## 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 M_{2} and O_{1} 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 http://pboweb.unavco.org/dmsdocs/Root 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 M_{2} and O_{1} 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.