Geophys. I . Int. (1990)102, 513-526
Local effects on tidal strain measurements at Esashi, Japan
T. Satol and J. C. Harrison2
'National Astronomical Observatory, * Mizusawa, Iwate 023, Japan
'GeodyMmics Corporation, 5520 Ekwill Street, Suite A , Santa Barbara, CA 93111, USA
Accepted 1989 December 15. Received 1989 October 19; in original form 1989 April 10
SUMMARY
Finite element modelling has been used to make a quantitative investigation of local
factors affecting the measurement of tidal strain a t the Esashi earth tide station.
Factors investigated, with the approximate magnitude of the perturbation produced
by each, are topography (12 per cent), regional geology (1 per cent), inhomogeneity
of the elastic properties of the granodiorite from which the strainmeter tunnels were
excavated (3 per cent), and the cavity effect (2 per cent). Correction for these
factors reduces the average discrepancy between the observed strains (observational
accuracy -1 per cent) and those predicted from the solid earth and ocean loading
tide from 23 to 8 per cent.
After such correction the ratios of observed to modelled amplitudes of the O1 and
M2 tides on each strainmeter are in fair agreement, but these ratios differ from
strainmeter to strainmeter; they are about 7 per cent too large on the N-S and
NE-SW strainmeters and 9 per cent too small o n the E-W instrument. These
studies illustrate the difficulties involved in relating strains measured in an
observatory such as Esashi to those representative of the surrounding region and
suggest that, with the level of detail and sophistication used, this relationship cannot
be modelled to much better than 10 per cent. Systematic differences between results
from the mid-point and free-end strainmeters imply that the effects of small-scale
rock inhomogeneities have not been modelled correctly.
Key words: Japan, local effects, tidal strain.
1 INTRODUCTION
Precise observations of tilt and strain for earth tide, tectonic
and earthquake prediction studies have often been made in
underground tunnels in order to attain a stable temperature
environment, to avoid the worst effects of rainfall and to
place the instrumentation in contact with competent rock in
which the observed tilt and strain may be more regionally
representative than those in near-surface, weathered layers.
Even so the measurements are affected by the shape of the
cavity, the surface topography and inhomogeneities in the
elastic properties of the surrounding rock (King & Bilham
1973; Hamson 1976). Corrections for these local influences
must be made before the observations are of regional
significance.
These corrections can in principle be calculated using
finite element techniques and important attempts to
evaluate and apply them have been made, by, for example,
* Formerly the International Latitude Observatory, Mizusawa
(ILOM).
Berger & Beaumont (1976), Takemoto (1981), and Emter &
Zurn (1985). The accuracy with which these corrections can
be made in practice depends on how well the relevant
geometries and distribution of elastic constants can be
determined and on the approximations that must be made in
order to make the finite element computations tractable.
This paper investigates the magnitude of local influences on
the strainmeters at Esashi and discusses the accuracy with
which the larger corrections can be made. Comparisons are
then made between the observed tidal parameters and those
expected from the solid earth and ocean loading tides after
correction for the local effects. This work is an advance on
previous studies in that the local effects are treated in
greater detail, the observation span was long enough to
reduce the effect of observational noise on the tidal
parameters to negligible proportions and the ocean tide
model, Schwiderski's (1980) model modified locally on the
basis of coastal observations, is superior to those used in
earlier studies.
The terminology and computational procedures of Berger
& Beaumont (1976) are used. The homogeneous tidal strain
513
514
T. Sat0 and J. C.Harrbon
is that which would be present on a smooth, elliptically
stratified, earth and is supposed to be equal to the average
large-scale strain in the vicinity of the observing site. This
large-scale strain field is modified by local irregularities and
the actual strain field near the earth's surface is called the
inhomogeneous strain.
2
THE OBSERVATIONS
The Esashi earth tide station (39"08'57"N, 14120'07"E) is
about 16 km east of Mizusawa in the southern part of Iwate
prefecture, Japan, and about 30 km from the Pacific coast. It
was designed specifically as a geophysical observatory and
observations there began in 1978. A triangle of observation
tunnels oriented N-S, E-W and NE-SW is accessed by a
lOOm long tunnel dug horizontally into the side of Mt
Abara (780m) (Fig. 1). None of the observation tunnels is
less than 60 m from the nearest point on the ground surface.
The extensometers use quartz tubes 3cm in diameter as
length standards. Each tube is supported by pairs of fine
stainless steel wires, 0.16 mm in diameter and 6 cm long, at
2 m intervals along its length. One end is fixed to a granite
pier and displacement transducers are attached at the
mid-point and free end. These displacement transducers are
differential transformers linear to better than 0.2 per cent in
their f l m m measuring spans (Tsubokawa & Asari 1979).
They are mounted on sliding stages which can be displaced
by means of differential micrometers for transducer
calibration and compensation of long-term drift. The
positioning of the sliding stages is reproducible to about
2 pm (Sato, Sato & Ooe 1980), which leads to a calibration
accuracy of about 1 per cent.
The transducer outputs are sampled digitally at 10 min
intervals, smoothed with Akaike's (1968) low-pass filter
which introduces negligible phase shift in the diurnal and
semi-diurnal tide bands, and resampled at 1hr intervals.
Low secular strain rates of about 2 x lO-'yr-'
reflect the
x
Unit i n m
Figure 1. Plane view of the observation tunnels of the Esashi earth
tide station. The north direction is indicated.
general stability of the instrumentation and site. The
instrumental characteristics of the strainmeters are summarized in Table 1. Esashi is one of about 30 similarly
instrumented observatories established to monitor tilt and
strain as part of the Japanese national program in
earthquake prediction.
The strain data were analysed using the Bayesian Tidal
Analysis Program Grouping Method (BAYTAP-G) described by Ishiguro et al. (1983), Ishiguro, Sato & Tamura
(1984), and Ishiguro & Tamura (1985). In this method the
observed time series can be decomposed into four .parts,
namely tides, drift, response to other variables which could
influence the observations and for which time series are
available, and random Gaussian noise. The solution is
obtained by minimizing the weighted sum of three
quantities; namely, the sums of the squares of (1) the
observational residuals, (2) the second differences of the
drift and (3) the differences of the tidal factors from their u
priori values. For given data, the weights are determined by
minimizing a Bayesian information criterion (ABIC)
proposed by Akaike (1980). The tides were computed using
Cartwright's (Cartwright & Tayler 1971; Cartwright &
Edden 1973) tables of the tidal potential and were combined
into 31 groups; 20 diurnal groups, 10 semi-diurnal and 1
terdiurnal. An amplitude and a phase factor were
determined for each group and the long-period tides were
treated as part of the drift.
A 3yr data span from 1984 January 1 to 1986 December
31 *as used. Much information about these observations is
given by Tsubokawa (1986) and Hosoyama (1987,1988),
including year-by-year tidal analyses using BAYTAP-G but
without any consideration of associated variables. The
results used in this paper were obtained from a new analysis
of the entire data set in which air pressure and temperature
observed in the tunnels were included as variables which
could influence the observed strain. For each, the time
series itself and two lagged series were used. Their inclusion
reduced the average rms residual for a single observation
from 0.33 to 0.18 nanostrain for the free-end strainmeters
and from 0.48 to 0.33 nanostrain for the mid-point
instruments.
Amplitudes, and phases relative to the local potential, of
the 0, and M, groups are given in Table 2, together with
the formal standard errors derived from the analysis. The
estimated errors in amplitude are all less than 1 per cent of
the observed amplitudes, and the phase errors are less than
0.5". This means that errors due to observational noise are
negligible for the purposes of this study, although systematic
instrumental effects such as calibration errors will not be
revealed by this analysis.
Table 1. Summary of the characteristics of the extensometers at the Esashi
station; F: free end, M: middle point.
Component
NS
EW
NE-SE
N 0.00 E
N 90.00 E
N 45.00 E
F
35. 77
35.69
50. 69
M
17.99
19.77
27.97
Azimuth (degrees)
Length
(m)
Resolution (10-1')
F
3
3
2
M
3
3
2
Local effects on tiahl strain
515
Table 2. Tidal amplitudes and phases, together with their standard deviations, as
estimated from the analysis of observations of the strain tides at Esashi from 1984
January 1 to 1986 December 31 with BAYTAP-G. A is the amplitude in
nanostrain and P the phase in degrees referred to the local potential; a positive
sign indicates a phase advance. M / F is the ratio of the strain amplitude at the
middle point to that at the free end. DP is the corresponding phase difference.
Middle point
Free end
P
A
EW
NE
P
DP
-1.5
0.4
4.15
0.03
0.4
0.4
1.09
1.9
0.02
11.38
0.01
-3.6
0.1
12.41
0.01
-4.6
0.4
1.09
- 1 .o
4.73
0.02
-8.2
4.71
0.02
-4.9
0.3
1 .oo
+3.3
0.2
3.02
0.01
-29.2
0.1
3.32
0.01
-32.6
0.1
1.10
-3.4
4.20
0.01
19.9
0.2
4.77
0.04
18.3
0.4
1.I4
-1.6
6.07
0.01
-26.3
0.1
7.73
0.01
-25.0
0.1
1.13
0.5
3.81
NS
A
MIF
The mid-point strains are systematically about 10 per cent
larger than these observed at the free ends as shown by the
M/F ratios in Table 2. Only for the 0, tide observed on the
E-W strainmeter are the mid-point and free-end strain
amplitudes in agreement. While the free end results would
normally be considered the more accurate owing to the
greater length of the extensometer, these differences are
much larger than can be accounted for in terms of
observational errors. Some studies were therefore made to
look for systematic instrumental effects which could result in
larger strains being observed at the mid-points.
The first possibility investigated is that the mid-point
displacement transducers are more sensitive than those at
the endpoints. The amplitudes given in Table 2 are based on
calibrations performed in October 1986 in which each
transducer was calibrated with its own micrometerpositioned sliding stage. These calibrations were repeated in
April 1988 using the same micrometer for all transducers.
The mean of the absolute values of the changes in measured
calibration over the 18 month interval was 0.4 per cent and
the largest single difference was 0.8 per cent. This check
eliminates the possibility of transducer calibration problems
at the 1 per cent level.
A second possibility concerns the effect of the finite
compliance of the supports of the quartz tube length
standards. Ground strain results in relative motion along the
length of the tube, between the hoops carrying the
suspension wires and the tube itself. As a result stresses are
induced in the tube which cause it to change length and
hence to underestimate the ground strain (Benioff 1935;
Balvadze et al. 1965; Takemoto 1975; Agnew 1986). As is
shown in the Appendix, it is possible to estimate these
compliances by freeing the fixed ends of the quartz tubes
and measuring the free periods of longitudinal oscillation.
Knowing these compliances it is then possible to estimate
the strain errors at the mid-points and the free ends. The
computations show (Table 3) that the strain reductions are
of the order of 1 per cent and that mid-point strains should
actually be slightly less than those observed at the free ends.
Although these effects are small, the observed strains listed
in subsequent tables have been corrected for finite support
compliance using the responses given in Table 3. Again
there is no explanation in instrumental terms for the larger
strains observed at the mid-points, and these are accepted as
real.
3 THE HOMOGENEOUS TIDE
The homogeneous tide is the sum of the solid earth and
Ocean load tides. The solid earth tide was computed using
Wahr’s (1981) computations of Love numbers for the
elliptical, rotating, elastic and oceanless earth model 1066A
(Gilbert & Dziewonski 1975).
The Ocean load tide was estimated using the computer
Table 3. Effect of suspension wires on the strain measured by each extensometer.
is the longitudinal free period of the extensometer as a pendulum. 6 is defined
in the Appendix. R is the ratio of the measured to the ground strain. F and M
indicate the free end and middle point of each extensometer, respectively.
Component
To (set)
6
R
F
M
NS
EW
NE
0.384 +I- 0.002
0.310 +I- 0.099
0.388 +I- 0.002
.0126
.0193
.0248
0.996
0.994
0. 992
0.994
0.991
0.989
516
T. Sat0 and J . C. Harrison
Figure 2. Mesh used for computation of the oceanic loading tides. The size of the small meshes is 5' in latitude and 7.5' in longitude and that
of the large meshes 1 degree in both directions. x marks the position of the Esashi station.
program GOTIC (Sato & Hanada 1984) which employs the
Naval Surface Weapons Center ocean tide models of
Schwiderski (1980) for the global oceans. This model was
modified along the south coast of Hokkaido and NE Honshu
to agree with the empirical values compiled by the
International Hydrographic Bureau (1966). In the Japan
Sea, which was not modelled by Schwiderski, the tidal
models of Tsukamoto & Nakagawa (1978,1980) were used.
The convolution integrals were performed using Green's
functions for the 1066A earth model computed with a
program kindly provided by S. Okubo & T. Endo (personal
communication). Close to the computation point GOTIC
uses the digital elevation file KS-110 of the Geographical
Survey Institute of Japan (1980) to lay out a fine
computation grid. Owing to the 30km distance of Esashi
from the closest ocean the finest mesh size (45" X 30) was
not used; a 7,5'x5' mesh was used in a 7 " ~ 7 "area
surrounding Esashi and a 1"X 1" grid over the rest of the
oceans (Fig. 2).
The body, ocean load and homogeneous strain tides along
the azimuths of the Esashi strainmeters are given in Table 4.
In most cases the contribution of the ocean load tide is
about 10 per cent of that of the body tide, but for the E-W
component of the M, tide the ocean load and body tides are
approximately equal.
4
THE FINITE ELEMENT MODELLING
The boundary conditions ensure that near the surface the
homogeneous strain tensor has only three independent
components. At any point, each component of the
heterogeneous strain can be expressed as a linear
combination of these three components, a relationship
which can be expressed in terms of a coupling matrix. The
Table 4. Amplitudes and phases of the strain tides on the extensometer azimuths
as predicted from Gilbert & Dziewonski's (1975) earth model 1066A and
Schwiderski's (1980) ocean tides. A is amplitude in nanostrain. P is phase in
degrees referred to the local potential; a positive sign indicates a phase advance.
Body tide
Oceanic bading
Homooe-
tide
NS
EW
NE
tide
A
P
A
P
A
P
0
1
4.16
0.0
0.63
-195.0
3.55
2.7
Mp
12.72
0.0
1.26
177.9
11.45
-0.2
0
1
6.77
0.0
0.77
-70.1
7.07
-5.9
M2
3.59
0.0
3.42
-84.9
5.17
-41.2
0
1
5.84
20.8
0.28
-53.1
5.93
18.1
M2
9.13
-26.8
0.35
-294.7
9.13
-24.6
Local effects on tidal strain
coefficients in this matrix are determined numerically by
computations with finite element models.
The elastic properties and irregular topographic surface of
a volume of rock larger than the size of the heterogeneities
of interest are represented in the model. Three computations are then made, in which its boundaries are displaced so
as to produce, respectively, overall pure N-S and E-W
extensional and NE-SW shear strains; the ratios of the
magnitudes of the components of strain induced at the
strainmeter sites to those of the applied strains are noted.
The lower boundary is fixed in the vertical direction and the
top is stress free. The SAP-IV (Bathe, Wilson & Peterson
1973) and ADINA (Bathe 1978) finite element programs
were used.
The resolution and accuracy of a finite element
computation increase as the element size is reduced, but so
do the demands on computer time and memory. These
considerations are particularly important with 3-D models.
Therefore some preliminary sensitivity studies were made
with 2-D plane strain models in order to determine the
element sizes and model dimensions needed in the definitive
3-D model. A two-stage zooming technique (Desai & Abel
1972) was also used. The first computation is made with a
large model and relatively coarse elements; an inner region
is then resampled using a finer mesh and deformed using
boundary node displacements determined in the first
computation.
-t
517
k
m
8.0km
Figure 3. Topography and contours of the ratio of local N-S strain
to the overall N-S strain applied to the model-the N-S strain to
N-S strain coupling factor-along a profile through the N-S access
and observation tunnels. These results were obtained by a 2-D finite
element computation using 488 elements.
4.1 Topography
Preliminary studies for the topographic computation were
made with a 24 km N-S profile along the line of the access
and N-S observation tunnels; the topography was modelled
realistically over the central 12 km but was flat over 6 km
buffer zones at the two ends. Uniform material properties
with Young's modulus of 5.4 x 10" N m-' and Poisson's
ratio of 0.27 were assumed; the horizontal dimensions of the
elements were about 100m near the centre, increasing
towards the ends so that a total of 488 was sufficient to fill
the area.
The magnitude of the N-S to N-S strain coupling factor
for the central 12 km section of this model (model A) is
shown in Fig. 3. It has a value of about 0.6 in the vicinity of
the highest point of the topographic profile (Mt Abara),
meaning that the actual N-S strain there is expected to be
only 60 per cent of the homogeneous strain. It is about 1.3
in the valley north of the observatory and vanes rapidly
through the outlined central region which contains the
Esashi observatory. A finer mesh with 25m elements near
its centre was constructed for this inner region and the
zooming method used to obtain a more detailed picture of
its deformation. The results for this model (B), Fig. 4,
indicate that the strain pattern has changed significantly with
a new region of strain magnification developing near the
entrance to the access tunnel. The smaller element size is
therefore essential.
The total size of the definitive model was determined by
investigating the sensitivity of the induced strains to
topographic variations some distance from the centre. A
flat-topped element was added to the south end of model B
and the boundary displacements applied directly to this
model, thus essentially shaving off the top of Mt Araba. The
I
-1
500m
-1
Figure 4. Enlarged view of the topography and the N-S strain to
N-S strain coupling factor in the area outlined in Fig. 3 after
application of the zooming technique. The position of the N-S
tunnels is indicated by the dotted line.
1.o
1.0
500m
1.0
-1.0
Figure 5. A flat-topped element has been added to the south of the
profile shown in Fig. 4 and the boundary displacements applied
directly to this model. The strain coupling factor along the tunnel is
identical to that shown in Fig. 4.
results, Fig. 5 , are essentially identical to model B in the
vicinity of the observatory, although there are significant
differences at greater depth.
On the basis of these pilot studies the 3-D model was
518
T. Sat0 and J . C. Harrison
N
VP
VS
P
2.5 1.4 2.6
5.1
2.8
2.6
5.L
3.1 2.6
5.4
3.1
E
T
2.6
ITrn
- 1
500m
1-
Figure 8. Cross-section along a profile containing the N-S tunnels
showing the tunnels, the material properties and the elements used
in the inner, fine-meshed, 3-D computation. The velocities are in
km s-l and densities in lo3kg m-’.
Figure 6. Topography of the area used in the 3-D finite element
computation. The zooming technique was used in the 500 x 500 m
inner square.
designed to cover the 5 X 5 k m area, for which the
topography is shown in Fig. 6, and to have a vertical extent
of 4 km. The 464 elements were arranged in four layers and
near the observatory the elements measured 100 x 100 m
horizontally by about 300m vertically. The 500 x 500m
inner area indicated in Fig. 6 was then examined in greater
detail using the zooming technique; the fine mesh model had
320 elements arranged in four layers with horizontal
dimensions of 25 X 25 m near the centre.
Uniform material properties identical to those used in the
2-D model were used in the larger model; the fine mesh
model, however, included a near-surface weathered layer
whose thickness and properties were determined by test
borings, Fig. 7. Seismic experiments were conducted in the
tunnels by Kitsunezaki (1980) prior to the start of regular
strain observations and the mean of the seismic velocities
thus determined was used to calculate the elastic properties
of the second layer. Seismic properties determined by
large-scale refraction work (Ichikawa 1969) were used for
the third and fourth layers. A cross-section of the inner
model along the N-S tunnel is shown in Fig. 8. Nodal
displacements determined for the fine model were
interpolated vertically to the level of the observatory tunnels
and used to determine the three components of horizontal
strain.
Contour plots for the coupling factors relating the N-S
linear, the NE-SW shear and the E-W linear strains to the
corresponding components of the homogeneous strain are
given in Fig. 9(a), (b) and (c) for a 100 x 100 m area; the
topography and a plan of the tunnels are shown in Fig. 9(d).
As expected from the 2-D modelling, the N-S coupling
factor decreases along the N-S tunnel with increasing
Weathering layer with corestone
Weathering layer
- Om
Gra nod io r i te
-40
-80
Figure 7. Profile of the weathered layers as determined from two test borings.
Local effects on tidal strain
4.2
519
Geological effects
Some elastic inhomogeneities were included in the
topographic modelling but geological effects also must be
included on two other scales, one larger and one smaller
than the topographic model.
Geological and seismological investigations of the crust in
NE Japan were camed out as part of the Japanese Upper
Mantle Project (UMP) (Japanese National Committee for
the UMP 1973). A seismic survey along a line from
Yokamachi to Kayakarikubo, carried out as part of this
project (Ichikawa 1969), passes right through Mt Abara and
the Esashi observatory. The seismic structure along this
profile (Fig. 10) formed the basis for choosing the elastic
constants in a 2-D finite element model, 90km in length and
45 km in vertical extent. The 11 geological units in the
section were combined into six groups whose seismic and
elastic parameters are given in Table 5.
A profile of the near-surface longitudinal strain ratio
derived from this model is given in Fig. 11. This ratio varies
from 1.1 to 0.8, the main feature being a zone of low strain
in the vicinity of Mizusawa associated with an increase in
crustal rigidity due to the abrupt termination of the 1-2 km
thick sedimentary sequence with V, = 2.8 km s-' in the west
against older material with V, = 4.8 km s-', to a local
increase in V, in the middle layer from 5.5 to 5.9 km s-l and
to a shallowing of the basal layer with V, = 6.2 km s-'. This
was recognized as an important discontinuity by Tsuboi,
Jitsukawa & Tajima (1956) on the basis of the steep gradient
in Bouguer gravity anomalies and named the MoriokaSirakawa tectonic line. The Esashi observatory, however, is
far enough east of Mizusawa to be only slightly affected by
this feature. Lateral variations of elastic properties near the
observatory itself are quite minor as the properties of the
granodiorite in which it is located are similar to those of the
t--room----)
Figure 9. Contour plots of the strain ratios at the Esashi station
obtained from the 3-D fine-meshed topographic model. Figs 9(a),
(b) and (c) show the strain ratios for the N-S linear, the NE shear
and the E-W linear strains respectively. The topography and the
positions of the tunnels are shown in Fig. 9(d).
distance from the entrance; this is also true for the NE-SW
shear strain which is reduced to about 90 per cent of the
corresponding homogeneous strain. On the other hand, the
coupling factor for E-W strain increases with distance into
the side of the mountain; it is fairly constant at about 80 per
cent along the length of the E-W strainmeter. The
off-diagonal terms in the full coupling matrices given in
Table 6 are of the order of 7 per cent.
140'
ch
I
n
A
m
R
pigpre 10. Seismic cross-section from which the distribution of elastic constants used in computing the regional geological effects was
determined. The position of the Esashi station is marked E.
520
T. Sat0 and J . C. Harrison
Table 5. Material properties adopted in the finite element model to estimate the
regional geological effect. V, is the P-wave velocity (kms-’), V, the S-wave
velocity (kms-’) and p the density (lo3kgm-3). E is Young’s modulus (lo9 in
MKS unit) and v Poisson’s ratio.
VP
0.9
0.8
t
I
0
E
P
VS
V
Group 1
2.8
1.4
2.1 8
11.4
0.33
Group 2
4.0
2.7
2.50
45.1
0.28
Group 3
4.95
2.75
2.54
49.0
0.28
Group 4
5.5
3.2
2.63
66.2
0.25
Group 5
5.85
3.4
2.68
76.5
0.25
Group 6
6.20
3.58
2.74
87.7
0.25
was modelled. Unfortunately Young’s modulus can be
determined directly only in the triangular prism of rock
enclosed by the strainmeter tunnels. In order to assign
values for the rest of the model, the distribution of this
modulus was supposed to be mirror-image symmetrical
about the NE-SW tunnel, its value was set equal to the
mean used in layer 2 of the topographic model in the outer
elements and it was varied smoothly in between. 5 x 5 m
elements were used in the central area. The resulting
transformation matrices are given in Table 6.
I
I
I
30
I
I
I
60
1
I
I
90
Figure 11. E-W strain to E-W strain coupling factor along the
profile of Fig. 10. The positions of the Esashi station and Mizusawa
are indicated by E and M respectively.
surrounding palaeozoic rocks. The net effect of the regional
geology is to reduce the E-W linear strain by a factor of
0.96.
There are, however, significant small-scale elastic
inhomogeneities in the immediate vicinity of the strainmeter
tunnels. These were explored seismically by Kitsunezaki
(1980). Small shots were exploded in the tunnel walls and
traveltimes of P- and S- waves to geophones installed along
the tunnels were measured. Eight shots were used. V, was
measured along 50 ray paths but V, could be measured
along only 20 of these owing to difficulties in picking the
S-wave arrival. The distribution of seismic velocity in the
horizontal plane containing the tunnels was obtained using a
tomographic inversion program written by R. Gross
(personal communication). There were insufficient S-wave
data to make an independent solution for V, and, as the
observations give no real evidence for variations in the
Vp-to-VS ratio, this ratio was held constant at 1.8-the
average of the experimental determinations.
The distribution of Young’s modulus thus obtained is
plotted in Fig. 12. Although Esashi had been selected as the
site for the observatory in the belief that the granodiorite
would form a homogeneous and coherent mass, Young’s
modulus in fact varies by f 1 4 per cent about its mean value.
For all strainmeters Young’s modulus is generally lower
between the fixed ends and mid-points than between the
mid-points and free ends; this is a possible explanation for
the larger strains observed at the mid-points.
A volume 100 x 100m horizontally by 128 m vertically
4.3
Cavity effects
The geometry of the tunnels at Esashi is complex but the
interactions between them are small over most of their
lengths because a cavity only affects the stress distribution in
its immediate vicinity and the tunnels, 2 x 2.5 m in
cross-section, are generally separated by tens of metres.
These sections behave as isolated tunnels for which there is
no cavity effect for strain measured along their lengths. Very
complicated behaviour is t o be expected at the corners of
the triangle where the tunnels intersect. However the
strainmeters d o not extend into these complex areas but are
confined to the straight, uniform, tunnels forming the sides
of the triangle. Therefore the cavity effects, averaged over
the total lengths of the strainmeters will be small and it was
considered adequate to estimate them from Harrison’s
(1976) computations for an isolated finite cylindrical tunnel.
The strain magnifications were estimated to be 1.01, 1.02
and 1.03 for the full length N-S, E-W and NE-SW
strainmeters. Much smaller effects are expected for the
strainmeters utilizing the mid-point transducers and for
these the cavity effects were neglected.
5
THE TOTAL EFFECT
In principle the effects discussed in the previous section
should be considered simultaneously in one large finite
element computation. Such an approach is impractical but,
owing to the different length scales involved, the several
effects are largely independent and their total effect can be
computed accurately enough by multiplying the separate
matrices, as was done by Berger & Beaumont (1976). It is
thus possible to relate the inhomogeneous strain at the
strainmeters to the homogeneous strain, to compute the
linear strains along the strainmeter axes and to find a matrix
Local effects on tidal strain
M
F
521
C
Qme 12. Distribution of Young's modulus in the plane of the Esashi tunnels estimated from the P-wave velocity data measured by
Kitsunezaki 1980. The units are lo9N m-2, the contour interval is 1 unit and the fine divisions on the coordinate axes represent 1 m. The
positions of the strainmeters are shown and their clamped (c), mid-points (m), and free ( f ) ends indicated. North is to the right.
which relates these linear strains to the homogeneous linear
strains in the N-S, E-W and NE-SW directions.
The extensions are represented by a column matrix
IENsEEWENEsWITand the horizontal strains by
lennencecelT;the tensor definition is adopted for the
magnitude of the shear strain enc. Then the strainmeter
extensions on the homogeneous model are given by
E=De
cos2A, 2cosA1sinA1 sin2Al
D = cos2A2 2cosA,sinA2 sin2A,
cos' A, 2 cos A, sin A, sin2A,
.
one for the full-length strainmeters and one for the
mid-point instruments. The coefficients in the two sets of
matrices are given in Table 6.
6 COMPARISON OF THE PREDICTED
STRAIN TIDES WITH OBSERVATIONS
The predicted homogeneous and inhomogeneous strain tides
are compared with those observed in Table 7 and,
graphically, in Figs 13 and 14. It is clear that the corrections
for the local effects have reduced the discrepancy between
the observed and predicted strains very significantly. For
example, for a perfect model the ratio of observed to
predicted strain amplitudes would be 1.00; for the
homogeneous tide the rms departure of this ratio from 1.00
is 0.25 while for the inhomogeneous tide it is reduced to
0.09.
For both the homogeneous and inhomogeneous tides the
prediction errors are correlated most strongly with the
azimuth of the strainmeter and less so with the choice of 0,
versus M, tide, or mid-point versus free-end strainmeter.
For the N-S strainmeter the mean amplitude ratio and
phase shift for the homogeneous tide are 1.09 and -3.6';
the inclusion of local effects improves these mean values
only slightly (to 1.07 and -2.2"), although the internal
consistency of the individual amplitude ratios has been
improved.
However, for the E-W strainmeter the amplitude ratios
for the homogeneous tide are much too small (30-40 per
Table 6. Summary of the transformation matrices. T is the topographic matrix, C
that for the cavity effect, G, that for the regional geological effect and G, that for
the rock inhomogeneity in the tunnels. N = CDG,TG,D-' transforms the homogeneous extensions to the inhomogeneous.
Free-en d
C
GI
1.003 - 0 . 0 0 4
0.999
-0.002
0.002
0.002
0
1.02
0
1.01
0
0
-0.002
-0.005
0.998
0.96 - 0 . 1 0
0.02
- 0 . 1 0 0.89 - 0 . 0 6
-0.05 -0.07
0.79
0
0
1.03
N
Gr
1
T
0
0
1
0
0
0
0
0.963
0.07 - 0 . 1 0
1.03
-0.01
0.82 - 0 . 0 7
- 0 . 0 5 - 0 . 0 8 0.82
Middle-point
1.036 - 0 . 0 0 4
1.043
-0.006
0.002 - 0 . 0 0 4
0
1
1
-0.000
-0.000
1.026
0
0
1
0.03
0.98 - 0 . 0 9
- 0 . 0 9 0.90 - 0 . 0 5
0.79
-0.06 -0.07
0
0
1
0
N
Or
0
0
T
C
GI
0
1
0
0.08 - 0 . 1 0
1.06
- 0 . 0 2 0.82 - 0 .0 8
- 0 . 0 5 - 0 . 0 7 0.85
0
0.019
0.963
Table 7. Comparison between the amplitudes and phases of the observed
tidal strains and those predicted by the homogeneous and inhomogeneous
models. (a) amplitudes (nanostrain) and phases relative to the local
potential for the free-end strainmeters; (b) amplitudes and phases for the
mid-point strainmeters; (c) observed/predicted amplitude ratios (A. ratio)
and phase differences (P. dif.) for the homogeneous model; and (d)
observed/predicted amplitude ratios and phase differences for the inhomogeneous model.
(a) Free-end slrainmelers
~
Tide
N-S
E-W
NE-SW
M2 N-S
E-W
NE-SW
01
Ampl.
Phase
3.55
7.07
5.93
11.45
5.17
9.13
2O.7
-5'.9
18O.1
-0O.2
-41O.2
-24O.6
(b)
Tide
N-S
E-W
NE-SW
M2 N-S
E-W
NE-SW
01
3.54
5.35
4.22
11.15
3.50
6.62
obsewed
-1O.2
-7O.8
21O.7
OO.6
-46".0
-25O.5
Ampl.
Phase
3.83
4.76
4.24
11.43
3.04
6.93
-1",5
-8'.2
19O.9
-3O.6
-29O.2
-26O.3
Mid-point strainmeters
Homogeneous
Ampl.
Phase
3.55
7.07
5.93
11.45
5.17
9.13
Inweneous
Ampl.
Phase
2".7
-5O.9
18O.1
-0O.2
-41O.2
-24O.6
Observed
Ampl.
Phase
Inhomogeneous
Ampl.
Phase
3.78
5.31
4.44
11.69
3.41
6.93
4.18
4.75
4.83
12.48
3.35
7.82
-0O.9
-8O.O
21O.3
OO.3
-47O.2
-25O.6
00.4
-40.9
18O.3
-4O.6
-32O.6
-25O.8
(c) Observedlhomogeneous strain
Free-end strainmeters
Azimuth
N-S
E-W
NE-SW
01
M2
A. ratio P. diff. A. ratio P. diff.
1.08
0.67
0.72
-4O.2
-2O.3
1O.8
(d)
Azimuth
N-S
E-W
NE-SW
1.00
0.59
0.76
-3O.4
12O.O
-1O.7
Mid-point strainmeters
01
A. ratio P. diff.
1.18
0.67
0.82
-2O.3
1O.O
OO.2
M2
A. ratio P. diff.
1.09
0.65
0.87
-4O.4
8".6
-1O.2
Observed/inhomogeneous strain
Free-end strainmeters
Mid-point strainmeters
01
M2
A. ratio P. diff. A. ratio P. diff.
A. ratio P. diff. A. ratio P. diff.
1.08
0.89
1.01
-0O.3
-0O.4
-1O.8
1.03
0.87
1.05
-4O.2
16O.8
-0O.8
01
1.11
0.90
1.09
M2
1".3
3".1
-3O.O
1.07 4 . 9
0.98 +14".6
1.13 -0O.2
Local effects on tidal strain
- 01
Free End
NS
7
Middle Point
EW
H
Figure W. Phasor plots of the observed (hollow circles) and
predicted inhomogeneous (filled circles) strain tides at the 0,
frequency. For explanation see inset in Fig. 14.
cent) and inclusion of the local effects does bring the
predicted tide into much better agreement with the
observations; the mean amplitude ratio is increased from
0.65 to 0.91. There are outstandingly large phase
discrepancies in the M, component which are made even
larger when the local effects are considered; the M2 E-W
strain tide is exceptional in that the ocean load contribution
is practically as large as that of the body tide, whereas it is
no larger than 15 per cent of the body tide in the other cases
considered (Table 4). The ocean load contribution is also
nearly in quadrature with the body tide and thus has a large
effect on phase. There is thus a strong suggestion that these
large phase discrepancies are due to errors in the ocean load
computations.
Consideration of the local effects has also brought the
predicted tide into significantly better agreement with
observations on the NE-SW strainmeter; the mean
amplitude ratio has been brought up from 0.79 to 1.07,
reducing the error from 21 to 7 per cent, and the internal
consistency of the individual ratios has been improved.
523
DISCUSSION
The topographic influence is the most important of the local
effects at Esashi; it is largest in the E-W direction where it
modifies the tidal amplitude by about 20 per cent. This
effectvaries significantly in horizontal distances comparable
to the strainmeter lengths (Fig.9), a consequence of the
relatively shallow (60-80 m) depth of the observatory
tunnels. It was necessary to employ small elements in the
finite element model in order to resolve these small-scale
variations. Even so, the changes are so rapid that errors in
positioning the observatory tunnels relative to the
topographic surface are significant.
The positions of the observatory tunnels are related by
conventional surveying to a satellite Doppler fix just outside
the entrance; this fix was made with a Magnavox model
MX1502 Geoceiver and has an estimated accuracy of 0.6m
in both the horizontal and vertical coordinates. The diagonal
elements in the T matrices of Table 6 vary most rapidly in
the vertical direction, about 3 per cent in 10m, so this
uncertainty in position translates to about 2 per cent in the
strain coupling coefficients.
Similar difficulties are to be expected when modelling the
topographic effects in any geophysical observatory consisting
of tunnels inside a steep-sided mountain. Unfortunately the
majority are of this type owing to the relative ease of
excavation, drainage and access as compared with sites that
are entirely below the surrounding terrain.
Elements of still smaller size were needed to model the
influence of the varying rock properties in the vicinity of the
tunnels, but the main uncertainty in this computation was in
mapping these variations. The seismic velocity could only be
determined with any confidence within the prism of rock
enclosed by the tunnels; survey problems and the difficulty
of estimating appropriate weathering corrections effectively
precluded the use of ray paths between the tunnels and the
outside surface. Furthermore, the interpretation of these
12
Free E n d
NS
M i d d l e Point
A
n
Out -phase
NE
(INSET)
2x10-9
H
Fcgure 14. Phasor plots of the observed (hollow circles) and predicted inhomogeneous (filled circles) strain tides at the M, frequency. Inset:
explanation of the phasor plots in Figs 13 and 14. The positive direction of the horizontal axis corresponds to zero phase relative to the local
tidal potential with phase lag increasing in the counter-clockwise direction. Phasors denoted by B, L and I indicate respectively the body tide,
the oceanic loading tide and the combined effectsof the topography geology and observing cavities. Thus the predicted inhomogeneous strain
tide is represented by the phasor formed by drawing a straight line from the origin to the filled circle.
524
T. Sat0 and J . C . Harrison
variations of seismic velocity, in terms of variations of
Young's modulus with no accompanying changes in density,
is questionable because they are correlated with the degree
of fracturing in the rock rather than with changes in
composition. Kitsunezaki (1980) counted the number of
visible cracks at a height of 1.3 m above the tunnel floor,
finding, for example, an average of 2.3 m-l along the N-S
tunnel between the ends of the strainmeter rod and a
somewhat higher density of 2.5 m-' between the fixed end
and mid-point. Very little is known about how such
fractures behave under tidal stresses; there may be an
important concentration of tidal strain across the fractures
with less deformation of the intervening rock.
Some measure of the success in modelling the effects of
these small-scale inhomogeneities may be obtained by
noting that the strains observed at the mid-point transducers
were 9 and 13.5 per cent larger, respectively, than those
observed with the full length N-S and NE-SW strainmeters.
However the ratios of observed to inhomogeneous tidal
strain differ by only 3.5 and 8 per cent; that is to say about
50 per cent of the difference in the observed tides has been
accounted for in the modelling.
Evans et af. (1979) reported 35 measurements of tidal
strain at 16 sites in Great Britain and concluded that the
large variability in observed amplitude was due to variations
in the elastic parameters with wavelengths of 100 m or less.
Their results did not show the consistency between the M2
and 0, amplitudes seen on the N-S and NE-SW
instruments at Esashi, and no attempts were made to
measure the elastic properties at the strainmeter sites. At
Esashi, a site chosen among other reasons for the uniformity
of the rock, significant unmodelled amplitude variations are
present within the lengths of the 30 m strainmeters, despite
an attempt to map the elastic inhomogeneities seismically
and to correct for them.
The ocean load tide is exceptionally large for the M2 tide
on the E-W strainmeter and is nearly in quadrature with the
body tide. This tide is also notable for showing exceptionally
large discrepancies in phase in Table 7, suggesting that the
ocean load computation is in error. This suggestion is
supported by the results of tidal tilt observations. Two
water-tube tiltmeters of different designs are installed
alongside both the E-W and N-S strainmeters (Shichi,
Okuda & Yoshida 1980; Sat0 et af. 1983). Data from the
tiltmeters designed at Mizusawa were analysed by Sasaki,
Sat0 & Ooe (1982) and 2yr of data from both these and
tiltmeters designed at Nagoya University were analysed by
Sat0 et al. (1986). The latter study showed that the results
from the two types of tiltmeter were consistent to within the
estimated accuracy of 0.07 milliarcsecond in amplitude and
1" in phase, and that the largest discrepancies (1
milliarcsecond in amplitude, 20" in phase) in comparisons
with the homogeneous tilt tide were in the M2 component
on the E-W tiltmeters.
The sensivity of the ocean load tide to the earth model
used in the calculation was investigated by repeating the
ocean load convolutions with Farrell's (1972) Green's functions. Changes to the computed load tides were trivialfor example the M, constituent of the E-W strain load tide
changed in amplitude and phase from 3.42 x
-84.9"
to 3.29 X
-83.6'. The major differences between
Green's functions computed from the various earth models
occur for loads close to the computation point and Esashi is
far enough (30km) from the ocean not to be greatly
affected. There remains the question of the effect of the
lateral changes in elastic structure associated with the
offshore subduction zone (Hasegawa, Umino & Takagi
1978). Beaumont (1978) investigated the effect of the lateral
changes in crustal structure across the continental margin of
Nova Scotia, Canada, on the tilt loading Green's functions.
The effects can be quite large but only locally in the
transition region and near the edge of the load. It is unlikely
that these lateral changes would have a major effect at
Esashi, but further investigation is required.
It is necessary to reduce the amplitude of the M, E-W
load tide by 1.2 X
or 33 per cent of its value in order to
remove the large phase discrepancy. If this figure is
indicative of the uncertainty in the other load tides, the
discrepancies between the observed and predicted inhomogeneous tides-but
not, of course, the differences
between the full- and half-length strainmeters-can be
explained by errors in the load tides; the mean magnitude of
the amplitude discrepancies in the comparisons with the
inhomogeneous tide in Table 7 is only 0.4 X
There remains the possibility of unmodelled large-scale
geological effects. Some computations were made with a
subduction zone model with elastic properties based on the
seismic model of Hasegawa et al. (1978) and the effects of
such a model were found to be small. The situation could be
more complicated than modelled if, for example, the mantle
were to exhibit anisotropic elastic properties. Tidal strain
data from more than one site, for example a profile across
the Tohoku region, would be helpful in distinguishing such
effects from those due to errors in the ocean tide models.
Despite extensive computation, it has not been possible to
demonstrate agreement between the observed and predicted
strain tides at Esashi to much better than 10 per cent. It is
difficult to pin down the source of the remaining
discrepancies. The general agrement, between amplitude
ratios for the 0, and M2 tides and the free-end and
mid-point strainmeters on each azimuth, suggests that the
dominant source of error is an azimuth-dependent error in
the coupling coefficients between local and regional strain,
such as would be produced by errors in the T or G, matrices.
On the other hand the large phase error in the E-W M2 tide
points to an error in the ocean load tide, while the failure of
the C and GImatrices to completely remove the amplitude
discrepancy between the free-end and mid-point observations indicates that the very local effects are not modelled
correctly.
Thus the strain measured at the Esashi station cannot
presently be related to the regional strain to much better
than 10 per cent; while this is not important in earthquake
prediction, where any indication of change in the regional
strain is useful, it does limit the utility of the measurements
where accurate measurement of the regional strain is
r e q u i r e b a s for instance in determining Love numbers.
ACKNOWLEDGMENTS
It is a pleasure to acknowledge the support of Dr T.
Tsubokawa, former director, and Dr K. Hosoyama, present
director of the ILOM. Most of the finite element
computations were made at the University of Colorado
Local effects on tidal strain
during a visit by TS which was supported by t h e Japanese
Ministry of Education. The whole-hearted support of the
L I K ~ Sstaff is gratefully acknowledged. Drs S. O k u b o and
T. Endo (now with the Schlumberger Corporation) of the
Earthquake Research Institute of Tokyo University
provided us with their computer program for calculating
tidal and loading Love and Shida numbers and Dr R. Gross
provided his program for inverting t h e seismic traveltimes in
the Esashi observatory experiment. Ocean tide d a t a from
the IHB were supplied by t h e Japanese Hydrographic
Department. Helpful discussions were held with Drs S
Manabe and K.Tanikawa of t h e ILOM.
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APPENDIX: EFFECT OF THE FINITE
COMPLIANCE OF THE EXTENSOMETER
TUBE SUSPENSION
The extensometer rods are supported by pairs of thin
stainless-steel wires, whose upper ends are attached to fixed
hoops which straddle the rod at 2 m intervals, and whose
lower ends are attached to metal rings through which the
rod passes. The ring, hoop and support wires are initially in
the same plane perpendicular to the rod, but longitudinal
motion of the rod causes the ring to move out of this plane
and the support wires to exert a horizontal component of
force opposing the motion. This force is responsible for the
compliance of the support. If the rod is freed and given a
small longitudinal displacement, it will oscillate with angular
velocity ( 0 )given by
0 2 =
longitudinal force /unit displacement
mass of tube
Assuming that all the support compliances (C) are equal
and that the supports are spaced a distance 1 apart,
= C/plA,
(Al)
where the tube is of cross-sectional area A and of density p .
Take the x axis along the length of the tube, directed from
the fixed to the free end. Assuming the quartz to be rigid
(its distortion will later be shown to be small), the effect of a
uniform strain e, = e is to displace a suspension wire
support hoop at x a distance ex, and thus produce a
horizontal force Cex on the tube. To simplify the
computations this force is considered to be uniformly
distributed over the length I, resulting a force Cexll per unit
length. The stress in the tube (of length L), found by
integration with the boundary condition of zero stress at the
free end, and division by the cross-sectional area A, is
1 Ce
2 IA
-- (LZ - 2)
and the strain
where Y is Young’s modulus. Representing CL’IYIA by 6,
the average strain between the fixed end and mid-point of
the quartz tube is (11124) be; and between the fixed and
free ends is (1/3)6e. The observed ground strains are thus
reduced by factors of [ l - (11/24)6] and [l - (1/3)6]
respectively. Use of equation (Al) allows 6 to be expressed
as
p( 0L)2/Y .
The values of 6, given in Table 3 together with the strain
reduction factors, are computed from the observed periods,
strainmeter lengths from Table 1 , and the values
p = 2650 kg m-’ and Y = 7.2 X 10” N m-’.