Effect of error in theoretical Earth tide on calibration of borehole strainmeters



[1] Since the installation of borehole strainmeters into the ground locally distorts the strain in the rock, these strainmeters require calibration from a known source which typically is the Earth tide. Consequently, the accuracy of the observed strain changes from borehole strainmeters depends upon the calibration derived from modeling the Earth tide. Previous work from the mid-1970s, which is replicated here, demonstrate that the theoretical tide can differ by 30% from the tide observed at surface-mounted, long-baseline strainmeters. In spite of possible inaccurate tidal models, many of the 74 borehole strainmeters installed since 2005 can be “calibrated”. However, inaccurate tidal models affect the amplitude and phase of observed transient strain changes which needs to be considered along with the precision of the data from the inherent drift of these borehole instruments. In particular, the error from inaccurate tidal model dominates the error budget in the observation of impulsive, sub-daily, strain-transients.

1. Introduction

[2] The Plate Boundary Observatory (PBO) has installed 74 strainmeters, in the western US, into 15.2 cm diameter boreholes between 100 to 200 meters below the Earth's surface. The objective of these measurements is to record small strain changes due to tectonic deformation, primarily from fault slip, and volcanic deformation resulting from intrusion. Each strainmeter consists of a vertical stack of four horizontal extensometers, each of 20 cm length within a stainless steel enclosure with three of the extensometers oriented at 60° intervals, and the fourth oriented at 90° from the second gauge. The combination of the borehole, the strainmeter, and the expansive grout, used to cement the strainmeter to the borehole, changes the strain that is in the rock to the strain that is observed by the instrument. Gladwin and Hart [1985] provide the theoretical background and guidance for the parameters that describe the coupling of the strainmeter to the surrounding rock. However, the parameters of the coupling model should be estimated by measuring the response at each extensometer from a known source of strain. One signal, which is typically and easily observed by the borehole strainmeters, is the Earth tide and this has been used by Hart et al. [1996] for one instrument and more recently by Roeloffs [2010] for a subset of PBO strainmeters. The strain from the Earth tide can be either measured independently of the borehole strainmeter, as was done by Hart et al. [1996], or the Earth tide can be predicted using one of the software packages currently available [e.g., Agnew, 1996, 1997], as was done by both Hart et al. [1996] and Roeloffs [2010]. With the exception of two PBO strainmeters, direct, independent measurement of the Earth tide at the strainmeter sites is not available. Consequently, any calibration or estimates of the parameters used to translate the observed strain into the strain in the host rock are dependent on the prediction of the Earth tide unless another well-known source of deformation is found and used.

[3] One problem in assessing the validity of the coupling model is that the theoretical tide is not known to the actual precision of the data [e.g., Beaumont and Berger, 1975]. This is replicated here using more recent data from long-baseline strainmeters (LBSM). Strain measured using the LBSMs, unlike the borehole strainmeters, is not modified by construction or installation of the strainmeter. In both the previous work and the analysis presented here, the theoretical models of the Earth tide can differ from the direct observation of the Earth tide at the surface using LBSMs. The differences can be as much as 3.5 nanostrain (0.0035 parts per million), which is a significant fraction of the 5 to 20 ns amplitude of the observed strain and more than the 0.1 ns nominal, observational error of tidal strain. Consequently, the evaluation of the quality of the calibration of the borehole strainmeter should incorporate the fact that the theoretical Earth tide is known, but not precisely known. As Berger and Beaumont [1976] demonstrate in a companion paper, the theoretical tide not only depends upon the deep structure of the Earth and the model of the ocean load both near and far from the coast, but also depends on nearby topographic and geologic effects.

[4] A rough calculation of the effect that the accuracy of the tidal model has upon the resolved strain change is as follows: If the transient strain observed from fault slip is 50 ns [Johnston and Linde, 2002; Roeloffs, 2010], then the error due to only knowing the tidal model at an accuracy of 10 to 20% level is of the order of 5 to 10 ns. On the other hand, the precision of borehole strainmeters is typically characterized by a random-walk process, for which the drift is scaled by the square root of time with a rough estimate of 100 ns/yr0.5 [Johnston and Linde, 2002]. Consequently, if the observed strain change occurred over a 10-day interval, the precision of the observed strain is 16 ns, or equivalent to the uncertainty of the tidal calibration. On the other hand, if the observed strain change occurred over a 1-hour interval, the precision of the observed strain is 1 ns, or much less than the uncertainty of the tidal calibration. Thus, with possible 50 ns signals, the accuracy of the tidal model dominates the error budget in the measurement of strain transients should they occur over intervals of less than 1-day. But, for transients over longer intervals, the background drift of the strainmeter dominates the measurement error.

[5] Below, I replicate the comparison of observed and model tidal strains done 35 years ago by Beaumont and Berger [1975] using more recent models and observations from long-baseline laser strainmeters (LSMs). I use the differences between the observed and modeled tidal strains as a measure of the accuracy of the tidal model that is used to calibrate the PBO borehole strainmeters. I show two examples of transient strain events to illustrate how the accuracy of the Earth tides affects the strain in the rock estimated from PBO strainmeter data and compare that accuracy with the precision of the observations.

2. Accuracy of Theoretical Tides

[6] Following the analysis by Beaumont and Berger [1975], I compare the theoretical tides with the observed tides from five different LSMs [Berger and Lovberg, 1970; Wyatt, 1989] and find significant differences (Figure 1). Currently, there are five locations in California that have at least one component of a LSM. The LSMs typically have baseline lengths in excess of a few hundred meters and each end-mount is optically anchored to several 10s of meter depth [Wyatt, 1989]. At one site, Piñon Flat, there are three components oriented at 0, 45, and 90 degrees. At three sites there are two perpendicular components. And at the fifth site there is only a single component. For each of 10 components of these strainmeters, I have estimated the observed O1 and M2 components of the tides and compared them with the predicted values of O1 and M2 using the SPOTL program [Agnew, 1996, 1997]. The differences between the observed and predicted tides were computed by subtracting their in-phase and quadrature components. The magnitude of the differences are plotted as a function of the magnitude of the observed strain (Figure 1a). In addition, at four sites the areal strain and the shear strains are compared (Figure 1b), where the areal strain is simply the addition of the two orthogonal components. At one of these four sites, Piñon Flat, there are three extensometers, so that two components of shear strain are determined. For the other three sites, consisting of only two perpendicular extensometers, only a single shear-strain component is determined from the difference between the two orthogonal extensometers.

Figure 1.

Difference between the modeled Earth tide (O1 and M2 constituents) and the observed Earth tide at five LSMs located in central and southern California. (a) The difference between the modeled and the observed tides for the 10 extensometers, which comprise the 5 LSMs, are plotted against the observed amplitudes of the tides for each extensometer. (b) The extensional strains are converted to areal strain, by summing perpendicular components; and to shear strain by taking differences of the perpendicular components at 4 of the 5 LSMs. At one of the four, Piñon Flat, a second shear strain can be determined since that site consists of three extensometers.

[7] In detail, each of the 10 components of each strainmeter was analyzed separately. Data from each site were analyzed for a 163-day period, starting 6 September 2008, and decimated to hourly samples (ftp://ncedc.org/pub/pbo/strain/processed/lsm). Using the strainmeter analysis software of Langbein [2010], the in-phase and quadrature components of 16 tidal frequencies, including O1 and M2, were estimated. Then, for the same period, the SPOTL program was used to create a time series of extensional, areal, and shear strains for the location of the strainmeter. The same tidal analysis that was done for the LSM data was repeated for the tidal time series. For the extensional components, these differences are shown in Figure 1a, while the areal and shear strains are shown in Figure 1b.

[8] For both the extensometer tides and the tensor component tides, the maximum tidal amplitude is 20 ns (for the areal M2 tide), but the differences between the observed and predicted tides range up to 5 ns. On the other hand, the standard error of the estimates of the observed strain is less than 0.1 ns, based upon the power-law nature of the background noise of these data [Agnew, 1986; Johnston and Linde, 2002]. In nearly all comparisons in Figure 1, the differences are a significant fraction, from 10 to 30%, of the observed tide. Thus, any tidal calibration of the borehole strainmeters could be systematically biased due to inadequate tidal models.

3. Calibration of Borehole Strainmeters

[9] Previous reports by Hart et al. [1996], and Roeloffs [2010] discuss the details of using the Earth tide to calibrate the borehole strainmeter. For this report, I have chosen the method described by Roeloffs [2010], in particular, her preferred methodology, to calibrate two PBO strainmeters, B004 and B073. B004 is located on the Olympic Peninsula, in Washington state, and detected transient strain event from the Cascadia subduction zone in May 2008. B073 is located near Parkfield, California, and detected a transient strain event due to a creep event on the San Andreas fault. Both of these events are discussed by Roeloffs [2010, also written communication, 2009], and I have reproduced those results in Figure 2.

Figure 2.

Transient strain changes for B004 and B073 for two different aseismic events. (a) The strain changes observed at B004 from April to July 2008, spans an aseismic slip event on the Cascadia subduction zone. The solid black trace replicates the strain changes estimated by E. A. Roeloffs (written communication, 2009) for this instrument. The theoretical tides from SPOTL were used to calibrate this strainmeter and to estimate the data in the black trace. Each gray trace is from the strain changes deduced from a separate calibration created by dithering the tides about those generated by SPOTL. In addition, the results have been high-passed filtered by removing a 30-day running average. (b) The strain changes observed at B073 from July though September 2008 spans a creep event in mid-August 2008. The arrow indicates the time of the creep event which was recorded at nearby creepmeters and other strainmeters. The cause of the other offset in mid-July is unknown.

4. Alternative Models of the Earth Tide

[10] To examine the impact of possible imprecision of the Earth tide models on the calibration of borehole PBO strainmeters and the interpretation of recorded strain transients, I have chosen to simulate possible Earth tide models that differ from the ones generated by SPOTL by amounts consistent with the differences shown in Figure 1. The results of these simulations are shown in gray in Figure 2 for B004 and B073.

[11] Specifically, a perturbed tidal model for three extensional components oriented at N0° E, N45° E, and N90°E were generated from the predicted tides at those azimuths. Each perturbed model was constructed by taking the tides predicted by SPOTL and adding a vector drawn from a uniform distribution, whose amplitude ranges between 0 and 3.5 ns for M2 and 0 and 3.0 ns for O1, and whose phase ranges between 0° and 360°. The three perturbed extensional components, were then transformed into areal and two shear strain components. A calibration matrix was estimated by fitting the observed tidal strains from the borehole strainmeter to the perturbed model of the tides using the method of Roeloffs [2010]. If the misfit between the observed tides and the perturbed tides was less than 2 ns, that perturbed model of the tides was deemed acceptable and the calibration matrix was applied to the borehole strainmeter data that spanned the time of each respective strain transient. A gray line for an acceptable estimate of the transient was added to the plots in either Figures 2a or 2b. The above simulation was repeated 1000 times, of which 15% had successful calibrations, resulting in a suite of different calibrations and corresponding estimates of the transient strain.

[12] For both transients, the simulations do reproduce the occurrence of these two events: the variations in May 2008 for B004 and the offset in mid-August 2008 for B073. However, the range of amplitudes of the simulations can be as large as the original transient itself.

[13] For B073, nearly all of the simulations of tidal calibrations reproduce the offset in mid-August 2008. But, the amplitude of the offset, when compared to the offset deduced from the original calibration, is highly variable. As an example, the original calibration indicates a 70 ns offset in areal strain. Yet, the simulations suggest that estimated offset could range from 30 to 140 ns. Using a rule of thumb that 25% of the range corresponds to one standard deviation, the variation between 30 to 140 ns suggests that the error due to the tidal model is 30 ns. On the other hand, examination of the typical drift is 2 ns over the one-hour duration of the creep event. This estimate of the repeatability, or precision, is based upon the background power spectrum of the data and its relation to drift [Agnew, 1992; Langbein, 2010]. Similarly, the comparisons between instrumental precision from background drift and possible error in the tidal model are made for the two shear components and these are listed in Table 1. The results show that the error from the tidal model is roughly an order of magnitude larger than the error from drift.

Table 1. Uncertainty due to Tidal Model and Precision of the Data
Standard Error 1-Hour Interval (ns)Error Using 25% Range From Simulationsa (ns)Standard Error 10-Day Interval (ns)Error Using 25% Range From Simulationsa (ns)
  • a

    See text for explanation.

Areal Dilatation2306040
Shear 2εns0.5253010
Diff. Shear εee − εnn0.625208

[14] The transient for B004 develops over a period of two weeks starting in early May 2008. By vertically shifting the trace of each of the simulated time series of this event in Figure 2a so that it overlays the early May time-series deduced from the original calibration, it becomes possible to assess the amount of variation in transient that is due to error in the tidal model and variation due to the drift or wander [Agnew, 1992] in the data. Comparison of the error due to drift of the instrument with the error in the tidal model (Table 1) shows that these two errors are nearly equivalent.

5. Discussion and Conclusions

[15] The results presented here demonstrate that the predicted tides match the observed strain tides from surface-mounted, LSMs at about the 2 ns level, or roughly 10 to 30% of the observed strain tide. Since the Earth tide is often used to calibrate borehole strainmeters, mis-modeling of the Earth tide will contribute to the error in any transient strain that might be observed on these borehole instruments. This possible systematic error is in addition to the repeatability, or precision, of the actual borehole strain measurements. That precision is often characterized as a random-walk process, for which the error accumulates in proportion to the square-root of the interval in time. Consequently, as shown in the above examples, offsets from earthquakes or creep events measured by borehole strainmeters are well resolved and have high precision. For large offsets or abrupt strain changes, the 10 to 30% error in the tidal model used to calibrate the strainmeter can both bias the estimate of the offset and be larger than the error due to instrumental drift. On the other hand, for more emergent transient signals that occur over longer periods, the error due to the precision of the measurements becomes comparable to the size of the bias due to an incorrect calibration. However, for any transient, the contribution from the precision due to instrument drift and the additional bias due to incorrect calibration need to be evaluated on a case-by-case basis.

[16] In the case of the 2εen shear component of B073, the estimated offset in mid-August is 35 ns (Figure 2b) with an error of 0.5 ns (Table 1); this is a signal-to-noise ratio of 70 to 1. However, in contrast, the error in shear strain due to the model for calibration could be 25 ns (Table 1); this effectively reduces the ratio of signal to noise to an unimpressive 1.4. Consequently, for rapid transients, inaccurate tidal models will be the limiting factor in the measurement of these types of strain changes

[17] Unlike borehole strainmeters, the surface-mounted, long-baseline instruments should not be affected by any coupling of the strainmeter to the Earth. Consequently the observations of the Earth tide by the five LSMs reported here should be the standard against which to compare the modeled tides. The differences between the tides predicted by SPOTL and the 10 components of LSM data clearly show that the models have errors of up to 30% of the observed strains (Figure 1). The SPOTL software includes not only the tide for the solid Earth, but the effects from ocean loading. The tidal model for the solid Earth is based upon a radially stratified, elastic Earth structure estimated from seismic observations. In addition to the tidal distortion of the solid Earth directly from the Sun and Moon, the Earth also distorts from the load from the ocean tide. The key ingredient of the load from the ocean is a model of the ocean basins. The ocean models are often described in terms of spatial resolution which, for the TPXO7.0 model employed by SPOTL is 0.25°. In addition, SPOTL provides some local models at higher resolution; 0.1° for the Sea of Cortez and Strait of Georgia/Juan de Fuca. The model Sea of Cortez provides additional adjustment for three of the LSMs in Southern California, while the Juan de Fuca model provides some adjustment for B004. The absence of corrections for loading from tides in Puget Sound could affect B004.

[18] Berger and Beaumont [1976] demonstrate that improvements in the tidal model can be made to bring the predictions closer to the observed tides. They list three possible improvements to the Earth tide model at a particular location: 1) Improvement in the ocean-tide models; 2) Departure from a radially stratified Earth, particularly in the vicinity of the strainmeter; 3) Distortion of the strain field due to topography. For the sites discussed in their paper, Berger and Beaumont [1976] demonstrated that they could achieve significantly improved match between the modeled and observed tides using finite element techniques. The finite element method is used to estimate the distortion from the local geology, through variations in the material properties, and topography. For this study, I have not attempted to estimate the contributions from the local structure and topography; nor, have I tried to make additional adjustments to the ocean load with higher resolution models of ocean than provided by SPOTL.

[19] The sensitivity of the modeled tides to local structure can be illustrated with SPOTL by calculating the change in predicted strain tides to the position of a strainmeter that is located near the coast. For example, B004, which is 10 km from the coast shows a 1 ns/km in shear, 2εen in location of the site. Higher sensitivity is seen for sites near the entrance to San Francisco Bay.

[20] In summary, the tidal analysis of current, high-precision, surface strainmeter data and comparison to the predictions from current, state-of-the-art, tidal modeling software, suggest that using strain-tide predictions can lead to potential bias in calibrating borehole strainmeters leading to inaccurate estimates of transient strain events. Berger and Beaumont [1976] show that the accuracy of the predicted tides should improved for the PBO strainmeters by inclusion of both the local topography, for which models already exist, and geologic structure, which can be problematic to obtain.


[21] Laser strainmeters are operated by Frank Wyatt and Duncan Agnew of UCSD. Borehole strainmeter data were collected through PBO, administered by UNAVCO, and processed by Kathleen Hodgkinson (UNAVCO).