h ,.o o.o -o.•

o.• -o.•,"_o.• ; •/o• / 0 50 KM

I I I I I I


pressed as

vl__ 2 « sin 0 to P = • [( )2 _ (sin 0) ] h. + ß . v j v •. J

and that for path OP' as

1 sin 0)2 ] «h. top' = [(v.+•v- )2 - ( v ß

+ • [(v1___)2 _ (sin 0)2 ] «h . v j j#i j

sin e + tx- Ax)

v

NE-corner

Fig. 11b. Layer 3 tabulation of diagonal elements for the resolution matrix for the stochastic inverse solution. See Figure 9b legend for details.

2.30 1.94 0.79 0.70 0.84

(.48) (.44) (.39) (.53) (.47)

SW-corner SE-corner

0.14 0.23 0.29 1.14 0.04 0.70 -1.51 -1.51 (.11) (.12) (.48) (.54) (.50) (.35) (.14) (.14)

0.08 -0.46 -0.35 -0.53 -0.63 -0.28 1.25 -0.47 (.15) (.48) (.66) (.60) (.66) (.62) (.49) (.22)

-0.46 2.16 0.80 0.43 0.17 -0.59 -0.21 0.84 (.50) (.65) (.71) (.73) (.74) (.66) (.68) (.40)

1.06 1.39 0.68 0.49 -0.70 -0.31 -2.50 1.47

(.47) (.69) (.74) (.77) (.77) (.74) (.62) (.49)

1.81 2.34 0.2g I o.s•l -1.44 -1.70 1.03 -0.61 (.47) (.70) (.77) (L_• (.79) (.71) (.58) (.33)

0.22 1.42 0.88 1.11 -0.64 -1.38 -1.99 -0.19 (.39) (.64) (.78) (.77) (.75) (.70) (.44) (.12)

0.04 -0.51 0.27 -0.15 -1.38 -1.74 -0.43 (.42) (.60) (.65) (.75) (.69) (.59) (.42)

0.35 -0.43 -0.30 •.24 -1.58 -1.71 -1.41 (.30) (.42) (.39) (.56) (.54) (.41) (.25)


In the text the change in travel time over the ray path contained in the i th layer is expressed as

d.

(•i + Adi )1---(1 + mi) -__1 V. V.

Ad. d.

- i ) Z(l+m +---m. Vo

This corresponds to

= 1 2 sin 0 top' - top [(•. + 6v. ) - ( v )2]-•hi

[ (vl__)2 sin 0 2 • sin 0 - . - ( v )]h i v Ax

We can identify (di/vi)m i as

z 1 _ (sin -- m. = [(v )2 )2] h. - • .+•v. v V. 1

-- [(vl___)2 -- (sin O)2]-•h. . v 1

h.6v.

= _[ (vl__)2 (sin O 2 -« l - V ) ] ß 3 1 v.

h. 6v. --

v. cos 0. v.

/v. = sin O/v is used. where Snell's law sin 0i. z Thus •/• is the travel tzme spent by the original

LAYER 5 (96-126 KM)

- 0.8 7-1.2 -I.2 -I.0 -0.8 .0 0.6 0.7 0.6

0.6/.2 -I./ -0.8 -0.6 - 0 4'•""• O.O. 0.2 - •

'- 1.0-- -0.8 -0.5 -02 04

o.d .-,., o., o., ,.o ,., ,., o.z o.• o.• •.o -o.z 0.0 • •.• o.z•-o.z

0.5 a• /-o.• -•.o -•..• •o.•, 1•.5 L •-P o.• 5

o• :o., -o.• _o? ,o • ,• ,, •o• / o. __--/ O 50 KM

I I I I I I


ray in the i th layer, and m. is the fractional change in slowness, which i• approximately equal to -6vi/v i. We also find the part due to path change as

Ad. ß sin 0

_ Ax (1 + m i) v V.

This part of the change, however, cancels out when we take into account the change in arrival time at the bottom layer. The change AT is equal to the travel time for the path P'P'• in Figure

NW-corner

0.14 0.04 0.16 0.93

(.11) (.20) (.45) (.54)

-0.48 -0.92 -0.70 -0.34

(.17) (.46) (.52) (.58)

-0.68 -0.29 1.18 1.03

(.28) (.49) (.65) (.63)

-1.16 1.65 0.59 1.54

(.34) (.46) (.59) (.71)

1.33 1.68 I .68 1.82

(.40) (.49) (.63) (.74)

0.50 1.21 0.62 0.24

(.39) (.55) (.60) (.62)

-0.27 0.53 -0.15 0.36

(.31) (.52) (.54) (.56)

0.54 -0,07 -0.32 -1.25

(.22) (.39) (.39) (.49)

0.51 0.17 -0.70

(.42) (.48) (.47)

0.71 1.25

(.62) (.59)

-0.13 0.13

(.64) (.66)

0.17 -0.55

(.71) (.72)

0.20 -0.10

(.74) (.75)

2• 0.67 (.74)

2.03 0.58

(,73) (.69)

0.58 -0.50

(.62) (.63)

-0.69 1..42

(.36) (.32)

-0.51 -0.17

(.52) (.51)

NE-corner

0.08 0.49 -2.45

(.52) (.46) (.42)

-0.25 1.25 1.04

(.67) (.61) (.51)

-0.26 -1.18 1.56

(.66) (.65) (.59)

-1.87 -2.95 0.09

(.71) (.61) (.57)

-2.04 -2.18 -0.34

(.72) (.61) (.56)

-1.59 -1.68 -1.43

(.64) (.60) (.54)

-0.63 -1.20 -1.56

(.50) (.45) (.39)

0.30 -0.34 -0.88

(.38) (.21) (.14)

0.49 -0.91 -0.46

(.44) (.23) (.14)

SW-corner SE-corner

Fig. 12b. Layer 4 tabulation of diagonal elements for the resolution matrix for the stochastic inverse solution. See Figure 9b legend for details.

Aki et al.- Three-Dimensional Structure of the Lithosphere 295

NW-corner NE-corner

-0.29 -0.77 -0.37 -0.25 0.26 1.48 2.32 1.32 0.13 (.26) (.52) (.57) (.59) (.68) (.66) (.67) (.58) (.59)

-0.40 -1.07 -0.89 -0.29 -0.97 -0.05 -0.28 0.75 0.85 (.26) (.44) (.5O) (.6O) (.61) (.66) (.61) (.68) (.63)

-1.59 0.31 0.52 0.43

(.51) (.39) (.63) (.67)

0.15 1.93 2.27 1.73

(.39) (.56) (.65) (.7O)

0.23 1.87 2.38 1.77

(.55) (.54) (.61) (.65)

0.52 1.00 1.11 1.73

(.53) (.59) (.58) (.67)

-0.14 0.42 0.00 -0.i5 (.55) (.55) (.54) (.50)

0.77 0.35 -0.57 -1.74 -1.12

(.39) (.41) (.46) (.45) (.23)

0.38 -0.74 -0.46 -1.76

(.39) (.52) (.51) (.57)

0.59 -1.16 -1.60 -2.07 0.37 (.71) (.69) (.67) (.68) (.63)

2.98 0.57 -2.80 -3.48 -0.35

(.71) (.69) (.73) (.67) (.65)

3•• 1.06 -1.95 -2.63 -1.97 (.70) (.67) (.62) (.69)

0.73 0.78 0.05 -1.62 -2.48

(.58) (.66) (.63) (.59) (.62)

-1.50 0.92 1.27 -0.39 -2.36

(.57) (.43) (.55) (.45) (.56)

0.76 1.43 0.12 -1.99

(.40) (.35) (.34) (.38)

-0.79 0.92 0.15 0.58 -1.28

(.51) (.49) (.48) (.28) (.34)

SW-corner SE-corner

Fig.13b. Layer 5 tabulation of diagonal elements for the resolution matrix for the stochastic inverse solution. See Figure 9b legend for details.

velocity u i AX -•x

\ \ \ \

,z) WAVE FRONT

Fig. 14. Geometry of ray path perturbation.

14. Since PP" is equal to Ax, we have sin 8

AT = tp,p,,- v Ax Thus we have

Ad.

--•(1 + m i) + AT = 0 V.

1

In general, we can neglect the effect of change in path because of the Fermat principle, which states that the travel time is stationary with respect to the perturbation of the ray path around the geometrical optics path.

Acknowledgment. We thank I. B. Ramberg, Uni- versity of Oslo, and H. Ramberg, University of Uppsala, for critical comments and suggestions on the Oslo graben geology and mantle diapirism. We also thank K. A. Berteussen, D. Newton, add G. Zandt for their assistance in the computations. This work was supported by the Advanced Research

Projects Agency monitored by the Air Force Office of Scientific Research through contracts F44620-75-C-0064 and F08606-74-C-0049• by the National Science Foundation under grant DES74- 22025• and by the Royal Norwegian Council for Scientific and Industrial Research. Portions of this research were carried out while one of

us (A.C.) was a visiting staff member at Lincoln Laboratory, M.I.T., and supported by the Advanced Research Projects Agency of the Department of Defense.

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(Received June 30, 1975• revised July 12• 1976,

accepted July 21, 1976.)

determination of the three-dimensional seismic structure ... · and experimental works are needed...

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