Input file specification

Fig. 1: Two obligatory input files and one additional file for the velocity model.

The user must provide a network inventory file (StationXML) and waveforms containing the recorded earthquake in MiniSEED format. Additionally, the user can provide a velocity model. If the velocity model is not provided, the P and S wave velocities at the source will be used for travel time calculations.

The application also requires P or S wave arrival times for time window selection. The user can either load these from a QuakeML file or pick them directly within the application.

⚠️ WARNING: Only one P and one S wave pick from each station are used. The user should pick P wave arrival times on the Z (vertical) component and S wave arrival times on either the E or N component.

Fig. 2: Loading arrival times from file. In this example 21 picks were correctly imported as points.

Fig. 3: Example of picking P phase arrival on a vertical EHZ channel of MOSK seismic station.

The user can also load event coordinates from a QuakeML file containing the event location (e.g., from the results of a phase association application) or manually enter them into the form. Latitude and longitude should be provided in degrees, while depth should be specified in kilometers.

Fig. 4: Loading event coordinates from the QuakeML file.

Input Parameters

Parameters of the Source

The user must define essential source parameters, including P and S wave velocities and the rock density in the source region. If an additional velocity model file is not provided, these values will be used to create a constant velocity model.

⚠️ WARNING: These parameters have a significant impact on the final results, so they should be selected carefully.

Time Window

The time window should be carefully chosen:

• It must be long enough to capture the signal of interest.
• It must not include other types of waves that could interfere with the analysis.
  ⚠️ WARNING: The software checks whether the time window for P-wave analysis includes the theoretical arrival time of the S wave. If so, the analysis is stopped to prevent contamination.

Frequency Range

The user needs to specify the minimum and maximum frequency for the analysis. Selecting a frequency range that is too wide or too narrow can negatively affect the accuracy of the results.

P or S Wave Analysis

The user can choose to perform the analysis for:

• P waves
• S waves
• Both, depending on the requirements of the study.

Using Station Elevations

The user has the option to include station elevations in the analysis.

Recommendation: If the velocity model is of poor quality, it is better to disable this option, in which case all elevations are set to 0 to avoid introducing errors.

Fig. 5: Basic input parameters.

Advanced Input Parameters

Users have access to several advanced options that can be adjusted to customize the analysis.

Taper

Users can select the length of a taper, which smooths the edges of time windows. This process helps manage potential signal artifacts and reduces spectral leakage during Fourier Transform calculations.

Signal-to-Noise Ratios (S/N)

Frequency Threshold: Users can set a minimum S/N ratio for frequencies. If the S/N ratio is below this threshold for a given frequency, that frequency is excluded from spectral fitting.

Time Window Threshold: For each time window, an average S/N ratio is computed. If this value does not meet the specified minimum spectral S/N ratio, the corresponding time window is excluded from the analysis.

Source Model

Users can choose between the Madariaga or Brune source models for calculating source size and stress drop, depending on their preference or the specifics of their analysis.

 Spectrum Model

The spectral fitting can be performed using either the Brune or Boatwright spectrum models, as specified by the user.

Norm

Users can select whether to use the L1 norm or L2 norm for spectral fitting.

Outlier Removal

A Z-score threshold can be defined to filter out extreme values of spectral parameters. If any of the parameters such as spectral level, corner frequency or quality factor exceeds this threshold, all parameters (for this seismic station) are excluded from the analysis for P or S waves.

Quality factor bounds

The user can also define quality factor range for spectral fitting. The user can also fix the range by providing the same lower and upper bound Q.

Fig. 6. Advanced input parameters.

Produced output

Files with the results

• csv – A file that includes the average values of all spectral parameters across all analyses, along with error factors (note: these are not the same as standard deviation).
• Separate CSV files are generated for each analysis, containing spectral parameters specific to each seismic station.
• JSON files are also created, containing the spectra values and the corresponding fitted spectra for all stations. These files are provided for analyses performed on P and/or S waves.
• log – A file containing application logs. The user can review this file to identify the reasons why results might be missing for certain seismic stations. It also provides details about the signal-to-noise (S/N) ratios for the stations. Additionally, the file records the input parameters used for the calculations.

Fig. 7: Example spectra of S-waves for a Mw~2.0 seismic event.

Data processing

Preprocessing

To prepare the signals for analysis, the data undergoes preprocessing in three steps:

• Removing the response + conversion to ground displacement,
• detrending,
• bandpass filtering with minimum and maximum frequencies provided in input parameters.

Time window

Time window is selected from the seismogram based on P or S wave pick and length of time window. Then it is tapered (i.e. smoothed) on both sides to avoid problems with signal transformation to frequency domain.

 Warnings:

• ⚠️ If the user decided to calculate spectral parameters for S waves and there is no S wave pick it will be automatically estimated based on P pick and velocity model.
• ⚠️ If a time window for P wave analysis is too long for any station (when it contains S waves) the software will not calculate spectral parameters for this station.

 Additionally, the noise time window with the same length is selected either directly before P wave arrival time or in case it is not present at the beginning of the provided time signal. It will be used for excluding stations which S/N ratio is smaller than a threshold value defined by the user and/or certain frequencies during spectral fitting.

Transformation to frequency domain

The software transforms the signals from all channels into the frequency domain using the Fast Fourier Transform (FFT), then calculates the root-mean-square for all channels in the stream to obtain the mean spectrum for each specific seismic station.

J-K integrals

The first method for estimating the low-frequency spectral level and corner frequency is the J-K integrals (Snoke, 1987).

Spectral fitting

The results from the J-K integrals are used as the initial solution for the spectral fitting method. This method employs the Nelder-Mead optimization algorithm to fit the obtained spectra, corrected for exponential attenuation, to a model spectrum — either Brune (1) or Boatwright (2). The software excludes frequencies from the calculations where the S/N ratio does not exceed the threshold value specified by the user.

This method returns not only optimal corner frequency and low-frequency spectral level but also quality factor (damping).

  (1)

  (2)

Where: Ω₀ – seismic moment, f_c – corner frequency, f – frequency.

Spectral parameters

Both the J-K integrals method and spectral fitting return the spectral level and corner frequency. Using these two values, along with some input parameters and constant values listed below, the software calculates the spectral parameters.

Constants depending on the source model:

K_BRUNE = 0.37

K_MADARIAGA_P = 0.32

K_MADARIAGA_S = 0.21

Radiation coefficients:

G_P = 0.52

G_S = 0.63

The calculated spectral parameters include the seismic moment (3), which can be directly converted to moment magnitude (4), source radius (5), and stress drop (6).

   (3)

Where: v – velocity (either of P or S waves), d – distance from the source, g – radiation coefficient (either for P or S waves)

  (4)

 (5)

Where: k – constant value depending on the source model (Brune or Marariaga for P or S waves)

(6)

Calculation of average results and their associated uncertainties for the event

As a result, the software returns spectral parameters for all stations. To calculate the average parameters for the seismic event, the software checks for outliers based on the Z-score defined by the user. If any parameter is identified as an outlier, the data from that station is excluded from the analysis.

In addition to the average value, an error factor is calculated assuming a log-normal distribution. The range of values within one standard deviation can be computed using formulas 7 and 8.

(7)

  (8)

Where: x_e – error factor, x – average value of a parameter.

References

• Kwiatek, G., P. Martínez-Garzón, G. Dresen, M. Bohnhoff, H. Sone, and C. Hartline (2015). Effects of long-term fluid injection on induced seismicity parameters and maximum magnitude in northwestern part of The Geysers geothermal field, Geophys. Res. Solid Earth 120, doi 10.1002/2015JB012362.
• Snoke, J. A. (1987). Stable determination of (Brune) stress drops, Seismol. Soc. Am. 77, 530–538.
• M Garcı́a Garcı́a, M.D Romacho, A Jiménez, Determination of near-surface attenuation, with κ parameter, to obtain the seismic moment, stress drop, source dimension and seismic energy for microearthquakes in the Granada Basin (Southern Spain), Physics of the Earth and Planetary Interiors,Volume 141, Issue 1,2004,Pages 9-26,ISSN 0031-9201, https://doi.org/10.1016/j.pepi.2003.08.006.
• Madariaga R., Dynamics of an expanding circular fault, Bull. seism. Soc. Am., 1976, vol. 66 3(pg. 639-666)

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