### Summary

A seismometer’s output is the convolution of the input signal and its time-domain instrument response equivalent, the impulse response:

Output(t) = Impulse(t) * Input(t)

The same would true for a second seismometer. If two coincident seismic channels x and y having the same orientation record the same input signal, then

Input(t) = OutputX(t) with ImpulseX(t) deconvolved = OutputY(t) with ImpulseY(t) deconvolved

The expressions for the two channels combine to give an empirical transfer function estimate based on the data describing the amplitude ratio and phase difference of the two channels.

If the metadata instrument response describes the instrument accurately, the corresponding metadata amplitude ratio and phase difference should equal the empirical transfer function estimated from the data.

This metric reports the ratio of the (data/metadata) gain (amplitude) ratios and (data – metadata) phase differences for the ever-present 6-second microseism signal. It also reports the MS coherence as a measure of input signal similarity. For channel pairs having MS coherence >= 0.999, two metadata responses that describe the instrument perfectly would have a gain ratio of 1 and phase difference of zero indicates that the metadata response is accurate.

When comparing horizontal channels, this metric compares a channel from the “primary” instrument with rotated a channel of matching orientation generated from both “secondary” instrument horizontal channels.

Bendat, J.S. and Piersol, A.G., 1971, ‘‘Random Data: Analysis and Measurement Procedures’‘, Wiley-Interscience, N.Y., 407p.

### Uses

For coincident sensors, the transfer function metric indicates how accurately the instrument response in the metadata describes the observed behavior of the instrument at a period of 6 seconds. After discarding values for which MS coherence < 0.999, the gain ratio and phase differences can be plotted over time for each channel pair to show relative instrument degradation or stability.

In addition, MS coherence can also be plotted with time for a channel pair. Channels that consistently have low coherence likely have hardware or timing problems.

This metric only has measurements for similar channels of coincident sensors and of those channels, only measurements whose ms_coherence is 0.999 or greater satisfy the assumptions of the method. Because of this, we expect this metric to be of limited use as a constraint for **rrds** requests.

### Data Analyzed

**Traces** – two coincident N.S.L.Cs (Network.Station.Location.Channel) with the same orientation per measurement

**Window** – 1 hour

**Data Source** – IRIS SEED archive

**SEED Channel Types** –
[BCDFHLM]H? | High Gain

### Algorithm

- Find stations with coincident sensors
- If horizontal channel orientations for primary and secondary sensor differ, rotate secondary sensor channels to match the primary
- For each primary and secondary channel pair x(t) and y(t)
- Calculate the complex transfer function observed in the data:
- Request an hour of data
- If sample rates differ, decimate the higher rate stream to the lower sample rate
- Demean and detrend both time series (x and y)
- Compute the complex transfer function of x and y from their cross-spectrum and power spectral density (PSD) to give the observed amplitude ratio of (y/x) and phase difference of (y –x):

For time series x and y, Complex transfer function TFxy(f) = Pyx(f) / Pxx(f) where Pyx(f) is the complex cross-spectrum of x and y Pxx(f) is the PSD of time series x

- Save the data amplitude ratio and phase difference averaged over 5-7 seconds period

- Calculate the corresponding metadata amplitude ratio and phase difference for both channels:
- Request both instrument responses for the same hour
- Calculate the metadata amplitude ratio and phase difference averaged over 5-7 seconds period
- Calculate the data/metadata gain ratio and phase difference:

For time series x and y, gain_ratio = average[ TFxy_amplitude(T=5-7s) ] / {average[amplitude_response_y(T=5-7s) / amplitude_response_x(T=5-7s)] } phase_diff = average[data_phasedifference(T-5-7s) – metadata_phase_difference(T=5-7s)]

- Calculate the magnitude squared (MS) coherence of both time series, saving the averaged MS coherence over 5-7 seconds period:

ms_coherence = |Pxy(f)|^2 / (Pxx*Pyy)

- Calculate the complex transfer function observed in the data:

### Metric Values Returned

**gain_ratio** = (data/metadata) gain ratio averaged over 5-7 seconds period

**phase_diff** = (data – metadata) phase difference averaged over 5-7 seconds period

**ms_coherence** = magnitude squared coherence averaged over 5-7 seconds period

**target** = traces from a station’s primary (X) and secondary (Y) instruments, labeled as N.S.LY:LX.CC:CCX.Q (Network.Station.LocationY:LocationX.ChannelPrefixY:ChannelX.Quality). The gain ratio represents the ratio of location codes LY/LX. CCX is the primary Channel name and CC is the prefix of the rotated secondary channel.

**start** – beginning of the data day requested (00:00:00 UTC)

**end** – end of the data day requested (truncated as 23:59:59 UTC)

**lddate** – date/time the measurement was made and loaded into the MUSTANG database (UTC)