Journal of Geophysical Research: Solid Earth

Tidal calibration of Plate Boundary Observatory borehole strainmeters: Roles of vertical and shear coupling



[1] A multicomponent borehole strainmeter directly measures changes in the diameter of its cylindrical housing at several azimuths. To transform these measurements to formation strains requires a calibration matrix, which must be estimated by analyzing the installed strainmeter's response to known strains. Typically, theoretical calculations of Earth tidal strains serve as the known strains. This paper carries out such an analysis for 12 Plate Boundary Observatory (PBO) borehole strainmeters, postulating that each of the strainmeters' four gauges responds (“couples”) to all three horizontal components of the formation strain tensor, as well as to vertical strain. Orientation corrections are also estimated. The fourth extensometer in each PBO strainmeter provides redundant information used to reduce the chance that coupling coefficients could be misleadingly fit to inappropriate theoretical tides. Satisfactory fits between observed and theoretically calculated tides were obtained for three PBO strainmeters in California, where the calculated tides are corroborated by other instrumentation, as well as for six strainmeters in Oregon and Washington, where no other instruments have ever recorded Earth tidal strain. Several strainmeters have unexpectedly large coupling coefficients for vertical strain, which increases the strainmeter's response to atmospheric pressure. Vertical coupling diminishes, or even changes the sign of, the apparent response to areal strain caused by Earth tides or deep Earth processes because near the free surface, vertical strains are opposite in sign to areal strain. Vertical coupling does not impair the shear strain response, however. PBO borehole strainmeters can provide calibrated shear strain time series of transient strain associated with tectonic or magmatic processes.

1. Introduction

[2] Borehole strainmeters are highly sensitive instruments uniquely capable of measuring deformation of the Earth's crust with subnanostrain (nstrain = 10−9) resolution over periods from hours to days. Borehole strain signals from aseismic fault slip [Linde et al., 1996], permanent fault displacement caused by earthquakes [e.g., Johnston et al., 2006], aseismic slip transients in subduction zones [McCausland et al., 2008], and volcanoes on the verge of eruption [e.g., Sturkell et al., 2006] demonstrate that several types of borehole strainmeters can record very small, but important, tectonic deformation. To fully utilize borehole strainmeter data, however, it is necessary to transform the strainmeter output to calibrated measurements of formation strain.

[3] The Plate Boundary Observatory (PBO) borehole strainmeters discussed in this paper are Gladwin Tensor Strainmeters (GTSMs), each consisting of four extensometers (referred to as “gauges”), that measure changes in diameter of the cylindrical strainmeter housing at different azimuths [Gladwin and Hart, 1985]. These measurements are easily scaled to linear strain by dividing by the instrument's diameter. But because the strainmeter and the grout in which it is emplaced have different elastic moduli from those of the surrounding formation and because each gauge deforms in response to strain perpendicular, as well as parallel, to its axis, this linear strain is not the same as that which the formation would have experienced were the strainmeter not there. Therefore, it is necessary to “calibrate” the strainmeter after it is installed. Calibration entails estimation of “coupling coefficients” that express the strainmeter output as linear combinations of formation strain components from a known source. These linear relations are then inverted to obtain a “calibration matrix” that gives formation strains as linear combinations of the strainmeter gauge outputs.

[4] Calibration is essential if borehole strainmeter data are to constrain models of tectonic or magmatic processes. A first-order issue is that the elongations of individual strainmeter gauges cannot be equated to linear elongations of the formation, because the strainmeter/grout inclusion deforms about twice as much in response to shear strain, as in response to areal strain [Gladwin and Hart, 1985]. The wall thickness and grout compressibility for each PBO strainmeter are chosen to achieve areal and shear strain response factors of 1.5 and 3, respectively, and the four gauges are built to have identical gains, leading to an “isotropic” calibration matrix that should be appropriate for all PBO GTSMs. However, it is evident from data that the calibration matrices differ significantly between individual installations.

[5] Earth tides provide the best characterized formation strains for calibrating borehole strainmeters. The procedure entails estimating the amplitudes and phases of the M2 and O1 tidal variations in the strainmeter output, and also calculating theoretical values for these quantities using software such as the SPOTL package [Agnew, 1996, 1997], which computes the strains caused by astronomical forces and ocean loading. Coupling coefficients can be obtained by numerically fitting the observed and theoretical tides. However, it is important to know whether the resulting coupling coefficients are physically reasonable, because if the models of marine tidal loading and/or Earth elasticity used to compute the tides are inaccurate, a misleading calibration could result. Since only borehole or laser strainmeters can resolve tidal strains, independent verification of the tidal calculations is rarely available.

[6] Data recorded by the PBO GTSMs generally have three features not observed in previously installed three-component borehole strainmeters of the same nominal design [Gladwin and Hart, 1985; E. Roeloffs et al., Review of borehole strainmeter data collected by the U.S. Geological Survey, 1985–2004, unpublished report to UNAVCO, Inc., 2004, available at Folder/Data Flow and Analysis/Strain Reports/strain_rpt_25may04.pdf]. First, the PBO GTSMs have large responses to atmospheric pressure changes. Second, when an isotropic calibration matrix is used to obtain formation strain time series, the phases of the M2 and O1 tides inferred from these time series typically differ by tens of degrees from theoretical phases. Third, for many of the strainmeters, the amplitudes of the areal strain tides are much smaller than expected. In addition, some colocated strainmeters have areal strain tidal phases that differ from each other by tens of degrees. These features have posed obstacles to calibrating the PBO GTSMs and using their data to constrain geophysical models.

[7] In this paper, it is shown that at locations where the theoretical tides are known to be approximately correct (i.e., there is little influence from ocean loading, and/or there is verification of the tides by other strainmeters), the tidal responses of the PBO GTSMs can be reconciled with the theoretical tides by estimating coupling coefficients of each gauge to all three components of the horizontal strain tensor, as well as to vertical strain. Orientation corrections are also needed for some strainmeters. The coupling coefficients scale the time histories of areal and shear strains inferred from the strainmeter data. Significantly, if vertical coupling is present, it can reverse the sign of a strainmeter's apparent coupling to areal strain. The coupling coefficients determined by the method presented here have clear physical interpretations, making it possible to judge whether their values are reasonable and whether the theoretical tides are correct.

2. Strainmeter Data and Tide Calculations

[8] Twelve PBO borehole strainmeters were analyzed (Table 1 and Figure 1). The selection criteria were (1) at least 6 months of data containing several 60 to 90 day periods suitable for determining the atmospheric pressure response coefficients and the tidal constituents and (2) a distance from tidal water bodies that greatly exceeds the borehole depth, in order to avoid the possible complication of vertical shear strains induced by direct tidal loading. Three of these strainmeters are in California, where tidal models are known to be approximately correct. Four are in western Oregon, where there have been no previous measurements of strain tides, but in locations far enough from the coast that tidal models are expected to be adequate. The remaining five strainmeters, on the Olympic peninsula of northern Washington, are subjected to complex marine tidal loads whose effects on strain have not been measured before.

Figure 1.

Map showing locations of the PBO borehole strainmeters discussed in this paper. (a) Strainmeters in Oregon and Washington. The distance between the two strainmeters in each of the colocated pairs B005–B007, B027–B028, and B035–B036, is less than the diameter of the symbol. (b) Strainmeters in southern California.

Table 1. Locations, Lithologies, Grout Type, Depth, and Measured CH0 Orientations for PBO Borehole Strainmeters Discusseda
StrainmeterLatitudeLongitudeLithologyGroutDepth (m)CH0 Azimuth
  • a

    SetGrout and Masterflow 1341 are manufactured by Master Builders Technologies, Inc.

B003 (FloeQuarry)48.0630−124.1412BasaltSetGrout170232.2
B004 (HokoFalls)48.2020−124.4270BasaltSetGrout164168.2
B005 (Shores 1)48.0595−123.5033BasaltSetGrout161319.7
B007 (Shores 3)48.0577−123.5040BasaltSetGrout140193.0
B018 (Delphi)46.9558−123.0211BasaltSetGrout226267.0
B027 (Lester 1)44.4973−122.9622Sandstone/mudstoneMasterflow 1341234316.3
B028 (Lester 2)44.4937−122.9638Sandstone/mudstoneMasterflow 1341241282.3
B035 (Grants Pass 1)42.5040−123.3830GraniteSetGrout226279.1
B036 (Grants Pass 2)42.5058−123.3817GraniteMasterflow 1341182258.9
B073 (Varian)35.9500−120.4700SandstoneSetGrout242270.1
B081 (Keenwild)33.7110−116.7140GraniteSetGrout2433.2
B084 (Pinyon Flat)33.6116−116.4564GraniteSetGrout157197.0

[9] Ten minute sampled time series of data from the four gauges of each strainmeter (Figure 2), and 30 min samples of atmospheric pressure, were downloaded from the Northern California Earthquake Data Center. The strainmeter gauge data were edited if necessary (for example, to remove outliers or artificial offsets from the data) and decimated to 30 min samples to match the sampling interval of the atmospheric pressure data. At this stage, the data represent fractional elongations of each of the four gauges of the 12 strainmeters. Preinstallation measurements show that relative gains for the four gauges of each strainmeter may differ by up to 20% (M. Gladwin, email communication, 2006). Rather than apply these gains to the data, they are incorporated into the coupling coefficients that are estimated in section 5.

Figure 2.

Plan view of the relative orientations of the four gauges in a PBO borehole strainmeter. (a) PBO notation. Data files refer to the gauges as CH0, CH1, CH2, and CH3. The orientation of CH0, ϕ0, is measured at the time of installation. The strainmeter is constructed so that CH1, CH2, and CH3 are 60°, 120°, and 150° counterclockwise (CCW) from CH0. The dots indicate the ends of the gauges at the specified orientations, but orientation of either end of the gauge may be used because only elongation is being measured. (b) Mathematical notation for the coordinate system with the x1 axis parallel to CH1. The y1 axis is 90° CCW from the x1 axis, so that it is parallel to CH3. The elongation of the ith gauge is referred to as ei. In coordinates with x1 aligned with CH1, it is convenient to have the polar coordinate angle of e0 be −120° and the polar coordinate angle of e2 be +120°. This results in the identification of e0, e1, e2, and e3 with the outputs of CH2, CH1, CH0, and CH3, respectively.

2.1. Estimation of Tides in the Data

[10] The strainmeter gauge time series were divided into records of approximately 60 days, overlapping when possible by 30 days. To determine the atmospheric pressure response of each gauge, the gauge outputs and pressure data were band-pass-filtered to exclude frequencies outside the 4–6 day band (E. Roeloffs, manuscript in preparation, 2009). Sixty day records during which there was a good correlation between the atmospheric pressure and the strainmeter gauge time series were identified. For these records, the atmospheric pressure response of each gauge was removed using the coefficients determined for the band-passed data (Figure 3). The corrected data were used for tidal analysis. All suitable 60 day records were analyzed for each of the 12 strainmeters. Because the instruments were installed at different times, and because some data were unuseable, the number of 60 day periods varied from four at B084, to 22 at B035. For B027 and B028, all data analyzed were within 6 and 10 months, respectively. For the other 10 strainmeters, the records analyzed span at least 1 year.

Figure 3.

Atmospheric pressure response coefficients for the individual gauges of the 12 strainmeters, determined as the linear regression coefficient between strainmeter and atmospheric pressure data band-passed to the 4–6 day band. The relative sizes of the atmospheric pressure response coefficients of the four gauges of each strainmeter are more stable in time than the absolute value of the response coefficients. Therefore, the numbers plotted are averages over time of the relative amplitudes among the four gauges, multiplied by the coefficient for CH1 from the time period with the highest correlation coefficient.

[11] The data from each strainmeter gauge were detrended by subtracting the sum of a decaying exponential and a quadratic polynomial fit to each 60 day window. The tidal analysis was then performed by least squares fitting sines and cosines at a list of known tidal periods to the strainmeter data. The harmonic time variation, f(t), of a particular tidal constituent can be expressed in two equivalent ways:

equation image
equation image

where Pk = equation image = ∣f(t)∣ is the amplitude and θk = arctan(qk/pk) is the phase. In equation (1a), the tidal variation is a linear function of pk and qk, which are the respective amplitudes of the cosine and sine terms (sometimes referred to as the “in-phase” and “quadrature” terms). Equation (1a) facilitates numerical fitting of the data and was used for all calculations described in this paper. Equation (1b) facilitates interpretation of the relationships among tide constituents and is used in the text. Phases in this paper are given relative to the tidal potential, assuming no ocean loads, calculated for the same time period using the FORTRAN code ertid [Agnew, 1996]. This tidal potential has the same phase as the theoretical areal strain tide if no ocean loading or local perturbations are acting. A positive phase, θk > 0, indicates that the peak extensional strain occurs after (lags) the peak potential.

[12] The tidal calibration uses only the two constituents M2 (period 12.42 h) and O1 (period 25.82 h). These constituents have large amplitudes and their periods differ enough that they can be easily separated from the purely diurnal and semidiurnal constituents, which are more susceptible to contamination by temperature variations and other noise. The averages over all records analyzed of the amplitudes and phases of these two constituents were used in the calibration process, and the variability among the values obtained from the different time periods was used to compute the variance in the tidal parameters (Table 2). The tidal amplitudes and phases typically vary 5% and 5°, respectively. Variation tended to be larger for stations where the data analyzed included the first few months after installation, and for individual gauges with small tidal amplitudes, which can result from azimuthal variation in the amplitudes of the formation strain tides. The tidal amplitudes, which are in units of linear strain of each gauge, vary from about 1 to 34 nstrain.

Table 2. Average Amplitudes and Phases of the M2 and O1 Tidal Constituents in the Strainmeter Gauge Dataa
StrainmeterGauge∣M2∣ (nstrain)Phase M2 (deg)∣O1∣ (nstrain)Phase O1 (deg)
  • a

    Error ranges are two standard deviations, over all time periods analyzed. Phases are relative to the tidal potential, calculated for the case of no loading.

B003CH08.11 ± 0.0816.9 ± 0.75.07 ± 0.2147.5 ± 1.2
B003CH14.70 ± 0.1675.7 ± 2.66.13 ± 0.41168.3 ± 2.5
B003CH213.30 ± 0.11−138.6 ± 0.56.77 ± 0.14−71.2 ± 2.3
B003CH36.38 ± 0.15−70.3 ± 0.87.92 ± 0.21−11.6 ± 2.4
B004CH09.65 ± 0.38−115.1 ± 0.89.09 ± 0.28−51.1 ± 1.6
B004CH110.79 ± 0.28−31.9 ± 0.611.17 ± 0.3015.8 ± 0.8
B004CH211.94 ± 0.5165.5 ± 1.09.98 ± 0.55154.7 ± 1.1
B004CH35.16 ± 0.14152.3 ± 1.16.62 ± 0.47−153.1 ± 1.5
B005CH09.40 ± 0.10−95.2 ± 0.44.97 ± 0.12−13.1 ± 1.8
B005CH17.97 ± 0.17−31.6 ± 1.25.47 ± 0.15−3.8 ± 2.2
B005CH217.15 ± 0.1398.8 ± 0.88.76 ± 0.36161.3 ± 1.7
B005CH36.60 ± 0.10115.2 ± 2.42.79 ± 0.15161.2 ± 3.3
B007CH016.01 ± 0.2077.9 ± 0.94.00 ± 0.23138.6 ± 3.0
B007CH115.44 ± 0.14−91.2 ± 1.57.73 ± 0.37−12.2 ± 2.4
B007CH25.43 ± 0.33−1.2 ± 1.22.24 ± 0.188.5 ± 6.2
B007CH312.60 ± 0.1169.2 ± 0.83.27 ± 0.12147.2 ± 3.0
B018CH026.38 ± 0.223.2 ± 0.410.26 ± 0.3731.9 ± 1.0
B018CH133.38 ± 0.74137.9 ± 0.96.51 ± 0.61−150.9 ± 6.7
B018CH223.05 ± 0.53−57.4 ± 0.67.90 ± 0.46−3.3 ± 3.5
B018CH330.96 ± 0.63−37.0 ± 0.89.20 ± 0.4720.0 ± 4.8
B027CH07.15 ± 0.10−64.7 ± 0.87.12 ± 0.12110.8 ± 1.4
B027CH111.48 ± 0.03137.6 ± 0.35.19 ± 0.16−90.5 ± 1.7
B027CH23.77 ± 0.06178.1 ± 0.73.75 ± 0.13−155.3 ± 1.8
B027CH36.29 ± 0.19−82.8 ± 1.16.78 ± 0.45137.0 ± 6.3
B028CH014.92 ± 0.08−37.8 ± 0.46.97 ± 0.3377.2 ± 1.0
B028CH15.16 ± 0.0811.2 ± 0.85.95 ± 0.119.9 ± 2.5
B028CH210.77 ± 0.08140.1 ± 0.17.00 ± 0.22−99.7 ± 1.0
B028CH31.55 ± 0.05−147.1 ± 1.83.03 ± 0.23−153.5 ± 3.6
B035CH023.61 ± 1.16−38.5 ± 2.216.35 ± 2.1953.7 ± 4.7
B035CH112.41 ± 0.3597.7 ± 1.311.77 ± 0.57−45.4 ± 1.9
B035CH24.45 ± 0.435.5 ± 3.86.72 ± 0.43−149.5 ± 4.3
B035CH316.88 ± 0.25−54.9 ± 1.810.32 ± 0.42106.0 ± 2.8
B036CH05.70 ± 0.12−24.4 ± 1.06.98 ± 0.2927.4 ± 1.5
B036CH119.29 ± 0.29142.6 ± 0.910.78 ± 1.06−129.4 ± 4.5
B036CH26.56 ± 0.14−80.0 ± 1.16.04 ± 0.21126.5 ± 4.7
B036CH312.19 ± 0.46−59.4 ± 0.79.34 ± 0.4181.1 ± 1.3
B073CH011.22 ± 0.15172.2 ± 1.54.08 ± 0.30169.9 ± 3.8
B073CH19.09 ± 0.60151.1 ± 1.55.70 ± 0.63−145.1 ± 3.4
B073CH20.98 ± 0.15−113.8 ± 5.63.66 ± 0.28161.7 ± 2.9
B073CH33.79 ± 0.08−163.4 ± 2.93.63 ± 0.26144.9 ± 3.4
B081CH019.08 ± 0.08−7.8 ± 0.37.72 ± 0.24−23.1 ± 1.5
B081CH114.50 ± 0.1010.2 ± 0.34.26 ± 0.1827.7 ± 1.0
B081CH26.12 ± 0.12−151.3 ± 0.93.84 ± 0.122.8 ± 1.1
B081CH35.86 ± 0.14−51.0 ± 1.66.98 ± 0.24−16.3 ± 1.8
B084CH02.75 ± 0.16−63.1 ± 6.75.40 ± 0.21−24.9 ± 1.8
B084CH117.24 ± 0.055.5 ± 0.23.46 ± 0.24−5.5 ± 0.7
B084CH23.80 ± 0.14−31.3 ± 0.76.28 ± 0.1421.7 ± 0.9
B084CH35.71 ± 0.26−135.1 ± 1.86.34 ± 0.213.9 ± 0.8

2.2. Theoretical Tides

[13] The theoretical amplitudes and phases of the M2 and O1 strain tides at each site (Table 3) were computed using the SPOTL package [Agnew, 1996, 1997]. The tide-generating forces modeled by SPOTL include gravitational forces imposed by the Sun and the Moon (the body tide), as well as loads from marine tides in the global ocean (Oregon State University TOPEX/Poseidon solution, Version 7.0, from, and, for sites in northern Washington, the Strait of Juan de Fuca and the Strait of Georgia (D. C. Agnew, email communication, 2005). SPOTL routines convolve marine tidal loads with a Green's function to obtain the strains induced at the Earth's surface, at each strainmeter's coordinates, and these strains are added to the body tide. The Gutenberg-Bullen Model A average Earth Green's function was used. Differences among the values obtained using the other Green's functions supplied with SPOTL are insignificant compared with variability of the tidal amplitudes and phases in the data.

Table 3. Theoretical Tides at the Strainmeter Locations, Computed Using SPOTLa
GTSMTide ContributionM2O1
equation imageEE + equation imageNNequation imageEEequation imageNN2equation imageENequation imageEE + equation imageNNequation imageEEequation imageNN2equation imageEN
Amp (nstrain)Phase (deg)Amp (nstrain)Phase (deg)Amp (nstrain)Phase (deg)Amp (nstrain)Phase (deg)Amp (nstrain)Phase (deg)Amp (nstrain)Phase (deg)
  • a

    The body tide is from gravitational forces imposed by the Sun and the Moon. “+Ocean” denotes the sum of the body tide and strains imposed by tides in the global ocean. “+Regional” denotes the sum of the “+Ocean” strains and strains imposed by regional water bodies, in this case the Strait of Juan de Fuca and the Strait of Georgia. Amp, amplitude.


[14] The aspects of the tidal calculations that are poorly known, and could lead to unrealistic theoretical tides, are detailed bathymetry, site geology, and topography. In this paper, when theoretical tides are referred to as “incorrect,” it is presumed that lack of such knowledge is the reason. Previously existing instrumentation verifies that the theoretical tides agree reasonably well with the observed tides at Piñon Flat (B084) [Hart et al., 1996] and Parkfield (B073) (E. Roeloffs et al., unpublished report, 2004). It is reasonable to assume that the theoretical tides are also appropriate for other sites at comparable distances from a relatively uncomplicated coastline (such as B027, B028, B035, B036, and B081). For the sites in northern Washington, however, the load tides are larger relative to the body tides, and no previous instrumentation has verified that the theoretical tides are correct.

[15] Local topography and geologic heterogeneity influence the formation tidal strains in the immediate vicinity of each strainmeter [Beaumont and Berger, 1975; Berger and Beaumont, 1976; Harrison, 1976]. However, these influences are difficult to calculate and to separate from uncertainties in strainmeter response. The approach taken here is to estimate the coupling coefficients needed to reconcile the observed and theoretical tides, and then to decide whether the theoretical tides adequately approximate the formation tidal strains, based on whether those coupling coefficients are reasonable.

3. Coupling Formulation

[16] This section describes how the elongations recorded by three or mores gauges of a strainmeter can be converted to measurements of the horizontal formation strain tensor. This formulation will be applied to the PBO strainmeters in section 4. Section 3.1 begins with the equation that describes the response of a single gauge to formation strain, and section 3.2 describes how this equation applies to Earth tides, strains from deep Earth sources, and to atmospheric pressure loading, with particular attention to the consequences of vertical coupling. Section 3.3 shows how the elongations of all the gauges in a strainmeter are expressed in common coordinates, and how the formulation introduced here reduces to the isotropic coupling formulation [Gladwin and Hart, 1985] that has been implemented by PBO for routine processing of borehole strainmeter data. Finally, section 3.4 shows how a matrix containing coupling coefficients can be inverted to obtain a “calibration” matrix. The calibration matrix gives the formation strain components as linear combinations of gauge outputs and is the final goal of the calibration procedure.

[17] Appendix A discusses similarities and differences between the formulation presented here and earlier analyses by Gladwin and Hart [1985] and Hart et al. [1996].

3.1. Response of a Single Strainmeter Gauge to Formation Strain

[18] A multicomponent borehole strainmeter consists of n (at least 3) horizontal extensometers (referred to as “gauges”), each of which measures changes in the inside diameter of the cylindrical strainmeter housing along a particular azimuth (Figure 2). Throughout this paper, the output of the ith gauge, ei, is thought of as a fractional elongation, or linear strain. ei generally is not equal to the elongation that would have occurred along the same line segment in the formation before the strainmeter was installed. Instead, the elongation of each extensometer is related to all horizontal components of the formation strain tensor. In Cartesian coordinates, xi, yi, aligned with the ith gauge (Figure 4), these three components are equation image, equation image, and equation image. As illustrated in Figure 4, positive increments of equation image will lengthen the ith gauge and positive increments of equation image will shorten that gauge. In a homogeneous, isotropic medium, changes of equation image would not affect the length of the ith gauge. However, the rock formation surrounding a borehole strainmeter is not necessarily homogeneous or isotropic, so it is assumed here that changes of equation image could, in fact, lengthen or shorten the ith gauge.

Figure 4.

Plan view diagram showing how the three components of the formation strain field, expressed in coordinates aligned with one strainmeter gauge, deform the strainmeter. (a) Undeformed strainmeter with ith gauge shown in bold. (b) Deformation of the strainmeter housing by positive increments of each component of the horizontal strain tensor. Positive increments of equation image lengthen the ith gauge, of equation image shorten the ith gauge, and increments of equation image do not affect the ith gauge's length in an ideal homogeneous, isotropic environment. (c) The representation of the horizontal strain tensor that is most useful in working with borehole strainmeter data, using areal strain ɛa = equation image + equation image, differential extension equation imageequation image, and engineering shear strain equation image.

[19] Although designed to measure only horizontal deformation, no borehole strainmeter is completely insensitive to vertical strain, ɛzz. Summing the responses to the horizontal strain tensor and the vertical strain leads to the following equation for gauge elongation:

equation image

In equation (2), equation image, equation image, equation image and Vi are dimensionless coupling coefficients to be determined. No terms are included for vertical shear strain because they are expected to vanish for Earth tides near the free surface in a uniform half-space. Earth tides are the only reference strains considered here, and they provide no information on possible coupling of the strainmeters to vertical shear strains.

[20] The coupling coefficients in equation (2) have clear physical meanings that imply certain constraints on them. The couplings to elongation parallel and perpendicular to the extensometer (equation image and equation image, respectively) are unquestionably needed. equation image is expected to be positive; >1 if the formation is stiffer than the strainmeter/grout inclusion, <1 otherwise. equation image should be negative, and smaller in absolute value than equation image. The engineering shear strain, equation image, is needed to describe the horizontal strain field completely, but nonzero equation image can arise only from nonuniformity or anisotropy, so they should be small compared with equation image and need not have similar values or even the same sign for all gauges. Coupling to vertical strain, ɛzzF, is included because the gauges of PBO borehole strainmeters contract when atmospheric pressure increases (imposing contractional vertical strain). The Vi are all expected to be positive and related to the atmospheric pressure response coefficients of each gauge, as explained in more detail in section 3.2.

[21] There are advantages to writing equation (2) in terms of the areal strain, ɛaF = equation image + equation image, which is invariant under rotation, and the differential extension, equation imageequation image, as shown in Figure 4c:

equation image

Equation (3a) has the form

equation image

In equation (3b), Di > Ci because equation image is always negative. Equation (3b) defines the coupling coefficients that are estimated in section 5 for the 12 PBO strainmeters.

[22] Relative “gauge weights,” gi, can be incorporated in equation (3b), most generally by multiplying both sides of that equation to obtain:

equation image

The gi can be subsumed into the unknown coupling coefficients, so there is no loss of generality by omitting them from the right-hand side of equation (4a). Depending on the purpose, the gauge weights can be defined in several ways. If the actual gauge gains are known, then the weights are their inverses and add no unknowns. If the gains are not known, various assumptions can be made so that equation (4a) is not underdetermined. In section 4.3, relative gauge weights are estimated assuming some constraints on the other coupling coefficients; there it is assumed that g1 = 1. On the other hand, the self-consistency relationships used in section 4 are most easily derived if equation (3b) is written as

equation image

by defining Di = D/gi, where D is a shear strain coupling coefficient common to all gauges. Equation (4b) is completely equivalent to equation (3b).

3.2. Effect of the Earth's Surface and Vertical Coupling on Strainmeter Response to Subsurface Sources, Earth Tides, and Atmospheric Pressure

[23] The vertical strain at the Earth's surface, produced by Earth tides or a source within the Earth (e.g., subsurface fault slip or magmatic upwelling), is related to the areal strain by the relation

equation image

in which ν denotes the formation Poisson ratio. Equation (5) holds because vertical stress must vanish at the Earth's surface for such sources. Relative to the source depth, or to the vertical wavelength of Earth tidal strains, the depth of the strainmeter boreholes (<300 m for all PBO borehole strainmeters) is negligible, so the strainmeter can be assumed to be at the Earth's surface. (This is one reason why a separate measurement of vertical strain has been viewed as unnecessary in a borehole strainmeter.) However, substituting equation (5) into equation (3b) yields

equation image

Equation (6) has the important implication that vertical coupling reduces the gauge's response to any areal strain that obeys a free surface boundary condition. Equation (6) also shows that if Vi > (ν/1 − ν)Ci, the response to areal strain may vanish or become negative, a situation that will be shown to occur for many of the 12 PBO strainmeters considered here (sections 4.1 and 5.35.5). A final implication of equation (6), critical to tidal calibration, is that when Earth tides are used as known formation strains, and coupling parameters to ɛaF, (equation imageequation image) and equation image are estimated, the result will provide only an apparent areal strain coupling coefficient,

equation image

It is not possible to determine Ci and Vi independently using Earth tides as the reference strain field.

[24] The strainmeter's response to atmospheric pressure is relevant because it contains information about the vertical coupling. Atmospheric pressure is a surface load so it does not obey equation (5). It is assumed here that this surface load is uniform over an area large compared to the strainmeter's depth. An increase of atmospheric pressure, Δp, imposes a contractional vertical stress, −Δσzz, that leads to both vertical and horizontal strains in the subsurface:

equation image
equation image

In equations (8a) and (8b), A, the ratio of contractional vertical strain to uniform surface pressure increase, is positive and has dimensions of compliance; b denotes the ratio of areal strain to vertical strain induced by atmospheric pressure. Three-component GTSMs in California, which have small responses to atmospheric pressure, record small contractional areal strains when atmospheric pressure increases, implying that b is also positive and is small compared to A (E. Roeloffs et al., unpublished report, 2004). The increment of gauge elongation can be written using (3b) and (8a) and (8b) as

equation image

Equation (9) shows that vertical coupling increases the gauge's response to atmospheric pressure, or to other surface loads. The gauge's atmospheric pressure response coefficient is

equation image

To determine whether all variation of the apparent areal coupling coefficient, equation imagei, among the gauges of a single strainmeter can be explained by differences in the degree of vertical coupling, the four atmospheric pressure response coefficients can be plotted versus equation imagei. If all gauges have the same areal coupling coefficient, Ci = C, then by eliminating Vi between equations (7) and (10) it can be shown that the amplitude of the atmospheric pressure response should decrease with increasing equation imagei according to [A(1 − ν)/ν]equation imagei ≈ 3Aequation imagei for ν = 0.25. Beavan et al. [1991] obtained a value of A = 0.42 nstrain/hPa, based on atmospheric pressure-induced fluctuations in groundwater levels in a sandstone.

3.3. Gauge Elongations in Geographic Coordinates and the Special Case of Isotropic Coupling

[25] In section 3.1, equation (3b) expresses the elongation of a single gauge in a coordinate system aligned along that gauge. To combine elongations from several gauges to measure horizontal tensor strain, the individual gauge responses must all be expressed in a single coordinate system x, y using

equation image

where ϕi is the angle between the ith gauge and an arbitrarily chosen x axis, measured counterclockwise. Equation (11) does not affect the areal strain, ɛaF, which is invariant under coordinate rotation.

[26] Substituting equation (11) into equation (3b) results in

equation image

An important point from equation (12) is that the response of the ith gauge to engineering shear strain in a coordinate system that is not aligned with that gauge does not vanish, even when the coupling coefficient equation image is small enough to neglect.

[27] Equation (12) reduces to the simpler “isotropic” coupling formulation used by PBO for automated processing where, for all gauges, equation image = 0, Vi = 0, and gauge weights, gi, can account for all the variability in the other coupling coefficients, so that there is a single shear coupling coefficient, D = giDi, and a single areal coupling coefficient, C = giCi. Under these special circumstances, and using x = east (E) and y = north (N), equation (12) reduces to

equation image

Equation (13) applies in the ideal situation where the formation is homogeneous and isotropic, for which Gladwin and Hart [1985] derived analytical expressions for areal and shear response factors, c = C/2 and d = D/2 , in terms of the relative elastic moduli of the formation, grout, and strainmeter housing. Typical values are c = 1.5 and d = 3. Equation (13) will be referred to as “isotropic” coupling. As shown later in this paper, equation (13) is not adequate for PBO strainmeters, but it is a useful approach in the absence of other knowledge about the coupling coefficients.

3.4. Calibration Matrices for Isotropic and More General Coupling

[28] For an isotropically coupled strainmeter with three gauges equally spaced 120° apart, one of which is oriented east-west, the strain components can be expressed in terms of the gauge elongations by writing equation (13) for each gauge and solving for the strain components:

equation image
equation image
equation image

For isotropic coupling, areal strain is a sum of gauge outputs, while differential extension and engineering shear are differences; this remains approximately true for more general coupling models. Equations (14a)(14b), and (14c) can be written in matrix form as

equation image

The matrix product on the left-hand side of equation (15) constitutes the “calibration matrix” for this idealized strainmeter. The first factor contains the coupling coefficients, which depend on only the elastic properties of the strainmeter, grout, and formation. The second matrix in the product depends on only the geometric arrangement of the gauges in the strainmeter.

[29] In the general case where the gauges are at arbitrary orientations, and where there are more than three gauges, the procedure is conceptually the same, but numerical inversion must be used to obtain the calibration matrix. The procedure, described here for four gauges, is straightforward to generalize to any number of gauges.

[30] First, the equations expressing the responses of all of the gauges to formation strain in common coordinates x, y (equation (13)) are assembled as the rows of a “coupling matrix,” C:

equation image
equation image

To solve for unknown formation strain components in terms of the observed gauge elongations, a “calibration matrix” S must be found such that

equation image

where I3×3 is the identity matrix. The calibration matrix S serves as an inverse for the matrix of coupling coefficients, C, which has more rows than columns for four or more gauges, and therefore is not invertible in the strict sense. S was computed here as the Moore-Penrose generalized inverse, implemented in the IMSL subroutine DLSGRR via singular value decomposition [Visual Numerics, Inc., 1997]. The Moore-Penrose generalized inverse yields the true inverse if the matrix is invertible, so that it can also be used for subsets of three gauges. For equation (17) based on four gauges, the resulting strains will be a least squares fit if all equations cannot be satisfied simultaneously, which could be the case for any strainmeter with more gauges than the three needed to uniquely determine the strain field. Using a generalized inverse enables equation (17) to be solved for any number of gauges.

[31] There is no simple algebraic factorization of C into separate matrices for elastic properties and gauge geometry, as can be done for isotropic coupling (equation (15)). C and S also depend on the arbitrary choice of coordinate system. Once S has been computed for any set of coordinates x, y, the formation strain tensor can be rotated to geographic (or other) coordinates.

[32] It should be kept in mind that when the Earth tides, which obey the free surface boundary condition, serve as the reference strains, the resulting calibration matrix is only appropriate for other signals that also obey that condition. Tectonic signals that originate from deep within the Earth have this property, but often there are also signals in strainmeter data caused by rainfall or other surface loading.

4. PBO Borehole Strainmeters: Preliminary Internal Consistency Checks

[33] This section shows how self-consistency checks for a PBO borehole strainmeter can be used to diagnose how its coupling coefficients differ from the isotropic case, to estimate the areal strain phases that it experiences in situ, and to estimate an orientation correction, before actually determining the coupling coefficients. These consistency checks reduce the possibility that an inappropriate tidal model would be masked by fitting coupling coefficients that differ greatly from isotropic coupling. The information from the self-consistency checks used in section 5 to guide the estimation of the coupling coefficients in equation (6).

4.1. Self-Consistency for Areal Strain

[34] The PBO borehole strainmeters can be thought of as having two subsets of equally spaced gauges: CH0, CH1, and CH2 oriented 120° apart, and CH1 and CH3, 90° apart (Figure 2). If the isotropic coupling model (equation (13)) is appropriate, then these two subsets provide two different gauge combinations, having only CH1 in common, that should be equal to each other, as well as to the areal strain, assuming the relative gauge weights are correct:

equation image
equation image

The gauge weights are not known a priori, and have proven difficult to estimate from the data. Fortunately, much can be deduced about the behavior of a PBO four-component borehole strainmeter by inspecting the linear combinations in equations (18a) and (18b) formed with equal gauge weights.

[35] Appendix B gives the somewhat lengthy, but straightforward, versions of equations (18a) and (18b) obtained with the more general coupling model (6), expressed as in equation (4b) using gauge weights and a single shear coupling coefficient D. The most helpful relations are obtained by neglecting all equation image, on the basis that they should be smaller than the other coupling coefficients (equations (B2a) and (B2b)). Under that assumption, the two gauge combinations on the left-hand side of equations (18a) and (18b), when properly weighted, differ only by multiplicative scalars, even if all the equation imagei are not the same. Therefore, they should yield time histories that are proportional to each other, as well as to formation areal strain, although possibly with opposite sign. For Earth tides, the phases of any tidal constituent obtained from these two expressions should either be the same, or differ by 180°, i.e.,

equation image

If the equation imagei differ among the gauges, the two gauge combinations (18a) and (18b) would yield different amplitudes, but these amplitude ratios should be the same for M2 and O1, and equal

equation image

Equations (19a) and (19b) can be tested for the 12 PBO strainmeters, as plotted in Figures 5a, 5b, and 5c. Nothing in these plots depends on orientation or on the values of the coupling coefficients. The theoretical areal strain phases, computed using SPOTL, are plotted for comparison in Figures 5a and 5b.

Figure 5.

Consistency checks assuming equal gauge weightings, and comparisons with theoretical tides. Areal strains are calculated using equations (18a) and (18b), assuming that all gauges have equal gains. Phases near −180° have had 360° added. (a) M2 phases. (b) O1 phases. (c) The ratio ∣(g1e1 + g3e3)/2∣/∣(g0e0 + g1e1 + g2e2)/3∣ for M2 and O1.

[36] Five of the 12 strainmeters (B027, B036, B073, B081, B084) approximately obey equation (19a): M2 and O1 tidal phases estimated from the two subsets of gauges are within 5° of each other (Figures 5a and 5b). For four of these five (B027, B073, B081, and B084), the amplitude ratio in equation (19b) is the same for O1 and M2 to within 12% (Figure 5c). Thus, B027, B073, B081, and B084 come close to agreement with both areal strain self-consistency relations without adjusting relative gauge weights. For the remaining seven strainmeters (B003, B004, B005, B007, B018, B028, B035), estimates from the two gauge subsets of the M2 and/or the O1 phases differ by at least 13°. At B018, the difference is 74° for M2 and for B003 the differences are 265° and 266° for M2 and O1, respectively.

[37] Figures 5a, 5b, and 5c contain evidence for strong vertical coupling at strainmeters B027, B036, and B073, where all of the areal strain phase estimates are within 25° of opposite in sign from the theoretical tides. Referring to equation (19b), one or more of the apparent areal coupling coefficients must be negative, which is most easily explained by large vertical coupling coefficients. Moreover, at B027 and B036, colocated strainmeters B028 and B035, respectively, have areal strain tides that are nearly in-phase with the theoretical tides, implying the degree of vertical coupling is unique to a particular instrument, installation, or small-scale geologic feature near the borehole. The sites for which the areal strain tide phases are approximately 180° out of phase with the theoretical tides (B027, B036, and B073) are among those with the largest atmospheric pressure response coefficients (Figure 3), consistent with equation (10).

[38] It is possible to solve for the relative weights of CH0, CH2, and CH3 (g0, g2, and g3) with respect to CH1(g1) by requiring both self-consistency relations for areal strain (equations (19a) and (19b)) to hold. The relative weights were restricted to being within ±30% of each other, to allow slightly more variation among gauges than the 20% stated by the strainmeter manufacturer. This nonlinear problem has three possible outcomes. The first possibility is a unique set of gauge weights, and these were found for B073 and B081. The next possibility is that more than one combination of gauge weights is equally satisfactory, and this was the outcome for eight of the remaining 10 strainmeters. Finally, no set of gauge weights within the allowed range provided an acceptable fit for B003 or B018. For B003 and B018, in addition to possible variation among the Ci, Di, and/or Vi of the four gauges, some other term must be important in the coupling model, such as significant coupling of one or more gauges to equation image. This situation cannot be attributed to the theoretical tides being incorrect or to an orientation error.

[39] For the 10 strainmeters that can satisfy (19a) and (19b) with gauge weight adjustments, Figure 6 shows the ranges of M2 and O1 areal strain phases obtained from the data as the relative gauge weights range within the limits that bring the areal strain phases from gauge subsets 012 and 13 to within a few degrees of each other. Figure 6 can be used to judge whether the areal strains experienced by each strainmeter are in phase with the theoretical areal strain tides. For B081 and B084, where the theoretical tides are known to be appropriate, the areal strain tidal phases experienced by the strainmeter are within a few degrees of the theoretical phases. For B073, if ±180° is added to the areal strain phases from equations (18a) and (18b), they come within 5° of the theoretical phases. For the two colocated strainmeter pairs B027–B028 and B035–B036 in Oregon, where the tidal model is plausibly correct, the observed areal strain phases agree with the theoretical phases as well as they do at B073, provided ±180° is added to the phases for B027 and B036. However, within both of these pairs, the observed phases differ up to 15° between the two colocated strainmeters.

Figure 6.

The ranges of M2 and O1 areal strain phases (vertical bars) obtained from the data as the relative gauge weights range within the limits that bring the areal strain phases from gauge subsets 012 and 13 to within a few degrees of each other. Gauge weights were constrained to ∣gi − 1∣ < 0.3. B003 and B018 are not shown because gauge weights in this range did not bring the areal strain phases from the two gauge subsets into agreement.

[40] For B003, B004, B005, and B007, in northern Washington, the observed and theoretical areal strain phases differ by 10° or more (Figures 5a, 5b, and 6). Only for B005 can they be reconciled with gauge weights within 30% of each other. A possibility is that the formation areal strain tides at B003, B004, and B007 include contributions from formation fluid pressure in which the tidal variations lag the tidal strain. The force on the strainmeter imposed by the combination of fluid pressure and solid rock strain would be equivalent to a reduced, phase-shifted areal strain. Unfortunately, the amplitude reduction cannot be determined using the methods of this paper, because it would be indistinguishable from reduced values of the equation imagei.

4.2. Self-Consistency for Differential Extension

[41] From the isotropic coupling model (14), it can be shown that two different gauge combinations should equal the differential extension in coordinates with x parallel to CH1:

equation image
equation image

Using the more general coupling model (6) adds a term proportional to the areal strain to both equations (20a) and (20b), even when all equation image can be neglected (Appendix B). If this term can be neglected (i.e., if the left-hand sides of equations (20a) and (20b) are equal), then the phases given by equations (20a) and (20b) may provide information about the instrument's orientation by comparing them with the theoretical tides rotated to coordinates parallel to gauge 1.

[42] Figure 7 shows, somewhat surprisingly, that the phases of equation imageequation image from equations (20a) and (20b) are generally more self-consistent than the areal strain phases. This implies that for most sites, the equation image and the differences among the Ci and Vi are small compared with D. Another implication is that whatever causes the areal strain phases to be inconsistent at sites such as B003, B004, B005, B007, and B018 is canceled out when gauge outputs are differenced to obtain shear strains. A possible explanation is a contribution from formation pore pressure that lags the areal strain.

Figure 7.

Differences between equation imageequation image phases calculated using CH0, CH1, and CH2 (equation (20a)) and CH1 and CH3 (equation (20b)). All gauge weights are assumed equal.

[43] The good agreement between equation imageequation image phases computed from the two gauge subsets justifies comparing the phases of equation imageequation image and equation image, calculated using unit gauge weights, with the theoretical phases for these shear strain constituents, rotated to gauge 1 parallel coordinates. Figure 8 summarizes the azimuths for which the observed shear strain tide phases equal the respective theoretical tide phases for all 12 strainmeters. Ideally, all shear strain tide phases for each strainmeter would agree with the theoretical tides for the same azimuth, and that azimuth would be the one measured at installation. Figure 8 shows that for most of these strainmeters, two or three of the four phases match near a single azimuth, while the remaining one or two match at different azimuths. At B004 and B073, the azimuth measured during installation agrees with the orientation for which the theoretical and observed shear strain phases match, while at B027 and B028, the observed shear strain phases agree better with a different orientation. Orientation corrections could be needed if, for example, magnetic minerals in the formation affected the magnetometer mounted on the strainmeter that is used to determine the orientation. The information in Figure 8 will be helpful in judging whether the orientation corrections obtained when inverting for coupling coefficients (section 5) are reasonable.

Figure 8.

Azimuths of CH1 for which the phases of equation imageequation image and equation image calculated from the tides in the strainmeter data, assuming equal gauge weights, best match the theoretical tides. Ideally, all shear strain tide phases for each strainmeter would agree with the theoretical tides for a single azimuth, and that azimuth would be the one measured at installation. Outliers, such as the O1 shear strain-derived azimuth for B073, are typically artifacts of the low observed amplitude of that tide constituent and/or of coupling that is poorly approximated by equal gauge weights. The azimuths measured at installation are also indicated, as are the azimuths obtained by inverting for coupling coefficients (“preferred” or “best” fits of Table 4).

[44] While the self-consistency tests do not actually yield coupling coefficients, they provide useful guidance as to whether the tidal model is appropriate, and, if so, what features in the coupling model are needed to reconcile the theoretical and observed tides. In particular, by considering the areal and shear phases separately, the need for an orientation correction is easily distinguished from the effects of unequal gauge weightings.

5. Tidal Calibration of the PBO GTSMs

[45] In this section, the coupling formulation of equation (6) is fit to the M2 and O1 tidal constituents for the PBO GTSMs, with guidance from the self-consistency checks of section 4. For nine of the 12 strainmeters, reasonable coupling coefficients were obtained that reconcile the observed and model tides. However, for three sites in northern Washington, the required coupling coefficients would be outside the ranges estimated from strainmeters where the theoretical tides are known to be approximately correct.

5.1. Procedure for Estimating the Coupling Coefficients

[46] For the ith gauge of each strainmeter, there are three unknown coupling coefficients (equation imagei = CiViν/(1 − ν), Di, and equation image), and the amplitudes and phases of the M2 and O1 tidal constituents as determined from the gauge elongation data provide four observations (Table 2). The theoretical tides are the “known” reference; as shown in Table 3, each of the strain components ɛa, ɛEE − ɛNN, and 2ɛEN was calculated for each site, for both the M2 and O1 tides, each of which has an amplitude and a phase. This information allows four linear equations to be written for each gauge by using equation (6). The equations for each gauge are written in coordinates with the x axis aligned with that gauge and the theoretical tidal strains are rotated to that coordinate system.

[47] Figures 9a and 9b illustrate the approach by plotting the amplitude and phase of the observed tidal elongations from each gauge, together with the amplitudes and phases of the strain constituents of the theoretical tides, expressed in coordinates parallel to that gauge. For each gauge, coupling coefficients are sought that allow the observed tidal elongation to be expressed as a vector sum of the theoretical areal strain, differential extension, and possibly engineering shear, imposed by the M2 and O1 tides. On the basis of equation (6), when the vertical coupling is small enough that the apparent areal strain coupling coefficient is positive, the expectation would be that each gauge's M2 and O1 phases would be intermediate to the corresponding phases of ɛaF and (equation imageequation image)F obtained from the theoretical tides, with the exact relationship depending on the relative sizes of the coupling coefficients. Figure 9a shows that for B081, all the gauges have this property. Figure 9b, for B073, shows that the apparent areal strain coefficients of all of the gauges are negative.

Figure 9.

Polar coordinate graphs comparing strainmeter gauge tides with the theoretical horizontal tidal strain components calculated using SPOTL. Radial axis is amplitude, and the angular coordinate represents the phase with respect to the tidal potential. For each gauge, the theoretical tides have been rotated so that they are expressed in a coordinate system with x parallel to that gauge's measured orientation. (a) Graphical depictions of the application of equation (6) to tidal calibration B081. The observed tides (star) lie between the vectors representing ɛa and equation imageequation image for all four gauges, implying that the observed tides can be expressed as a weighted vector sum of ɛa and equation imageequation image using positive weights (coupling coefficients). (b) Same as Figure 9a except for B073. To express the vectors of observed tides as vector sums of ɛa and equation imageequation image, negative coupling coefficients to ɛa are required for all four gauges.

[48] Rotation of the theoretical tides to gauge-parallel coordinates requires an orientation. If the orientation is assumed to be known, then the coupling coefficients of each gauge are determined independently. Using the sine and cosine terms of each tide results in four linear equations for each gauge, which can be solved using least squares. Alternatively, the orientation correction can also be treated as an unknown. The orientation correction makes the equations nonlinear and links the solutions for the individual gauges into a system with 13 unknowns and 16 observations.

[49] One complication is that when the orientation correction is an unknown, there is a tradeoff between the equation image and the Di if equation image is a free parameter for all four gauges. This is because when rotating the theoretical tides to gauge-parallel coordinates, the relative amplitudes of (equation imageequation image) and equation image depend on the rotation angle. Therefore, different combinations of azimuth and coupling coefficients can provide identical fits to the tidal observations, but would not yield the same calibration matrices or strain time series.

[50] The following strategy was adopted to handle the orientation ambiguity and to avoid overfitting the observations. For the first fit, all of the equation image were set to zero, which allows the equation imagei and the Di to be estimated for all the gauges, as well as a unique orientation. To identify a situation where an erroneous orientation correction might facilitate a fit to an inappropriate tidal model, the orientation obtained is compared with Figure 8. This first fit is listed as “equation image = 0” in Table 4. Then, to obtain the “preferred” fits in Table 4, successive fits were done, each allowing all of the equation image to vary within a larger percentage of the average value of the Di. For these fits, the azimuth was fixed to the value obtained with equation image = 0, and measures of the quality of fit were computed as functions of the allowable spread between the four values of Di. Significant improvements in the fit are not obtained beyond a certain spread in the Di (typically 30%) or increase in the allowed ratio of the equation image/Di (typically 25%). The preferred fits in Table 4 were obtained by restricting the Di and equation image to these ranges. For each fit, Table 4 lists Davg, which is the average of the four Di, and gauge weights, gi, defined by gi = Davg/Di. The gi may differ more than the allowed spread in the Di, for example if there is one gauge whose Di accounts for most of the difference from the average.

Table 4. Coupling Coefficients Determined by Fitting Equation (6) to the Tide Constituents of Each Strainmeter Excluding B018a
GTSM FitParameterParameter ValueGaugeAdjusted Azimuth (deg)giequation imageiequation imageM2 MisfitO1 Misfit
  • a

    Two fits are shown for each strainmeter. The first has all equation image constrained to be zero; the “preferred” fit has the Spread(Di) and ∣Hxyi/Di∣ constrained to the maximum range that significantly improved the fit, where Spread(Di) = [max ∣DiDj∣]/Davg. ΔAz denotes the azimuth correction; gi = Davg/Di and Emin is the minimum value of the total proportional misfit as defined in equation (22). Fits labeled “best” shown for B005 and B007 are considered unsatisfactory and are interpreted to represent matches to incorrect theoretical tides.

equation image = 0ΔAz34.8CH1207.01.676−0.2210.0000.6213.21.3522.1
equation image = 0ΔAz−5.4CH1102.80.5610.2990.0000.242.21.6414.7
equation image = 0ΔAz15.7CH1275.42.2630.2590.0000.435.40.417.5
equation image = 0ΔAz21.2CH1154.20.4771.1230.0000.744.80.233.0
equation image = 0ΔAz−30.1CH1226.20.908−0.0930.0000.918.00.519.9
equation image = 0ΔAz31.1CH1253.40.9150.2440.0000.5811.21.0417.4
equation image = 0ΔAz11.6CH1230.71.1090.5680.0001.7013.81.6914.9
equation image = 0ΔAz3.8CH1202.70.929−0.4650.0001.728.90.807.4
equation image = 0ΔAz3.2CH1213.30.797−0.3660.0001.2613.90.9115.6
equation image = 0ΔAz8.9CH1312.11.1430.4420.0000.916.30.4911.4
equation image = 0ΔAz24.7CH1161.71.2290.5090.0000.362.10.3811.0

[51] The appropriateness of the fit was assessed by comparing the spread in the Di among the gauges and the ratios of equation image/Di with the values obtained for B081, B084, and B073, where the tidal model is known to be approximately correct. The degree to which variations in the equation imagei could be explained by variations in the atmospheric pressure response coefficient was investigated. Plots of the same type as Figures 9a and 9b were also used to assess the plausibility of the fits.

[52] The equations were solved numerically using the IMSL quasi-Newton optimization routine DBCONF [Visual Numerics, Inc., 1997] to minimize the overall proportional misfit, E, defined as

equation image

In equation (21), eik denotes the elongation of the ith gauge for the kth tidal constituent as estimated from the data, and equation imageik is the corresponding estimated elongation based on a trial set of coupling coefficients and azimuth correction; eik and equation imageik are both expressed using sine and cosine terms and the absolute values are calculated as described following equation (1b).

[53] DBCONF allows ranges to be specified for each unknown. All initial fits were done requiring −0.5 ≤ equation imagei ≤ 2.0, 0.1 ≤ Di ≤ 4.0, and ∣equation image/Di∣ ≤ 0.0001 and with a free orientation. All these solutions converged to values of equation imagei and Di strictly between the limits; the results are listed in Table 4, labeled “equation image = 0.” To obtain the “preferred” fit, also listed in Table 4, the orientation was fixed at the value determined for the “equation image = 0” fit, and the amount of variability among the Di and the equation image was limited, so some of those coefficients are at the limits of their allowed ranges. On the basis of the uncertainties in Table 2, a general goal was to fit both the M2 and O1 tides on all gauges to an absolute misfit of 0.5 nstrain or a proportional misfit of 5% of the observed amplitude, whichever was smaller; this goal was not met for at least one tidal constituent on one gauge of most strainmeters.

[54] Sections 5.25.5 describe the results of estimating the coupling coefficients for the 12 PBO strainmeters under study.

5.2. B081 and B084 in Southern California: Low Vertical Coupling, Verified Tidal Model, and Good Fit of Theoretical and Observed Tides

[55] As discussed in section 4.3, the phases of the observed areal strain tides at station B081 (Keenwild) agree fairly well with the theoretical tides using the isotropic coupling coefficients. This good agreement is consistent with laser strainmeter measurements at the relatively nearby site Piñon Flat that indicate the SPOTL-calculated tides are approximately correct there [Hart et al., 1996].

[56] The results of fitting equation (6) are in Table 4. With the equation image terms constrained small, several absolute misfits exceed 0.5 nstrain, and CH1 O1 and CH2 M2 have proportional misfits of 11.4% and 13.5%, respectively. However, the 8.9° orientation correction appears reasonable based on Figure 8. Figure 10 is a plot illustrating how the measures of misfit decrease as the differences among the Di are allowed to increase. The fit continues to improve significantly until the Di are permitted to differ from the average by 15%. Allowing nonzero values of equation image does improve the fit, but the misfits do not decrease significantly if equation image/Di is allowed to increase beyond 0.15. The nonzero values of equation image seem to be most needed for CH1; even with all gauges coupled to equation image, misfits > 10% remain for CH0 O1 and CH2 M2.

Figure 10.

Quality of fit measures for B081 as functions of the allowed spread of the coupling parameter Di among the four gauges, for solutions with all equation image = 0 and for the “preferred” fit where ∣equation image∣ was permitted to be as large as 0.3Davg. The total proportional misfit E (defined in equation (21)) and the maximum proportional misfit are relative to amplitudes of the observed tides. The maximum proportional and absolute misfits refer to the particular tide constituent and gauge least well matched by the coupling coefficients.

[57] On the basis of its low atmospheric pressure response coefficients (Figure 3), B081 is expected to have very little vertical coupling, yet both fits in Table 4 yield equation image2 less than half of the other equation imagei. Numerical experiments to test how closely the data could be fit with all of the equation imagei within a tighter range showed that the full range of equation imagei is needed. More specifically, all attempts at fitting the B081 tides yielded equation image2 < 0.1, whereas the other equation imagei were all larger than 0.2. Figure 11 is a graph of the atmospheric pressure coefficients versus the equation imagei. For B081, the variation among the equation imagei is not explained by variations in atmospheric pressure coefficients. In particular, CH2 does not have the highest atmospheric pressure coefficient.

Figure 11.

Atmospheric pressure coefficients versus apparent areal strain coupling coefficients, by instrument and gauge. Lines are fit to all four gauges for B027 and B073 and to the three gauges with the most negative equation imagei for B003 and B036.

[58] The calibration matrix for producing the formation strains from the gauge outputs, using the tidally calibrated coupling coefficients from the preferred fit based on equations (16) and (17), are listed in Table 5, together with the corresponding matrix for isotropic coupling with c = 2C = 1.5 and d = 2D = 3.

Table 5. Matrix for Combining Gauge Elongations of Strainmeter B081 Into Formation Strains in a Coordinate System With the x Axis Parallel to CH1
Strain ComponentB081 “Preferred” FitIsotropic Coupling, gi = 1, 2C = c = 1.5, 2D = d = 3
equation image−0.0360.493−0.178−0.309−0.0740.370−0.074−0.222
equation image−0.3200.2130.6500.202−0.3840.0000.3840.000

[59] B081 recorded a small coseismic strain step at the time of the M5.5 Chino Hills earthquake on 29 July 2008, 100 km from B081. Figure 12 shows time series of strain inferred from the B081 gauge data using the two coupling matrices shown in Table 5. The time histories are quite similar for the two sets of coupling coefficients. This similarity suggests that the effect on the time series of using equal gauge weightings may be unimportant, as long as the areal and shear coupling coefficients are adjusted to agree with the tides. Figure S1 in the auxiliary material compares the strain field computed from source parameters of the Chino Hills earthquake with the observed strain steps. The ɛEE − ɛNN strains at B081, inferred using either isotropic or tidally calibrated coupling coefficients, agree well with the calculation, but the other calculated strain components disagree in sign with the observations. The strain signals could be affected by local deformation triggered by dynamic strains.

Figure 12.

Time series for strainmeter B081 showing the coseismic step from the M5.5 Chino Hills earthquake on 29 July 2008. All strain time series were computed from gauge data that have had atmospheric pressure effects, tidally induced variations, and trends removed as described in section 2.1. (a) Time series computed using the “preferred” coupling coefficients based on tidal calibration (Table 4). (b) Time series computed assuming isotropic coupling with equal gauge weights, c = 2C = 1.5, and d = 2D = 3.

[60] The same fitting procedure was followed for station B084 (Pinyon), and the results are tabulated in Table 4. Davg for B084 (1.0) is comparable to that of B081 (1.2), and the equation imagei have a comparable spread and a similar range of values. B081 and B084 are the only two strainmeters considered in this paper for which the vertical coupling is low enough that the equation imageiCi, revealing intrinsic variability of the Ci.

5.3. B073 in Central California: High Vertical Coupling, Verified Tidal Model, and Good Fit of Observed and Theoretical Tides

[61] Station B073 (Varian) is an interesting example because its vertical coupling is so strong that the areal strain M2 and O1 phases are within 4° of being exactly opposite in phase from the theoretical areal strain tides (Figures 5a, 5b, and 6). As discussed in section 4.2, this is consistent with high vertical coupling causing the equation imagei to become negative. The polar plots in Figure 9b also indicate that a negative equation imagei is needed for every gauge, and the same result was obtained in numerically fitting the observations. The coefficients in Table 4 show that with equation image = 0, all tides of CH1 and CH2 are misfit by more than 10%. The overall fit improves for ∣equation image/Di∣ up to 0.45, considerably larger than for B081 or B084. Including the equation image coefficients changes Davg by about 10%, but has little effect on the equation imagei, which are all negative. Figure 11 shows that for B073, the atmospheric pressure coefficients decrease in amplitude linearly with increasing equation imagei. The slope of this relation would correspond to A = 4.0 nstrain/hPa, where A is the compliance relating vertical strain to surface pressure.

[62] Figure 13 shows data from B073 that includes a strain change caused by a creep event on the San Andreas fault on 17 August 2008. In the time series obtained using the isotropic coupling, the creep event appears to produce contractional areal strain. However, when the tidal calibration is used, the areal strain changes sign because all the equation imagei are negative. Figure S2 shows plots of the strain field calculated from a simple model of the creep event. Although a detailed analysis of this event is outside the scope of this paper, plausible simple models of the event indicate that the areal strain change at B073 was positive, consistent with the “preferred” tidal calibration.

Figure 13.

Time series for strainmeter B073 showing response to a creep event on the San Andreas fault on 17 August 2008. All strain time series were computed from gauge data that have had atmospheric pressure effects, tidally induced variations, and trends removed as described in section 2.1. (a) Time series computed using the “preferred” coupling coefficients based on tidal calibration. (b) Time series computed using the isotropic coupling model with equal gauge weights, c = 2C = 1.5, and 2d = D = 3.

[63] Of the 12 strainmeters analyzed in this paper, B081, B084, and B073 can be considered “reference strainmeters” for which the theoretical tides have been independently verified. The range of coupling coefficients needed to reconcile the observed and theoretical tides for these three instruments provide guidelines for evaluating the coupling coefficients obtained for other strainmeters. The degree of misfit between the observed and theoretical tides for other strainmeters should also be comparable to that found for these three “reference” strainmeters, if the theoretical tides are correct.

5.4. Colocated Strainmeter Pairs in Oregon: Differences in Vertical Coupling Between Closely Spaced Strainmeters

[64] Two colocated pairs of strainmeters in Oregon (B027 and B028; B035 and B036) are no closer to the ocean than B081 or B084, and the coastline is relatively uncomplicated. So it is plausible that the theoretical tides are also correct for these sites. B027 and B028 are 400 m apart, and B035 and B036 are 450 m apart. For practical purposes, the theoretical tides are the same for both colocated strainmeters of each pair.

[65] B027 has much higher atmospheric pressure response coefficients than B028, indicating that B027 has greater vertical coupling. Figures 5a, 5b, and 6 show that the areal strain phases for B027 are within a few degrees of the theoretical phases ±180°, in contrast to B028, for which no ±180° correction is needed. This implies that, when isotropic coupling is assumed, these two colocated strainmeters will appear to record areal strain tides of opposite sign.

[66] Fitting the observed and theoretical tides requires orientation corrections of −30.1° at B027 and 31.1° at B028, which are consistent with the shear strain phases shown in Figure 8. The misfits between observed and theoretical tides for B027 and B028 are larger than for the three reference strainmeters. Allowing nonzero equation image reduces the overall proportional misfit, E, by only 10% for both of these sites. CH3 of B028 is particularly poorly fit, but it is also the smallest constituent (Table 2). The estimated coupling coefficients are similar to those of the California strainmeters, with Davg = 1.0 for B027 and 1.1 for B028.

[67] The apparent areal coupling coefficients, equation imagei, are all negative for B027. They are linearly correlated with the atmospheric pressure response coefficients; the slope corresponds to A = 3.4 nstrain/hPa, similar to B073 (Figure 11). For B028, three of the equation imagei are positive, but equation image3 is very small and negative. The areal strain response of B028 CH3 is almost completely canceled out by vertical coupling, suggesting this gauge should be omitted when combining gauge outputs to obtain formation areal strain. The equation imagei for B028 do not correlate as closely with the atmospheric pressure response as do those for B073 or B027, but they spread no more than, and are comparable in value to, those at B081. B027 and B028 demonstrate that first-order differences in the areal tide phases between two colocated strainmeters can be attributed to different degrees of vertical coupling, and first-order differences among the shear strain phases can be explained by orientation corrections of tens of degrees. Overall, the fits to the theoretical tides at this location are achieved with coupling coefficients similar to those of the reference strainmeters. The only indication that the theoretical tides may not be correct is that misfits are larger than for the reference strainmeters.

[68] B036 has higher atmospheric pressure response coefficients than colocated B035, indicating higher vertical coupling at B036. High vertical coupling would also explain why areal tide phases for B036 are almost 180° different from the theoretical phases (Figures 5a and 5b). If ±180° is added to the B036 phases, then both B035 and B036 have areal strain phases within about 10° of the theoretical phases, but about 15° different from each other. Neither B035 nor B036 match the theoretical tides well with equation image = 0, but good fits were obtained when this parameter was allowed to vary. The value of Davg = 2.4 at B035 is higher than for any of the other sites, including B036 (Davg = 1.6). Compared to the strainmeters previously discussed, the equation imagei for B035 scatter more and correlate less well with the atmospheric pressure response coefficients (Figure 11). However, the equation imagei for B036 do correlate with atmospheric pressure and the slope of the relation corresponds to A = 4.1 nstrain/hPa, similar to B073 and B027.

[69] At this writing, no signals unequivocally of tectonic origin have been recorded at B027, B028, B035, or B036, so there is no good example of the effect of tidal calibration on the time series.

5.5. Northern Washington: Difficulty Matching Theoretical Tides and Possible Pore Pressure Effects

[70] The five PBO sites B003, B004, B005, B007, and B018 are of special interest because they have recorded strain transients associated with the episodic tremor and slip (“ETS”) episodes that appear to originate on the Cascadia subduction thrust at depths of 35 to 40 km [Dragert et al., 2001; Melbourne et al., 2005]. At these sites, the theoretical tidal strains are dominated by load tides from the Pacific Ocean and the Straits of Georgia and Juan de Fuca (Table 3). Comparison of the strainmeter tides with the theoretical tides is further complicated by the finding (section 4.1) that B003, B004, B007, and B018 have areal strain tide phases further from self-consistency than any of the other eight strainmeters (Figures 5a and 5b). On the other hand, B004, B005, B007, and B018 have equation imageequation image phases that are just as self-consistent as the other strainmeters (Figure 7), and the azimuths for which the different shear strain phases best agree with the theoretical tides are fairly closely grouped (Figure 8).

[71] It is shown in section 4.1 that adjusting gauge weights can bring the areal strain phases at B004 and B007 no closer to the theoretical phases than 14°. In other words, for B004 and B007, the shear strains are self-consistent and agree in phase with the theoretical shear strains, but the areal strains measured by the instruments lag behind the theoretical areal strains. Moreover, neither areal strain phase at B007 agrees with the corresponding phase at colocated B005; this disagreement is unrelated to the theoretical tides. It appears that at B004 and B007, the time series from all the gauges include a term that cancels out when the gauge outputs are differenced to form shear strains. This signal might be caused by tidal variations in formation pore pressure that lag the areal strain tide.

[72] B003 and B004 can both be reconciled with the SPOTL tides, with the fits being much better with nonzero equation image, and parameter ranges generally close to those of other strainmeters (Table 4). The orientation correction (28.8°) at B003 is consistent with Figure 8. For B004, as at B035, the four equation imagei are not correlated with atmospheric pressure coefficients and they span a greater range than at other sites (Figure 11). At B003, three of the four equation imagei lie near the linear variation with atmospheric pressure estimated for B073 and B028.

[73] For the colocated pair B005 and B007 (the PBO “Shores cluster”), the observed and modeled tides cannot be made to agree as closely as at other sites. It was not possible to obtain a good fit even with widely varying Di and ∣equation image/Di∣ = 0.4 or more (Table 4). The “best” fits included orientation corrections that appear unnecessary in the context of Figure 8, and still had proportional misfits of 30% and/or absolute misfits exceeding 2 nstrain for some constituents. The largest misfits are on B005 CH2 and CH3, and on B007 CH0, which is closely aligned with B005 CH2. These results suggest that the tides experienced by B005 and B007 are not the same as the theoretical tides. A probable reason why B005 and B007, as well as B018, cannot be reconciled with the theoretical tides is their proximity to Puget Sound and its associated tidal waterways, which are not included in the regional tidal model.

6. Concluding Discussion

[74] The coupling formulation proposed in equation (6), which allows each gauge to be coupled to all components of the horizontal formation strain tensor, as well as to vertical formation strain, successfully reconciles the model and observed tides at nine of the twelve strainmeters considered. Key features of the coupling coefficients are as follows: (1) the shear coupling coefficients vary 20–30% among the gauges, and range from 0.8 to 2.5, consistent with the range of shear strain response factors calculated by Gladwin and Hart [1985] based on isotropic axisymmetric models of the strainmeter, grout, and borehole; (2) the apparent areal strain coupling coefficients for many of the strainmeters are reduced, or even changed in sign, by vertical coupling; and (3) coupling of each gauge to engineering shear strain, equation image, is helpful in reconciling the observed and model tides.

[75] The fourth gauge of each PBO strainmeter enables self-consistency checks to be performed, which provide qualitative information about the strainmeter coupling and orientation before actually determining the coupling coefficients. Analysis of “reference” strainmeters, in locations where there is independent information about the tidal model, provides information about how closely the observed and theoretical tides should match, as well as on the range of coupling coefficients to be expected. Graphical representations of the tides are also helpful in judging the appropriateness of the tidal model. Two features of the strainmeter coupling are discussed in more detail below.

6.1. Evidence for Vertical Coupling

[76] The nearly 180° difference between observed and theoretical areal strain tide phases at strainmeters with high atmospheric pressure response coefficients can be explained if the gauges in the PBO borehole strainmeters change length in response to vertical formation strain, despite being designed to respond only to horizontal deformation. Vertical coupling is undesirable because it (1) degrades the response to areal strains of tectonic origin, (2) enhances the response to surface loading, such as that caused by snow and rain, and (3) requires that different calibration matrices be used for tidal and tectonic strains, as opposed to strains caused by surface loads.

[77] The reason why the PBO GTSMs have strong vertical coupling is unknown, but should be determined. Three-component GTSMs that have been monitored in California for 20 years or more, including two within a few kilometers of B073, have little or no response to atmospheric pressure (E. Roeloffs et al., unpublished report, 2004). It may be relevant that, for the strainmeters for which the apparent areal strain coupling coefficients are linearly related to the atmospheric pressure response coefficients, the inferred ratio of vertical strain to surface pressure, A, ranges from 3.4 to 4.1 nstrain/hPa, almost an order of magnitude larger than the value of 0.42 nstrain/hPa obtained by Beavan et al. [1991]. Each PBO strainmeter gauge is housed in a cylindrical steel chamber 0.38 m long, which is grouted to the formation along its length and separated from adjacent gauges by brass spacers of slightly smaller diameter. Coupling to vertical strain could be through the grout-strainmeter bond, or, alternatively, individual gauges could be deformed vertically between the spacers. The degree of vertical coupling does not appear to be related to the type of grout used or the depth of installation (Table 1), and the existence of colocated pairs that differ in the degree of vertical coupling rules out local geology as the critical factor.

6.2. Possible Influence of Pore Fluid Pressure

[78] Phase differences between the observed and theoretical areal strain phases for three strainmeters in northern Washington may be related to formation pore pressure. The possibility that pore pressure could interfere with tidal calibration of borehole strainmeters was raised by Segall et al. [2003]. In a homogeneous, isotropic porous elastic medium, if no fluid flow can take place (“undrained” conditions), pore pressure is expected to vary in proportion to the areal strain tide. Although the presence of the pore pressure will modify the forces applied to the strainmeter, the only effect on the analysis described in this paper would be to change the values of equation imagei.

[79] It is common for formation fluid pressure tides to differ in phase from the areal strain tides if fluid flow can occur on the diurnal and semidiurnal time scales of the tides [see, e.g., Roeloffs, 1996]. Assuming the fluid pressure acts equally on all gauges, the net result would be to change the amplitudes and phases of the areal strain tides experienced by the strainmeter. The shear strain tides would be much less affected.

[80] Pore pressure influence on the strainmeter could induce frequency dependence in the areal strain coupling coefficients, because more complete pressure equilibration would occur during transients lasting weeks, such as the Cascadia slow slip events, than on the time scale of the tidal calibration signal. Further work on this topic is warranted, especially as it pertains to the coupling coefficients of the northern Washington strainmeters.

Appendix A:: Relation to Previously Published Strainmeter Coupling Formulations

[81] In a strainmeter that is isotropically coupled, each gauge's response can be described with two coupling coefficients that are the same for all gauges (equation (13)). Gladwin and Hart [1985] introduced the expression for the elongation of a single gauge in response to formation strain under homogeneous, isotropic conditions (equation (9) of Gladwin and Hart) and presented expressions for the areal and shear response factors for that situation (equations (10) and (11) of Gladwin and Hart). The first multicomponent borehole strainmeters, installed in the early 1980s, were three-component GTSMs, and they were assumed to be isotropically coupled (equation (13)). Tidal calibration was used to estimate the areal and shear response factors.

[82] The equations relating elongations to strains for isotropic coupling are easily expressed in matrix form (equation (15)) as the product of a diagonal matrix whose entries are the two coupling coefficients, a geometry matrix describing how the gauges are arranged in the strainmeter, and, if needed, an orientation matrix that relates the coordinate system used for the strains (e.g., E-N, or fault-perpendicular and -parallel) to the orientation of the strainmeter (generally, to one of its gauges). Equation (15) formed the original conceptual basis for understanding strainmeter calibration, and led to the concept of “instrument strain”:

equation image
equation image
equation image

If coupling is really isotropic, then the gauges can be thought of as making n measurements across a single circular cross section of the strainmeter-grout inclusion as it deforms into an ellipse, and the instrument strain is the strain tensor that describes that ellipse. Equation (15) can then be written as

equation image

The coupling coefficients appear in a “coupling matrix” relating instrument strain to formation strain. (The term “far-field strain” was used by Gladwin and Hart [1985] in the same way that “formation strain” is used in this paper.)

[83] As described by Hart et al. [1996], by the mid-1990s it was recognized that isotropic coupling did not always reconcile the observed and theoretical tides, especially the shear strain tides, to within the expected errors. They addressed this issue in two ways. First, they postulated that “local” strain (formation strain in the immediate vicinity of the strainmeter) could be perturbed by local geology and/or topography not included in theoretical tides. The theoretical tides would represent “remote” strain, representative of a larger spatial scale. Second, retaining the conceptual view that strainmeter calibration is equivalent to finding a matrix relating instrument strains and (local) formation strains, they sought more general matrices to take the place of the diagonal matrix of equation (A2). To distinguish the two effects, they compared theoretical tides with strain tides observed by a long-baseline laser strainmeter (LSM) at Piñon Flat Observatory (PFO) in southern California, which showed that there were small, but significant, differences between the theoretical and observed tides. However, the isotropic coupling matrix was still unsatisfactory in reconciling the LSM tides with the tides observed by a three-component GTSM at PFO.

[84] The approach used by Hart et al. [1996] to arrive at a more general coupling formulation does not include the possibility of vertical coupling. A further difference from the approach used here is that, instead of first estimating the behavior of individual gauges, Hart et al. [1996] solve equation (17) directly for the entries of the calibration matrix S. Each row of S consists of the n calibration parameters that multiply the respective elongations of the n strainmeter gauges in the linear combination that forms one of the three components of the formation strain tensor. If the M2 and O1 tide constituents are used as reference strains, their amplitudes and phases provide four independent “knowns.” Therefore S can be found directly for the three-component borehole strainmeter at PFO, as well as for the four-component PBO borehole strainmeters.

[85] There are two drawbacks to the approach of directly solving for S. First, unlike the approach presented in this paper, it cannot be extended to n > 4 gauges if tides are used as the reference strains: there are always only four knowns (amplitude and phase for each of two tide constituents), but there are n unknown coefficients. Second, the physical meanings of the entries of S are less clear than the coupling coefficients of individual gauges, which makes it difficult to discriminate between an incorrect tidal model and a strainmeter for which coupling is strongly nonisotropic.

[86] Hart et al. [1996] investigate the nature of the strainmeter coupling by inverting S (which is square for a three-component strainmeter) to obtain C, which they refer to as the “transducer coupling matrix” and which plays exactly the same role as C in equation (16b) of this paper. Assuming there is no vertical coupling, the method of this paper and the method of Hart et al. [1996] should yield essentially the same matrix C for a three-component strainmeter. If all the equation image vanish, equation (16a) reduces to equation (A10) of Hart et al. [1996], which gives the expected form of C for variation of coupling along the borehole (noting that ci and di of their equation (A10) are actually “response factors,” and equal 2Ci and 2Di, respectively, of this paper). Equation (16a) could also be written in the form of equation (A11) of Hart et al. [1996], for coupling of a strainmeter to anisotropic rock, although equation (16a) can describe a more general situation.

[87] The explicit expressions for the entries of the matrix C (equation (16a)) show that factoring out orientation and geometry matrices to isolate an intrinsic instrument-strain-to-formation-strain coupling matrix (K of Hart et al. [1996]) is not a natural decomposition, especially if the equation image are nonzero (although such a factorization could be performed numerically). For this reason, the concept of “instrument strain” is ill-defined for nonisotropic coupling. If the responses of all of the gauges are different, then they cannot be viewed as measuring the same cross section at different azimuths; in fact, the differing responses among the gauges imply that the deformed cross sections at the depth of each gauge are not the same. Therefore, “instrument strains” obtained, for example, from equations (A1a), (A1b), and (A1c) may not actually constitute a tensor. A practical implication is that strainmeter gauge elongations should always be converted to formation strain before applying coordinate rotations.

Appendix B:: Self-Consistency Relations for the General Coupling Formulation

[88] If equation (6) is used to describe the coupling of each gauge to formation strain, the gauge combinations that would give the areal strain for an isotropically coupled strainmeter (equations (18a) and (18b)) become

equation image
equation image

Equations (B1a) and (B1b) become simpler if the equation image are neglected, which is reasonable because these coefficients are expected to be small and to possibly differ in sign among the gauges, thereby canceling out in expressions for areal strain. Equations (B1a) and (B1b) reduce to

equation image
equation image

The quantities in brackets in equations (B2a) and (B2b) are all scalars, which may be positive or negative, depending on the degree of vertical coupling. This implies that when applied to Earth tide constituents, if the gauge weights are correct, both equations should give phases that are either the same, or differ by ±180°, even if the other coefficients differ among the gauges. Also, if the gauge weights are correct, the ratios of the amplitudes of these two quantities should be the same for both the M2 and O1 tidal constituents. The two preceding statements, which will be referred to as the “self-consistency checks for areal strain,” should hold regardless of the theoretical tides. If, in addition, the theoretical tides are correct, then their phases should either agree with, or differ by ±180° from, the phases obtained from equations (B2a) and (B2b).

[89] To form the gauge combinations of equations (20a) and (20b), which give the differential extension in gauge 1 parallel coordinates if the strainmeter is isotropically coupled, the shear strain components all need to be rotated to CH1-parallel coordinates using equation (11). Then equations (20a) and (20b) generalize to

equation image
equation image

The respective expressions for the differential extensions differ from their isotropic counterparts (equations (20a) and (20b)) by terms proportional to the areal strain and to the engineering shear, both of which differ in phase from the differential extension. If the equation image and the differences among the Ci and Vi can be neglected, those terms vanish and equations (B3a) and (B3b) and (20a) and (20b) become identical. Under those conditions, the two equations (B3a) and (B3b) should yield the same phase, which would be the phase of (equation imageequation image) experienced by the strainmeter, regardless of the theoretical tides.


compliance for vertical strain from surface load (equation (8a)).


ratio of horizontal strain to vertical strain induced by atmospheric pressure (equation (8b)).


areal strain response factor of Gladwin and Hart [1985]; c = 2C.


dimensionless areal strain coupling coefficient common to all gauges (equations (4b) and (13)).


dimensionless areal strain coupling coefficient for the ith gauge (equation (3b)).

equation image

apparent areal strain coupling coefficient for ith gauge, CiViν/(1 − ν) (equation (7)).


matrix by which strains are multiplied to obtain gauge elongations (equations (16a) and (16b)).


areal strain response factor of Gladwin and Hart [1985]. d = 2D.


dimensionless shear strain coupling coefficient for all gauges (equations (4b) and (13)).


dimensionless coupling coefficient of ith gauge to equation imageequation image (equation (3b)).


average of the four Di.


fractional elongation of the ith gauge (section 3.1).


(full size) overall proportional error between observed and theoretical tides (equation (21)).


function of time.

g0, g1, g2, g3

relative gauge weights (dimensionless) (equations (4a) and (4b)).

equation image

dimensionless coupling coefficient of ith gauge to equation image (equation (2)).

equation image

dimensionless coupling coefficient of ith gauge to equation image (equation (2)).

equation image

dimensionless coupling coefficient of ith gauge to equation image (equation (2)).


identity matrix.


coefficient of cosine term in kth tidal constituent (equation (1a)).


amplitude of kth tidal constituent (equation (1b)).


coefficient of sine term in kth tidal constituent (equation (1a)).


matrix by which elongations are multiplied to obtain strains (equation (17)).




dimensionless coupling coefficient of ith gauge to ɛzzF (equation (2)).

x, y, z

spatial coordinates of a right-handed Cartesian coordinate system in which x and y are horizontal and z is vertical, with the positive direction upward.


Cartesian coordinate direction parallel to ith gauge (section 3.1).


Cartesian coordinate direction perpendicular to ith gauge (section 3.1).


change in atmospheric pressure (section 3.2).


strain; individual components are defined in section 3.1.


formation areal strain; ɛaF = ɛxxF + ɛyyF in any Cartesian coordinates x, y.


orientation or phase angle.


Poisson ratio.


vertical stress.


orientation or phase angle.


frequency of kth tidal constituent, radians per unit time.




pertaining to the formation.




[90] I acknowledge EarthScope and its sponsor, the National Science Foundation, for providing data products used in this study. I also thank the UNAVCO PBO borehole strainmeter team for their thorough and prompt answers to numerous questions. Helpful discussions with Duncan Agnew and Michael Gladwin are gratefully acknowledged. Detailed reviews by John Beavan, Nick Beeler, Kathleen Hodgkinson, John Langbein, Michael Lisowski, Wendy McCausland, and an anonymous reviewer provided invaluable guidance for improving the manuscript. Some figures were prepared using the gnuplot (copyright, Free Software Foundation) and GMT [Wessel and Smith, 1998] software packages. This work was supported by the National Earthquake Hazards Reduction Program.