Bulletin of the Seismological Society of America. Vol. 63, No. 3, pp. 937-958. June 1973
P R E D I C T I O N AND SUPPRESSION OF L O N G - P E R I O D N O N P R O P A G A T I N G
SEISMIC NOISE
BY ANTON ZIOLKOWSKI
ABSTRACT
Approximately half the noise observed by long-period seismometers at LASA
is nonpropagating; that is, it is incoherent over distances greater than a
few kilometers. However, because it is often strongly coherent with microbarograph data recorded at the same site, a large proportion of it can be predicted by
convolving the microbarogram with some transfer function. The reduction in noise
level using this technique can he as high as 5 db on the vertical seismometer and
higher still on the horizontals. If the source of this noise on the vertical seismogram
were predominantly buoyancy, the transfer function would be time-invariant. It is
not. Buoyancy on the LASA long-period instruments is quite negligible. The noise
is caused by atmospheric deformation of the ground and, since so much of it can
be predicted from the output of a single nearby microbarograph, it must be of very
local origin. The loading process may he adequately described by the static deformation of a fiat-earth model; however, for the expectation of the noise to be finite,
it is shown that the wave number spectrum of the pressure distribution must be
band-limlted.
An expression for the expected noise power is derived which agrees very well
with observations and predicts the correct attenuation with depth. It is apparent
from the form of this expression why it is impossible to obtain a stable transfer
function to predict the noise without an array of mierobarographs and excessive
data processing. The most effective way to suppress this kind of noise is to bury
the seismometer: at 150 m the reduction in noise level would he about 10 db.
INTRODUCTION
It has been known for some time that the noise recorded by long-period seismometers
is composed of two parts: propagating noise, so-called because it is coherent over distances of at least 100 kin, and nonpropagating noise, so-called because it is incoherent
over distances greater than a few kilometers. Because the long-period seismometers at the
Montana Large Aperture Seismic Array (LASA) are placed so far apart, the nonpropagating component of the noise appears to be spatially disorganized. It has been
shown by Haubrich and Mackenzie (1965) and Capon (1969) among others, using
coherence spectra, that the nonpropagating component of the noise is correlated with
microbarograph recordings at the same site at times when this component is significant.
Since this component contributes at least 40 per cent of the noise up to 50 per cent of the
time at LASA (Capon, 1969), it seemed worthwhile to consider the possibility of eliminating it using the knowledge that the output of a given seismometer will be correlated with
the output of a nearby microbarograph.
The process of forming a coherence spectrum is a linear one, a coherence spectrum
being the amplitude spectrum of a cross-correlation function. Any coherence in the
frequency domain indicates that part of one time series is linearly dependent on the other
time series. This is not to say that the physical process which relates the two time series
is linear, merely that there is a linear component which, therefore, can be removed with a
linear time-domain filter. If the physical process relating the two series & linear, it is
937
938
ANTON ZIOLKOWSKI
possible to predict with any accuracy one wants the part of one time series which depends
on the other time series.
THE PREDICTION PROCESS
We can divide the seismometer output as a function of time, S(t), into two parts: the
part, N(t), which is correlated with the microbarograph output and, therefore, caused by
local pressure fluctuations in the atmosphere; and the other part, E(t), not correlated
with the microbarograph output, and, therefore, containing any signals and noise caused
by all other uncorrelated effects. Thus
S(t) = N(t) + E(t) .
(1)
It is possible to define an atmospheric pressure fluctuation, P'(t), in the vicinity o f the
seismometer and microbarograph which would produce a unit positive impulse at time
P'(t)
I
Microboro(::jrap~
~h
I
t=O
-
I
t4
e~
Seismomet
Zlt)
=
I
t
(o)
P'(t)
I
I) Microbarograp~
I
I
I
t=O
t =o
=
t4
Seismometer~~
I(t-r)
I
I
[
t
Produces Artificial
Delay of T
t
t =0
(b)
FIG. 1. (a) A possible situation if the microbarograph impulse response is minimum-delay: the
postulated pressure fluctuation P'(t) produces a spike at time t = 0 in the microbarograph.output and
some response I(t) in the seismometer output. (b) If the microbarograph impulse response is not minimum-delay, P'(t) can be devised so as to produce an almost perfect spike at time t = z in the microbarograph output; the consequent output l(t) in the seismometer can be artificially delayed by an amount
"t'.
t = 0 in the output of the microbarograph, and some other respofise,/(t), in the output
o f the seismometer (see Figure la). This postulated pressure fluctuation, P'(t), is the
inverse of the microbarograph impulse response and is strictly realizable, in the sense that
it does not exist before t = 0, only if the microbarograph impulse response is minimumdelay (see, for example, Robinson, 1967). In practice it is possible to obtain as good an
LONG-PERIOD
NONPROPAGATING
SEISMIC
NOISE
939
approximation as is desired to the inverse ofa nonminimum-delay response by introducing
a sufficiently long delay, z, into the output so that the impulse appears at time t = t.
Of course, now it is impossible to derive a causal relationship between the microbarograph output, M(t), and the correlated noise component, N(t), without also introducing
an artificial time delay of at least v into the output of the seismometer (see Figure lb).
Thus, allowing for these delays, we may write
N(t) = ~ ~ M(~) I ( t - ~)d~. .
(2)
If we define the cross-correlation function to be
B(t-l)dr,
~bAdl) = lira ~
(3)
T-+oo
we can obtain the cross-correlation, qSSM, of the seismometer output with the microbarograph output
1
r
CsM(S) = lira ~ j" r { j ' ~ M ( ~ ) I ( t - ~)d~}M(t-s)dt
T -.'-co
1
r
+ lim ~1" TE(t)M(t-s)dt,
(4)
T ---~oo
using equations (1) and (2). From the definition of E(t) we see that the second integral in
equation (4) must be zero for all values of s, and by making the substitution q = t - ~ - s
the first integral can be transformed to give
(5)
Cs~(s) = f~ ~ ¢ ~ ( ~ ) I ( ~ +s)d~.
For digital data this equation can be written as
qbsM(s)=
Cl)mm(q)I(q+S)
~
(6)
r / ~ -- o0
for all integer values of s, where
1
qbA~(l) =
T
lira
~ A(t)B(t-l).
T ~oo 2T+ 1 t = - r
In practice S(t) and M(t) are series of finite length, say N points, and the transfer function
I(t) is a transient which needs to be computed for only n points. In this case equations (6)
become
n-I
Osm(S) =
~
q~mr~(r/)l(r/+s),
s = 0, l ..... n-- 1
(7)
I/=0
where Cl)sm(S) and @uu(q) can be estimated using the equation
1
N-
I/I
*An(l) -- N-Ill t~o A(t)B(t-l)
(8)
(Blackman and Tukey, 1958). Negative values of r/need not be included in equations (7),
for ~MU(r/) is a symmetric function and I(t) is zero for negative values of t. Equations (7)
are the well-known system of normal equations and can be solved efficiently for I(t) by
Levinson's method (1949). Having obtained an approximation to I(t) (the quality of the
approximation depends on the quality of the estimates of (I)su and ~MM and improves
with any increase in the number of points, n, for which I(t) is computed), we can compute
an approximation to N(t) using equation (2) and subtract this from the seismic signal,
940
ANTON ZIOLKOWSKI
S(t), to obtain an approximation to E(t) which should now have a better signal-to-noise
ratio than S(t).
It may be worth noticing that there is nothing inherent in this process which confines
its application to the prediction and removal of pressure effects in seismic data. Whenever
one time-dependent quantity is a linear function of some other time-dependent quantity,
the dependence can be determined. This technique could be used, for example, possibly
with great advantage, to remove both temperature and pressure variations from earthstrain records. Some care would have to be taken, when removing more than one effect,
to remove any linear dependence of the second process on the first.
AN EXAMPLE
The above technique was applied to a noise sample beginning at 17h 20m 00s on
August 23, 1967, recorded on the 3-component long-period seismometer at A0 at LASA.
Capon (1969) showed for this sample that the nonpropagating component contributed
nearly 10 db more to the noise measured by the vertical seismometer in the 20- to 40-see
period range than did the propagating component. At this time, there were two microbarographs operating at A0, one with a wind filter on the input and one without. The
wind filter has the effect of removing all frequencies in the seismic band out of the microbarograph data and, for the purposes of demonstrating the use and validity of the above
technique, adulterates the data sufficiently to render it entirely useless. All calculations,
therefore, were performed on the "uncensored" data obtained with the naked microbarograph.
Figure 2 shows the autocorrelation function, qbMM(t), the cross-correlation function,
¢bs~t(t), and the transfer function I(t) with the artificial time-delay z, for each of the components, (a) vertical, (b) north-south and (c) east-west. Figure 3 shows the microbarograph output (1), the seismometer output (2), the predicted noise (3), and the seismometer
output with the predicted noise subtracted (4), for each of the three seismometer components. It is clear that a considerable reduction in noise level has been achieved. The
gain in the process, in reducing trace 2 to trace 4 in Figure 3, is shown in Figure 4 in db
as a function of frequency for each seismometer component, the peak gain in each case
being at about 40 sec. No special significance should be attached to the figure of 40 sec.
Under different weather conditions we would not necessarily expect the noise to peak
at the same frequency.
To demonstrate that all of the seismic noise linearly related to the microbarograph output has been removed, spectra are shown in Figure 5 of the coherence between the
microbarograph and seismometer outputs (1), and the microbarograph output and the
seismometer output with the predicted noise subtracted (2), for the three seismometer
components. We can see that the measured coherence (2) in each case in Figure 5 indicates
with 95 per cent confidence that everything in the seismometer output which was coherent
with the microbarograph output has been eliminated.
THE COUPLING PROCESS
It is of some interest to try to understand the physics of the coupling between the
atmosphere and the seismometer. It could be argued that the understanding is not really
required because the linearly-related component can be removed by filtering; all that is
required is to determine how linear the process is. However, it may be cheaper and
simpler to reduce the noise by a more careful isolation of the instrument, if that is all that
is required, rather than to install microbarographs and do processing. Therefore, if the
LONG-PERIOD NONPROPAGATING SEISMIC NOISE
I
941
(o)
t=o
r
Vertical
~SM(t)
I(t)
T
I
I
I
(b)
~-0
North-South
<~MMlt)
I(t)
J
r
I
(c)
t=O
I
East-West
<]~SM(t)
I(t)
I
-200
-100
I
100 200
TIME, t (sec)
I
300
- -
400
Fig. 2. The autocorrelation function, ~MM(t), of the microbarograph output, the cross-correlation
function, ~su(t), between the seismometer and microbarograph outputs and the transfer function, l(t),
derived from equations (7) for (a) vertical, (b) north-south and (c) east-west components.
i Fb°r
(2)
o m~
t
(3) ~
(4)
200
00 rap.
(o1 Vertical Component
i p.bor
(2)
00 m~
1000
t
(3)
1000
(4)
(b) North-South Component
(f)
/~bor
,ooo
-
(3) ~ ' ~ ~ / ~ / ~ / ~ / g A ~ ~ ~ / / / ~
t000
¢/- 50OmF
0
fO00
(4) ~ / ~ / V ' v ~ ~ J ~ / ~ ~
" ' ' V ~
(c) East-West Component
[
o
I
I
2
I
l
4
n
I
L
6
I
8
J
[~AV - 5oo inF.
'J
_0
I
iO
TIME (rain)
FIG. 3. Data from LASA site AO beginning 17h 20m 00s on August 23, 1967. For each of the three
components of the seismometer, (a) vertical, (b) north-south and (c) east-west, are shown the seismometer
output (2)~ the predicted noise in that output (3), and the seismometer output minus the predicted noise
(4). In each case (1) is the microbarograph output at Ag.
LONG-PERIOD NONPROPAGATINGSEISMICNOISE
8
943
t
m
v
Z
6
0
F123
b.I
fie
LO 4
O3
m
o
(b) N o r t h - S o u t h
z
/
!
2
(c) East-West
/
(a) Vertical
~_ Signa~
Band
0
i
l
0.01
~
,
1
r
i ,Jl
0.05
i
,
i
O.l
FREQUENCY
0.5
(Hz)
FIG. 4. The gain in noise reduction in db as a function of frequency in converting trace (2) to trace (4) in
Figure 3 for each seismometer comoonent.
t.0"
l
[
o.e/
I
r
I
I
I
~ Signal |
--'1 Band I'--
I
I
~
I
/
_
I
[
I
l
r
I
Signal l~
~l Band
~
0.6
_
t
_
t
0.4
I
0.01
(3.1
0.01
0.1
0.1
0.0t
FREQUENCY (Hz)
FREQUENCY (Hz)
FREQUENCY (Hz)
(a) Vertical Component
(b) North-South Component
(c) East-West Component
Fig. 5. From the data shown in Figure 3 are the computed coherencies between the microbarograph
and seismometer outputs (1) and the microbarograph output and seismometer output minus the predicted
noise (2) for each seismometer component, (a) vertical, (b) north-south and (c) east-west. There is at least
95 per cent confidence associated with any coherence above 0.3.
944
ANTON ZIOLKOWSKI
coupling process is understood, the best method of reducing this kind of noise can be
determined.
There are two ways in which variations in atmospheric pressure can produce responses
in a seismometer: via buoyancy in the instrument, and by elastic deformation of the
ground which is then measured in the usual way. As it happens, the buoyancy effects on
the Montana LASA long-period instruments are absolutely negligible. This can be shown
by a variety of techniques.
The first indication that the measured noise is not buoyancy is that it shows up as
strongly on the horizontal instruments as it does on the verticals. The horizontal instruments cannot respond to changes in atmospheric pressure unless these give rise to tilts
in the ground. In any case, great care was taken at LASA in the installation of the
instruments to isolate them from any possible buoyancy effects.
Each of the three long-period seismometers at every subarray site at LASA is housed
in a case. For vertical seismometers, this case has an air-tight seal, the case then having a
time constant of at least 8 hr for pressure variations; in addition, all three cases are enclosed in a sealed vault which has a time constant of the order of 40 rain (Gudzin and
Hennen, 1967). Therefore, any atmospheric pressure fluctuations of the order of 40 sec
would be so severely attenuated by this protection that any resultant instrumental
buoyancy would be undetectable. The crucial experiment to verify this was performed
by Gudzin (1972) when employing identical seismometers in setting up the Tonto Forest
Observatory. A vertical long-period instrument in its sealed case was put in a room which
was subjected to internal pressure variations of 1 mbar amplitude; no motion in the
seismometer was detected. Since the maximum 40-sec atmospheric pressure fluctuations
are expected to be only of the order of 100 pbar we would, therefore, expect to be able
to rule out buoyancy as a means of coupling.
Finally, if the predictable portion of the noise shown in Figure 2 were predominantly
caused by buoyancy in the seismometer, and if the impulse responses of the microbarograph and the seismometer remained stable, one would expect that the transfer functions,
I(t), calculated for the noise samples shown in Figure 3, would serve equally well to
predict the nonpropagating component of the noise on some other day when that component was again dominant. That the application of the previously computed transfer
functions to the output of the same microbarograph at A0 to predict the noise in the
same seismometers at A0 did not prove successful (see Figure 6) for data which Capon
(1969) had shown to have a large nonpropagating component is confirmation that buoyancy, at A0 at any rate, is quite negligible. We are thus forced to conclude that the
observed noise is caused by atmospheric deformation of the ground.
Since the nonpropagating noise is incoherent over distances greater than a few kilometers, we might expect the source of such noise to have a coherence distance of the same
order. Herron et al. (1969) have shown that the coherence of background atmospheric
pressure fluctuations decreases approximately as the logarithm of distance for period
bands of 8 to 64 min, the coherence increasing with increasing period. Since a significant
coherence at 8-min period was obtained by Herron et al. (1969) only at distances less than
3 kin, we can assume that the coherence distance, the maximum distance at which there is
significant coherence, for periods of the order of 40 sec is substantially less than 3 km.
Herron et al. (1969) have also shown that there is a very strong correlation, greater than
0.8, between the wind speed, averaged over a given period, and the power density of
pressure fluctuations between 10 and 50 sec during that period. The resulting picture of
atmospheric pressure disturbances is one of turbulence: the local pressure pattern,
referred to a tYame of reference moving relative to the ground at the average wind speed,
changes substantially in the time it takes the reference frame to move a distance greater
(~)
4
__1205p,bar
(2)
(3)
(4)
(a) Vertical Component
-5O
~
(~)
25
~bar
0
100o
(2)
500 m/~
0
-~ t000
(3)
(4)
(b) North-South Component
50
(t)
{2)
,
. I
0 0
(4)
m/a.
(c) East-West Component
I
0
J
I
2
I
~
4
M
t
I
6
i ,r
8
I
10
TIME (rnin)
FIG. 6. Data from LASA site A0 beginning 19h 50m 00s on July 1, 1967. As in Figure 3, for each of
the three components of the seismometer, (a) vertical, (b) north-south and (c) east-west, are shown the
seismometer output (2), the predicted noise in that output (3), and the seismometer output minus the
predicted noise (4). In each case (1) is the microbarograph output, but the predicted noise has been
calculated using the transfer functions shown in Figure 2.
946
ANTON ZIOLKOWSKI
than the coherence distance--typically somewhat less than 3 km for periods of the order
of 40 sec.
In the example given above, a large proportion of the noise was found to be predictable
from a single microbarograph located near the seismometer. Since we have concluded
that the noise is caused by atmospheric ground loading and since the microbarograph
measures atmospheric pressure variations coherent only within 3 km or less, we deduce
that most of the observed nonpropagating noise is of very local origin, that is, generated
by pressure variations within 3 km or less.
ESTIMATION OF NOISE POWER
Given this situation, we may try to describe it with a simple model of the Earth. It is
clear that the curvature of the Earth is irrelevant to this problem, so we may approximate
the Earth by a homogeneous elastic half-space. Possibly, it would be an improvement to
consider a multilayered half-space; but this will not be done here, partly because the
author would like only to arrive at some qualitative understanding of the problem and
some order of magnitude answers, and partly because it is quite beyond either his enthusiasm or capability to do so. Wind speeds are two or three orders of magnitude less than
the speeds of seismic waves; we therefore incur negligible error in considering the
atmospheric deformation of the ground as a problem in statics. Only the vertical component of displacements in response to normal stresses on the surface will be considered;
effects such as the wind blowing on trees and other obstacles which can produce displacements are ignored.
We may use the Green's function given by Landau and Lifshitz (1970) to find the
vertical displacement, u~(0, 0, z), at the point (0, 0, z) in the medium
u,(O, O, z) =
f ~ fo~ l + v I 2 ( l _ v ) z 2 1
_ ~ _ oo-2~-E
r +•
P(x, y)dxdy ,
(9)
where the surface of the half-space is the xy plane, z is positive into the medium, v is
Poisson's ratio, E is Young's modulus, P(x, y) is the pressure distribution and r =
(x 2 +y2 +z2)1/2. If we assume that P(x, y) is a stationary, white, random, two-dimensional
noise function whose autocorrelation is, therefore, a delta function, we can find an
expression for the expectation of uz2 (see Appendix 1)
E{uZ} =2r c ~°° F
k,
t-
kzz2
Jo L(Ro~Uzb'/~ (Ro~+ zb ~ ]
|q2
RodRo ,
(1 O)
where k s = (1 - v2)/xE and k 2 = (1 + v)/2rtE. This integral diverges at the upper limit
for any finite value of z and diverges at the lower limit when z --- 0. This means that we
would expect to get infinite noise from such a model. Since infinite noise is not observed,
something must be wrong with the model.
Other workers who have tried to estimate the power of the noise by atmospheric
loading of a half-space have also arrived at solutions which blow up in this way. Haubrich
(1970), for example, estimates the noise at the surface of an infinite half-space by dividing
the surface into an infinite number of segmented concentric rings, each segment having
the same area, and then adding up the contributions from all the segments. The atmospheric pressure acting on a given segment is assumed to be uniform and to vary with time
independently of the pressure on any other segment, the power of the variations over each
segment being the same. Instead of having to evaluate an integral which does not
converge, Haubrich finds he has to sum a series which does not converge. Undeterred,
LONG-PERIOD NONPROPAGATING SEISMIC NOISE
947
he explains, "it doesn't diverge very fast," and includes only the inner 1,000 rings, thus
ignoring the remainder which unfortunately contribute infinitely more to the calculated
noise than the ones he includes. By taking an estimate of u~2 from observations, he
deduces the segment size.
Savino et al. (1972) have made improvements to Haubrich's model by including depth
dependence. They estimate the segment size from measurements of coherence of earth
noise and then adjust the number of rings to force the calculations to agree with observations. They, too, igriore all the segments in rings beyond a certain radius which would, if
included, contribute infinitely more to the calculated noise than the ones they choose to
consider.
In an entirely different approach, Sorrells (1971) assumes that the wind-induced
pressure field is a plane wave which propagates at wind speeds. His model, therefore,
does not allow for any decay of coherence with distance perpendicular to the direction
of propagation. However, Sorrells's solutions for the displacements at a given frequency
(equations 21 and 22) do show that the displacement will become infinite at any depth
whenever the frequency is zero. This is exactly the same as including the zero-wave number
component in statics.
Consider a 2-dimensional static sinusoidal pressure distribution, rr~z, on the surface
of the half-space, where
a ~ = P cos (2n~x) cos (2rq~y)
(11)
and c~ and (/are wave numbers. The vertical displacement, u~, at a depth z is given by
Fung (1965)
a ~ [2(1uz = 4rc/~c
v)+27rcz]e-2~c~
where p is the rigidity of the half-space and c =
of the strain, @(u~)/#z, at a depth z, is
c3(u~) _
c3z
(12)
(~z+f12)i/2. The vertical component
a ~ (1 -2v+2~cz)e-2=cL
2p
(13)
When the wave number, c, is zero, the vertical component of strain, @(uz)/#z, is constant
right to the bottom of the half-space. But there is no bottom to the half-space. Therefore,
the integral of the strain, u~, is infinite. If we had been using a spherical earth model this
would not have happened. However, since the observed noise is generated so locally
that the curvature of the Earth must be irrelevant to the problem, we must filter out this
impermissible d.c. component.
The divergence of the integral in equation (10) at the lower limit when z = 0 is caused
by the inclusion of point forces. If we let P(x, y) in equation (9) become P~5(x)5(y), a
point force at the point (0, 0, 0), we obtain for the displacement
Uz-
(1 + v) ( 3 - 2 v ) P ~ ,
2~E
z
(14)
which is infinite when z = 0. By defining the autocorrelation function of P(x,y),
E{P(Xl, Yl)P(x2, Y2)}, to be equal to the delta-function PZ@(x~-x2)6(y~-yz), we are
specifying that the pressure distribution be a stationary, white, random distribution of
point forces with zero mean. Every point at the surface of the half-space, therefore,
expects to be sitting under a point force. If we disallow point forces, which are not
physically realizable, by filtering out the infinitely large wave numbers, we can avoid the
divergence which occurs in equation (10) at the lower limit of the integral when z = 0.
948
ANTON ZIOLKOWSK!
It should be noticed that Haubrich's approach of dividing the surface of the half-space
into cells of equal area each subjected to an independent uniform pressure load is
equivalent to low-pass filtering the pressure distribution in wave number space.
The spectrum of atmospheric pressure variations rises at approximately 7 db/octave
toward the long periods (Gossard, 1960). Herron et al. (1969) point out that these longperiod variations generally travel at about the same speed as the frontal movement of
storms (10-15 m/sec). Therefore, the wave-number spectrum of these variations rises
at about the same rate toward the long wavelengths. Farrell (1972) has shown that the
t000
"0
t00
0
E
t,0
E
E
I.i.I
Q
_.1
t.0---
o.t
o.oot
I
I L IIILII
I
0.01
I i ill11L
i
0.t
I [tLhl
t.0
FREQUENCY (Hz)
FIG. 7. Long-period system transfer function.
Green's functions for the loading of a homogeneous elastic sphere, in the limit of decreasing distance from the load, reduce to the solutions for a point load on a homogeneous
elastic half-space. He also shows that, close enough to the point load, a layered earth
model responds in the same way as a homogeneous sphere that has the properties of the
topmost layer. In order to justify our use of the half-space model to describe the loading
process, therefore, we must be convinced that the long-wavelength atmospheric pressure
disturbances do not produce observable nonpropagating seismic noise and can be
filtered out of the solution. It can be seen from Figure 7 that the LASA long-period
instrument response falls off at approximately 20 db/octave. Since the nonpropagating
noise can be rising at only about 7 db/octave, the sensitivity of the seismometer to this
kind of noise is decreasing at about 13 db/octave. It is permissible, therefore, to use a
LONG-PERIOD NONPROPAGATING SEISMIC NOISE
949
half-space model to describe the atmospheric loading process provided the longwavelength disturbances, which do not contribute to the noise, are filtered out of the
calculations.
To avoid the convolution integrals which would arise from attempting to band-pass
filter the Green's function, we will consider the problem in wave-number space (Appendix
2). The d.c. component is removed by application of a filter of the form [1-e-~C],
where e is the wave number, and the point forces are removed by application of a filter
of the form e-~L These filters have been chosen for convenience (they lead to simple
integrals) to demonstrate the necessity for band-limiting the wave-number spectrum;
slightly different answers would doubtless be obtained with equally arbitrary, less convenient filters which have different cutoff characteristics. With this approach we find the
expectation of u~2 to be (see Appendix 2)
=
[(1 - va)/i + 2fez(1 - v)I2+nZzZI3]
(15)
where
,
[-4(2rcz + r/) (y + 2az + r/)- 1
Z'=--'Oge L
[==
J'
yz
2(2fez + q) (7 + 4rcz + 2q) (Y+ 2rcz + ~/)'
y2 [Tz + 6y(2rcz + r/) + 6(2ztz + r/)2]
I3 = 4(2zrz + r/) 2 (7 + 47rz --k2q) z (Y+ 21rz+ q)z
and p Z is the mean-square pressure deviation per unit wave number squared. From
equation (15) we see that I 1 becomes infinite if 7 is infinite or if q and z are both zero. In
other words, the expectation of u~2, E{Uz2}, is finite providing the d.c. term and point
forces are eliminated.
Let us now consider the estimation of u=2 at the surface of a half-space for realistic
data. Equation (15) becomes, for z = 0
E{U2} =
Pt 2( 1 - v2) log e F4r/(7 + r/)-]
27r/,z
L( 7 j j.
(16)
In this equation we can replace P1 z by pZ. 2z, where pZ is the mean-square atmospheric
pressure deviation and 22 is the mean-square wavelength of the atmospheric pressure
distribution. In reality, however, we are not dealing with a static situation: uz, P and 2
are time-dependent quantities. Nevertheless, we may still find this formula satisfactory
if we are prepared to make some more assumptions. We have already noted the high
correlation, observed by Herron et al. (1969), between the wind speed averaged over a
given period and the power density of pressure fluctuations between 10 and 50 sec during
that period. This suggests that we could replace 22 by V 2 T 2, where V 2 is the mean-square
wind speed and T is the period at which there is maximum coherence between the
seismometer and microbarograph outputs. If we further assume that the power of the
pressure fluctuations at any point averaged over time is the same as the power of the
pressure fluctuations at any time averaged over space, we can replace p2 by P2(t).
Equation (16) then becomes
E{u2(t)} =
p2(t). V2(t). TZ(1 - v2),
[4q(y + q)~
(17)
950
ANTON ZIOLKOWSKI
We can test this formula using the data from the example used above. Estimates of
u~a(ti, p2(t) and V2(t) were made using the formula
a2(t ) = I r a2(t) d t ,
Jo
z
(18)
where ~ was about 2 hr 20 rain. T is taken to be the peak in the coherence spectrum
between the vertical seismometer and the microbarograph. ~/and V are the shortest and
longest wavelengths which contribute to ob~rvable noise on the vertical seismometer
and were estimated in the following way. The smallest and largest periods to which the
seismometer can respond were chosen to be those at which the sensitivity of the instrument is down a factor of 10 from the peak. These periods are about 9 and 80 sec (see
Figure 7). t/and y are then assumed to be equal to these periods multiplied by the lowest
and highest recorded wind speeds, 5 and 10 m/sec, in the time period T. So t/ = 45 m
and ~ = 800 m. This is not inconsistent with the measurements of coherence made by
Herron et al. (1969). From Landers (in preparation) measurements of P- and S-wave
velocities in the sediments at LASA, Poisson's ratio, v, is estimated to be about 0.3 and
# is estimated to be about 3.5 x 10 ~° dyne cm -2. Using equation (18), the following
estimates were made: p2(t) = 62.6 dyne 2 c m - 4 ; V2(t) = 37 m 2 sec -2
Inserting all these numbers into the right-hand side of the formula 17 and taking
T = 40 sec from Figure 5, we find E{uz 2} = 723 x 10 -18 m 2. From equation (18) and
the seismometer output we find u~2 = 764 x 10-~8 m 2.
The agreement between the estimated and observed noise is obviously too good to be
true. Uncertainties exist in the estimates of v and # (Landers, in preparation) which may
affect the result by a factor of 2; the logarithmic factor is somewhat arbitrary in that it is
derived from arbitrary assumed filters. It should also be remembered that some of the
noise measured is propagating noise. However, for this sample of data, Capon (1969)
showed that the nonpropagating component contributed nearly 10 db more in the 20to 40-sec period band than did the propagating component; therefore, only about
70x 10 -18 m 2 of the observed noise can be propagating. The formula will probably
hold good within less than an order of magnitude.
It is interesting from a practical viewpoint to find how much improvement could be
obtained by burying the seismometer. Making the same assumptions as for the case of
z = 0 above, equation (15) becomes
E{u}} =
PZ(t). V2(t). T 2
27r#z
[(1 - v2)Ix + 2~zz(1 - v)I 2 + 7rZzZlal .
(19)
Figure 8 shows E{u= z} plotted as a function of z using the same data for Pa(t), I~2(t),
etc. as above. It would seem from this example that the noise-power of this particular
nonpropagating noise, generated by atmospheric pressure disturbances with wavelengths
of the order of 45 to 800 m, could have been reduced an order of magnitude by burying
the seismometer 150 m.
Sorrells et al. (1971) describe a very careful experiment to measure noise caused by
local atmospheric pressure changes. The two 3-component seismometers, one at a surface
site and one buried 183 m in a salt mine, were completely isolated from any measurable
buoyancy effects. During calm periods the noise spectra of the two vertical instruments
are virtually identical. During a windy period when the rms pressure fluctuation was
12 #bars and the mean wind speed was 7.8 m / s e c the noise level in the surface instrument increased by 10 db in the 20- to 100-sec band, whereas that in the mine instrument
remained the same. This noise condition is approximately the same as in the example
951
LONG-PERIOD NONPROPAGATING SEISMIC NOISE
used throughout this paper; in particular, the wavelengths of the pressure distribution
are about the same--perhaps a little longer. Figure 8 predicts that at 183 m, one would
expect to get a little over l 1 db decrease in noise level. The results of Sorrells et al. (1971)
thus add some independent confirmation of the validity of equation (19).
50
t000
m
b
~00
20
°J::l..
E
u
04
W
>
la.I
i
t0
~0
_J
W
i1co
"10
i
i
I
0
~
I
200
I
I
400
I
I
600
I
I
800
I
I
~000
1
o
12oo
DEPTH OF BURIAL, z(m)
Flo. 8. E{u}z ~ calculated from equation (19) as a function o f z using the following data: P2(t) = 62.6
d y n e 2 c m - 4 ; V2(t) = 37 m 2 s - 2 ; T = 4 0 s ; / 1 --- 3.5× l0 t° dyne c m - 2 ; v = 0.3; r/= 4 5 m ; ), = 8 0 0 m .
A N IMPROVED PREDICTION TECHNIQUE?
Equation (19) shows clearly why the transfer function, I(t), found for the example
described above was not suitable for predicting the noise in a later sample of data: the
relationship between P(t), the atmospheric pressure, and u(t), the displacement caused by
atmospheric deformation of the ground, is not constant. There is a time-dependent
weighting factor which depends on wind-speed, V(t), and period at which the noise peaks,
T. In fact, the weighting factor will depend on 2(t), the rms wavelength present at time t.
However, if one makes the first-order assumption that 2(0 is proportional to V(t) for a
narrow-band seismometer, (obviously 2(0 is independent of the seismometer characteristics, but the wavelengths which can cause observable noise do depend on the seismometer) and, if one also assumes that any one point at the surface is as good as any other
for measuring V(t), one might just expect to do a little better with the prediction technique
by weighting the microbarograph output, M(t), with the anemometer output, A(t), at
the same site. Of course, the responses of the microbarograph and anemometer are
entirely different; therefore, the two time series M(t) and A(t) would have to be deconvolved to remove the instrument responses and get back to the inputs P(t) and V(t).
952
ANTON ZIOLKOWSKI
This argument may seem a little tenuous. After much experimentation with the data
and showing that careful weighting of the deconvolved microbarograph output with the
deconvolved anemometer output produced a time series which was no improvement over
the original, unadulterated, microbarograph output, the author has reluctantly come to
the conclusion that this argument is, indeed, tenuous, if not erroneous. If one had an
instrument which measured wavelength as a function of time, 2(t), the conclusion might
have been different. To obtain estimates of 2(t)would require an array ofm icrobarographs.
CONCLUSION
Although on some occasions, a large proportion of the nonpropagating noise observed
on a long-period instrument at LASA is strongly coherent with the output of a nearby
microbarograph and can, therefore, be predicted and eliminated, the transfer function
which converts the microbarograph output to the predicted noise is not a stable function
of time and must be calculated anew everytime. Attempts at stabilizing the transfer
function by weighting the deconvolved microbarograph output with the deconvolved
anemometer output at the same site have failed.
Several lines of argument, in particular the lack of stability of the computed transfer
function over time, indicate that the noise is caused by atmospheric deformation of the
ground and not buoyancy in the seismometer.
Most of this nonpropagating noise appears to be generated very locally--within 3 km
or less of the seismometer--by pressure disturbances in the atmosphere moving at the
ambient wind velocity. The process can, therefore, be described approximately using a
half-space model for the Earth and static loading theory. In order to keep the calculations
in accordance with observations, the wave-number spectrum of the pressure disturbances
must be band-limited. The model may then be used to derive a formula, equation (19),
for the estimation of the noise power produced by such band-limited pressure disturbances
which agrees well with observations. The decay of this noise with depth predicted by the
model agrees very well with independent measurements made by Sorrells et al. (1971).
There would have been some future in predicting and eliminating this noise using the
theory presented in the section on the prediction process in this paper had it been possible
to obtain a stable transfer function. Since this appears to be out of the question, due to the
turbulence of the fickle wind, there remain two ways of suppressing this kind of noise.
One method is to bury the seismometer at a depth of 150 m, where the reduction in noise
level should be about 10 db. The other method is to employ an array of seismometers
placed no closer together than 1 km. To achieve the same reduction in noise level as
would be obtained with a single buried instrument would require 10 surface instruments.
ACKNOWLEDGMENTS
I have been fortunate in having had the benefit of many useful suggestions and helpful criticism from
staff members of the Seismic Discrimination Group of Lincoln Laboratory. In particular, I would like
to thank Dr. John Filson, Dr. Clint Frasier, Dr. Bruce Julian, Dr. Richard Lacoss and Dr. Tom Landers.
REFERENCES
Abramowitz, M, and I. A. Stegun (1964). Handbook of Mathematical Functions, U.S. Dept. of Commerce
National Bureau of Standards Applied Mathematics Series 55.
Blackman, R. B. and J. W. Tukey (1959). The Measurement of Power Spectra, Dover Publications, New
York.
Bracewell, R. (1965). The Fourier Transform and Its Applications, McGraw-Hill, New York,
LONG-PERIOD NONPROPAGATING SEISMIC NOISE
953
Capon, J. (1969). Investigation of long-period noise at the Large Aperture Seismic Array, J. Geophys.
Res. 74, 3182-3194.
Fung, Y. C. (1965). Foundations of Solid Mechanics, Prentice-Hall, Englewood-Cliffs, New Jersey.
Gudzin, M. G. (1972). Private communication.
Gudzin, M. G. and F. M. Hennen (1967). Final Report, Project VT/6701 LASA LP System. Geoteeh
Report No. 67-17.
Haubrich, R. A. (1970). The origin and characteristics of microseisms at frequencies below 140 cycles
per hour, Mono de I'UGGL
Haubrich, R. A. and G. S. MacKenzie (1965). Earth noise, 5 to 500 millicycles per second, 2. Reaction of
the Earth to oceans and atmosphere, J. Geophys. Res. 70, 1429-1440.
Herron, T. J., I. Tolstoy, and D. W. Kraft (1969). Atmospheric pressure backround fluctuations in the
mesoscale range, J. Geophys. Res. 74, 1321-1329.
Landau, L. D. and E. M. Lifshitz (1970). Theory of Elasticity, Pergamon, Elmsford, New York.
Landers, T. E. (Paper in preparation).
Levinson, N. (1949). The Wiener rms error criterion in filter design and prediction. Appendix B of
Wiener, N., 1949 in Extrapolation, Interpolation and Smoothing of Stationary Time Series, Wiley,
New York.
Robinson, E. A. (1967). Statistical Communication and Detection with special reference to Digital Data
Processing of Radar and Seismic Signals, Griffin, London.
Savino, J., K. McCamy, and G. Hade (1972). Structures in earth noise beyond twenty seconds--A
window for earthquakes, Bull. Seism. Soc. Am. 62, 141-176.
Sorrells, G. G. (1971). A preliminary investigation into the relationship between long-period seismic
noise and local fluctuations in the atmospheric pressure field, Geophys. J. 26, 71-82.
Sorrells, G. G., J. A. McDonald, Z. A. Der, and E. Herrin (1971). Earth motion caused by local atmospheric pressure changes, Geophys. J. 26, 83-98.
LINCOLN LABORATORY
MASSACHUSETTSINSTITUTEOF TECHNOLOGY
LEXINGTON, MASSACHUSETTS02173
Manuscript received October 13, 1972
APPENDIX 1
Estimation of Noise Power Using the Green's Function
The vertical displacement, uz(O, O, z), at the point (0, O, z) in a half-space subjected to a
pressure distribution P(x, y) is given by
u~(0, 0, z) =
f~z f2°°
-~ ql+v[-2(1-v)
_L ~~- + r ~ J
P(x,y)dxdy,
(A1)
(Landau and Lifshitz, 1970), where the surface of the half-space is the xy plane, z is
positive into the medium, v is Poisson's ratio, E is Young's modulus and r = (x 2 +y2 +
z2) 1/2. We can immediately write down the expectation of Uz2
where
k 1 = (1 - v2)/~zE,
k2 = (1 + v)/2~zE,
r I = (x12+y12+z2) ~ and
r2 = (x22+yz2+Z2) W2.
(A2)
954
ANTON ZIOLKOWSKI
In equation (A2) the only quantity on the right-hand side which is not deterministic is
[P(xl, Yt) P(x2, Y2)]- We can rewrite this equation, therefore, as
E(uz2} = Ioo
foo f~o fro (kl..t k2z2~(kl
k2z2\
~1 f j \ r 2 + ~ 2 3 ) E{P(xl'yl)
-o~ -oo -oo -~ \171
P(X2'y2)}dXldyldX2dy2"
(A3)
In this equation E{P(xl,yl)P(x2,Y2) ) is the autocorrelation function of P(x,y) in two
dimensions. If we assume that P(x, y) is a stationary, white, random, 2-dimensional noise
function we can then define E{P(xl, Yl)P(x2, Y2) } = P2t~(xl-x2)6(yl-y2). In other
words, the autocorrelation of P(x, y) will be a 2-dimensional delta-function. It is now
convenient to change coordinates. Let
X' -- x l --X2
2
'
y, _ Yl--Y2
2
'
Xl -I-X2
XoThen
r1 =
2
, Yo =
Yl +Y2
2
(Relative coordinates)
(center of mass coordinates).
[(Xo-}-x')2-k-(yo-}-y')2-kz2]1/2,
r2 = [(Xo -- x') 2 q- (Yo --2') 2 -1-z2] 1/2
and
(~ ~ ~ ;~ /kl k2z2\/kl
E{u:2} = J_ooJ_oJ_oJ_®t -r'- + . w Tr, )trz- +
k2z2\_2
r2 ) 1-' <$(x')8(y')dx'dy'dxodyo.(A4)
.-TT
The integrals over x' and y' vanish except for x' = 0 and y' = 0. Therefore,
E{u, z) =
oo,'r°°- oo\(k-l+rok'E~3~2P-Tro
J dx° dY°'
(AS)
foo_
where
ro = (Xo2 +Y0 2 "-I-Z2)1/2.
We can now change to circular polar coordinates since we have circular symmetry.
Let
Ro = (Xg'-[-y02)1/2, 0=tan-a(Y--°~.
kXo}
Then
ro 2 = Ro 2 d-z 2
and
f2.f [
e(u?) = Jo Jo L(go
= 27z
k2z2 q2
+(Ro2
RodRodO
kz
k z 2 -12
0 (Ro2+Z2)l/2+(Ro2+Z2)3~
R°dR°
(A6)
APPENDIX 2
Estimation of Noise Power by Synthesizing Fourier Components
We can express the pressure distribution on the surface of a half-space as an infinitely
broad spectrum of components of the form
LONG-PERIOD NONPROPAGATING SEISMIC NOISE
azz(C~,//) = P~ (~, fl) cos (2gc~x + ~b,(c0) cos (2n//y + ~b2(/~) ) ,
955
(A7)
where c~ and fi are wave numbers and qS~ and @2 are phase angles which are functions of
wave number. By following the same procedure as in Fung (1965), one can find the displacement response at a depth z to one component
u~(~,//) -- °'z~(~x'~) [2(1--v)+2nCZ] e -zÈ~ ,
(A8)
4npc
where e = (cd +//2)i/2. The response to all components is therefore
u~ =
f~i~a~(~'fl)[2(l-v)+2ncz]e-2'~C~d~d//
--o0
--~
4n/~-~
(19)
where the dimensions of Gz(~, fl), and, therefore, PI(~,//) are stress.length. All the
wave numbers are included. It can be verified easily (see Appendix III), by setting
PI(~, fl) to a constant and ~b1 and ~2 to zero for all values o f ~ and//, that equation (19)
reduces to the Green's function when the pressure distribution is a point force.
We will now band-limit the wave-number spectrum using the filter (1 - e - ~ ) to remove
the d.c. component and the filter e - ~ to remove the point forces; ? and ~/are positive
numbers. If we then write the expression for G~ in exponential form, thus making Uz
complex, we find the band-limited response
Uz = ~- ~ - ooPt(cq//)f(c) e zÈi(~x+/~y)e i(o' (') + ~ (~)) dc~d//,
(A 10)
where
f(c)=
[2(1 -v)+2ncz] [ 1 - e -re] e -2"cz e -"c .
4ripe
We may now find the expectation of us 2 or uzu~*, where the asterisk(*) denotes complex
conjugate. Thus
E{u~Uz*) = E{I~_o~I~_o~S~ooI~_o~Pl(~,, fll)P,(c~2, fi2)f(cl)f(c2)
x exp [2ni(~ix+flty-c~2x-fl2y)]
x exp [i(q51(cq)+q52(//0-qbl(e2)-~b2(f12)] dcqdBld~edB2}.
(A1 1)
where cl = (cq 2 +//12) 1/2 and c2 = (~z 2 +//z2) 1/2. All the filtering is contained inf(cl) and
f(cz). We can make an estimate of Gu~* by assuming that the pressure distribution at any
time, before filtering, is a stationary white random noise, as in Appendix 1. We can
achieve this situation by applying the following restrictions to parameters which depend
on c~ and ]3We assume that P~(~,//) = / 1 , a constant, for all e and//.
We assume that q51(c0 is a white, random noise with zero mean which can take values
between - n and n; the probability that qS~ has a value between 0 and rio is equal to
aO/2n.
We assume that q52(//) is another random noise with the same statistics as qS~(~).
However, we further assume that q51 and q5a are completely uncorrelated, that is
E{~bl(¢0q52(¢2)} is zero and ~ can range from - o o to oo.
We may now write E{Gm.* } as
E{uiu~*} = I ~_~I ~_o~I~_~I ~_~ Pla f(q)f(c2) exp [2ni([el - c¢21x+ [//~ - fl2]Y)]
x E{exp [i(c~(~)+dp2(//l)-d?,(%)-~z(f12))]}
d~d//~dc~2d//2 .
(A12)
ANTON Z1OLKOWSKI
956
When el ~ e2 and/~1 ~ /~2 ~bl(el), qS~(e2), ~b2(fll) and q52(f12)are independent quantities
such that
E{ exp [i(q51(ex) + qSz(fll) - qbl(ez) - q52(f12))]}
= E{ exp [i$1(e0]} • E{ exp [i~bz(fl0] ) . E{ exp [-i~b,(e2)]}. E{ exp [-i~bz(f12)] }
=0.
However, when el = e2 and/~1 = f12,
E{ exp [i(q51(ex) + qSz(fll)- qSx(e2)- ~b/(fl/))]} = 1 .
Therefore,
E{ exp [i(q51(cq) + qSz(f10 - ~bl(e2) - q52(f12))l} = 6(ea - e2) 6(fll - f12).
It now becomes convenient to change coordinates such that
e'
_
e ~ 2- ~ ' /3'
_
~1-/3~
2
~o-
el + ez
[~o-"
2
31 + 3 z
2
whence
E{UzUz* } = S ~_oo~~_oo~~_ooS~ oo e l 2f ( c l ) f ( c 2 ) exp [4rti(e'x + fl'y)] 6(e') 6(fl') de' dfl' de0 dflo.
(A13)
Therefore,
E(u~uz*} = SE ~-~ ~ Pa 2 f2(Co) de ° dflo
(A14)
where c o = (Co + flo) 1/2. We may change coordinates again by putting 0 = tan-1 (flo/eo).
Equation (A14) then becomes
E{uzuz*}
=
r~,~r®
J0
J0 P12f2(co)codcodO
= 2re ~ P 1 2 f 2 ( c o ) c o de o .
(A15)
This integral may be evaluated by standard methods (Abramowitz and Stegun, 1964) to
give
E { u z u , } = PI"--T[(1- v2) 11 + 2(1 - v) rcz 12 + ~2Z2133
(AI 6)
2/rf12
where
F4( 2nz + ~/)(Y + 2rcz + r/)]
= --loge /
_l'
72
Iz = 2(27rz + r/) (Y + 4nz + 2q) (y + 2zrz + q)
and
72 [72 + @(2rcz + r/) + 6(2rcz + r/) 2]
/3 = 412nz + q]2 [7 + 47rz + 2q] 2 [Y+ 2nz + q]z •
Equation (A16) becomes infinite if either 7 is infinite or r/and z are both zero. In order
that the expression for the noise converges for a half-space model, the pressure distribution must be band-limited in wave-number space.
LONG-PERIOD NONPROPAGATING SEISMIC NOISE
957
APPENDIX 3
Synthesization of the Green's Function.for a Half-Space from Fourier Components
The normal stress at the surface of a half-space may be written as
az~ = S~oS_~oo Pl(c~, fl) cos (Drex+qSt(c0) cos (27rfly+(o2(fl))dedfl,
(A17)
where P1 is a function of e and fl and ~bl(7) and q52(fl) are phase angles. The vertical
displacement, u~, at a depth z is given by
uz =
foo y ,
--oo
--oo
pl(c~,fl)[2(1-v)+2rccz]e-2=C=c°s(2~x+gP,(~))c°s(2~flY+qa2(fl))dadfl
4relic
(A18)
where c = (~2+fi2)1/2. (See Appendix 2).
If we now make P~ a constant for all a and fl and put ~bl(a) = 0 and ~bz(a) = 0 we
can write equation (A17) as
azz = S~-o~*-oo Pa cos (2m~x) cos (2~rfiy) d~ dfl
= Re[~-~5°2-~ P1
e-2ni(=x+#Y)d~
tiff].
(A19)
This has circular symmetry in ~, fl space and can be written as a Hankel Transform. We
put e 2 = ~2+fl2 and x 2 + y 2 = r 2. Then
a= = Re[~o ~o P1 e-2~i ..... ~o-~o) ededO]
= Re[Sy Px(I2o~ e -2~'~°~° dO) cdc]
= 27r ~ el Jo(2zcer) ede
= P~
6(r)
7zr
- P,a(x,y),
(A20)
(Bracewell, 1965). Thus, the pressure distribution becomes a point force at the origin.
Equation (A 18) becomes
u= =
Re - -
L2rc " 2p J-ooJ-~o . - j
-
-
e -2=i''x+ay)
e-
2ncz
d~ d f l ] .
(A21)
This also has circular symmetry in ~, fl space and can, therefore, be written as a Hankel
transform.
4~/~u~
=pi
R e [ f ~ f z "o[ 2 ( l - v ! + 2 r c c z ]
.. I- ~.oo[-20-v)+2~ez]
= "eLJo l
= Re 2zr
L
;
°-2 .... ,~. -=., . . . . .
jo
.Jo e
2(1-v)+27zcz
do
..... ,o-~) cdcdO]
e
]
o dO} cde]
e_2,~ZJo(2ncr) ede
]
= 27zj'~o 2(1 - v) e- 2~tczcJo(2~cr)cdc + 27r~ 2rcz e- 2ncz Jo(27rcr)cdc
2(1 - v)
z2
= (Z 2 + r 2) 1/2 ~ (72 _[_r 3)3/2
(A22)
958
ANTON Z1OLKOWSKI
(Bracewell, 1965). Therefore,
P1 [-2(l-v)
z2-1
where R = (x z + y2 + zZ)~/z. This is the Green's function