Bayesian polarity determination

1. A device for determining a focal point source type of an earthquake by assigning to an unknown onset of a signal in a noisy seismic time series a probability of that onset having a signed excursion away from a trace mean, the device comprising:

one or more means arranged to:

define a polarity at a given time sample in the noisy seismic time series by reference to a sign of an amplitude difference between extrema immediately before and immediately after the given time sample;

define a positive polarity probability density function by combining the polarity at the given time sample with a statistical weight based on the amplitude difference between the extrema immediately before and immediately after the given time sample, marginalized over estimated noise in the noisy seismic time series;

define a negative polarity probability density function consistent with the positive polarity probability density function and a requirement that instantaneous positive and negative probability values sum to unity;

take the respective products of the positive and negative polarity probability density functions with an onset probability density function defining a probability that the onset of the signal occurs at a given time sample;

marginalize the respective products of the positive and negative polarity probability density functions with the onset probability density function over time to estimate final probabilities that the onset has a positive or a negative polarity; and

using the estimated final probabilities to determine the focal point source types of fractures or faults in a rock formation in response to earthquake events.

2. The device according to claim 1 wherein the probability density function, P(+|σ meas ), for a positive polarity (+), given a polarity function pol(t) that defines the polarity at time t, an amplitude change (Δ(t)) between successive extrema, and an estimated noise standard deviation σ meas is a cumulative distribution of the product of polarities and amplitude changes.

3. The device according to claim 1 , wherein one or more processors are arranged to define the polarity, define the positive polarity probability density function, define the negative polarity probability density function, take the respective products, and marginalize the respective products for each of plural noisy seismic time series.

4. The device according to claim 1 , wherein the noisy seismic time series is a seismogram.

5. The device according to claim 4 , wherein the seismogram is a waveform of a pressure wave, a shear wave, a horizontally polarized shear wave or a vertically polarized shear wave.

6. The device according to claim 1 which is a computer system, the one or more means being one or more processors of the computer system.

7. The device according to claim 1 which is a dedicated electronic hardware device.

8. A method for determining a focal point source type of an earthquake by assigning to an unknown onset of a signal in a noisy seismic time series a probability of that onset having a signed excursion away from a trace mean, the method comprising:

defining a polarity at a given time sample in the noisy seismic time series by reference to a sign of an amplitude difference between extrema immediately before and immediately after the given time sample;

defining a positive polarity probability density function by combining the polarity at the given time sample with a statistical weight based on the amplitude difference between the extrema immediately before and immediately after the given time sample, marginalized over estimated noise in the noisy seismic time series;

defining a negative polarity probability density function consistent with the positive polarity probability density function and a requirement that instantaneous positive and negative probability values sum to unity;

taking the respective products of the positive and negative polarity probability density functions with an onset probability density function defining a probability that the onset of the signal occurs at a given time sample;

marginalizing the respective products of the positive and negative polarity probability density functions with the onset probability density function over time to estimate final probabilities that the onset has a positive or a negative polarity; and

using the estimated final probabilities as a constraint when performing source inversion on the noisy seismic time series and thereby determining the focal point source types of fractures or faults in a rock formation in response to earthquake events.

9. The method according to claim 8 , wherein the probability density function, (P+| meas ), for a positive polarity (+), given a polarity function pol(t) that defines the polarity at time t, an amplitude change (Δ(t)) between successive extrema, and an estimated noise standard deviation σ meas is a cumulative distribution of the product of polarities and amplitude changes.

10. The method according to claim 8 , wherein the method is performed repeatedly for each of plural noisy seismic time series.

11. The method according to any claim 8 , wherein the noisy seismic time series is a seismogram.

12. The method according to claim 11 , wherein the seismogram is a waveform of a pressure wave, a shear wave, a horizontally polarized shear wave or a vertically polarized shear wave.

13. The method according to claim 11 , comprising a preliminary step of performing seismic testing using one or more hydrophones, geophones, accelerometers, and/or distributed acoustic sensing to obtain the seismogram.

14. A method for performing a hydraulic fracturing operation in a well, the method comprising:

injecting fluid into the well to create cracks in downhole rock formations;

simultaneously obtaining microseismic data from the formations, the microseismic data including one or more seismograms;

performing the method of claim 11 on the one or more seismograms; and

using the estimated final probabilities to determine the focal point source types of cracks created in response to the injection of the fluid.