3. Specifying and Using Response Information

–By Mike Hagerty & Adam Ringler

3.1. Theory of Instrument Response

In geophysics, we often work with instrument responses in the frequency domain as they are simpler to combine and manipulate For instrument responses specified in the time domain, we must first transform the response of the frequency domain using one or more transforms. Depending on the application, one or more of the following transforms may be used: The Fourier Transform, the Laplace Transform and the z-Transform.

3.1.1. The Fourier Transform

Introduction

The Fourier Transform (\(t \rightarrow \omega\)) is defined by

\[X(\omega)=\frac{1}{2\pi}\int_{-\infty}^{\infty}x(t)e^{-j\omega t}dt\]

while the Inverse Fourier Transform (\(\omega \rightarrow t\)) is given by

\[x(t)=\int_{-\infty}^{\infty}X(\omega)e^{+j\omega t}d\omega\]

There are different conventions in use for the Fourier Transform. The conventions differ in which transform (forward vs. reverse) has the positive sign in the exponent, and which transform is scaled by \(\frac{1}{2\pi}\). Some authors prefer to scale each transform by \(\frac{1}{\sqrt{2\pi}}\) instead. What is important is that forward and reverse transforms must have exponents that are opposite in sign, and the product of the scalefactors must equal \(\frac{1}{2\pi}\).

Discrete Time Fourier Transform (DTFT)

In the Fourier transform pair above, both time (\(t\)) and frequency (\(\omega\)) are continuous parameters. In contrast, for signals sampled discretely in time, we may define the related Discrete Time Fourier Transform (DTFT) as

\[X(\omega)=\frac{1}{2\pi}\sum_{n=-\infty}^{\infty}x[n]e^{-j\omega n}\]
\[x[n]=\int_{0}^{2\pi}X(\omega)e^{+j\omega n}d\omega\]

where \(n\) is the discrete sample number, and \(\omega\) is still continuous.

Discrete Fourier Transform (DFT)

And finally, when both time and frequency are discrete, we define the Discrete Fourier Transform (DFT) pair by

\[X[k]=\frac{1}{N}\sum_{n=0}^{N-1}x[n]e^{-j2\pi kn/N}\]
\[x[n]=\sum_{k=0}^{N-1}X[k]e^{+j2\pi kn/N}\]

Note that the popular Fast Fourier Transform (FFT) is a particular implementation of the DFT.

3.1.2. The Laplace Transform

Introduction

The Laplace Transform is defined by

\[X(\sigma,\omega)=\int_{-\infty}^{\infty}x(t)e^{-\sigma t}e^{-j\omega t}dt\]

If we make the substitution, \(s=\sigma + j\omega\), this becomes

\[X(s)=\int_{-\infty}^{\infty}x(t)e^{-s t}dt\]

Each point in the complex s-plane is associated with a frequency, \(\omega\) and an exponent \(\sigma\). Thus, each point in the s-plane describes a sinusoid of frequency \(\omega\) that is either exponentially growing (\(\sigma>0\)) or exponentially decaying (\(\sigma<0\)) with time.

Note that the Laplace transform evaluated along the imaginary axis (where the attenuation parameter, \(\sigma=0\)), reduces to the complex Fourier transform, \(X(\omega)\).

The Laplace transform at point \(s\) is a measure of the similarity between the input signal, \(x(t)\), and the corresponding exponentially growing/decaying sinusoid. A large value of \(X(s)\) corresponds to a strong similarity between the input signal and the sinusoid \(e^{-(\sigma + j\omega)t}\), indicating a strong presence of the sinusoid in the input signal.

In practice, we are often only interested in causal signals that begin at \(t=0\). Using the unit step function,

\begin{equation*} u(t)=\begin{cases} 1 & t\ge0\\ 0 & t<0 \end{cases} \end{equation*}

we may ensure causality by writing \(x(t)=u(t)x(t)\) , so that the Laplace Transform becomes

\[X(s)=\int_{0}^{\infty}x(t)e^{-s t}dt\]

Poles and Zeros

Suppose we have a data processing system (e.g., analog sensor + datalogger) that can be characterized by the linear differential equation,

\[a_{2}\ddot{y}(t)+a_{1}\dot{y}(t)+a_{0}y(t)=b_{2}\ddot{x}(t)+b_{1}\dot{x}(t)+b_{0}x(t)\]

where \(x(t)\) is the input signal (e.g., the ground motion), \(y(t)\) is the output signal (the signal recorded) and \(a_{k}\) and \(b_{k}\) are constant (time-invariant) coefficients. If we assume the system is causal, so that the signals + derivatives are all 0 for \(t<0\) , then the Laplace Transform of the equation gives

\[a_{2}s^{2}Y(s)+a_{1}sY(s)+a_{0}Y(s)=b_{2}s^{2}X(s)+b_{1}sX(s)+b_{0}X(s)\]

or

\[(a_{2}s^{2}+a_{1}s+a_{0})Y(s)=(b_{2}s^{2}+b_{1}s+b_{0})X(s)\]

From this we can write the system transfer function relating the output to the input as

\[H(s) = \frac{Y(s)}{X(s)}=\frac{b_{2}s^{2}+b_{1}s+b_{0}}{a_{2}s^{2}+a_{1}s+a_{0}}\]

or more generally,

\[H(s) =\frac{\sum_{k=0}^{M}b_k s^n}{\sum_{k=0}^{N}a_n s^n}\]

This is the coefficient representation of the transfer function. It represents the transfer function as the ratio of two polynomials. The roots of the numerator polynomial are called ‘zeros’, while the roots of the denominator polynomial are called ‘poles’.

Often, for analog stages, it is more convenient to factor the transfer function in terms of these poles and zeros:

\[H(s)=\frac{\Pi_{k=1}^{M} (s-z_{k})} {\Pi_{k=1}^{N} (s-p_{k})}\]

where \(z_{k}\) are the M zeros of the system, and \(p_{k}\) are the N poles.

Because the coefficients of the numerator and denominator polynomials are real, the corresponding roots (poles and zeros) must occur in complex conjugate pairs.

Thus, the poles and zeros are either real or form pairs that are symmetric with respect to the real axis in the complex \(s\)-plane. In addition, it can be shown that for systems that are stable and causal, the poles all have real parts \(\le 0\).

Recall that the Laplace transform variable is given by \(s=\sigma+j\omega\). Along the imaginary axis, \(\sigma=0\) and hence \(s=j\omega\). Thus, we may express the complex frequency response of the analog stage by calculating its polezero expansion

\[H(f)=A_0\frac{\Pi_{k=1}^{M} (s-z_{k})} {\Pi_{k=1}^{N} (s-p_{k})}\]

where \(s=j2\pi f\) [rad/s] or \(s=jf\) [Hz].

Thus, given the poles and zeros of an analog stage, in order to properly calculate the stage frequency response, we must know the units of \(s\) used to calculate the poles and zeros.

In StationXML, these units are specified by the PzTransferFunctionType element within the PolesZerosType response stage:

<Stage number="1">
<PolesZeros>
   ...
   </OutputUnits>
      <PzTransferFunctionType>LAPLACE (RADIANS/SECOND)</PzTransferFunctionType>
      <NormalizationFactor>1.0</NormalizationFactor>
      <NormalizationFrequency unit="HERTZ">1.0</NormalizationFrequency>

where the possible values for PzTransferfunctionType are:

  1. “LAPLACE (RADIANS/SECOND)”

  2. “LAPLACE (HERTZ)”

  3. “DIGITAL (Z-TRANSFORM)” (Discussed in next section)

Note also the <NormalizationFactor> with unit “HERTZ”. These units are distinct from those used to identify \(s\) above. The <NormalizationFrequency> specifies the frequency (in Hz) at which the polezero transfer function is normalized. The recommended practice is to choose a value of normalization factor, \(A_0\), that normalizes the polezero expansion to unity at the specified normalization frequency, \(f_n\):

\[|H(f_n)| = 1.0\]

3.1.3. The z-Transform

Introduction

The z-Transform is defined by

\[X(z)=\sum_{n=0}^{\infty}x[n]z^{-n}\]

where

\[ \begin{align}\begin{aligned}z=re^{j\omega}\\z^{-n}=r^{-n}e^{-j\omega n}\end{aligned}\end{align} \]

Notice that on the unit circle, where \(|z|\equiv |r|=1\) and \(z=e^{j\omega}\) , the z-transform reduces to the discrete Fourier transform (DTFT):

\[X(e^{j\omega})=\sum_{n=0}^{\infty}x[n]e^{-j\omega n}\]

The z-transform measures the similarity between the input signal \(x[n]\) and the signal \(z^{-n}\).

\(z^{-n}\) represents exponentially increasing (for r < 0) or decreasing (r > 0) sinusoids. e.g., \(e^{-j\omega n}\) is a sinusoid with angular frequency \(\omega\) [radians/sample] that expands with sample number n.

Thus, the location (value) of \(z\) in the complex plane controls what \(z^{-n}\) looks like.

The fractional or angular frequency, \(\omega\) [radians/sample] is related to the linear frequency of the sinusoid through

\[2\pi[radians/cycle]=\omega[radians/sample]\cdot N[samples/cycle]\]

so the number of samples/cycle is given by

\[N=\frac{2\pi}{\omega}samples/cycle\]

and this corresponds to a period of \(T=N\Delta t\) [seconds],

where \(\Delta t\) is the sampling interval (secs) and is related to the sampling rate by: \(f_{s}=\frac{1}{\Delta t}\). Then the frequency of oscillation is given by \(f=\frac{1}{T}=\frac{1}{N\Delta t}=\frac{f_{s}}{N}\) [Hz]

In other words, as the angle in the complex z-plane goes from \(\omega=0\) to \(\omega=\pi\), the linear frequency goes from \(f=0\) to \(f=f_{Nyq}\) [Hz], where the Nyquist frequency, \(f_{Nyq}=\frac{f_s}{2}\) [Hz].

Thus, in implementing the frequency response of the z-transform (e.g., when calculating the response of a FIR filter), it is common to write it in a way that removes the dependency on the actual sample rate, or

\[X(e^{j\omega})=\sum_{n=0}^{\infty}x[n]e^{-j 2\pi n \frac{f}{f_s}} = \sum_{n=0}^{\infty}x[n]e^{-j 2\pi n f \Delta t}\]

Difference Equations

z-transforms of linear time-invariant (LTI) systems described by difference equations play an important role in signal processing.

The general form of a difference equation is:.

\[\sum_{k=0}^{N}a_{k}y[n-k]=\sum_{k=0}^{M}b_{k}x[n-k],\]

where \(a_{0}\ne0\) (the coefficient of y[n] can’t be zero)

Taking the z-transform of both sides,

\[\sum_{k=0}^{N}a_{k}z^{-k}Y(z)=\sum_{k=0}^{M}b_{k}z^{-k}X(z)\]

or

\[Y(z)=\frac{\sum_{k=0}^{M}b_{k}z^{-k}}{\sum_{k=0}^{N}a_{k}z^{-k}}X(z)\]

From this we can write the system transfer function

\[H(z)=\frac{Y(z)}{X(z)}=\frac{\sum_{k=0}^{M}b_{k}z^{-k}}{\sum_{k=0}^{N}a_{k}z^{-k}}\]

The transfer function is the z-transform of the system impulse response, \(h[n]\), or

\[H(z)=\sum_{n=0}^{\infty}h[n]z^{-n}\]

The transfer function can also be factored in terms of poles and zeros (for \(b_{0}\ne0\))

\[H(z)=\frac{b_{0}}{a_{0}}\frac{\Pi_{k=1}^{M}(1-c_{k}z^{-1})}{\Pi_{k=1}^{N}(1-d_{k}z^{-1})}\]

where \(c_{k}\) are the M zeros of the system, and \(d_{k}\) are the N poles.

For a system to be both stable and causal, its poles must lie inside the unit circle, or \(|d_{k}|<1\) for \(k=1,N\).

z-Transform Frequency Response

How does the location of the poles and zeros of the z-transform influence the complex frequency response, \(H(f)\) ?

We start by only considering the magnitude response, \(|H(f)|\).

The z-transform only exists within a region of the complex z-plane where the infinite sum [eqn X] converges. We call this region the Radius of Convergence (ROC) of the system.

If our system, described by difference equations, is stable, then the ROC must include the unit circle, \(|z|=1\) where

\[ \begin{align}\begin{aligned}H(z)=\frac{Y(z)}{X(z)}=\frac{b_{0}}{a_{0}}\frac{\Pi_{k=1}^{M}(1-c_{k}z^{-1})}{\Pi_{k=1}^{N}(1-d_{k}z^{-1})}\\H(e^{-j\omega})=\frac{b_{0}}{a_{0}}\frac{\Pi_{k=1}^{M}(1-c_{k}e^{-j\omega})}{\Pi_{k=1}^{N}(1-d_{k}e^{-j\omega})}\end{aligned}\end{align} \]

The magnitude of the product is equal to the product of the magnitude, thus

\begin{eqnarray} |H(e^{-j\omega})| &=& \frac{|b_{0}|}{|a_{0}|}\frac{\Pi_{k=1}^{M}|(1-c_{k}e^{-j\omega})|}{\Pi_{k=1}^{N}|(1-d_{k}e^{-j\omega})|} \\ &=& \frac{|b_{0}|}{|a_{0}|}\frac{\Pi_{k=1}^{M}|e^{-j\omega}(e^{j\omega}-c_{k})|}{\Pi_{k=1}^{N}|e^{-j\omega}(e^{j\omega}-d_{k})|} \\ |H(e^{-j\omega})| &=& \frac{|b_{0}|}{|a_{0}|}\frac{\Pi_{k=1}^{M}|(e^{j\omega}-c_{k})|}{\Pi_{k=1}^{N}|(e^{j\omega}-d_{k})|} \end{eqnarray}

In other words, as we traverse the unit circle through circular ‘frequency’, \(\omega\), from \(0-2\pi\), the magnitude of the response depends on the distance between the point on the unit circle, \(e^{j\omega}\), and the zeros, \(|e^{j\omega}-c_k|\), as well as the distance between the point and the poles, \(|e^{j\omega}-d_k|\), or

\[|H(e^{-j\omega})| = \frac{|b_{0}|}{|a_{0}|}\frac{\Pi_{k=1}^{M}|distance-to-zeros|}{\Pi_{k=1}^{N}|distance-to-poles|}\]

Thus, \(|H(e^{j\omega})|\) is small when \(e^{j\omega}\) is near the zeros and it is large when \(e^{j\omega}\) is near the poles.

Examples
Example 1

Consider a system with zeros at \(z=1,-1\) and poles at \(z=0.95e^{\pm j\pi/4}\), with response function

\[|H(e^{-j\omega})| = \frac{|(e^{j\omega}-1)||(e^{j\omega}+1)|} {|(e^{j\omega}-0.95e^{j\pi/4})||(e^{j\omega}-0.95e^{-j\pi/4})|}\]
_images/z-transform_fig1a.png
_images/z-transform_fig1b.png

Poles near the unit circle push the magnitude response up at those frequencies, while zeros near the unit circle pull it down; if the zero is actually on the unit circle, then it forces the magnitude response to be exactly 0 at that frequency.

Example 2

Here’s an example pass-band filter comprised of 8 poles and 8 zeros. We can predict from the position of the poles and zeros that the frequency response will be 0 at \(\omega=0\) and will be maximum near \(\omega=\frac{\pi}{2}\).

_images/z-transform_fig2.png

3.1.4. FIR-IIR Filters

Introduction

As we’ll see in the next section, each of the multiple stages that comprise an instrument response can be thought of as a filter that modifies the amplitude and phase of the original signal (e.g., ground motion) in some way.

In fact, to truly understand instrument response and data processing in general, it is necessary to have some familiarity with digital signal processing.

There are two categories of discrete-time filters that we routinely encounter in seismology:

  1. FIR filters (Finite Impulse Response)

  2. IIR filters (Infinite Impulse Response)

Both filters can be constructed using difference equations, hence, they are often represented in terms of their z-transforms.

FIR filters can be written as:

\[y[n]=\sum_{k=0}^{M}b_k x[n-k]\]

while IIR filters can be written as:

\[y[n]=\sum_{k=0}^{M}b_k x[n-k] + \sum_{k=1}^{N}a_k y[n-k]\]

FIR filters can be thought of as a sum of weighted values of past inputs, \(x[n-k]\) (the so called moving average filter). IIR filters have this same moving average component, but also offer the possibility of feedback, since the current output \(y[n]\) can also depend on a weighted combination of past outputs, \(y[n-k]\).

For a finite input impulse, the subsequent impulse response of a FIR filter is finite. However, because of the dependence on past outputs, the impulse response of the IIR filter is, at least in theory, infinite; it continues long after the input signal has finished.

In the FIR case, the system function, found by taking the z-transform of the difference equation, can be written

\[H(z)=\sum_{k=0}^{M}b_k z^{-k}\]

while for the IIR case, the system function is

\[H(z)=\frac{\sum_{k=0}^{M}b_k z^{-k}} {\sum_{k=0}^{N}a_k z^{-k}}\]

where \(a_0=1\).

The system functions can be factored in terms of their poles and zeros as

\begin{eqnarray} H_{FIR}(z) &=& b_0{\Pi_{k=1}^{M}(1-c_k z^{-1})} \\ H_{IIR}(z) &=& \frac{b_0{\Pi_{k=1}^{M}(1-c_k z^{-1})}} {\Pi_{k=1}^{N}(1-d_k z^{-1})} \end{eqnarray}

Thus, the FIR filter has arbitrary zeros, but only has poles at the origin (\(z=0\)). However, poles (or zeros for that matter) at the origin don’t affect the frequency response since they are located a fixed distance (\(|z|=1\)) from the unit circle.

In contrast, the IIR filter may have both zeros and poles at arbitrary locations, making them especially flexible.

The corresponding impulse responses are found by taking the inverse z-transform of the system functions,

\begin{eqnarray} h_{FIR}[n] &=& \begin{cases} b_n & 0\le n\le M \\ 0 & else \end{cases} \end{eqnarray}

Thus the FIR impulse response is given by the difference equation coefficients, \(b_k\), themselves, and the impulse response dies after \(M\) terms.

The impulse response of the causal parts of the IIR filter can be written as

\begin{eqnarray} h_{IIR}[n] &=& \sum_{k=1}^{N}A_k(d_k)^{n}u[n] \end{eqnarray}

where \(u[n]\) is the unit step function (\(u[n]=1,n\ge 0\)).

Because of the geometric series \(d_k^{n}\), the IIR impulse response decays but never actually reaches zero.

FIR vs IIR

The primary distinguishing factor between FIR and IIR filters is this:

FIR filters are guaranteed to have a linear phase response, which is much easier to deal with, while IIR filters have non-linear phase response.

Some pros and cons of each filter type is summarized below.

FIR Filters:

  • Pros
    • Can be designed using optimization techniques to match a desired magnitude/phase response

    • Allow for arbitrary magnitude/phase response

    • Allow for linear or zero phase response (no distortion)

    • Are always stable

  • Cons
    • Can require a large number of coefficients (e.g., \(M\approx 100\)) to achieve desired accuracy, particularly for steep filters.

IIR Filters:

  • Pros
    • Can be implemented very efficiently - fewer coefficients than FIR for comparable frequency selective filter accuracy (e.g., \(M\approx N\approx 8\))

    • Filtering is fast

  • Cons
    • Generally can’t use optimization techniques to design

    • Better approach is to start from a well-known analog filter design and transform it to discrete-time filter.

    • Limited to frequency selective filters (e.g., bandpass, high-pass, etc)

    • Phase is nonlinear (will always cause phase distortion within the passband)

    • Zero phase filters are impossible to implement exactly (you can get this by filtering forward + backward, but this can’t be implemented in real-time!)

In spite of the cons listed above, there are some instances where IIR filters are preferred. For instance, for implementing maximally flat selective filters (e.g., Butterworth bandpass filters) or for modeling the behavior of systems with feedback.

Nevertheless, the vast majority of filters encountered in seismic metadata are anti-alias filters used at each decimation stage of the digitizer, and the digital anti-alias filters most commonly used are linear phase FIR filters that produce a constant time shift.

Hence, in what follows we will concentrate on FIR filters.

Classification of FIR Filters

FIR filter frequency response can be written

\[H(e^{j\omega})=\sum_{k=0}^{M}b_{k}z^{-k}=\sum_{k=0}^{M}b_{k}e^{-j\omega k}=\sum_{k=0}^{M}h[k]e^{-j\omega k}\]

where in the last expression, we identify the filter coefficients \(b_{k}\) as the inpulse response values: \(h[k]=b_k\) to show that the output of the FIR filter is the convolution of the input signal \(x[n]\) with the filter impulse response.

It can be shown that the FIR filter response has generalized linear phase of the form,

\[H(e^{j\omega})=A(e^{j\omega})e^{-j(\omega\alpha+\beta)}\]

where \(A(e^{j\omega})\) describes the real amplitude, \(\beta\) is a constant phase factor, and \(\alpha\) is the constant group delay.

A consequence of this constant group delay (also called phase delay) is that the shape of the input waveform is not changed; all frequencies are delayed the right amount so that they add together in the same way to form the output signal. The resulting output signal has the same shape as the input signal but is delayed in time.

Some general observations about FIR filters are:
  • FIR filters contain as many poles as they have zeros.

  • The number of zeros (poles), \(M\), is called the order of the FIR filter

  • All the poles are located at the origin (inside the unit circle), hence FIR filters are said to be stable.

  • These poles don’t affect the magnitude of the frequency response, only the phase.

Note that a filter of order M has length M+1.

FIR filters with generalized linear phase are often divided into 4 types depending on whether the order M is even or odd, so that the number of points is either odd or even, and whether the impulse response (=FIR coefficients) exhibits even or odd symmetry about the middle point.

FIR filters with symmetrical impulse response are often called two-sided or acausal. As a consequence of the symmetry of the filter impulse response, the onsets of very impulsive signals (with energy at frequencies near the Nyquist cut-off for the FIR filter), may be contaminated by precursory (=acausal) oscillations.

Type I: M even

M even + even symmetry about the midpoint M/2

Note that in this case, there will be M+1 (odd) points in the filter and M/2 will fall on an index right in the middle:

\[h[k]=h[M-k],0\le k\le M\]

We can write out the frequency response and use symmetry to simplify,

\begin{eqnarray} H(e^{j\omega})&=&\sum_{k=0}^{M}h[k]e^{-j\omega k} \\ &=&h[0]+h[1]e^{-j\omega\cdot1}+h[2]e^{-j\omega\cdot2}+...+h[M-1]e^{-j\omega\cdot(M-1)}+h[M]e^{-j\omega\cdot M} \\ &=&e^{-j\omega\frac{M}{2}}\Big[h[0]e^{+j\omega\frac{M}{2}}+h[1]e^{-j\omega\cdot1}e^{+j\omega\frac{M}{2}}+...+h[M-1]e^{-j\omega\cdot(\frac{M}{2}-1)}+h[M]e^{-j\omega\frac{M}{2}}\Big] \\ &=&e^{-j\omega\frac{M}{2}}\Big[h[0]e^{+j\omega\frac{M}{2}}+h[M]e^{-j\omega\frac{M}{2}}+h[1]e^{-j\omega\cdot1}e^{+j\omega\frac{M}{2}}+...+h[M/2+1]e^{-j\omega\cdot1}+h[M/2]\Big] \\ &=&e^{-j\omega\frac{M}{2}}\Big[h[0](e^{+j\omega\frac{M}{2}}+e^{-j\omega\frac{M}{2}})+h[1](e^{+j\omega(\frac{M}{2}-1)}+e^{-j\omega(\frac{M}{2}-1)})+...+h[M/2-1](e^{+j\omega}+e^{-j\omega})+h[M/2]\Big] \\ &=&e^{-j\omega\frac{M}{2}}\Big[h[0]2cos(\frac{M}{2}\omega)+h[1]2cos((\frac{M}{2}-1)\omega)+...+h[M/2-1]2cos(\omega)+h[M/2]\Big] \\ H(e^{j\omega})&=&e^{-j\omega\frac{M}{2}}\sum_{k=0}^{M/2}a[k]cos(\omega k) \end{eqnarray}

where \(a[0]=h[M/2],a[1]=2h[M/2-1],...,a[M/2]=2h[0]\).

In general, \(a[0]=h[\frac{M}{2}]\), and \(a[k]=2h[\frac{M}{2}-k],k=1,...,\frac{M}{2}\).

The \(a[k]\) coefficients are real, hence the sum is real, and the response satisfies the generalized linear phase property:

\[H(e^{j\omega})=A(e^{j\omega})e^{-j(\omega\alpha+\beta)}\]

Hence for Type I, the amp is: \(A(e^{j\omega})=\sum_{k=0}^{M/2}a[k]cos(\omega k)\), while the phase term is: \(e^{-j\omega\frac{M}{2}}\) and the corresponding group delay is: \(\alpha=\frac{M}{2}\).

Type II: M odd

M odd + even symmetry about the midpoint M/2

Note that in this case, there will be M+1 (even) points in the filter, hence the symmetry mid-point falls between two sample points.

\[h[k]=h[M-k],0\le k\le M\]

By similar algebra as above, we can write the frequency response as

\[H(e^{j\omega})=e^{-j\omega\frac{M}{2}}\sum_{k=1}^{\frac{(M+1)}{2}}b[k]cos(\omega(k-\frac{1}{2}))\]

where \(b[k]=2h[(\frac{(M+1)}{2}-k],k=1,...,\frac{(M+1)}{2}\).

Thus, this system also has group delay \(\alpha=\frac{M}{2}\).

Type III/IV anti-symmetric

Type III (M even) and Type IV (M odd) FIR filters exhibit anti-symmetry about the midpoint: \(h[k]=-h[M-k]\).

As a result, their expansions reduce to summation of sine functions and can’t be used to implement low-pass filters, hence they aren’t used for anti-alias filtering.

Practical Concerns

Thus, we normally use FIR filters of type I or II for anti-alias filtering. Because of their symmetry, only half the coefficients need to be stored in the metadata.

In StationXML, a symmetric filter can be represented using a FIR response stage, with sub-element indicating the symmetry (odd/even).

In contrast, a non-symmetrical FIR can only be stored in a more general Coefficients response stage, which retains all of the coefficients.

In practice, even symmetric FIR filter coefficients are often stored in a Coefficients response stage.

This is how the FIR response is calculated in ObsPy, which uses the venerable evalresp C code underneath the hood. Note that in evalresp, this type of filter is termed FIR_ASYM, meaning it can handle both symmetric (about the mid-point) and non-symmetric FIR coefficients. All of the coefficients are used in the expansion to calculate the filter response.

In contrast, IIR filter coefficients can’t be stored in a FIR response stage, since it only allows for numerator coefficients. IIR filter coefficients can be stored in a Coefficients response stage. However, IIR responses are very sensitive to round-off errors in the values of the stored coefficients and can become unstable. Therefore, many IIR filters are instead stored as a PolesZeros response stage of type ‘D’ (digital) and are expanded in terms of the poles and zeros of the z-transform as discussed above.

3.1.5. Convolution

Introduction

As we’ll see in the next sections, a geophysical sensor (e.g., seimometer) connected to a datalogger that digitizes and records the input signal (e.g., ground motion), represents a linear time-invariant (LTI) system. We can thus model the overall effect of the instrumentation on the input signal as a linear combination of stages representing each component of the instrumentation. The stages are connected sequentially so that the output of stage 1, representing the sensor, forms the input of stage 2, which might represent either a pre-amplifier or a digitizer. As the input signal passes through each stage, we say that it is “convolved” with the impulse response of that stage, to form the output signal that then becomes the input signal for the subsequent stage.

Convolution is a mathematical operation between two functions. For instance, if function \(f(t)\) represents the input signal to a stage, and function \(g(t)\) represents the impulse response of the stage, then the output of the stage is the convolution between \(f(t)\) and \(g(t)\).

Given two functions \(f(t)\) and \(g(t)\) defined for all \(t\ge 0\), their convolution at time \(t\) is defined by:

\[(f*g)(t)=\int_{0}^{t}f(t-\tau)g(\tau)d\tau\]

where \(*\) represents the convolution operator.

Suppose that f and g are piecewise continuous and of exponential order. Then

\[L[f*g]=L[f]L[g]\]

Where \(L\) is the Laplace Transform operator.

Proof

If we extend the functions \(f\) and \(g\) to be 0 for \(t<0\), then the integral above is the same as

\[(f*g)(t)=\int_{0}^{\infty}f(t-\tau)g(\tau)d\tau\]

i.e., for \(\tau>t\), \((t-\tau)<0\) and \(f(t-\tau)=0\).

So we can write the Laplace Transform as

\[L[f*g](s)=\int_{0}^{\infty}\int_{0}^{\infty}f(t-\tau)g(\tau)d\tau e^{-st}dt\]

Interchanging the order of integration gives

\[L[f*g]=\int_{0}^{\infty}\int_{0}^{\infty}f(t-\tau)e^{-st}dtg(\tau)d\tau\]

Substitute \(u=t-\tau,du=dt\),

\[L[f*g]=\int_{0}^{\infty}\int_{0}^{\infty}f(u)e^{-s(u+\tau)}du\cdot g(\tau)d\tau\]

or

\begin{eqnarray} L[f*g](s) &=& \int_{0}^{\infty}f(u)e^{-su}du\cdot \int_{0}^{\infty}g(\tau)e^{-s\tau}d\tau \\ &=& L[f]L[g] \end{eqnarray}

In other words, the Laplace Transform of the convolution of \(f\) and \(g\), is equal to the product of the Laplace Transform of \(f\) times the Laplace Transform of \(g\). This holds true for all of the “frequency” transforms (Fourier, Laplace, z).

It is for this reason that most instrument response calculations are performed in the frequency domain, by multiplying the frequency response of subsequent stages (or filters) together.

3.1.6. References

  • Haykin, S. and B. Van Veen. Signals And Systems. Wiley, Somerset, New Jersey, 2002.

  • Oppenheim, A.V. and R.W. Schafer. Discrete-Time Signal Processing. Prentice Hall, Englewood Cliffs, New Jersey, 1989.

  • Ringler, A. T. and P. Bastien (2020). A brief introduction to seismic instrumentation: Where does my data come from?, Seis. Res. Lett., 91 (2A), 1074-1083

  • Ringler, A. T. and J. R. Evans (2015). A quick SEED tutorial, Seis. Res. Lett., 86 (6), 1717-1725.

  • Scherbaum, F. Of Poles and Zeros: Fundamentals of Digital Seismology. Kluwer Academic. 2nd Edition. 2001.

  • Steim, J.M. Theory and Observations - Instrumentation for Global and Regional Seismology. In: Gerald Schubert (editor-in-chief) Treatise on Geophysics, 2nd edition, Vol 1. Oxford: Elsevier; 2015. p. 29-78.

3.2. Practical Instrument Response

3.2.1. Introduction

Geophysical data are recorded by an instrument that imparts its own signature onto the data. When the data are later analyzed, one of the first steps is to remove the effect of the instrumentation used to record it, the so-called instrument response. This is typically done in the frequency domain, by dividing the complex Fourier Transform of the data by the complex Fourier Transform of the instrument response.

\[Z(f)=\frac{X(f)}{I(f)}\]

where \(X(f)\) is the Fourier Transform of the recorded time series, \(I(f)\) is the Fourier Transform of the instrument response, and \(Z(f)\) is the Fourier Transform of the data with the instrument response removed.

So how does one obtain the Fourier Transform of the instrument response? Very often, this is calculated by combining the information describing each stage of the instrument response in specific formats.

3.2.2. Sensor Response as a Linear Sequence of Stages

_images/stage_sequence.png

A recording system (sensor + datalogger) represents a linear, time-invariant system. As such, the total response of the system (= the instrument response) can be calculated by linearly combining the response of each individual stage in the system. In the time domain, the operator that represents linear combination in this way is convolution, however, it’s difficult to visualize the result of convolving several stages together. Fortunately, in the frequency domain, the operator that links the individual stage responses together is multiplication, and it’s trivial to combine stage responses together.

The schematic shown in the figure above represents a generic ideal of the instrument response as a sequence of stages. Where each stage is implemented in the hardware, e.g., whether it physically resides in the sensor or the datalogger (or whether these are integrated into a single unit) is not specified.

A more specific description of instrument response particular to most seismic instrumentation is this: The ground motion (typically velocity or acceleration) is “input” to the seismic sensor which outputs continuous voltage (an analog signal) proportional to the input in some way. This continuous voltage could then be amplified, either by an external preamplifier (+ filter possibly) or, more commonly, by circuitry within the datalogger itself. Next, the continuous signal is sampled by the ADC (analog-to-digital conversion) circuit of the datalogger, resulting in discrete data samples.

Typically, the sampling is done over a sequence of stages where the first stage highly oversamples the input data. Each subsequent stage is a combination of low-pass filter, typically implemented using a FIR filter, followed by decimation of the data stream by some decimation factor. This anti-alias FIR filter is necessary at each decimation step to avoid aliasing of energy above the Nyquist frequency, which would contaminate the signal of interest. This cascade of filter/decimate stages begins at the high sample rate (e.g., 102400 samples per second for the Reftek RT130) and continues, with typical integer decimation factors (2,4,5,8,10,16 etc) at each step, until the final desired output sample rate is reached. Thus, the input units of the first sensor stage is the ground motion (e.g., m/s), while the input units of the first datalogger stage is Volts. After the ADC, the input/output units for each subsequent stage is Counts.

3.2.3. Stage 1: The Analog Sensor

The first stage of the response often represents the effect of an analog sensor (e.g., seismometer, microphone, etc), which takes as input a physical quantity (e.g., ground motion in \(\mu ms^{-1}\), air pressure in \(Pa\), temperature in \(^{\circ}C\), etc.) and outputs Volts.

We need some way to represent how this sensor stage works and what distortion, if any, it applies to the underlying time series (the input physical quantity).

Commonly, the analog sensor stage is stored as a sequence of poles and zeros of the Laplace Transform (see Laplace Transform description above) along with associated scale factors.

Recall that the Laplace transform variable is given by \(s=\sigma+j\omega\). Along the imaginary axis, \(\sigma=0\) and hence \(s=j\omega\). Thus, we may express the complex frequency response of the analog stage by calculating its polezero expansion

\[H(f)=A_0\frac{\Pi_{k=1}^{M} (s-z_{k})} {\Pi_{k=1}^{N} (s-p_{k})}\]

where \(s=j2\pi f\) [rad/s] or \(s=jf\) [Hz] , \(z_k\) are the \(M\) zeros and \(p_k\) are the \(N\) poles. \(A_0\) is the normalization factor, typically chosen so that \(|H(f_n)|=1.0\) where \(f_n\) is the normalization frequency.

Thus, given the poles and zeros of an analog stage, in order to properly calculate the stage frequency response, we must know the units of \(s\) (Hz or rad/s) used to calculate the poles and zeros using the expansion above.

With the normalization factor \(A_0\), the polezero expansion results in a complex frequency response with magnitude = 1.0 at the normalization frequency. For seismometers whose response is flat to ground velocity, the normalization frequency is typically chosen somewhere within the flat part of the response spectrum. For broadband sensors, it is also considered good practice to select a normalization frequency lower than two times the lowest sampling frequency. For example, if you are sampling VHZ data at 0.1 sps, then you want to describe \(A_0\) at a frequency \(\lt 0.05\) Hz.

Thus, the poles and zeros give the shape of the sensor response, but not the gain (see Fig. X below).

Analog polezero examples

Below are the poles and zeros for two broadband seismometers (STS-1 and STS-2) and a short-period sensor (L-22D). All have a response that is flat to velocity within some frequency band, which is controlled by the location of the poles and zeros in the \(s\)-plane.

Sensor: Streckeisen STS-2 (3rd Generation)

Gain:

1500 [V/m/s]

Freq of gain:

1.0 [Hz]

A0 normalization:

3.4684E+17

Freq of normalization:

1.0 [Hz]

Poles:

Zeros:

real (rad/s)

imag (rad/s)

real (rad/s)

imag (rad/s)

-0.037

-0.037

0.0

0.0

-0.037

+0.037

0.0

0.0

-15.54

0.0

15.15

0.0

-97.34

-400.7

-176.6

0.0

-374.8

0.0

-463.1

-430.5

-97.34

+400.7

-463.1

+430.5

-520.3

0.0

-10530.0

-10050.0

-10530.0

+10050.0

-13300.0

0.0

-255.097

0.0

Sensor: Streckeisen STS-1

Gain:

2400 [V/m/s]

Freq of gain:

0.02 [Hz]

A0 normalization:

3.94858E+03

Freq of normalization:

0.02 [Hz]

Poles:

Zeros:

real (rad/s)

imag (rad/s)

real (rad/s)

imag (rad/s)

-0.01234

+0.01234

0.0

0.0

-0.01234

-0.01234

0.0

0.0

-39.18

+49.12

-39.18

-49.12

Sensor: Sercel L-22D

Gain:

87.9 [V/m/s]

Freq of gain:

10.0 [Hz]

A0 normalization:

1.0

Freq of normalization:

10.0 [Hz]

Poles:

Zeros:

real (rad/s)

imag (rad/s)

real (rad/s)

imag (rad/s)

-8.884

+8.887

0.0

0.0

-8.884

-8.887

0.0

0.0

Notice from the table above that the poles always appear in complex conjugate pairs (in the degenerate case this is represented by a single real pole). This will always be the case, since the poles and zeros ultimately relate to a polynomial expression with real coefficients that describes the seismometer response. In addition, notice that all of the poles have a negative real part. This is required for stability reasons as discussed in the theory section above.

The poles and zeros have units in the complex s-plane. For the examples given (and for most analog stages), <PzTransferFunctionType> is “LAPLACE (RADIANS/SECOND)”, hence the poles and zeros have units of rad/s.

When viewing the total instrument response as a plot of amplitude and phase versus frequency, most of the shape is controlled by the polezero expansion of the analog sensor stage.

_images/analog_stage_comparison.png

Fig. X Plot of polezero expansion of analog sensor for Streckeisen STS-2 (green) and Sercel L-22D (red) [see table for polezero values used].

In the figure above we plot the analog stage polezero expansion for a broadband seimometer (STS-2) and a short-period seismometer (L-22D). In both cases, the A0 normalization frequency, \(f_n\) is located within the flat part of the spectrum, and each response has a corner frequency, \(f_c\) below which the magnitude response rolls off. The corner frequency is the frequency at which the response magnitude is -3dB below the flat part of the spectrum.

Given the poles and zeros, one can determine the low-frequency corner frequency, \(f_c\), from the magnitude of the lowest frequency pole in the s-plane.

For instance, for the STS-2 example shown, the lowest frequency pole is at \(-.037 \pm .037j\) in the complex s-plane, and the corresponding corner frequency is:

\[ \begin{align}\begin{aligned}\omega = \sqrt{.037^2 + .037^2} = \sqrt{2} \times .037 = .052326 rad/s\\f_c = \frac{\omega}{2\pi} = 0.0083279 s^{-1} \rightarrow T_c = 1/f_c = 120 sec\end{aligned}\end{align} \]

As expected, the STS-2 sensor has a corner period of 120 s.

By similar reasoning, the corner frequency of the short-period L-22D sensor is

\[ \begin{align}\begin{aligned}\omega = \sqrt{8.884^2 + 8.887^2} = 12.566 rad/s\\f_c = \frac{\omega}{2\pi} = 2.0 Hz\end{aligned}\end{align} \]

For many applications the exact instrument response is not needed and it is sufficient to calculate a single scalefactor to convert from recorded COUNTS to ground motion (e.g, M/S). For instance, if the signal of interest only contains energy within the flat part of the spectrum (e.g., band-limited signal), then we might be able to ignore the polezero shape altogether and compute and overall scalefactor (to go from COUNTS to M/S) for the sensor + datalogger.

For instance, if the STS-2 sensor discussed above were connected to a generic Reftek RT-130 datalogger, we can calculate an approximate forward scalefactor:

\[1500 [V/m/s] \times 6.291290\times 10^{5} [Counts/V] = 9.4369\times 10^{8} [Counts/m/s]\]

While this is often done to do a quick conversion to ground velocity, several caviats must be mentioned.

  1. Very few signals are truly bandlimited this way and we’re essentially applying the wrong correction factor to the signals outside the bandwidth

  2. Often, the frequency at which the sensor response is normalized, is not the same as the frequency at which the datalogger response is normalized. For instance, the Reftek RT-130 is normalized at 0.05 Hz, while the STS-2 is normalized at 1.0 Hz. This matters because the amplitude response of the datalogger filters are not perfectly flat, hence to compute the overall instrument response (sensor + datalogger), the datalogger gain often has to be recalculated at the frequency of the sensor normalization.

  3. It ignores the phase response, which can be very important for modelling waveforms, etc.

Alternatively, when simple scaling is insufficient, it is necessary to use all of the response stages to compute the exact instrument response. When the datalogger normalization frequency is different from the sensor normalization frequency, the datalogger response is recalculated at the sensor normalization frequency and the new sensitivity, equal to the product of each stage amplitude response at this normalization frequency, is stored in the StationXML <InstrumentSensitivity> element.

Converting s = rad/s to/from Hz

Recall the polezero expansion of the Laplace Transform of the transfer function,

\[H(s)=A_{0}\frac{\Pi_{k=1}^{M}(s-z_{k})}{\Pi_{k=1}^{N}(s-p_{k})}\]

Consider a system with 1 zero and 2 poles (forming a complex conjugate pair) in the complex \(s\)-plane (rad/s). Say \(z_{1}=z_{r}+jz_{i}\), \(p_{1}=p_{r}+jp_{i}\) and \(p_2=\bar{p_1}\), where each term \(z_r, z_i, p_r, p_i\) has units of rad/s.

Then

\begin{eqnarray} H(s=j\omega)&=&A_{0}\frac{j\omega-(z_{r}+jz_{i})}{(j\omega-p_{1})(j\omega-\bar{p_{1}})} \\ &=&A_{0}\frac{2\pi[jf-(z_{r}^{'}+jz_{i}^{'})]}{(2\pi)^{2}[(jf-p_{1}^{'})(jf-\bar{p_{1}^{'}})]} \\ \end{eqnarray}

where the primed quantities equal the unprimed quantities divide by \(2\pi\), e.g., \(z_r^{'}=\frac{z_r}{2\pi}\)

We can write this as

\[H(s=jf)=A_{0}^{'}\frac{[jf-(z_{r}^{'}+jz_{i}^{'})]}{[(jf-p_{1}^{'})(jf-\bar{p_{1}^{'}})]}\]

where \(A_{0}^{'}=\frac{2\pi}{(2\pi)^{2}}A_0\).

In general, we can write

\[H(s=jf)=A_{0}^{'}\frac{\Pi_{k=1}^{M}(jf-z_{k}^{'})}{\Pi_{k=1}^{N}(jf-p_{k}^{'})}\]

where \(A_{0}^{'}=A_{0}\cdot(2\pi)^{M-N}\) and where \(M\) is the number of zeros and \(N\) is the number of poles.

Thus, if we are given a polezero stage with units of rad/s (\(\omega\)), we can convert this stage to units of Hz and maintain the proper normalization by first dividing each pole and zero by \(2\pi\), and then scaling the normalization factor \(A_{0}\) by \((2\pi)^{M-N}\).

Conversely, if we are given a polezero stage with units of Hz (\(f\)), we can convert it to units of rad/s by first multiplying the each pole and zero by \(2\pi\), and then scaling the given normalization factor \(A_{0}^{'}\) by \((2\pi)^{N-M}\).

For example, the poleszeros of the Streckeisen STS-I sensor are shown in the table below for units = Hz (<PzTransferFunctionType>LAPLACE (HERTZ)</PzTransferFunctionType>)

Sensor: Streckeisen STS-1 poleszeros given in Hz

Gain:

2400 [V/m/s]

Freq of gain:

0.02 [Hz]

A0 normalization:

1.0001869E+02

Freq of normalization:

0.02 [Hz]

Poles:

Zeros:

real (Hz)

imag (Hz)

real (Hz)

imag (Hz)

-0.0019639

+0.0019639

0.0

0.0

-0.0019639

-0.0019639

0.0

0.0

-6.2357

+7.8177

-6.2357

-7.8177

3.2.4. Stage 2: The Pre-Amplifier

Not all response sequences have a pre-amplifier. When present, it may be implemented using an analog circuit (V->V) or it may be a digital circuit integrated within the analog-to-digital (datalogger) recorder itself.

For example, an analog pre-amplifier stage with a gain of 8 would be represented as:

<Stage number="2">
   <StageGain>
      <Value>8</Value>
      <Frequency>1</Frequency>
   </StageGain>
</Stage>

where the <Frequency> is normally chosen to be the same as the normalization frequency, \(f_n\) in stage 1.

An external pre-amplifier stage could have an associated filter, either implemented using a polezero or coefficient representation.

Often, the preamplifier is integrated with the datalogger and merely adds a uniform scalefactor to the instrument response.

Warning

Using place-holder pre-amps at stage 2 may not be good practice.

Some entities (e.g., the Nominal Response Library or NRL) always include a pre-amp at stage 2 in order to standardize the numbering of response stages (e.g., so that the datalogger response always begins at stage 3). If no pre-amp was actually present, then a place-holder stage with gain=1 is used.

However, there is disagreement about whether all responses should have such a place-holder stage or whether it is better practice to have the response stages more faithfully correspond to the equipment that is actually used.

3.2.5. Stage 2+: The Datalogger

The datalogger, or analog-to-digital converter (ADC) has two main functions: 1. To digitize the analog signal (Volts) coming from the sensor and 2. To output the digitized signal (digital counts) at the desired sample rate(s).

These functions are typically achieved by first highly oversampling the analog signal, and then passing it through a sequence of filter/decimate steps to achieve the desired output sample rate(s).

Each filter/decimate step is represented by a stage in the StationXML response, representing the effects of low-pass filtering (typically implemented with an anti-alias FIR filter) and decimation to a lower sample rate.

FIR anti-alias filter

The recommended practice for storing FIR filters is to normalize the filter response at a specified frequency:

\begin{eqnarray} G(f)&=&S_d H_c(z)\Big|_{z=e^{j2\pi \Delta t f_s}} \\ &=&S_d\sum_{k=0}^{M}b_k z^{-k} \Big|_{z=e^{j2\pi \Delta t f_s}} \\ &=&S_d\sum_{n=0}^{M}b_k e^{-j 2\pi f_s k \Delta t} \end{eqnarray}

where \(b_k\) are the FIR coefficients, \(M\) is the filter order, \(\Delta t\) is the sample rate [seconds], and \(f_s\) is the frequency at which the filter is normalized to have a gain of \(S_d\).

The Quanterra QDP380/QDP680 family of dataloggers employ the 64-element FIR filter described in Table X as a digital anti-alias filter in the stage 4 decimation from 40Hz down to 20 Hz.

The FIR coefficients, \(b_k\) are plotted in Fig. X

_images/Qx80_FIR_64.png

Fig. X Plot of filter coefficients for Qx80 FIR filter

The FIR filter has 64 coefficients (order M = 64-1 = 63) and is symmetric about the midpoint (which lies in between samples 32 and 33). Hence it is a FIR Type II symmetric filter.

Decimation

In the normal process by which the datalogger holds and samples information, followed by decimation, a time delay is often introduced into the recorded trace (see Fig. X). This delay, if known, can be stored in the <Delay> element. For example, a delay of 1.0 seconds would be stored as:

<Response>
   <Stage>
      <Decimation>
         <Delay>1.0</Delay>
      </Decimation>
   </Stage>
</Response>

while any applied time correction, e.g., to cancel out the delay, can be stored in:

<Response>
   <Stage>
      <Decimation>
         <Correction>1.0</Correction>
      </Decimation>
   </Stage>
</Response>

An anti-alias FIR filter normally introduces a positive delay into the recorded trace as indicated in the figure. If this delay is removed from the data, e.g., by introducing a negative offset -x prior to recording, then the positive value +x is stored in the <Correction> element.

_images/decimation_delay.png

3.3. StationXML Response Examples

3.3.1. Broadband sensor

3rd generation Streckeisen STS-2 sensor + Reftek RT130 datalogger

StationXML Show/Hide

<?xml version='1.0' encoding='UTF-8'?>
<FDSNStationXML xmlns="http://www.fdsn.org/xml/station/1" schemaVersion="1.1">
  <Network code=..>
    <Station code=..>
      <Channel code=.. locationCode=..>
        ...
        <SampleRate>40.0</SampleRate>
        <Response>
          <InstrumentSensitivity>
            <Value>941864732.693</Value>
            <Frequency>1.0</Frequency>
            <InputUnits>
              <Name>M/S</Name>
              <Description>Velocity in Meters per Second</Description>
            </InputUnits>
            <OutputUnits>
              <Name>COUNTS</Name>
              <Description>Digital Counts</Description>
            </OutputUnits>
          </InstrumentSensitivity>
          <Stage number="1">
            <PolesZeros>
              <InputUnits>
                <Name>M/S</Name>
                <Description>Velocity in Meters per Second</Description>
              </InputUnits>
              <OutputUnits>
                <Name>V</Name>
                <Description>Volts</Description>
              </OutputUnits>
              <PzTransferFunctionType>LAPLACE (RADIANS/SECOND)</PzTransferFunctionType>
              <NormalizationFactor>3.4684e+17</NormalizationFactor>
              <NormalizationFrequency unit="HERTZ">1.0</NormalizationFrequency>
              <Zero number="0">
                <Real>0.0</Real>
                <Imaginary>0.0</Imaginary>
              </Zero>
              <Zero number="1">
                <Real>0.0</Real>
                <Imaginary>0.0</Imaginary>
              </Zero>
              <Zero number="2">
                <Real>-15.15</Real>
                <Imaginary>0.0</Imaginary>
              </Zero>
              <Zero number="3">
                <Real>-176.6</Real>
                <Imaginary>0.0</Imaginary>
              </Zero>
              <Zero number="4">
                <Real>-463.1</Real>
                <Imaginary>-430.5</Imaginary>
              </Zero>
              <Zero number="5">
                <Real>-463.1</Real>
                <Imaginary>430.5</Imaginary>
              </Zero>
              <Pole number="0">
                <Real>-0.037</Real>
                <Imaginary>-0.037</Imaginary>
              </Pole>
              <Pole number="1">
                <Real>-0.037</Real>
                <Imaginary>0.037</Imaginary>
              </Pole>
              <Pole number="2">
                <Real>-15.64</Real>
                <Imaginary>0.0</Imaginary>
              </Pole>
              <Pole number="3">
                <Real>-97.34</Real>
                <Imaginary>-400.7</Imaginary>
              </Pole>
              <Pole number="4">
                <Real>-97.34</Real>
                <Imaginary>400.7</Imaginary>
              </Pole>
              <Pole number="5">
                <Real>-374.8</Real>
                <Imaginary>0.0</Imaginary>
              </Pole>
              <Pole number="6">
                <Real>-520.3</Real>
                <Imaginary>0.0</Imaginary>
              </Pole>
              <Pole number="7">
                <Real>-10530.0</Real>
                <Imaginary>-10050.0</Imaginary>
              </Pole>
              <Pole number="8">
                <Real>-10530.0</Real>
                <Imaginary>10050.0</Imaginary>
              </Pole>
              <Pole number="9">
                <Real>-13300.0</Real>
                <Imaginary>0.0</Imaginary>
              </Pole>
              <Pole number="10">
                <Real>-255.097</Real>
                <Imaginary>0.0</Imaginary>
              </Pole>
            </PolesZeros>
            <StageGain>
              <Value>1500.0</Value>
              <Frequency>1.0</Frequency>
            </StageGain>
          </Stage>
          <Stage number="2">
            <StageGain>
              <Value>1.0</Value>
              <Frequency>0.05</Frequency>
            </StageGain>
          </Stage>
          <Stage number="3">
            <Coefficients>
              <InputUnits>
                <Name>V</Name>
                <Description>Volts</Description>
              </InputUnits>
              <OutputUnits>
                <Name>COUNTS</Name>
                <Description>Digital Counts</Description>
              </OutputUnits>
              <CfTransferFunctionType>DIGITAL</CfTransferFunctionType>
              <Numerator>1.0</Numerator>
            </Coefficients>
            <Decimation>
              <InputSampleRate unit="HERTZ">102400.0</InputSampleRate>
              <Factor>1</Factor>
              <Offset>0</Offset>
              <Delay>0.0</Delay>
              <Correction>0.0</Correction>
            </Decimation>
            <StageGain>
              <Value>629129.0</Value>
              <Frequency>0.05</Frequency>
            </StageGain>
          </Stage>
          <Stage number="4">
            <Coefficients>
              <InputUnits>
                <Name>COUNTS</Name>
                <Description>Digital Counts</Description>
              </InputUnits>
              <OutputUnits>
                <Name>COUNTS</Name>
                <Description>Digital Counts</Description>
              </OutputUnits>
              <CfTransferFunctionType>DIGITAL</CfTransferFunctionType>
              <Numerator>0.000244141</Numerator>
              <Numerator>0.000976562</Numerator>
              <Numerator>0.00244141</Numerator>
              <Numerator>0.00488281</Numerator>
              <Numerator>0.00854492</Numerator>
              <Numerator>0.0136719</Numerator>
              <Numerator>0.0205078</Numerator>
              <Numerator>0.0292969</Numerator>
              <Numerator>0.0393066</Numerator>
              <Numerator>0.0498047</Numerator>
              <Numerator>0.0600586</Numerator>
              <Numerator>0.0693359</Numerator>
              <Numerator>0.0769043</Numerator>
              <Numerator>0.0820312</Numerator>
              <Numerator>0.0839844</Numerator>
              <Numerator>0.0820312</Numerator>
              <Numerator>0.0769043</Numerator>
              <Numerator>0.0693359</Numerator>
              <Numerator>0.0600586</Numerator>
              <Numerator>0.0498047</Numerator>
              <Numerator>0.0393066</Numerator>
              <Numerator>0.0292969</Numerator>
              <Numerator>0.0205078</Numerator>
              <Numerator>0.0136719</Numerator>
              <Numerator>0.00854492</Numerator>
              <Numerator>0.00488281</Numerator>
              <Numerator>0.00244141</Numerator>
              <Numerator>0.000976562</Numerator>
              <Numerator>0.000244141</Numerator>
            </Coefficients>
            <Decimation>
              <InputSampleRate unit="HERTZ">102400.0</InputSampleRate>
              <Factor>8</Factor>
              <Offset>0</Offset>
              <Delay>0.00013672</Delay>
              <Correction>0.00013672</Correction>
            </Decimation>
            <StageGain>
              <Value>1.0</Value>
              <Frequency>0.05</Frequency>
            </StageGain>
          </Stage>
          <Stage number="5">
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_images/sts-2_rt130.png

3.3.2. Broadband sensor

Streckeisen STS-1 sensor (360 s) + Quanterra Qx80 datalogger (80)

StationXML Show/Hide

<?xml version='1.0' encoding='UTF-8'?>
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_images/sts-1_Qx80.png

3.3.3. Short-period sensor

Geotech GS-13 short-period sensor + Quanterra Quanterra Qx80 datalogger

StationXML Show/Hide

<?xml version='1.0' encoding='UTF-8'?>
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_images/gs-13_Qx80.png

3.3.4. Short-period sensor

Sercel L-22D short-period sensor (Rc=5470 Ohms, Rs=20000 Ohms) + Reftek RT72A-08 24-bit datalogger, 1 stream, 100 Hz, gain 32

StationXML Show/Hide

<?xml version='1.0' encoding='UTF-8'?>
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_images/l-22d_rt72a-08.png

3.3.5. Accelerometer

Kinemetrics FBA-3 + Kinemetrics Etna

StationXML Show/Hide

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              <Numerator>-0.00617528</Numerator>
              <Numerator>-0.00755632</Numerator>
              <Numerator>0.00248027</Numerator>
              <Numerator>0.00713658</Numerator>
              <Numerator>0.00045073</Numerator>
              <Numerator>-0.00574386</Numerator>
              <Numerator>-0.00239384</Numerator>
              <Numerator>0.00388706</Numerator>
              <Numerator>0.00333858</Numerator>
              <Numerator>-0.00200474</Numerator>
              <Numerator>-0.00343633</Numerator>
              <Numerator>0.00041425</Numerator>
              <Numerator>0.00293458</Numerator>
              <Numerator>0.00070846</Numerator>
              <Numerator>-0.00210965</Numerator>
              <Numerator>-0.0013206</Numerator>
              <Numerator>0.00121105</Numerator>
              <Numerator>0.00148249</Numerator>
              <Numerator>-0.00042439</Numerator>
              <Numerator>-0.00131798</Numerator>
              <Numerator>-0.00014412</Numerator>
              <Numerator>0.00097203</Numerator>
              <Numerator>0.00046456</Numerator>
              <Numerator>-0.00057662</Numerator>
              <Numerator>-0.00056541</Numerator>
              <Numerator>0.00022852</Numerator>
              <Numerator>0.00050938</Numerator>
              <Numerator>1.884e-05</Numerator>
              <Numerator>-0.00036979</Numerator>
              <Numerator>-0.00015306</Numerator>
              <Numerator>0.00021076</Numerator>
              <Numerator>0.00019097</Numerator>
              <Numerator>-7.641e-05</Numerator>
              <Numerator>-0.00016463</Numerator>
              <Numerator>-1.156e-05</Numerator>
              <Numerator>0.00010896</Numerator>
              <Numerator>5.221e-05</Numerator>
              <Numerator>-5.186e-05</Numerator>
              <Numerator>-5.686e-05</Numerator>
              <Numerator>9.78e-06</Numerator>
              <Numerator>4.172e-05</Numerator>
              <Numerator>1.156e-05</Numerator>
              <Numerator>-2.73e-05</Numerator>
              <Numerator>-3.362e-05</Numerator>
              <Numerator>-1.681e-05</Numerator>
              <Numerator>-2.26e-06</Numerator>
              <Numerator>1.55e-06</Numerator>
              <Numerator>6e-07</Numerator>
            </Coefficients>
            <Decimation>
              <InputSampleRate unit="HERTZ">400.0</InputSampleRate>
              <Factor>2</Factor>
              <Offset>0</Offset>
              <Delay>0.17</Delay>
              <Correction>0.17</Correction>
            </Decimation>
            <StageGain>
              <Value>1.0</Value>
              <Frequency>1.0</Frequency>
            </StageGain>
          </Stage>
        </Response>
      </Channel>
    </Station>
  </Network>
</FDSNStationXML>
_images/kinemetrics_etna_fba-3.png

3.3.6. YSI 44031 thermistor

The Berkeley Digital Seismic Network (BDSN) seismometers, use a Yellow Springs Instrument Co. (YSI) 44031 thermistor to monitor the temperature of the seismometer. The thermistor response has been determined by measuring its voltage output as a function of input temperature. It has been calibrated within a range of temperatures from -5C to 68.59C.

The resistance of the thermistor is a non-linear function of the temperature and its response can be described by a polynomial.

In order to model the response within 0.2 degrees C accuracy, a MacLaurin polynomial with 11 coefficients:

\[Temp(V)=\sum_{n=0}^{10} a_n V^{n}\]

The coefficients are given in Table 1.

\(a_n\)

value

\(a_0\)

0.12505E+02

\(a_1\)

0.13824E+02

\(a_2\)

0.41039E+01

\(a_3\)

0.12932E+01

\(a_4\)

0.18741E+01

\(a_5\)

0.17250E+01

\(a_6\)

-0.61021E+00

\(a_7\)

-0.10540E+01

\(a_8\)

0.13974E+00

\(a_9\)

0.39061E+00

\(a_{10}\)

0.95345E-01

Because this is a polynomial response, the corresponding StationXML looks a little different than the usual responses (e.g., for seismometers). Instead of a InstrumentSensitivity element, there is an InstrumentPolynomial element. In addition the analog stage is represented by a Polynomial stage. The Polynomial stage and the InstrumentPolynomial stage both contain all of the MacLaurin coefficients, however, in the InstrumentPolynomial stage, those coefficients have been scaled by the datalogger sensitivity to give units of Counts instead of Volts.

How the InstrumentPolynomial was calculated

The InstrumentPolynomial stage looks a lot like the Polynomial stage except that the overall system gain has been incorporated into the polynomial coefficients.

The overall system gain is just the product of the individual stage gains:

\[g0=\Pi_{n=0}^{N} gain_n\]

where \(g0\) is the system gain.

Then the \(n^{th}\) coefficient of the MacLaurin series is scaled by the inverse \(n^{th}\) power of the system gain:

\[a^{\prime}_n=\frac{a_n}{(g0)^{n}}\]

For the example shown, the system gain is \(g0=838860.80\) so that the scaled coefficients are:

coefficient

value

\(a^{\prime}_0\)

0.12505E+02

\(a^{\prime}_1\)

1.64795e-05

\(a^{\prime}_2\)

5.83199e-12

\(a^{\prime}_3\)

2.19077e-18

\(a^{\prime}_4\)

3.78471e-24

\(a^{\prime}_5\)

4.15279e-30

\(a^{\prime}_6\)

-1.75122e-36

\(a^{\prime}_7\)

-3.60588e-42

\(a^{\prime}_8\)

5.69904e-49

\(a^{\prime}_9\)

1.89904e-54

\(a^{\prime}_{10}\)

5.52585e-61

A complete StationXML Response element is shown below for the YSI-44301 thermistor attached to a Reftek RT130 datalogger sampling at 40Hz.

StationXML Show/Hide

  <?xml version='1.0' encoding='UTF-8'?>
     <FDSNStationXML xmlns="http://www.fdsn.org/xml/station/1" schemaVersion="1.1">
        <Network code=..>
        <Station code=..>
           <Channel code=.. locationCode=..>
              ...
              <SampleRate>40.0</SampleRate>

        <Response>
           <InstrumentPolynomial name="InstrumentPolynomial">
              <Description>None</Description>
              <InputUnits>
              <Name>degC</Name>
              <Description>TEMPERATURE in Celsius</Description>
              </InputUnits>
              <OutputUnits>
              <Name>COUNTS</Name>
              <Description>Digital Counts</Description>
              </OutputUnits>
              <ApproximationType>MACLAURIN</ApproximationType>
              <FrequencyLowerBound unit="HERTZ">0.0</FrequencyLowerBound>
              <FrequencyUpperBound unit="HERTZ">0.01</FrequencyUpperBound>
              <ApproximationLowerBound>-5.02</ApproximationLowerBound>
              <ApproximationUpperBound>68.59</ApproximationUpperBound>
              <MaximumError>0.072</MaximumError>
              <Coefficient>12.505</Coefficient>
              <Coefficient>1.64794921875e-05</Coefficient>
              <Coefficient>5.83199266657175e-12</Coefficient>
              <Coefficient>2.1907660147785217e-18</Coefficient>
              <Coefficient>3.784714809535227e-24</Coefficient>
              <Coefficient>4.1527864425849766e-30</Coefficient>
              <Coefficient>-1.7512168159552436e-36</Coefficient>
              <Coefficient>-3.605880325679582e-42</Coefficient>
              <Coefficient>5.699037789738209e-49</Coefficient>
              <Coefficient>1.8990406231916714e-54</Coefficient>
              <Coefficient>5.525847819332687e-61</Coefficient>
           </InstrumentPolynomial>
           <Stage number="1">
              <Polynomial name=" SENSOR RESPONSE   ">
              <InputUnits>
                 <Name>degC</Name>
                 <Description>TEMPERATURE in Celsius</Description>
              </InputUnits>
              <OutputUnits>
                 <Name>V</Name>
                 <Description>Volts</Description>
              </OutputUnits>
              <ApproximationType>MACLAURIN</ApproximationType>
              <FrequencyLowerBound unit="HERTZ">0.0</FrequencyLowerBound>
              <FrequencyUpperBound unit="HERTZ">0.01</FrequencyUpperBound>
              <ApproximationLowerBound>-5.02</ApproximationLowerBound>
              <ApproximationUpperBound>68.59</ApproximationUpperBound>
              <MaximumError>0.072</MaximumError>
              <Coefficient>12.505</Coefficient>
              <Coefficient>13.824</Coefficient>
              <Coefficient>4.1039</Coefficient>
              <Coefficient>1.2932</Coefficient>
              <Coefficient>1.8741</Coefficient>
              <Coefficient>1.725</Coefficient>
              <Coefficient>-0.61021</Coefficient>
              <Coefficient>-1.054</Coefficient>
              <Coefficient>0.13974</Coefficient>
              <Coefficient>0.39061</Coefficient>
              <Coefficient>0.095345</Coefficient>
              </Polynomial>
           </Stage>
           <Stage number="2">
              <StageGain>
              <Value>1.0</Value>
              <Frequency>0.0</Frequency>
              </StageGain>
           </Stage>
           <Stage number="3">
              <Coefficients name=" DIGITIZER">
              <InputUnits>
                 <Name>V</Name>
                 <Description>Volts</Description>
              </InputUnits>
              <OutputUnits>
                 <Name>counts</Name>
                 <Description>Digital Counts</Description>
              </OutputUnits>
              <CfTransferFunctionType>DIGITAL</CfTransferFunctionType>
              <Numerator>1.0</Numerator>
              </Coefficients>
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              <Offset>0</Offset>
              <Delay>0.0</Delay>
              <Correction>0.0</Correction>
              </Decimation>
              <StageGain>
              <Value>838860.8</Value>
              <Frequency>0.0</Frequency>
              </StageGain>
           </Stage>
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                 <Name>counts</Name>
                 <Description>Volts</Description>
              </InputUnits>
              <OutputUnits>
                 <Name>counts</Name>
                 <Description>Digital Counts</Description>
              </OutputUnits>
              <CfTransferFunctionType>DIGITAL</CfTransferFunctionType>
              <Numerator>0.000244141</Numerator>
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              <StageGain>
              <Value>1.0</Value>
              <Frequency>0.0</Frequency>
                        </StageGain>
           </Stage>
           <Stage number="5">
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                 <Name>counts</Name>
                 <Description>Volts</Description>
              </InputUnits>
              <OutputUnits>
                 <Name>counts</Name>
                 <Description>Digital Counts</Description>
              </OutputUnits>
              <CfTransferFunctionType>DIGITAL</CfTransferFunctionType>
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                 <Description>Volts</Description>
              </InputUnits>
              <OutputUnits>
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                 <Description>Digital Counts</Description>
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              </Decimation>
              <StageGain>
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              <Coefficients name=" DECIMATION">
              <InputUnits>
                 <Name>counts</Name>
                 <Description>Volts</Description>
              </InputUnits>
              <OutputUnits>
                 <Name>counts</Name>
                 <Description>Digital Counts</Description>
              </OutputUnits>
              <CfTransferFunctionType>DIGITAL</CfTransferFunctionType>
              <Numerator>0.000244141</Numerator>
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                 <Name>counts</Name>
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              </InputUnits>
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                 <Name>counts</Name>
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              </Coefficients>
              <Decimation>
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              <Delay>0.0</Delay>
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              </Decimation>
              <StageGain>
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              <Frequency>0.0</Frequency>
              </StageGain>
           </Stage>
           <Stage number="11">
              <Coefficients name=" DECIMATION">
              <InputUnits>
                 <Name>counts</Name>
                 <Description>Volts</Description>
              </InputUnits>
              <OutputUnits>
                 <Name>counts</Name>
                 <Description>Digital Counts</Description>
              </OutputUnits>
              <CfTransferFunctionType>DIGITAL</CfTransferFunctionType>
              <Numerator>-1.09889e-05</Numerator>
              <Numerator>-1.99798e-05</Numerator>
              <Numerator>-3.29668e-05</Numerator>
              <Numerator>-4.39561e-05</Numerator>
              <Numerator>-4.79522e-05</Numerator>
              <Numerator>-4.09589e-05</Numerator>
              <Numerator>-1.8981e-05</Numerator>
              <Numerator>1.8981e-05</Numerator>
              <Numerator>6.7932e-05</Numerator>
              <Numerator>0.000118881</Numerator>
              <Numerator>0.000158842</Numerator>
              <Numerator>0.000174826</Numerator>
              <Numerator>0.000157843</Numerator>
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              <Numerator>-0.000178822</Numerator>
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              <Numerator>3.29668e-05</Numerator>
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              <Numerator>0.000310691</Numerator>
              <Numerator>0.000263737</Numerator>
              <Numerator>0.00013087</Numerator>
              <Numerator>-6.09391e-05</Numerator>
              <Numerator>-0.00026074</Numerator>
              <Numerator>-0.000408593</Numerator>
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              <Numerator>0.000155845</Numerator>
              <Numerator>0.000438563</Numerator>
              <Numerator>0.000623379</Numerator>
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              <Numerator>0.000456546</Numerator>
              <Numerator>0.000108891</Numerator>
              <Numerator>-0.000315686</Numerator>
              <Numerator>-0.000694309</Numerator>
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              <Numerator>-0.00085415</Numerator>
              <Numerator>-0.000533469</Numerator>
              <Numerator>-7.99164e-06</Numerator>
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              <Numerator>0.00108792</Numerator>
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              <Numerator>-0.000484518</Numerator>
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              <Numerator>-0.00290311</Numerator>
              <Numerator>-0.0026174</Numerator>
              <Numerator>-0.00148752</Numerator>
              <Numerator>0.000215785</Numerator>
              <Numerator>0.002014</Numerator>
              <Numerator>0.00335166</Numerator>
              <Numerator>0.00376825</Numerator>
              <Numerator>0.00304597</Numerator>
              <Numerator>0.0013037</Numerator>
              <Numerator>-0.001009</Numerator>
              <Numerator>-0.0032208</Numerator>
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              <Numerator>-0.0047233</Numerator>
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              <Numerator>2.49751e-05</Numerator>
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              <Numerator>-1.99798e-05</Numerator>
              <Numerator>-1.09889e-05</Numerator>
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              </Decimation>
              <StageGain>
              <Value>1.0</Value>
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</FDSNStationXML>
_images/YS-44301-thermistor.png

3.3.7. Setra 270

Setra 270 Pressure Transducer

This example was lifted from [62] Response [Polynomial] Blockette section (p.85) of the SEED manual (v.2.4).

The Setra Model 270 Pressure Transducer response is given as a polynomial response with 2 coefficients, valid for input pressure between 600-1100 mbar.

\[Pressure(V)=\sum_{n=0}^{1} a_n V^{n}\]

where \(a_0=600\) and \(a_1=100\)

Volts

mbar

0.0

600

1.0

700

2.0

800

3.0

900

4.0

1000

5.0

1100

e.g., over this voltage range (0-5V), the input (mbar of pressure) is a linear function of the output (Volts).

The Response element for the Setra sensor alone consists of a Polynomial stage and an InsrumentPolynomial Stage

Warning

This part is not finished!

StationXML Show/Hide

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_images/YS-44301-thermistor.png