Tidal calibration of plate boundary observatory borehole strainmeters

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH: SOLID EARTH, VOL. 118, , doi:1.19/1jb9651, 1 Tidal calibration of plate boundary observatory borehole strainmeters Kathleen Hodgkinson, 1 John Langbein, Brent Henderson, 1 Dave Mencin, 1 and Adrian Borsa 1 Received July 1; revised 9 November 1; accepted 1 December 1; published 1 January 1. [1] The Plate Boundary Observatory, the geodetic component of the EarthScope program, includes 74 borehole strainmeters installed in the western United States and on Vancouver Island, Canada. In this study, we calibrate 45 of the instruments by comparing the observed M and O 1 Earth tides with those predicted using Earth tide models. For each strainmeter, we invert for a coupling matrix that relates the gauge measurements to the regional strain field assuming only that the measured strains are linear combinations of the regional areal and shear strains. We compare these matrices to those found when constraints are imposed which require the coupling coefficients to lie within expected ranges for this strainmeter design. Similar unconstrained and constrained coupling matrices suggest the instrument is functioning as expected as no other coupling matrix can be found that better reduces the misfit between observed and predicted tides when the inversion is unconstrained. Differences imply a coupling matrix with coefficients outside typical ranges gives a better fit between the observed and predicted tides. We find that of the strainmeters examined have coupling matrices for which there is little difference between the constrained and unconstrained inversions. If we allow a greater divergence in the shear coupling coefficients and consider the possibility that one gauge may not function as expected, the discrepancies between the unconstrained and constrained coupling matrices are resolved for a subset of the remaining strainmeters. Our results also indicate that most of the strainmeters are less sensitive to areal strain than expected from theory. Citation: Hodgkinson, K., J. Langbein, B. Henderson, D. Mencin, and A. Borsa (1), Tidal calibration of plate boundary observatory borehole strainmeters, J. Geophys. Res. Solid Earth, 118, , doi:1.19/1jb Introduction [] Between June 5 and October 8, UNAVCO installed 74 borehole strainmeters as part of the Plate Boundary Observatory (PBO), the geodetic component of the Earthscope program. Borehole strainmeters are designed to perform well at periods of minutes to weeks, and their purpose in the Observatory is to capture strain transients that fall between the detection envelopes of GPS and seismology [Silver et al., ]. With this objective, the strainmeters were deployed in arrays where it was thought such short-term strain transients might be occurring: the Cascadia Subduction Zone, the Mendocino Triple Junction, along the San Andreas and San Jacinto faults, in the Eastern California Shear Zone, at Mount St. Helens and at Yellowstone (Figure 1). The instruments are fulfilling this 1 UNAVCO, 65 Nautilus Drive, Boulder, Colorado, USA. U.S. Geological Survey, Menlo Park, California, USA. Institute of Geophysics and Planetary Physics, Scripps Institution of Oceanography, UC San Diego, MC 5, La Jolla, California, USA. Corresponding author: K. Hodgkinson, UNAVCO, 65 Nautilus Drive, Boulder, Colorado, USA. (hodgkinson@unavco.org) 1. American Geophysical Union. All Rights Reserved /1/1JB9651 expectation by recording multiple Episodic Tremor and Slip events in the Pacific Northwest [Wang et al., 8; Hawthorne and Rubin, 1; Dragert and Wang, 11], aseismic creep events in central California [Langbein, 1; Roeloffs, 1]; coseismic deformation [Barbour and Agnew, 11] and tsunami-related strain signals [Hodgkinson et al., 1]. [] Before strain data can be incorporated into any geophysical model that represents tectonic or volcanic processes an instrument response, or coupling matrix, that relates the measurements to the regional strain field is required. The aim of this study is to calibrate PBO strainmeters using the Earth tides as a reference signal. We first determine the coupling matrices using a completely general method where we assume only that the measurements are linear combinations of the regional areal and shear strains. We then compare these matrices with those determined when we impose constraints on the inversion requiring the coupling coefficients to lie within ranges expected given the instrument design and found in previous studies. [4] Three types of reference signals have been used in previous work for borehole strainmeter calibration: seismic waves [Grant and Langston, 9], laser strainmeter measurements [Beaumont and Berger, 1975; Hart et al., 1996; Langbein, 1] and the modeled Earth tides [Hart et al., 1996; Roeloffs, 1]. Neither the laser strainmeter nor the seismic methods are feasible for calibrating most of the PBO borehole strainmeter network at this time because there 447

2 B97 B1 B96 B98 B9,1, 11 B4 B B5,6,7 B14 CASCADIA A SUBDUCTION ZONE B18 B941 B B4 MENOCINO O TRIPLE JUNCTION B B1 B5,6 B B B95 B45 B94 B7,8 B4 B9-1 MT ST HELENS B1 B B -115 YELLOWSTONE B6 B8 B [5] The PBO borehole strainmeter network consists of Model 1 Gladwin Tensor Strainmeters (GTSM1) which are 1.5 m in length, weigh ~5 kg and have a resolution of.1 or.5 nanostrain depending on the instrument generation [Gladwin, 1984]. They were installed in 15-cm diameter boreholes at depths of 1 to 5m.PBO GTSM1 s collect data continuously at rates of samples per second (sps), 1-sps, and 1-minute intervals [Anderson et al., 6]. GTSM1 strainmeters consist of four horizontal extensometers, about 9 cm in diameter, stacked vertically within each instrument. Each gauge consists of three parallel capacitance plates. The gap between two adjacent plates is fixed while the third plate moves as the borehole deforms. Gauge elongation is derived from a comparison of the fixed gap and the distance between the middle and free plate. Looking down the instrument, the second and third gauges are rotated 6 and 1 counter-clockwise to the first (bottom) gauge, respectively, while the fourth gauge is rotated counterclockwise to the third (Figure ). 5 SAN JUAN BAUTISTA km B57 B65 B58 B67 B7B78 B75 B76 B79 PARKFIELD B EASTERN CALIFORNIA SHEAR ZONE B916 MOJAVE B918 B91 B ANZA B81, B8 B84, B86 B87, B88 B9-11 Figure PBO borehole strainmeters (solid red circles) examined in this study with sufficiently large M and O 1 SNR for tidal calibration. 1 open red circles indicate strainmeters where the RMS difference between observed and modeled gauge measurements was greater than.84 ns. Orange traces indicate faults younger than 15, years, USGS [1]. The Anza network spans the San Jacinto Fault while the Parkfield network spans the San Andreas. are no laser strainmeters outside California, and the seismic calibration method requires an array of broadband seismometers to be deployed around the strainmeters. In contrast, the Earth tides provide a reference signal against which most PBO strainmeters can be compared. Not only are the signals relatively easily predicted using programs such as SPOTL [Agnew, 1996] but they also lie in the frequency band in which strainmeters are designed to perform well. Langbein [1] has pointed out that the tidal calculations are not always accurate, even in areas tens of kilometers from coastlines but currently they remain the only reference signal against which most of the strainmeters can be calibrated. Hart et al. [1996] presented a technique to calibrate a - gauge strainmeter using the real and imaginary, or in-phase and quadrature, components of the Earth tides. Roeloffs [1] used the Earth tides to calibrate 1 PBO strainmeters and investigated the coupling of vertical strain into the horizontal strain measurements. In this paper, we document the tidal signals recorded by 57 of the 74 PBO strainmeters with a suitably high signal-to-noise ratio (SNR) in the tidal frequency band and suggest a calibration matrix for 45 where we have reasonable agreement between the recorded and model-calculated tides. 5. Calibration Methodology [6] To calibrate borehole strainmeters, we wish to find a set of coupling coefficients that relate the gauge measurements to the regional strains. By regional strains, we mean the strains computed over an area that is large compared to any heterogeneities in the immediate surrounding of the strainmeter that might affect its output. Here, we assume plane stress and an x-y coordinate system where the x-axis is eastwards and the y-axis northwards. Then, E EE + E NN is the areal strain (E A ), the engineering shear strain (E S )is given by E EN and E EE E NN is the shear strain that corresponds to east-west stretching and north-south compression (E D ). Roeloffs [1] refers to the E EE E NN shear strain as differential extension. In the most general case, the gauge measurements recorded by the ith gauge, e i, can be expressed as e i ¼ a i1 E A þ a i E D þ a i E S (1) where the a ij terms represent the areal and shear straincoupling coefficients. In equation (1), the only assumption is rock gauge 4 15 cm gauge 6 6 GTSM1 gauge gauge 1 87 mm grout stainless steel instrument casing Figure. Schematic cross section of a GTSM1 strainmeter (not to scale). The gauges, represented by arrows, are stacked vertically and are isolated from each other within the instrument. The inside diameter of the instrument is ~87 mm. 448

3 that the gauge measurements are linear combinations of the areal and shear strains. For a four-gauge strainmeter, the a ij coefficients form a four by three matrix, which we shall refer to as the general coupling matrix, A. [7] A coupling matrix can be determined by comparing the real (in-phase) and imaginary (quadrature) components of the M and O 1 tides recorded by each gauge and the corresponding components of the tides as predicted by tidal models (equation ), 6 4 e M;R 1 e M;I 1 e O1;R 1 e O1;I 1 e M;R e M;I e O1;R e O1;I e M;R e M;I e O1;R e O1;I e M;R 4 e M;I 4 e O1;R 4 e O1;I a 11 a 1 a 1 M;R M;I O1;R O1;I a ¼ 1 a a E A E A E A E A 6 74 M;R M;I O1;R O1;I 4 a 1 a a 5 E D E D E D E D 5 E M;R a 41 a 4 a S E M;I S E O1;R S E O1;I S 4 () where superscripts R and I refer to the real (in-phase) and imaginary (quadrature) components of the tide, respectively. The M and O 1 tides are used because they have the largest amplitudes outside the 4- and 1-hour period bands, which can be contaminated with thermal signals. Equation () is an over-determined problem with sixteen observations versus twelve model values. The Moore-Penrose pseudo inverse of the coupling matrix is referred to as the calibration matrix and is the matrix applied to the gauge measurements to calculate regional areal and shear strain. [8] An alternative view that provides more insight to the calibration is as follows; assume the strain s i along an axis oriented at θ i counterclockwise from the x-axis is given as s i ¼ :5½E A þ E D cosðθ i ÞþE S sinðθ i ÞŠ () Jaeger and Cook [1976]. This is the strain that would occur along an axis oriented at θ i if the strainmeter was not installed at all. When a strain gauge is installed in a borehole, however, it responds primarily to strain along and perpendicular to its length. If the strain perpendicular to its axis is denoted as p i where p i ¼ :5½E A E D cosðθ i Þ E S sinðθ i ÞŠ (4) and the gauge sensitivity to perpendicular and parallel strains are w per and w par,respectively,(w par > andw per < ) then the total gauge strain is e i = w par s i w per p i. Rearranging gives n e i ¼ :5 w par w per þ w par þ w per EA ð ES sinðθ i Þ or, in the terminology of Hart et al. [1996] ÞþÞ w par þ w per ð ED Þcosðθ i Þ (5) e i ¼ :5ðCE A þ DE D cosðθ i ÞþDE S sinðθ i ÞÞ (6) where C is the areal coupling coefficient and D the shear strain coupling coefficient. C and D are introduced because the combined elastic properties of the strainmeter and grout never perfectly match the surrounding rock. Gladwin and Hart [1985] found typical values of 1.5 and for the C and D values, respectively, based on models of the strainmeter, grout, and borehole. [9] Equation (6) assumes that C and D are constant regardless of the orientation or location of the gauge and represents the isotropic case. There is good reason, however, to believe that a more general coupling model than that suggested by equation (6) applies. Irregularities in the borehole, variations in rock properties, grout, and local topography are all factors that would cause each strain gauge to have different sensitivities to changes in the areal and shear strains. In this case, the C coefficient could differ for each gauge, and the D coefficient differs not only for each gauge but possibly also for E D and E S. Equation (6) then becomes e i ¼ :5ðc i E A þ d i1 E D cosðθ i Þþd i E S sinðθ i ÞÞ (7) where c i, d ij, and d ij, are the areal and shear coupling coefficients. Inspection of equation (5) shows that if the gauge responds only to strain parallel and perpendicular to it then d i1 and d i cannot differ. Hart et al. [1996] suggested that the d ij values could differ by % based on typical variations of the elastic properties of rock while Roeloffs [1] proposed they could differ if the gauges respond to another component of the strain field such as engineering shear strain in coordinates parallel and perpendicular to each gauge. [1] Using gauge 1 as the reference gauge (counter clockwise rotation positive), equation (6) expanded to four gauges becomes, e 1 c 1 d 11 cosðθ 1 Þ d 1 sinðθ 1 Þ 6 e e 5 ¼ 1 e 4 c d 1 cosððθ 1 þ 6ÞÞ d sinððθ 1 þ 6ÞÞ c d 1 cosððθ 1 þ 1ÞÞ d sinððθ 1 þ 1ÞÞ5 c 4 d 41 cosððθ 1 þ 15ÞÞ d 4 sinððθ 1 þ 15ÞÞ [11] The a ij components of equation () can be identified with the components of the coupling matrix in equation (8). We note that while the a ij components of equation () are completely independent the second and third columns of the coupling matrix in equation (8) contain the strainmeter orientation. [1] In this study, we wish to examine the difference between the general coupling matrices obtained via a least squares inversion of equation () which we shall refer to as the nonconstrained general coupling matrix A NC, and those found when we solve equation (8) but constrain the c i and d ij values to lie within ranges we expect of the strainmeters based on the previous analyses of Gladwin and Hart [1985], Hart et al. [1996], and Roeloffs [1]. Specifically, we require 1 < c i < 4,.1 < d ij < 5 and also that the d ij coefficient pairs for each gauge do not differ by more than %. Although the orientation of the strainmeter is measured at installation, it can often be erroneous due to the magnetic properties of the rock or malfunction of the instrument compass; we therefore consider it an unknown too. We first solve (equation 8) using 1 < c i < 4,.1 < d ij < 5 and constraining the d ij values to differ by no more than % for each gauge. We refer to this matrix as the fully constrained coupling matrix,a FC. We then remove the constraint that the d ij values cannot differ by more than % (still using 1 < c i < 4,.1 < d ij < 5) and refer to this as the loosely constrained coupling matrix, A LC. [1] Equation (8) was solved using the Nelder-Mead downhill simplex method [Nelder and Mead, 1965], which finds the minimum of a function of several variables. In this case, the function to be minimized was the Root Mean Square Error (RMSE) of the differences between the real E A E D E S 5: (8) 449

4 and imaginary components of the observed gauge tides, e ij, (right hand side of equation ) and the predicted gauge tides, ^e ij, when the computed coupling matrix is applied to the predicted strains, vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X i¼4 X j¼4 u e ij ^e ij t i¼1 j¼1 RMSE ¼ (9) 16 [14] The inversion required a set of starting values, which were c i = d ij = 1, while θ 1 was set equal to the value measured at installation. [15] To quantify the difference between the A NC and A FC matrices, we calculate the Root Mean Square difference between A þ NC A NC and the identity matrix (I) where A þ NC is the Moore-Penrose pseudo inverse of A NC. We refer to this as the RMSI, vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X i¼ X j¼ A þ NC A u FC ij I ij t i¼1 j¼1 RMSI ¼ (1) 9 [16] For each strainmeter, we verified that A þ NC A NC ¼ I. If A FC A NC, it follows that A þ NC A FC I and the RMSI should be close to zero. The difference between the A NC and A LC matrices is calculated similarly. We compute the RMSI rather than simply difference the coupling matrices because it not only allows us to quantify the difference between matrices for each strainmeter but also allows comparison between strainmeters.. Tidal Analysis [17] The M and O 1 amplitudes were extracted from the strain time-series for each gauge using BAYTAP-G [Tamura et al., 1991]. For 65 of the 74 strainmeters, we used to 4 months of continuous data beginning January 9. For nine strainmeters, the data windows were 1 year to months in length because of data outages or instrument problems. The raw 1-sps data were decimated to a 5-minute-interval data set using causal minimum-phase FIR filters [Agnew and Hodgkinson, 7] and converted to units of strain. Abrupt steps such as coseismic offsets or instrument artifacts have to be removed from the data before tidal analysis. This was done by visually inspecting the 5-minute time-series, flagging data that should be excluded (e.g., periods of site maintenance) and picking times of possible offsets. The strain data, barometric pressure data collected at the site, times of possible offsets and a list of data to exclude were input to CleanStrain + [Langbein, 9], which produced a list of offsets and associated errors. The clean strain and barometric pressure data were then low-pass filtered and decimated to 1-hour sampling interval for input to BAYTAP- G. BAYTAP-G uses a Bayesian modeling procedure to estimate the amplitudes and phases of the tidal constituents, a long-term trend, a noise component and a barometric response coefficient. Fifty-seven strainmeters had an average SNR greater than 5 for the M tide and 5 for O 1.Theamplitudes and phases of the M and O 1 tidesfrom each gauge of these strainmeters form the set of observables used to calibrate the strainmeters (Supplement Table 1). The 17 strainmeters with SNR values less than 5 were excluded from further analysis. [18] Because the GTSM1 strainmeter has two sets of equally spaced gauges, gauges 1, and are 1 apart while gauges and 4 are 9 apart (Figure ), there are two sets of gauge combinations that can be used to calculate areal and shear strain. For the isotropic case (equation 6) and assuming each gauge is equally sensitive to both vertical and horizontal strain it can be shown using equation (6) that, 1 ð e 1 þ e þ e Þ ¼ CE A and 1 ð e þ e 4 Þ ¼ CE A. [19] Likewise, it can be shown that for the shear strains ð e e 1 e Þ ¼ DE D and 1 ð e e 4 Þ ¼ DE D. [] The phase and amplitudes of the tides determined using the three-gauge combination should be similar to that of the two-gauge combination if the strainmeter is installed in an isotropic medium, and the gauges are equally sensitive. Differences indicate deviation from this case. Comparison of the - and -gauge measurements has the benefit of being independent of any tidal models. Seventeen of the 57 strainmeters examined had M phases that differed by less than 6 and had amplitudes ratios of between.8 and 1. for areal strains, while 4 strainmeters had similar differences and ratios for M shears strains (Supplemental Table ). Eleven strainmeters were self-consistent within these ranges for both areal and shear M strains, and only 6 strainmeters (B8, B84, B1, B6, B91, B97) were self-consistent in both the M and O 1 tidal bands and both areal and shear. This suggests that most strainmeter installations are not representative of the isotropic case. We note that 1 of the strainmeters have similar (within %) amplitude ratios for both M and O 1 tides for areal and shear strain. [1] The predicted M and O 1 tides for E A, E D, and E S were calculated at each site using NLOADF [Agnew, 1997]. The ocean loads were computed using the TPXO7. global model plus regional load models for the Gulf of California, the Strait of Juan de Fuca and the San Francisco Bay Area. These were then added to the solid Earth tides to form the total calculated tidal signal (Supplement Table ). We used these predicted amplitudes and phases to generate a -year time series for input to BAYTAP-G. The amplitudes and phases determined by BAYTAP-G from the synthetic time series form the reference set of amplitudes and phases to be compared with the observed gauge tides. 4. Results [] We first assessed the appropriateness of the tidal model for the 57 strainmeters with a good SNR by examining the RMSE (equation 9) of the observed and predicted gauge tides using the general nonconstrained coupling matrix A NC.We anticipated that the largest source of error in the tidal models would be unmodeled loading signals caused by inlets or estuaries. Aside from the Yellowstone strainmeters which are installed near Yellowstone Lake, the Mojave and Anza strainmeters (Figure 1) are the farthest inland of all the PBO strainmeters and are most likely to be free of any complicated coastal or lake loading signal. Reasonable agreement between the laser strainmeter at Pinyon Flat in Anza and modeled tides also gives us some confidence in the models [Agnew and Wyatt, ]. The RMSE of these strainmeters ranged from.7 to 45

5 Table 1. Comparison of Coupling Matrices for PBO Borehole Strainmeters (BSM): RMSE (Equation 9), RMSI (Equation 1), θ 1 is the Calculated Orientation of Gauge 1 (Counter Clockwise Positive) The install column contains the orientation measured during installation where available. Strainmeters are sorted by category and then by name. Lines which are italicized show the comparison of the A NC and A FC matrices after adjustments: a) the d ij spread constraint was relaxed to 4%, b) one gauge was omitted and d ij spread constraint = % and c) one gauge was omitted and the d ij spread constraint = 4%. These strainmeters become Category 1 instruments with these adjustments. The following BSMs not calibrated because the unconstrained A solution RMSE was too large. These BSMs are plotted as open circles in Figure 1. The RMSE is given in parentheses. B5 (.9), B6 (.76), B7 (1.4), B9 (.4), B1 (6.69), B11 (4.1), B1 (.47), B18 (.81), B (1.88), B8 (1.1), B97 (1.1), B98 (5.8). + Installation orientation unknown 451

6 Table. Calibration Matrices. Moore-Penrose Pseudo Inverse of the Coupling Matrix A NC. 45

7 Table. Calibration Matrices for Selected Category and Category Strainmeters. The Calibration Matrix for the Three Gauge Solutions Are the Inverse of the Three-Gauge Coupling Matrix A NC. BSM Calibration matrix d ij spread constraint = %, omit one gauge B Omit gauge B Omit gauge B Omit gauge B Omit gauge B Omit gauge d ij spread constraint = 4%, omit one gauge B Omit gauge B Omit gauge B Omit gauge B Omit gauge BSM Calibration Matrix B Omit gauge B Omit gauge Moore-Penrose Pseudo Inverse of the A FC matrix B B B B B B B B nanostrain (Table 1); we considered values less than twice the maximum i.e., <.84 nanostrain to indicate that modeled tides are acceptable as a reference calibration signal. [] Eleven strainmeters had an RMSE greater than the upper acceptable limit; all are situated near bodies of water (open circles in Figure 1 and notes to Table 1). Strainmeters B9, B1, B11, B1, and B98 are on Vancouver Island and lie within a few hundred meters of the coastline. B97, also on Vancouver Island, is at the northeast end of a fjord that opens to the Pacific. B is less than km from the Oregon Pacific coast and 5 m from a river. The - and -gauge combination areal phases of these strainmeters differ by less than 7 and all but B11 have a - and -gauge areal amplitude ratio between.8 and 1. which suggests that the strainmeters are functioning in a self-consistent manner and the large RMSE are likely due to incorrect tidal models. While tidal models are provided for the Straits of Georgia and Juan de Fuca in NLOADF, there are none for Puget Sound, which may explain the large RMSE at B18. B8, the only inland strainmeter with a large RMSE, is installed on the shores of Yellowstone Lake. B944, however, is just as close to the lake and has a low RMSE, which suggests a problem with the B8 data rather than the tidal models for the area. B8 also has large phase differences when comparing the - and -gauge phase measurements. B5 and B7 are installed within a few hundred meters of each other and about 9 km from the coastline of the Strait of Juan de Fuca. Not only are the RMSE values high for these sites but also the - and -gauge areal phase differences are quite large (> 9 ). Strainmeters with an RMSE greater than double the upper acceptable limit are probably not suitable for calibration using current Earth tide and ocean load models. Although the B6 RMSE value is within the acceptable range (.76), it is relatively high and we choose to exclude it along with its colocated strainmeters, B5 and B7. [4] The 11 coastal strainmeters and B8 were excluded from further analysis leaving 45 strainmeters suitable for tidal calibration. It is important to note that this does not imply that the coastal strainmeters are not functioning well, rather the tidal models may not be adequate in these areas. This is especially the case where the - and -gauge strain combinations have similar phase and amplitudes. The RMSE gauge misfit of the A NC, A LC,andA FC coupling matrices, are presented in Table 1. The areal coupling coefficients for each matrix type are plotted in Figure. The shear coupling coefficients determined for each matrix are shown in Figure 4, with the sine and cosine term included, and in Figure 5, which shows the d ij values. The Moore-Penrose pseudo inverses of the A NC matrices are given in Table. 5. Discussion [5] The areal coupling coefficients, a i1 (equation ) and c i (equation 8), change little for all three sets of calibration matrices (Figure ). As found by Roeloffs [1], many of the strainmeters have a negative areal coupling coefficient on at least one gauge. We find this to be the case for of the 45 strainmeters examined and that in general, the areal coupling coefficients tend to be lower than the value of 1.5 suggested by the modeling of Gladwin and Hart [1985]. This suggests that for this particular strainmeter design, the areal coupling is weak and interpretation of the areal strain data should be done with this in mind. [6] Roeloffs [1] demonstrated that the areal coupling coefficients, c i, determined through tidal calibration should be considered apparent areal coupling coefficients, i.e., 45

8 Coupling Coefficient 1 1 Areal Coupling Coefficients A. Gauge 1 Coupling Coefficient 1 1 B. Gauge Coupling Coefficient 1 1 C. Gauge Coupling Coefficient 1 1 D. Gauge 4 Category 1 Category Category B14 B7 B8 B B B9 B4 B58 B65 B67 B7 B76 B79 B8 B84 B86 B9 B B916 B917 B918 B91 B B6 B45 B78 B81 B87 B88 B1 B B6 B91 B96 B94 B95 B941 B944 B B4 B4 B1 B5 B57 B75 Strainmeter Figure. Areal coupling coefficients for each gauge for each strainmeter. Circles denote the A NC matrix (a i1 ) component. Squares and triangles represent the areal component (.5*c i )forthea LC and A FC matrix components respectively. Upper horizontal dashed line in each plot indicates the suggested areal coupling coefficient of.5*c =.75. Vertical dashed lines separate Category 1, Category, and Category strainmeters. n c i ¼ C i V i 1 n (11) where C i is the true areal coupling coefficient, V i a coefficient that indicates the gauge sensitivity to vertical strain and n is Poisson s ratio. As pointed out by Roeloffs [1], if V i is large enough, c i will be negative. The barometric response coefficients (Supplement Table 1) provide an indication of the instrument sensitivity to changes in vertical strain. When the magnitude of the barometric response is greater than 5 ns/kpa, the areal coupling coefficient tends to be negative (Figure 6); 71% of the negative values correspond to a barometric response of magnitude 5 ns/kpa or greater. [7] Figures 4A 4D are a graphic representation of the shear strain components of the coupling matrix. The A NC matrix components are shown as circles, the A LC components as squares and the A FC components as triangles; red and blue colors represent the E D and E S shear components, respectively. Overlying shapes of the same color in Figure 4 indicates similarity of the three types of coupling matrices. The differences in orientation between the A LC and A FC solutions are less than 5 for 5 of the 45 strainmeters examined (Figure 4E). The A LC and A FC orientations differ from the installation measurement by less than 1 for 1 and of the strainmeters, respectively (Table 1). Since there is a possibility that the strainmeter orientation measurement made at time of installation can be erroneous, the magnitude of the difference between the measured and calculated orientation is not problematic. More insight on the shear coupling coefficients can be gained if the matrix components are plotted without the orientation term (Figure 5). As in Figure 4, squares represent the A LC solution and triangles represent the A FC solution. For an isotropic setting similar shapes would cluster about the same value. The hashed horizontal line indicates the D = value suggested by theory. Inspection of Figure 5 shows that when the d ij spread constraint is removed, the d ij value of several strainmeters tends to become very small perhaps indicating over fitting of the data. [8] For of the 45 strainmeters examined, there is almost no difference between A NC, A LC and A FC. The RMSI differences (equation 9) are all less than.1 (Table 1), and little change in matrix components is observed in Figure 4. This suggests that these strainmeters could be considered as functioning according to the model described in equation (8) and with coupling coefficients falling within expected ranges. No better solution is found, in terms of lower RMSE, when all constraints are removed. We consider these Category 1 strainmeters (Table 1). For these strainmeters, we suggest using the Moore-Penrose pseudo inverse of the A NC matrix to combine the gauge measurements into areal and shear strain (Table ). 454

9 Coupling Coefficient Coupling Coefficient Coupling Coefficient Coupling Coefficient Differential Extension and Shear Strain Matrix Components A. Gauge 1 B. Gauge C. Gauge D. Gauge 4 Category 1 Category Category B14 B7 B8 B B B9 B4 B58 B65 B67 B7 B76 B79 B8 B84 B86 B9 B B916 B917 B918 B91 B B6 B45 B78 B81 B87 B88 B1 B B6 B91 B96 B94 B95 B941 B944 B B4 B4 B1 B5 B57 B75 Change ( ) B14 B7 B8 B B B9 B4 B58 B65 B67 B7 B76 B79 B8 B84 B86 B9 B B916 B917 B918 B91 B B6 B45 B78 B81 B87 B88 B1 B B6 B91 B96 B94 B95 B941 B944 B B4 B4 B1 B5 B57 B75 E. Change in orientation of Gauge 1 for thea LC and A FC Solutions Category 1 Category Category Figure 4. The differential extension and shear strain components for each gauge. Circles denote the A NC matrix components. Squares represent the A LC solution and triangles the A FC solution. (A D) Blue represents the a i and.5d i1 cosθ i components while red represents the a i and.5d i sinθ i components. (E) Change in orientation of gauge 1 (counter clockwise + ve). Vertical dashed lines separate Category 1, Category, and Category strainmeters. [9] For 16 strainmeters, the fully constrained coupling matrix, A FC,(c i and d ij range constrained plus the % d ij spread constraint) differed from the nonconstrained, A NC,matrixwith an RMSI greater than.1. The A LC coupling matrices for these sites (c i and d ij range constrained), however, were very similar to the A NC matrices. We refer to these as Category instruments. For the remaining seven strainmeters, both A LC and A FC differed from A NC with an RMSI of more than.1. Unlike Category strainmeters, the differences were not reconciled when the % d ij spread constraint was removed. These are termed Category strainmeters. The average d ij value of the A FC coupling matrix (triangles in Figure 5) for these strainmeters is higher,.7, than those found for Category 1 and Category strainmeters where they are. and.1, respectively. These average values are lower than the value of suggested by Gladwin and Hart [1985] but are reasonably close compared to the areal coupling coefficients where we find 4 of the 45 strainmeters examined have areal coupling coefficients less than % of the suggested value of 1.5 on at least one gauge. [] To examine whether the change in RMSE between the A FC and A NC solutions was significant for Category and Category strainmeters, we added Gaussian noise to the tidal measurements and reran the inversions. This process was repeated 1 times: each time, we calculated the difference between the sum of the squared residuals for the A FC and A NC solutions divided by the sum of the squared residuals of the A NC solution. Then, we determined the ratio that 95% of the solutions fell below. This was compared to the ratio calculated from the actual observations. If the observed ratio was less than the 95% level, the reduction in misfit could be considered as not being significant. We repeated this procedure to compare the A LC and A NC solutions and then the A FC and A LC solutions. Using this method of evaluation, the reduction in RMSE was found to be significant only for B. 455

10 Differential Extension (d i1 ) and Shear Strain Coefficients (d i ) A. Gauge 1 Coefficient Coefficient Coefficient Coefficient B. Gauge C. Gauge D. Gauge 4 Category 1 Category Category B14 B7 B8 B B B9 B4 B58 B65 B67 B7 B76 B79 B8 B84 B86 B9 B B916 B917 B918 B91 B B6 B45 B78 B81 B87 B88 B1 B B6 B91 B96 B94 B95 B941 B944 B B4 B4 B1 B5 B57 B75 Figure 5. The differential extension and shear strain coupling coefficients for each gauge. Triangles represent the A FC solution and squares the A LC solution (the % d ij spread constraint removed). Blue represents d i1 and red d i. Dashed horizontal lines the suggested D = value. Vertical dashed lines separate Category 1, Category, and Category strainmeters. Barometric Response Coefficient Versus Apparent Coupling Coefficient (a i1 ) Barometric Response (ns/kpa) coefficients 1 coefficients 9 coefficients coefficients Areal Coupling Coefficient (a i1 ) Figure 6. Barometric response coefficients versus a i1 (A NC matrix) for each gauge of each strainmeter listed in Table 1. Number of coefficients that fell within each quadrant is indicated. Black dashed line indicates the least squares fit to the data. [1] To explore the discrepancy between the A NC and A FC matrices for the Category and Category strainmeters, we investigated relaxing the d ij spread constraint and considered the possibility of one gauge not functioning as expected. For Category, instruments removing the % d ij constraint reconcile the A NC and A FC matrices, but completely removing any constraint on the spread of the d ij values could lead to over fitting. We found that if the spread constraint was relaxed to 4%, the A NC and A FC matrices of B, B6, B96, and B95 had RMSI differences of.1 or less, i.e., these become Category 1 strainmeters where A NC A FC. We suggest that the calibration matrix of these strainmeters 456

11 should be the Moore-Penrose pseudo inverse of the A NC matrix (Table ) given that the only change to the model is allowing the range of the d ij values to increase to 4%. [] Since only three gauges are needed to calculate areal and shear strains, we explored the possibility that one gauge might be causing the difference between A NC and A FC by repeating the inversions for the remaining strainmeters using all possible combinations of three of the four gauges and the % d ij spread constraint. We found that for five of the remaining strainmeters (B4, B57, B81, B1, B91) omitting one gauge resulted in a coupling matrix that had coefficients that fell within expected ranges, had d ij that differed by no more than %, and a better solution could not be found via an unconstrained inversion i.e., these also become Category 1 strainmeters where A NC A FC. We repeated the inversions for the remaining 14 strainmeters, again omitting one gauge each time, but relaxing the d ij spread constraint to 4%. We found in this case B1, B6, B45, B78, B87, and B941 produced A FC coupling matrices very similar to the A NC matrix. It is possible that a gauge within each of these strainmeters may not function according to equation (7). In these cases, we suggest using the inverse of the three gauge coupling matrix omitting the suspect gauge (Table ). [] There are eight strainmeters (B, B4, B5, B75, B88, B, B94, and B944) for which we found A FC and A LC matrices that were significantly different from the A NC matrix. These differences could not be attributed to one problem gauge or to the d ij spread. For B94 and B944, the differences are resolved only when the d ij spread constraint is removed entirely but for the others even this did not reconcile the difference. There is ambiguity as to which matrix should be used to convert gauge measurements to areal and shear strain for these strainmeter as both matrices yield similar misfits. We cannot distinguish between these matrices using tidal data alone, and in the absence of independent measurements, we present both the Moore- Penrose pseudo inverse of the A NC matrix (Table ) and the A FC matrices (Table ) as viable calibration matrices. [4] Roeloffs [1] uses a different technique to extract the amplitudes and phases from the strain data forming a different set of observable against which to calibrate the strainmeters. Using the observables given by Roeloffs [1] and inputting that to the calibration procedure used here serves as a comparison of the methodology and inversion techniques used. In doing this, we find the A NC matrices for B7, B5, B7, B81, and B84, and the A FC coupling matrices of B6 and B8 differ from the matrices of Roeloffs [1] with an RMSI of less than.11 but differences are larger for B (RMSI of A FC =.8) and B4 (RMSI of A FC =.14). 6. Conclusion [5] Based on the large misfit of the observed and the predicted tides, we find that strainmeters located within a few kilometers of the coastline are not suitable for tidal calibration. This is most likely due to the complexities of modeling localized tidal loads near the coastline and not a reflection on the quality of the data from these instruments. [6] We find that many of the 45 strainmeters determined suitable for tidal calibration have negative areal coupling coefficients on at least one gauge. There is a correlation between barometric response coefficients, an indicator of sensitivity to vertical strain changes, and negative areal coupling coefficients; where response coefficients are greater than 5 ns/kpa, the strain gauge is more likely to have a negative areal coupling coefficient. A large barometric response coefficient could suggest that the strainmeter is increasingly sensitive to vertical strain, and this could lead to a nosier areal strain data set. We find that the areal coupling coefficients are significantly less than the 1.5 factor suggested by Hart et al. [1996]. Any interpretation of the areal strain measurements from this instrument set should take into consideration that these particular strainmeters have a decreased sensitivity to areal strain and that a component of vertical strain is coupled into the horizontal strain measurements. [7] For of the 45 strainmeters examined, the calibration matrices determined via an unconstrained comparison of the recorded tides and the areal and shear tides predicted by models are consistent with those determined when the inversion is constrained such that the coupling coefficients must fall within expected ranges given the instrument design. These strainmeters seem to function according to the model presented in equation (8), have values of c i and d ij that lie within expected ranges, and d ij values within % of each other. No better matrix can be found for these strainmeters when all constraints are removed. [8] If one is prepared to accept that one gauge within a strainmeter may not be functioning as expected and the d ij values may diverge by as much as 4%, then an additional fifteen of the strainmeters examined produce very similar coupling matrices via unconstrained and constrained inversions. For the remaining eight strainmeters, we cannot explain the difference between the unconstrained and constrained coupling matrices by allowing a larger spread in d ij values or by omitting one gauge in the inversion. A calibration method independent of the tides, e.g., seismic analysis is needed to distinguish between these two matrices. The validity of the tidal calibrations could be tested in the future using several methodologies including detailed structural modeling of the nearby crustal structure to obtain a better model of the Earth tide at the strainmeter, comparison of seismic waves recorded on a local network of seismometers and the strainmeter [Grant and Langston, 9], and dislocation modeling of local earthquake data. [9] Acknowledgments. We would like to acknowledge the work of Mike Gottlieb, Warren Gallaher, Wade Johnson, Chad Pyatt and Liz Van Boskirk, the team of Borehole Strainmeter Engineers at UNAVCO who ensure the collection of a high-quality strain data set. We thank John Beavan, Evelyn Roeloffs, Karen Luttrell and an anonymous reviewer for helpful reviews. This material is based on data and engineering services provided by the Plate Boundary Observatory operated by UNAVCO for EarthScope ( and supported by the National Science Foundation (no. EAR-58 and EAR-7947). References Agnew, D. C. (1996), SPOTL: Some programs for ocean-tide loading, SIO Ref. Ser. 96-8, 5 pp., Scripps Institution of Oceanography, La Jolla, CA. Agnew, D. C. (1997), NLOADF: A program for computing ocean-tide loading, J. Geophys. Res., 1(B), , doi:1.19/96jb458. Agnew, D. C., and K. M. Hodgkinson (7), Designing compact causal digital filters for low-frequency strainmeter data, Bull. Seismol. Soc. Amer., 97, 91 99, doi:1.1785/

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