Pore pressure sensitivities to dynamic strains: Observations in active tectonic regions

Abstract

Triggered seismicity arising from dynamic stresses is often explained by the Mohr‐Coulomb failure criterion, where elevated pore pressures reduce the effective strength of faults in fluid‐saturated rock. The seismic response of a fluid‐rock system naturally depends on its hydromechanical properties, but accurately assessing how pore fluid pressure responds to applied stress over large scales in situ remains a challenging task; hence, spatial variations in response are not well understood, especially around active faults. Here I analyze previously unutilized records of dynamic strain and pore pressure from regional and teleseismic earthquakes at Plate Boundary Observatory (PBO) stations from 2006 to 2012 to investigate variations in response along the Pacific/North American tectonic plate boundary. I find robust scaling response coefficients between excess pore pressure and dynamic strain at each station that are spatially correlated: around the San Andreas and San Jacinto fault systems, the response is lowest in regions of the crust undergoing the highest rates of secular shear strain. PBO stations in the Parkfield instrument cluster are at comparable distances to the San Andreas Fault (SAF), and spatial variations there follow patterns in dextral creep rates along the fault, with the highest response in the actively creeping section, which is consistent with a narrowing zone of strain accumulation seen in geodetic velocity profiles. At stations in the San Juan Bautista (SJB) and Anza instrument clusters, the response depends nonlinearly on the inverse fault‐perpendicular distance, with the response decreasing toward the fault; the SJB cluster is at the northern transition from creeping‐to‐locked behavior along the SAF, where creep rates are at moderate to low levels, and the Anza cluster is around the San Jacinto Fault, where to date there have been no statistically significant creep rates observed at the surface. These results suggest that the strength of the pore pressure response in fluid‐saturated rock near active faults is controlled by shear strain accumulation associated with tectonic loading, which implies a strong feedback between fault strength and permeability: dynamic triggering susceptibilities may vary in space and also in time.

1 Introduction

Quantifying the response of fluid‐saturated rock to applied stress is critical to understanding frictional behavior of faults [Beeler et al., 2000] and mechanisms of dynamic earthquake triggering [Hill and Prejean, 2015], since increases in pore fluid pressure reduce the effective strength of a fault. Water pressure changes in response to seismic waves have been recorded in wells tapping aquifer systems for many decades [e.g., Leggette and Taylor, 1935; Eaton and Takasaki, 1959], but measuring quasi‐static pore pressure response in the Earth requires some knowledge of the deformation signal. In the case of seismic waves, for example, the size of the pressure response is generally expected to correlate with the level of seismic energy density [Wang et al., 2009].

According to the theory of linear poroelasticity [Wang, 2000], the pressure perturbation in an undrained medium is proportional to the dilatational stress imposed on it [Roeloffs, 1996]. Consequently, direct estimates of the local hydromechanical properties affecting the proportionality between pressure and stress can be made with measurements of both fluid pressure and either the associated ground motion, with knowledge or assumption of phase velocity [Cooper et al., 1965; Liu et al., 1989; Brodsky et al., 2003; Geballe et al., 2011], or the dynamic strains directly [Ohno et al., 1997; Kano and Yanagidani, 2006; Kitagawa et al., 2011]. Despite a rich observational history of seismic waves in groundwater level time series, little is known about spatial variations in well response in regions with active faults [Wang and Manga, 2009], mainly because of the expense of drilling and instrumenting scientific boreholes.

The borehole strainmeters in the Plate Boundary Observatory (PBO) offer an unprecedented opportunity to test for variations in pore pressure response around active faults in a variety of tectonic provinces. Here I analyze the pore pressure response to seismic waves at all 23 PBO borehole stations in which pore fluid pressure is recorded. These stations are located along the North American/Pacific plate boundary from the Anza region in Southern California to Vancouver Island, British Columbia. Woodcock and Roeloffs [1996] found a logarithmic relationship between peak water heights and the associated peak ground displacements at a water well in a fractured rock aquifer in southwest Oregon; the PBO stations are in a similar hydrogeologic environment, and we expect the PBO stations to behave similarly.

The scaling relationships I develop here, for the PBO systems, are based on dynamic strains from regional and teleseismic waves, corroborated by tidal and spectral analyses, and are the first of their kind. All PBO systems show a pore pressure response to strain, but some show a markedly reduced response. Within clusters of instruments having characteristic dimensions much smaller than the wavelengths of the seismic waves, I find that spatially correlated variations in response are apparently related to strain accumulation on nearby faults.

2 Instruments and Data

In this study I compare direct observations of dilatational strains from seismic waves and Earth tides with the associated pore fluid pressure changes. As we will see later (and has been discussed previously by Roeloffs [2000]), observations of pore pressure or water level are not simple proxies for dilatational strain; rather, they are complementary measurements that add information useful for interpretation.

The horizontal strain tensor is measured with a Gladwin [1984] style borehole strainmeter (BSM) at depths from 150 to 250 m I low‐pass filter the linearized, raw −20 Hz strain records to obtain 1 Hz records of instrumental strain (see Barbour and Agnew [2011] for details) and then high‐pass filter the records with a seventh‐order Butterworth filter based on coefficients for a corner frequency of 1 mHz, 30 dB attenuation in the stopband, and 0.5 dB allowable ripple in the passband; note that the same high‐pass filter is applied to the pore pressure records. This filter does not remove spurious signals introduced into the surface wave band by the power system [Barbour and Agnew, 2011], which means that in practice, the detection threshold for seismic strains is on the order of 10⁻⁹. Filtered instrumental strains are then converted to tensor strains in the rock using gauge coupling coefficients derived from tidal calibration studies [Roeloffs, 2010; Hodgkinson et al., 2013]. The tensor strain components are then rotated into a transverse‐radial (θ − r) coordinate system using the back azimuth from the station to the seismic source. Volume strains can be derived from areal strains by assuming that the BSM operates in an elastic solid under plane stress conditions (traction‐free surface), but here I consider only the effect of areal strains. The areal strains are obtained from the sum of uniaxial extensions in the transverse and radial directions (E_θθ+E_rr). In general, this is an accurate representation of strains associated with the seismic wavefield at teleseismic distances [Agnew and Wyatt, 2014].

The pore pressure (PP) sensing system is relatively simple by comparison: an open section of the borehole, typically ≈50 m above from the strainmeter (see Table 1), is packed with high‐permeability sand (rather than with the relatively impermeable grout elsewhere). This sand‐packed region is probed by a screened section of a rigid, 2 inch inner‐diameter tube, permitting pressure measurement of the surrounding fluid‐saturated rock in the tube. A pressure transducer, submerged and affixed rigidly at a known depth below the top of the borehole casing, measures the absolute pressure of the fluid inside the tube. Unlike the BSM, the PP sensor does not require in situ calibration: the nominal calibration is sufficiently accurate for these purposes, and the data logger records transducer output at a maximum of 1 Hz. The depth of the PP sensor is sufficiently below the piezometric surface [Freeze and Cherry, 1979] to allow for seasonal variation in pressure head associated with aquifer recharge cycles. The small diameter of the tube (which extends from just below the screen to the surface) ensures that wellbore storage and inertial effects, common in water height measurements of seismic waves [e.g., Cooper et al., 1965], are negligible. Stations tapping artesian (naturally flowing) aquifers have a pressure containment system inside the tube above the transducer (see Table 1).

Table 1. Plate Boundary Observatory Boreholes With Pore Pressure Instrumentation

		Installation						Depths
			First BSM	First PP (Nb^b The number of records available at 1 Hz sampling. )			Elevation	PP	BSM
Region	Site	Cap Typea^a PP systems are contained with either • an inflatable packer inside the sensing tube, ⊕ a cap at the top of the tube, or ∘ uncapped—fluids exposed to atmosphere. ×: Inflatable packer failed on 2010:190 (9 July).	(20 Hz)	(1 Hz)	Longitude	Latitude	(m)	(m)	(m)
Cascadia	B010	•	2005:269	2009:280 (26)	−123.4513	48.6502	5	174	198
	B012	•	2005:264	2009:196 (30)	−125.5420	48.9246	13	149	169
	B011	•	2005:256	2009:215 (28)	−123.4482	48.6495	22	204	224
	B005	•	2005:200	2010:010 (24)	−123.5033	48.0595	303	144	161
	B004	⊕	2005:166	2010:010 (23)	−124.4270	48.2019	30	141	166
	B003	•	2005:172	2010:010 (24)	−124.1409	48.0624	285	152	169
	B001	•	2005:180	none	−123.1314	48.0431	237	137	152
Mendocino	B022	⊕	2006:034	2009:321 (25)	−123.9310	45.9546	10	134	220
	B028	∘	2007:078	2009:321 (25)	−122.9638	44.4937	140	219	240
San Juan Bautista	B066	•	2007:158	2009:148 (31)	−121.5922	36.8575	67	210	235
	B067	•	2007:163	2009:148 (28)	−121.5655	36.7650	126	140	158
	B058	•	2007:137	2009:196 (26)	−121.5808	36.7995	114	142	166
Parkfield	B076	∘	2006:286	2008:290 (35)	−120.4248	35.9398	445	183	197
	B078	•	2006:285	2008:290 (35)	−120.3452	35.8377	387	165	181
	B073	•	2006:291	2008:290 (36)	−120.4717	35.9467	535	226	241
	B079	•	2006:286	2008:290 (36)	−120.2057	35.7157	437	164	180
Anza	B088	⊕	2007:027	2010:096 (19)	−116.6205	33.3749	1404	137	160
	B084	•	2006:169	2008:290 (36)	−116.4564	33.6116	1271	135	158
	B087	∘	2006:168	2010:096 (20)	−116.6027	33.4955	1139	94	161
	B082	•×	2006:161	2008:290 (36)	−116.5960	33.5982	1375	192	242
	B086	∘	2006:168	2010:094 (21)	−116.5310	33.5575	1392	216	244
	B081	•	2006:166	2009:148 (29)	−116.7142	33.7112	1467	212	243
	B946	∘	2010:204	2011:068 (12)	−116.5925	33.5373	1429	134	147

• ^a PP systems are contained with either • an inflatable packer inside the sensing tube, ⊕ a cap at the top of the tube, or ∘ uncapped—fluids exposed to atmosphere. ×: Inflatable packer failed on 2010:190 (9 July).
• ^b The number of records available at 1 Hz sampling.

I analyzed signals from a catalog of 54 earthquakes that occurred between early 2006 and mid‐2012, with magnitudes (M_W) ranging from 4.5 to 9.0 and source depths from 5.4 to 65 km; these are listed in Table S1 in the supporting information. In order to ensure that the selected events allow for observable dilatational strains spanning multiple orders of magnitude, I chose events from the Advanced National Seismic System (ANSS) catalog following a nonlinear relationship between magnitude and epicentral distance in degrees (Δ), based roughly on signal‐to‐noise detection thresholds for the BSMs [Barbour and Agnew, 2012]:

(1)

where here I use M_min = 5.9 (see also Figure 1) and δM = 1. Because of data gaps, inadequate sampling rates, and other factors, the number of events with high‐frequency pore pressure records at particular boreholes ranges from 12 to 36, and in one case (station B001), only a single event was recorded. Table 1 lists some station details and the number of 1 Hz records available. The first event with any observable high‐frequency pore pressure record is the 2009:203 M_W7.4 earthquake located near the northern coast of Papau, Indonesia. The number of pore pressure records is ≥17 for all events after 2009:321.

Figure 1

Magnitude versus distance ranges for the earthquakes used in this study. Horizontal bars show the range of distances to PBO stations for each earthquake (distinguished by colors); their levels have been randomly shifted (slightly) to minimize overlap. The dashed line shows the criteria used to select earthquakes from the ANSS catalog (equation 1), which I require to be valid for at least one station. The two labeled earthquakes represent exceptions to the selection criteria: (1) the 2011 M_W4.5 San Juan Bautista, which was used to test for directivity effects (see text), and (2) the 2011 M_W9 Tohoku‐oki, which was used for spectral analyses (see Figure 12).

I also included two events that do not follow the magnitude‐distance relationship in equation 1: (1) the 2011 M_W4.5 San Juan Bautista (SJB) earthquake in Northern California, to examine the potential for directivity effects at local distances [Seekins and Boatwright, 2010], and (2) the teleseismic 2011 M_W9 Tohoku‐oki earthquake, offshore of Japan [Simons et al., 2011], because the associated Rayleigh waves are approximately harmonic at these epicentral distances and have strongly polarized strain components that can be used in cross‐spectrum analyses (described later).

3 Relating Dynamic Pressures to Dynamic Strains

The principal assumptions generally made when relating pore pressure variations to dilatational strains in a well‐aquifer system are that (1) the formation is a half‐space in a state of plane stress, where the free surface is traction free; (2) fluids flow in the radial direction, within a confined aquifer of infinite radial extent [Hsieh et al., 1987; Rojstaczer, 1988a; Kitagawa et al., 2011]; and (3) fluid flow is assumed to follow Darcy's law, where fluid flux, q, in a porous medium is proportional to the pressure gradient, ∇p:

(2)

where k is the permeability of the aquifer and η is the kinematic viscosity of the fluid. The excess pressure p is obtained from p ≡ P − ρgz, the difference between the absolute pressure at the transducer, P, and the hydrostatic level defined by the density of the fluid, ρ, gravitational acceleration, g, and the depth of the transducer, z. In this equation, the permeability of the aquifer is assumed to be isotropic, an assumption that is violated when flow occurs predominantly within fracture networks [Rojstaczer et al., 2008; Ingebritsen and Manning, 2010].

Strain oscillations accompanying seismic waves often have periods too short and/or wavelengths too long to permit or induce significant fluid flow, so that fluid pressure responds in an “undrained” manner [Wang, 2000]. When the response is undrained, the pressure is expected to be related to volume strain E_kk through Skempton's coefficient [Rice and Cleary, 1976; Rojstaczer and Agnew, 1989] and the undrained bulk modulus κ_u or the shear modulus, μ and ν_u, the undrained equivalent of Poisson's ratio:

(3)

Assuming plane stress conditions, the vertical strain is related to areal strain, E_A = E₁₁ + E₂₂, by

(4)

and thus equation 3 can be expressed as

(5)

which implies that if ν_u = 1/3, a reasonable assumption for crustal rock [Wang, 2000], then

(6)

The pressure response to strains from the solid Earth tides, and atmospheric loading, is also expected to be undrained and equivalent to equation 6, owing to the large wavelengths of the loading signals. Figure 2 shows pressure and strain time series at station B084 in Anza before, during, and after the 2010:094 M_W7.2 El Mayor Cucapah earthquake [Wei et al., 2011], which occurred ∼200 km from the Anza network (Table 1); these records demonstrate how an undrained pore pressure response is linearly proportional to areal strain from frequencies at least as low as atmospheric pressure changes to at least as high as strong regional body waves, as we should expect. But, as I shall show, the linear proportionality of excess pressure to areal strain (e.g., equation 6) does not accurately describe all of the observations reported here.

Figure 2

Pore pressure records at station B084 in Anza before, during, and after the 2010 M_W7.2 El Mayor Cucapah earthquake [Wei et al., 2011]. (a) Excess pore pressure is linearly proportional to strain signals occurring over a wide frequency band, including coseismic (static) changes, daily to weekly atmospheric pressure changes, and daily to subdaily Earth tides. When the contributions from all known strain signals are removed from the original record, the residual pore pressure shows a pronounced postseismic transient that is most likely associated with pore fluid diffusion. (b) Pore pressures associated with higher‐frequency seismic waves (tens of seconds or shorter) are also linearly proportional to strain. At these frequencies tidal and atmospheric effects are negligible. (c) The proportionality between pressure and strain at seismic frequencies is −1.18 ± 0.01 GPa ε⁻¹ (R² = 0.96) with a Pearson correlation coefficient of −0.98; these values are consistent with estimates from cross‐spectrum analyses (shown later). Other stations do not show a similar linear response though. (d) Simplified diagram of a typical PBO BSM and PP borehole (see section 2 for a description).

4 Empirical Scaling Results

To form scaling relationships which predict pressure from strain, I use areal strains in the rock, E_A, measured by the strainmeter, and excess pore pressure at the transducer, p. Specifically, I obtain peak values from analytical envelope functions of the band‐pass‐filtered (10⁻³ Hz to 1 Hz) pressure and strain time series, calculated using the Hilbert transform [see Bracewell, 1986, chapter 13]; in doing so, I can account for any time lags between the pressure and strain signals, since they may not be perfectly out of phase, and any potential bias associated with coseismic signals [Barbour et al., 2015]. To account for possible departures from a purely proportional (undrained) response, I explore a more general relationship whereby excess pressure is assumed to be proportional to a power, d, of areal strain. The functional form of this generalized scaling relationship between pressure and strain is

(7)

or

(8)

where parameters can be estimated by generalized least squares regression or by explicit maximization of likelihood expressions. But this equation is only valid for c > 0, which is satisfied by using peak values of strain and pressure envelope functions, which are always positive. The equation is also only valid for 0 ≤ d ≤ 1, where if d = 1, the response is undrained (as described previously) and if d = 0, the response is effectively null; that is, pressure changes are completely independent of the size of the imposed strain. A value of d < 0 is unphysical in this situation. It is important to note that although the exponent d is scale independent, the modulus depends on the normalization used. Consequently, the coefficient should always be viewed as an apparent modulus, and any variations in it should be interpreted in the context of the strain and pressure normalizations.

In order to verify the quality of the estimates of , and d, I compare results from iteratively reweighted (“robust”) least squares—a penalized maximum likelihood method—with equivalent results from constrained (d≥0) least squares and with the posterior distribution of a Markov chain Monte Carlo simulation using a Gibbs sampler—a linear regression using Bayesian statistics [cf. Smith and Roberts, 1993]. In all cases the apparent modulus ( ), and strain exponent (d), can be determined with high statistical confidence, even for the relatively small data sets. For future studies, however, it is important that more data at high dynamic strains be considered, since these may help reduce standard errors in the parameter estimates. I find that all of the pore pressure systems show some response to dynamic strains associated with seismic waves, which is generally expected (e.g., equation 5). In Table 2 I report the coefficients determined by robust least squares; Table S2 in the supporting information gives all of the estimates for each station.

Table 2. Strain‐to‐Pressure Response Coefficients for PBO Borehole Stations

	Bootstrap Ratioa^a Nonparametric bootstrap of the population mean of the logarithm of the ratio of observed peak pressure to peak strain (1000 resamples), given with 95% confidence intervals.		Regressionb^b Results based on robust regression of the observed peak pressure against the observed peak strain (equation 8) using iteratively reweighted least squares and a Huber‐type loss function, given with standard errors of the parameter estimates (σ). Note that is an apparent modulus which is normalization dependent (see text).
	(log  Pa ε^−1)		(log  Pa ε^−1)
Station	Mean	95%			d	σd
B088	9.852	0.062	9.787	0.255	0.989	0.034
B084	8.966	0.110	8.922	0.133	0.982	0.018
B005	9.711	0.123	9.447	0.571	0.957	0.073
B073	9.408	0.102	9.128	0.219	0.948	0.028
B076	10.120	0.134	9.066	0.379	0.851	0.048
B067	9.005	0.103	7.733	0.400	0.823	0.052
B078	10.035	0.113	8.509	0.608	0.805	0.074
B011	8.727	0.154	7.350	0.461	0.799	0.064
B012	9.076	0.104	7.371	0.411	0.774	0.052
B010	9.985	0.153	8.232	1.019	0.773	0.128
B022	8.941	0.197	6.307	0.751	0.671	0.097
B004	9.378	0.167	6.317	0.627	0.606	0.079
B058	8.824	0.138	4.866	0.521	0.503	0.065
B087	8.557	0.206	3.849	0.867	0.374	0.115
B066	9.815	0.193	4.369	0.992	0.306	0.124
B003	9.221	0.256	2.774	1.079	0.206	0.132
B082	8.456	0.211	2.143	0.739	0.151	0.099
B081	8.190	0.310	1.553	0.698	0.090	0.095
B079	9.094	0.207	1.507	0.905	0.061	0.112
B946	8.441	0.320	1.269	1.428	0.049	0.189
B086	8.371	0.347	1.344	0.578	0.043	0.078
B028	8.937	0.241	1.239	0.792	0.038	0.100

• ^a Nonparametric bootstrap of the population mean of the logarithm of the ratio of observed peak pressure to peak strain (1000 resamples), given with 95% confidence intervals.
• ^b Results based on robust regression of the observed peak pressure against the observed peak strain (equation 8) using iteratively reweighted least squares and a Huber‐type loss function, given with standard errors of the parameter estimates (σ). Note that is an apparent modulus which is normalization dependent (see text).

The earthquake catalog I use does not support testing for temporal variations in response coefficients, which might be expected if there are transient enhancements in permeability [Elkhoury et al., 2006; Manga et al., 2012]. Although it is difficult to determine if the observational scatter is associated with permeability enhancements, the magnitude‐distance relationship used to select events ensures that energy densities of the associated seismic waves are below empirical thresholds compiled by Wang and Manga [2009].

In the case of SJB earthquake, there is no measurable difference in local response from waves that are focused along the San Andreas Fault, but this effect may only be significant in strain measurements for more energetic earthquakes at similar distances.

Figure 3 shows the ratio between pressure and strain obtained for each event (light tick marks) at each station, which is a first‐order estimate of the size of the pore pressure as a function of strain. Confidence intervals for the mean value of this ratio are calculated by nonparametric bootstrap resampling of the logarithm of this ratio, and these show significant variations across the network. Stations with relatively small confidence regions (e.g., B088) exhibit the expected linear, undrained response. For the Anza stations, in Southern California, ratios based on BSM observations are comparable to ratios found with the predicted strains from the scaling relationships in Agnew and Wyatt [2014]: this confirms that BSM calibration coefficients are sufficiently accurate since the strain prediction equations were derived independently from long‐baseline laser strain measurements at the Piñon Flat Observatory (PFO) in Anza. In Table 2 I report the ratios and their confidence intervals and the apparent moduli; in Figure S1 in the supporting information I compare them visually.

Figure 3

Ratios of peak values of excess pore pressure and areal strain from seismic waves at PBO boreholes, estimated from envelope functions (see section 4), in units of Pa ε⁻¹. The square point ranges, labeled and colored by geographical region, show 95% confidence intervals and population means, from nonparametric bootstrap resampling; the circle point ranges for the Anza instrument cluster show a similar bootstrap result, instead using the predicted areal strains from Agnew and Wyatt [2014], a scaling relationship based on dynamic strains recorded by the long‐base laser strainmeters at the Piñon Flat Observatory, where station B084 is located. Station B001 has only a single record with a 1 min sampling rate.

The bootstrap mean ratios vary from 10^8.2 to 10^10.1 Pa ε⁻¹ (0.158 to 12.6 GPa ε⁻¹), which is a surprising result given that the average distance between stations is small relative to the principal wavelengths of the surface waves. For example, in Anza, station pairs such as B946 and B084 are separated by only ∼16 km, which is small compared with surface wave wavelengths of at least ∼50 km for a ∼20 s Rayleigh wave. Despite the station separation being more than a factor of 3 smaller than the principal wavelength, the mean ratios at these two stations differ by a factor of 3.4 (Table 2). Such a large variation in response is not likely due to wavefield effects. It is worth noting that while station B084 is cemented to competent rock at PFO, B946 is cemented to cataclastic rock (A. Allam, personal communication, 2014).

Some amount of scatter seen in the data may be a consequence of simply taking peak values, since to some degree the results depend on how the signals are processed (e.g., filter details); but another likely explanation might be that wave refraction along the path of propagation induces significant, nonzero, transverse extensional strains, as Agnew and Wyatt [2014] found in some long‐base laser strain records. Strain envelopes might thus be slightly distorted by wave interference, and from some amount of coupling with differential extension (defined as E_θθ−E_rr), a component of strain which involves no areal or volume change, which may lead to considerable observational scatter. Despite this, the independent ratio‐based and regression‐based measures of seismic response confirm the strong spatial variations in ratios seen across the PBO network and within instrument clusters having characteristic interstation distances that are generally smaller than the principal wavelengths of the surface waves. In the following section I examine spatial variations within the Parkfield and Anza clusters, which show an apparent relationship with distance to nearby strike‐slip faults.

5 Variations in Response Around Active Faults

Due to limited station coverage, it is unclear how the responses at stations in Northern California, Oregon, Washington, and on Vancouver Island (British Columbia) are related to active deformation occurring along the Cascadia subduction system or near the transition between the San Andreas Fault and the Cascadia system at the Mendocino Triple Junction. In contrast, instrument clusters in Central and Southern California—San Juan Bautista, Parkfield, and Anza—show variations in response that appear to depend on their location relative to the San Andreas and San Jacinto strike‐slip fault systems.

5.1 San Juan Bautista and Parkfield Clusters

The spatial pattern of response coefficients in the San Juan Bautista and Parkfield instrument clusters, in Central and Northern California (Figure 4), appears to be related to fault‐parallel variations in shear strain rates along the San Andreas Fault (SAF), reflected by variations in dextral creep rates. Figure 5 shows the observations, the maximum likelihood estimates of the scaling relationship, and the Bayesian posterior distribution discussed above, separated by station. The geometry of the these clusters facilitates inspection of variations in response mostly at similar distances from the fault, and Figure 6 shows regression coefficients along a great circle roughly parallel to the strike of the fault, comparing them to interseismic creep rates compiled by Tong et al. [2013].

Figure 4

(left) Map of PBO borehole stations and Quaternary faults in Central and Northern California. Triangles are stations with only a BSM, and circles are stations with a BSM and PP. Squares show the spatial extents of the focus maps to the right. (right) Focus maps of stations along the San Andreas Fault, at San Juan Bautista (top), and Parkfield (bottom). Circles are double‐difference relocated seismicity (scaled by magnitude) since 2000. Inverted triangles are locations of geodetic estimates of average dextral creep rates and their 2σ uncertainties [from Tong et al., 2013]. In the Parkfield map, the largest earthquake shown is the 2004 M_W6 earthquake [Bakun et al., 2005]; diamonds are deep tectonic tremors since 2000 [Nadeau and Guilhem, 2009], and squares are locations of the SAFOD borehole, the Little Cholame pore pressure station (operated by the U.S. Geological Survey (USGS)), and the two long‐base laser strainmeters at CHL (topographic images ©2015 Google Inc.).

Figure 5

Scaling relationships between excess pore pressure and areal strain at (top row) San Juan Bautista and (bottom row) Parkfield PP‐BSM stations. Each frame shows peak values in high‐pass‐filtered envelope functions of strain and pressure (as in Figure 3), along with best fitting regression lines from (1) iteratively reweighted (robust) least squares using a Huber‐type loss function [Huber and Ronchetti, 2009] and corresponding residual standard error (RSE), (2) constrained (nonnegative) least squares, and the (3) 95% coverage of posterior draws from a Gibbs‐type Markov chain Monte Carlo (MCMC) simulation of the scaling relationship (filled region) and the corresponding standard deviation of the random noise component (σ).

Figure 6

Spatial variations in regression coefficients response at the Parkfield and San Juan Bautista instrument clusters (shown in Figure 5). (first panel) Map of PBO borehole stations and the main surface trace of the San Andreas Fault in a UTM projection, relative to a great circle passing through the fault near Parkfield; the origin is at Highway 46 near Cholame. Circles are PBO BSM+PP stations, and triangles are PBO BSM‐only stations. The star marks the epicenter of the 2004 M_W6 earthquake, and grey dots give locations of tectonic tremor for reference. (See also Figure 4). Regression coefficients from equation 7: (second panel) (an apparent modulus) and (third panel) d (the strain exponent) shown with their standard errors (±2σ). (third panel) Geodetic estimates of dextral creep rates (±2σ) from Tong et al. [2013], which closely match geologic‐based estimates [e.g., Titus et al., 2011].

There is a significant reduction in both the apparent modulus and the strain exponent of the pore pressure response that appears to depend directly on creep rates. At Parkfield, for example, where the stations are roughly equidistant from the fault, the response is extremely small in the locked zone, whereas along the creeping section the response is strong. This type of spatial pattern suggests a dependence on rates of secular shear strain accumulation across the fault, which in the case of the SAF can vary significantly along strike: profiles of strain are expected to be broader in the locked section than in the creeping section, where they are expected to be focused at the fault owing to shallow creep. Gradients in geodetic velocity profiles [Shen et al., 2011; Tong et al., 2013] highlight this expected pattern.

5.2 Anza Cluster

The spatial pattern of response coefficients in the Anza instrument cluster, in Southern California (Figure 7), also appears to be related to the major active fault system nearby: the San Jacinto Fault (SJF) system. The geometry of the Anza cluster facilitates inspection of spatial variations in the fault‐perpendicular direction, and Figure 8 shows the observations and regressions separated by station. Figure 9 shows a representative set of time series from the 2011 M_W9 Tohoku‐oki earthquake, including tensor strain components at station B084 and pore pressure records at stations B084, B087, and B081, which are sorted by their proximity to the main trace of the SJF; these show increasingly complex frequency dependence toward the fault, which spectral analyses (described later) also confirm.

Figure 7

(left) Map of PBO borehole stations and Quaternary faults in Southern California. Triangles are stations with only a BSM, and circles are stations with a BSM and PP. The square shows the spatial extent of the focus map to the right. (right) Focus map of stations in the Anza area: open triangles are stations with only a BSM, and circles are stations with a BSM and PP. Small diamonds are double‐difference relocated seismicity from Yang et al. [2012]; large diamonds are locations of M_W ≥ 5 events since 1981. The dynamic strain scaling relationship based on observations from the three long‐base laser strainmeters (squares) at the Piñon Flat Observatory [Agnew and Wyatt, 2014] was used in Figure 3 (topographic images ©2015 Google Inc. and INEGI).

Figure 8

Scaling relationships between excess pore pressure and areal strain at Anza PP‐BSM stations. See Figure 5 for a description.

Figure 9

Time series illustrating the varying pore pressure response to teleseismic strains from the 2011 M_W9 Tohoku‐oki earthquake within the Anza instrument cluster. (a) Tensor strain components at station B084, rotated such that “radial extension” is uniaxial extension in the direction of wave propagation; these are highly representative of strain time series from other stations. (b) Pore pressure responses at stations B084, B087, and B081, sorted by their approximate distance to the San Jacinto Fault. Note the decreasing amplitude and increasing frequency dependence: at B081 only the Rayleigh waves induce a visible pressure perturbation on this scale. (c) Zero‐lag correlations between pore pressure and areal strain at each station in Figure 9b, with a reference line (equation 3) of 1 GPa ε⁻¹, comparable to the El Mayor Cucapah example shown in Figure 2.

Figure 10 shows regression coefficients for the Anza cluster as a function of distance from the fault and compares them to results for the Parkfield and San Juan Bautista clusters. Stations located within 5 km of an actively creeping section of the San Andreas Fault (namely, B073, B076, B078, and B066) are shown with muted colors, since shear strain rates at those stations are likely dominated by shallow creep, rather than deep slip elsewhere. It is clear that a reduction in response occurs over multiple kilometers away from the principal fault surface, as compared to stations farther away, and appears to vary smoothly (within observational error); this again suggests a connection to interseismic strain rates in the crust.

Figure 10

(a) Observed variations in regression‐based response coefficients (equation 7) at stations in the Anza, Parkfield, and San Juan Bautista instrument clusters (depicted in Figures 5 and 8) as a function of distance from the nearest fault. For the Anza cluster, this distance is to the nearest major segment in the San Jacinto Fault (SJF) system, which includes either the Coyote Creek, San Jacinto, Hot Springs, or Buck Ridge fault; otherwise, the distance is to the San Andreas Fault (SAF). The apparent modulus, , is shown in the top frame, and the strain exponent, d, is shown in the bottom frame; all values are shown with ±2σ uncertainty. Stations located within 5 km of an actively creeping segment of the fault are shown with muted colors. (b) An illustration of the physical mechanism proposed to explain the observed patterns. Fractures in the crust are distributed away from the fault following a power law [e.g., Savage and Brodsky, 2011] but are only hydraulically conductive if they are critically stressed within the local stress state [e.g., Barton et al., 1995]. Continuous shear strain imposed by tectonic loading (∂E₁₂/∂t) promotes aperture dilation [e.g., Willis‐Richards et al., 1996], assuming strain rates are greater than chemical and mechanical healing rates; the spatial profile in strain can be strongly affected by shallow creep. Hydromechanical coupling between fracture aperture and flow rate (e.g., the “cubic” law) leads to spatial variations in effective permeability and mechanical compliance, with strong enhancements located near the fault [e.g., Chester and Logan, 1986]. The associated increases in hydraulic diffusivity have the effect of shifting the purely undrained portion of the frequency‐dependent response to longer periods [e.g., Rojstaczer, 1988a], which predicts that the pressure response to strain will be reduced near the fault.

Recent geodetic studies [Lindsey et al., 2013] confirm the lack of shallow creep along the San Jacinto Fault; hence, the spatial pattern in strain accumulation there is expected to be broad. Even though strain rates on the SJF are of the order of the southern portion of the SAF [Lindsey and Fialko, 2013], the SJF has at least an order of magnitude less cumulative displacement than the SAF [Dickinson, 1996] and is structurally complex [Zigone et al., 2014]. This implies that spatial variations in pore pressure response across the SJF are aggregate effects from multiple fault strands, since the spatial pattern of strain rates may not be associated with a principal fault surface.

5.2.1 Tidal and Spectral Response

Even though scaling relationships between pressure and strain based on seismic observations are robust, it is instructive to verify them using a different set of strain signals and alternative methods of analysis. I do this for the well‐studied cluster of instruments in Anza by analyzing the time‐varying response to tidal strains and the cross spectrum of the response to dilatational strains associated with Rayleigh waves.

5.2.1.1 Tidal Response

The volumetric strain sensitivity of an aquifer to solid Earth tides is considered to be equivalent to equation 5 [Rojstaczer and Agnew, 1989], and here I analyze pore pressure data at tidal frequencies for stations B082 and B084 from 2007:109 to 2011:299, estimating the amplitude and phase of the primary tidal constituents M2 and O1, the lunar semidiurnal and solar diurnal cycles, using nonoverlapping 30 day windows. Tidal amplitude estimates are compared to the predicted areal strains from a solid Earth model corrected for ocean loading effects [Agnew, 1997]. Figure 11 shows the strain and pressure tidal amplitudes from these analyses, along with weighted least squares estimates of the strain‐to‐pressure proportionality constant and 95% confidence intervals: 4.5 ± 1.3 GPa ε⁻¹ and 12.5 ± 0.3 GPa ε⁻¹ for B082 and B084, respectively. These values may be biased high, though, since they do not account for effects of unconstrained specific storage [Wang, 2000].

Figure 11

Pore pressure response to tidal strains at stations B082 and B084, in the Anza instrument cluster. Data points are time‐varying amplitude estimates of the principal diurnal (O1) and semidiurnal (M2) constituents between the time shown. Dashed lines are reference curves, and thick lines are best fitting curves after excluding spurious estimates (lighter colored points): 4.5 ± 1.3 and 12.5 ± 0.3 GPa ε⁻¹, respectively, shown with 95% confidence prediction intervals. For B082, 15 of 54 estimates during this time period are deemed spurious, whereas at B084 only 3 of 54 are deemed spurious. The greater variability in pore pressure response and number of spurious points at B082 is due to daily and subdaily effects associated with anthropogenic fluid extraction [Barbour and Wyatt, 2014].

Next I estimate the relative compressibility of rock at B082 and B084. Combining the P wave velocities from geophysical logs with scaling relationships from Brocher [2005], I estimate median porosities to be 16% and 9%, respectively. (Figure S2 in the supporting information shows the distribution of values for the V_P‐based estimates of rock density and porosity for each station.) Assuming that the pore space of the formation is saturated with water having a compressibility of 4.4 × 10⁻¹⁰ Pa⁻¹, these porosity estimates imply bulk moduli of 3.8 GPa and 12.4 GPa, respectively, which are comparable to seismic‐based estimates, although not precisely equivalent.

5.2.1.2 Spectral Response

Even though surface waves are not stationary signals in a strict sense, they are approximately harmonic over short periods of time (quasi‐stationary), and cross‐spectrum (csd) estimation can be used to calculate an empirical transfer function as a function of frequency, R(f), which maps the amplitude spectrum of strain, S_ε, to the amplitude spectrum of pore pressure, S_p [e.g., Hsieh et al., 1987; Rojstaczer, 1988b]. The statistical description of this mapping is

(9)

where is a normally distributed random effect. For a given csd based on the covariance between strain and pressure signals, S_εp, the coherence spectrum is

(10)

where S_ε is the power spectral density (psd) of the strain signal, S_p is the psd of the pressure signal, and f_N is the Nyquist frequency (one half the sampling frequency). The absolute gain of R—the admittance—is

(11)

And finally, the phase spectrum Θ is the argument of the csd or the ratio of the real (ℜ) and imaginary (𝔍) components of S_εp:

(12)

which is bound within −π ≤ Θ ≤ π. See the appendix for details on how G(f) and Θ(f) are estimated in practice.

Figure 12 shows csd estimates for time series from the 2011 Tohoku‐oki earthquake, at Anza stations B081, B087, B084, and B088 (see Figure 9); values with statistically insignificant coherence are shown as muted colors. Statistically significant admittance and phase spectra are relatively flat over the surface wave frequency band, indicating that pressure is related linearly to strain: the undrained assumption is reasonable at most frequencies, although the applicable frequency band narrows closer to the fault.

Figure 12

Sine multitaper cross‐spectrum analyses of pore pressure and areal strains from the 2011 M_W9 Tohoku‐oki earthquake, where strains are treated as “input” to the physical system, producing pore pressure perturbations (“output”). See section 5.2.1, and the appendix, for details on cross‐spectrum estimation. (top) Coherence spectra, with grey bands highlighting the spurious noise sources identified by Barbour and Agnew [2011]; statistically significant estimates are highlighted by thick lines. (middle) Admittance spectra representing the frequency‐dependent “gain” between pressure and strain, expressed in GPa ε⁻¹. (bottom) Unwrapped phase spectra in degrees of lag between pressure and strain; a value of 180° corresponds to a pressure signal exactly out of phase with a strain signal, as the example in Figure 2 suggests.

Signals from station B087 are notable exceptions to the expected linear pressure‐strain relationship: frequency‐dependent admittance and shifted phase spectra (a roughly 60° greater lag in pressure from the expected lag of 180°) indicate that the response there may be influenced by fracture geometry [Bower, 1983]. And the sensing volume (tube) at B087 is not sealed, which could make the analysis prone to biases from atmospheric and inertial effects.

Radial flow models are commonly used to estimate hydraulic and mechanical properties of the rock [e.g., Ohno et al., 1997]; however, in this case, spurious noise signals in the strain records at surface wave frequencies [Barbour and Agnew, 2011] create artificial coherence dropouts at short periods (Figure 12) and distort the amplitude and phase spectra nearby though spectral leakage. Unfortunately, because important details of the spectral response are affected, it is difficult to obtain robust parameter estimates using surface waves, but I calculate the elastic moduli of rock based on admittance estimates in the undrained frequency band at each station, over a range of Skempton's coefficients (see Table 3). It is apparent that the elastic properties of the surface rock are reduced substantially in the vicinity of the fault, which is consistent with tidal and regression results. It is also worth noting that while spectrum‐based estimates of poroelastic moduli agree with bootstrap estimates of the mean ratio of pressure to strain (Figure 3), they only agree with regression‐based results when adjusted for the associated strain exponent (d). In other words, since regression‐based results give an apparent modulus, affected by strain normalization, any comparisons with such should account for d, especially when d << 1. For example, the ratio of uncertainties in pore pressure and areal strain is when c and d are known.

Table 3. Elastic Moduli as a Function of Skempton's Ratio, B, Estimated From the Cross Spectrum

				Elastic Modulib^b Calculated for ν = 1/4,νu = 1/3.
	Band	Gaina^a Estimate with standard error (below, in parentheses) in the frequency band shown—the undrained frequency band.		Bulk, κu	Shear, μ
Station	(s)	(GPa ε^−1)		(GPa)	(GPa)
B088	7–500	7.750	B = 0.1	116	70
		(0.010)	0.3	39	23
			0.5	23	14
			0.7	17	10
			0.9	13	8
B084	8–265	1.019	B = 0.1	15.3	9.2
		(0.006)	0.3	5.1	3.1
			0.5	3.1	1.8
			0.7	2.2	1.3
			0.9	1.7	1.0
B087c^c May be contaminated by fracture geometry (see Figure 12).	16–73	0.373	B = 0.1	5.59	3.35
		(0.025)	0.3	1.86	1.12
			0.5	1.12	0.67
			0.7	0.80	0.48
			0.9	0.62	0.37
B081	17–26	0.044	B = 0.1	0.664	0.398
		(0.003)	0.3	0.221	0.133
			0.5	0.133	0.080
			0.7	0.095	0.057
			0.9	0.074	0.044

• ^a Estimate with standard error (below, in parentheses) in the frequency band shown—the undrained frequency band.
• ^b Calculated for ν = 1/4,ν_u = 1/3.
• ^c May be contaminated by fracture geometry (see Figure 12).

5.3 Effects of Faulting in Response

The effects of earthquakes on near‐surface rock include enhancements in both mechanical compliance (reductions in elastic shear modulus or “rigidity”) and permeability; these enhancements tend to decay with distance in the fault‐perpendicular direction [Chester and Logan, 1986]. The history of slip on a fault is an effective predictor of the spatial density of macroscale damage features (fractures) around it [Savage and Brodsky, 2011], where decay rates in density scale with the maturity (cumulative displacement) of faults. But fractures in crystalline rock are not necessarily hydraulically conductive, unless their orientation within the local stress field places them near the frictional failure (critically stressed) state: then they are generally always hydraulically conductive [Barton et al., 1995], at high and low confining stresses [Ito and Zoback, 2000; Townend and Zoback, 2000]. And because the stress state of the crust is modified by faulting [e.g., Hardebeck and Michael, 2004; Hickman and Zoback, 2004; Cochran et al., 2009], it is possible that spatiotemporal variations in fault‐rock permeability depend on the time history of fault slip.

As Figure 10 shows, the response near active strike‐slip faults is related to distance from the fault. Based on this set of observations, for the San Andreas Fault in Central California and the San Jacinto Fault in Southern California, I propose that the apparent relationship between pore pressure response and distance is associated with secular shear strain accumulation: significant reductions in response may be the result of effective permeability enhancements due to mechanical propping (via shear displacement) of critically stressed (hydraulically conductive) fractures. This mechanism for shear‐enhanced crustal permeability has been proposed before to explain certain structural features [e.g., Sibson, 1996] but has previously never been observed directly with reductions in water pressure response. Further work, however, must be done which foregoes using fault distance (a proxy for strain) in favor of geodetic‐based strain estimates: nuances of strain accumulation patterns can be accounted for, so that stations near creeping faults can also be considered.

A connection between secular strain patterns across faults and permeability enhancement might be expressed in microseismicity patterns if mechanical propping is not consistent with chemical healing rates, if new fractures form or if existing fractures propagate. But significant differences between the spatial distributions of seismicity and occurrence rates on the SAF and the SJF make this a very difficult feature to test for. It is also unlikely that tectonic tremor reflects this connection, especially since locations associated with the SAF are generally confined to the crustal root [Shelly and Hardebeck, 2010], and so far, only events triggered by the 2002 M_W7.8 Denali earthquake have been identified for the SJF [Wang et al., 2013].

Lastly, there is the issue of the level of reduction in elastic modulus. As was mentioned previously, the regression‐based results represent an apparent modulus; hence, the true reductions in modulus are likely multiple orders of magnitude smaller than those seen in Figure 10, for example. But in consideration of recent tomographic imaging [Allam and Ben‐Zion, 2012; Allam et al., 2014], observations from dense seismic array deployments [Zigone et al., 2014], space geodetic studies [Lindsey et al., 2013], inferences from poroelastic modeling [Barbour and Wyatt, 2014], and the data presented here, it is highly likely that significant reductions in modulus are ubiquitous around the San Jacinto Fault. More work is needed, though, to firmly establish the magnitudes and spatial extent and of these reductions.

6 Conclusions

I have used seismic waves to show that pore pressure in formations tapped by PBO borehole instruments responds to dynamic strains with different strengths at different locations. I find variations in strength across the entire PBO network, with the strongest variations occurring within instrument clusters having characteristic dimensions much smaller than the principal seismic wavelengths.

Focusing on observations from instrument clusters around strike‐slip faults in California, I find an apparent relationship between the spatial variation in response and secular shear strain accumulation rates, which result at Parkfield and San Juan Bautista from variations in creep rate and at Anza from fault‐perpendicular distance. For the Anza cluster, tidal and cross‐spectrum analyses confirm the effective decrease in pore pressure response seen in the time series (e.g., Figure 9) for stations near the fault.

Variations in pore pressure response along the San Andreas Fault, as observed at the Parkfield and San Juan Bautista clusters, suggest varying susceptibilities to dynamic stress triggering by pore pressure perturbation, but the depths to which these patterns of reduced response persist have yet to be investigated.

The results presented here also establish that with sets of high‐frequency pore pressure and strain measurements made at strategic locations around active faults, we may gain direct insight into the interactions between dynamic stresses and the behavior of shallow, fluid‐saturated rock.

Acknowledgments

Strain data were obtained from Incorporated Research Institutions in Seismology web services (http://service.iris.edu/irisws/), and pore pressure data were obtained from the UNAVCO archive (http://pore.unavco.org/pore/). The Parkfield earthquake catalog was obtained from the Northern California Earthquake Data Center (http://dx.doi.org/10.7932/NCEDC) through their composite earthquake catalog http://www.ncedc.org/ncedc/catalog-search.html), and tremor locations were obtained from the TremorScope online catalog (http://seismo.berkeley.edu/research/recent_tremor.html). I used the bayesm software package (http://cran.r-project.org/package=bayesm, version 2.2) to do the Markov chain Monte Carlo computations, the ggmap software package (http://cran.r-project.org/package=ggmap, version 2.4) to download topographic images from the Google Static Maps API, and the kitagawa software package (http://cran.r-project.org/package=kitagawa, version 2.1) with Bob Parker's cross‐spectrum estimation program, cross, for spectral analyses. Websites were last accessed in July 2015. I thank Xiaopeng Tong for providing San Andreas Fault creep rate data. Input from Mike Gottlieb, Wade Johnson, and Dave Mencin on the instruments and their installations have been invaluable. Constructive comments from Duncan Agnew, Steve Hickman, Art McGarr, and Frank Wyatt strengthened the focus of this work. This paper ultimately benefited from critical reviews by Evelyn Roeloffs and David Shelly (U.S. Geological Survey) and two anonymous reviewers. I thank the Associate Editor, Mike Poland, for his additional comments. Support for this work came from NSF grants EAR10‐53208 and EAR12‐51568 and the USGS Mendenhall Postdoctoral Fellowship program.

Appendix A: Cross‐Spectrum Estimation

In this appendix are details of the cross‐spectrum estimation method used to calculate empirical transfer functions of pore pressure from areal strain. I use the adaptive sine multitaper method of Prieto et al. [2009], which adjusts the optimal number of tapers at each frequency to minimize the mean square error, the sum of squared bias, and variance [see also Barbour and Parker, 2014].

Prior to spectral estimation, the input series is band‐pass filtered to have maximum energy density in the desired frequency band, 10⁻³ Hz to 1 Hz, and to minimize the bias in the spectral estimates. The strain and pressure series are then aligned in time using cross correlation, and the cross spectrum is estimated; the effect of removing the delay is added back to the phase spectrum.

Uncertainties in the real and imaginary components of the transfer function are uncorrelated and have equivalent variance; hence, standard errors of the estimates are well approximated using the coherence, γ², and the number of tapers K applied to the spectral density estimates at each frequency:

(A1)

Priestley [1981] shows that a combination of K and the coherency (γ) is distributed as a central F distribution:

(A2)

From this statistic it follows that the probability that the absolute coherency is greater than some threshold λ (conditional on K) is

(A3)

and in this paper I use 95% coverage probabilities to test for significance.

Standard models for the response of an aquifer system to harmonic waves (e.g., long‐period seismic waves) are based on a homogeneous, isotropic flow model where the waves induce strain in a confined aquifer (one having aquitards above and below it) and fluid flows radially (horizontally) into and out of a well penetrating the aquifer. The solution for seismic waves causing fluctuations in an open well was first given by Cooper et al. [1965], with subsequent improvements in accuracy given by Hsieh et al. [1987], Rojstaczer [1988a], and Liu et al. [1989], which account for geometric effects from the well and aquifer; Kitagawa et al. [2011] adapted the Hsieh et al. [1987] model to a sealed well. The Rojstaczer [1988a] and Kitagawa et al. [2011] models are appropriate for radial flow induced by quasi‐static or harmonic volume strains at open and sealed wells, respectively.