Here I show how the modeling tools in kitagawa can be used to study actual data. Specifically, I will show records of strain and pore-pressure from borehole stations in the Plate Boundary Observatory in a manner similar to the approaches taken in @barbour2011 and @barbour2015.

library(kitagawa)
## Loaded kitagawa (2.2.2) -- Spectral response of water wells

Pore Pressure Changes from Teleseismic Strains

We first load the psd package, which includes a suitable dataset for this example. In particular, we're interested in assessing the frequency-dependent relationship between pore pressure \(p\) and areal strain \(E_A\)[Relative changes in borehole diameter, which can be related to volume strain in the rock] during the seismic portion of the record.

library(psd)
## Loaded psd (1.0.1) -- Adaptive multitaper spectrum estimation
data(Tohoku)
toh_orig <- with(subset(Tohoku, epoch=='seismic'), {
  cbind(
    scale(1e3*areal, scale=FALSE), # scale strain to nanostrain, remove mean
    scale(1e2*pressure.pore, scale=FALSE) # scale hPa to Pa, remove mean
  )
})
colnames(toh_orig) <- c('input','output')
toh.dat <- window(ts(toh_orig), 100, 2400)

Note how the records of this earthquake – the 2011 \(M_W 9\) Tohoku-Oki earthquake some thousands of kilometers away – are very nearly a mirror image of each other: plot of chunk unnamed-chunk-3 This indicates that the pore pressure response can be modeled as a convolution of an input signal (dynamic strain) and transfer function (\(p = G \star E_A\)). It also says that energy carried by the seismic wavetrain is focused predominately at long periods and very nearly harmonic; this is consistent with the theory of linear poroelasticity, which predicts that \[ p \approx - \frac{4}{3} B \mu E_A \] assuming an undrained Poisson's ratio of \(1/3\), where \(B\) is the Skempton's coefficient and \(\mu\) is the elastic shear modulus of the fluid-saturated rock. In this case the (scalar) proportionality implied by the timeseries is -2.265 \(GPa / \epsilon\), but we will see how this is actually frequency dependent. plot of chunk unnamed-chunk-4

Cross-Spectrum Estimation

First let's use Don Percival's package sapa to estimate a cross spectrum between pressure and strain, treating strain as the input to the system and pressure as the output:

library(sapa)
k <- 2*130
toh.cs <- sapa::SDF(toh.dat, method='multitaper', n.taper=k, sampling.interval=1)
print(toh.cs)
## Cross-Spectral Density Function estimation for toh.dat
## ------------------------------------------------------
## Cross-SDF labels          : S11 S12 S22 
## Length of series          : 2301 
## Sampling interval         : 1 
## Frequency resolution (Hz) : 0.0002172968 
## Centered                  : TRUE 
## Recentered                : FALSE 
## Single-sided              : TRUE 
## Method                    : Multitaper 
## Number of tapers          : 260 
## Taper: sine
##    Number of points: 2301 
##    Number of tapers: 260 
##    Normalized: TRUE

plot of chunk unnamed-chunk-6

Estimating the Response: Coherence, Admittance, and Phase

If the results of the SDF computation give a matrix of complex spectra \([S_{11}, S_{12}, S_{22}]\), the coherence spectrum \(\gamma^2\) can be calculated by \[ \gamma^2 = \frac{\left|S_{12}\right|^2}{S_{11} S_{22}}, \] the admittance spectrum (or gain) \(G\) can be calculated from \[ G = \gamma \sqrt{S_{22} / S_{11}}, \] and the phase spectrum \(\Phi\) can be calculated from \[ \Phi = \arg{S_{12}} \]

f <- as.vector(attr(toh.cs, 'frequency'))
lf <- log10(f)
p <- 1/f
lp <- log10(p)
S <- as.matrix(toh.cs)
colnames(S) <- attr(toh.cs, 'labels')
S11 <- S[,'S11']
S12 <- S[,'S12']
S22 <- S[,'S22']
Coh <- abs(Mod(S12)^2 / (S11 * S22))
G <- abs(sqrt(Coh * S22 / S11))
Phi <- atan2(x = Re(S12), y = Im(S12))
Phi2 <- Arg(S12)
all.equal(Phi, Phi2)
## [1] TRUE

As @priestley1981 shows, the multitaper coherency spectrum (\(\gamma\)) can be described by an \texit{F} distribution: \[ \frac{2 k \gamma}{(1-\gamma)} \sim F(2,4k) \] Hence, the probability that the absolute coherency is greater than \(c\) is \[ P(|\gamma| \geq c, k) = (1 - c^2)^{k-1} \]

gam <- seq(0.001, 1, by=0.001)
gamrat <- 2 * gam / (1 - gam)
Pgam <- pf(k*gamrat, 2, 4*k)

plot of chunk unnamed-chunk-10

The standard error in the admittance follows from the coherence spectrum: \[ \sqrt{(1 - \gamma^2)/k} \]

G.err <- sqrt((1 - Coh) / k)

We can safely assume that the spectral density estimates for periods longer than \(\approx 100\) seconds will be either spurious, or lacking in seismic energy, so we will exclude them.

csd <- data.frame(f, p, lf, lp, Coh, G, G.err, Phi = Phi * 180 / pi)
csd.f <- subset(csd, p <= 100)
is.sig <- csd.f$Coh > coh.99

plot of chunk unnamed-chunk-13

This is now implemented in the function cross_spectrum. In comparison with a Welch-type CSD – calculated by setting k=NULL, the sine multitaper result is more accurate across the full frequency band, and does not degrade at low frequencies:

TohCS <- cross_spectrum(toh.dat, k=50, verbose=FALSE)
TohCS_welch <- cross_spectrum(toh.dat, k=NULL, verbose=FALSE) # turn off k to get a Welch overlapping csd
plot(Admittance ~ Period, TohCS, col=NA, log='x', main="Pore Pressure from Strain: Tohoku", xlab="Period, sec")
lines(Admittance ~ Period, TohCS_welch, col='salmon')
lines(Admittance ~ Period, TohCS, lwd=2)

plot of chunk unnamed-chunk-14

References