Source Parameters

Energy – moment relationship

Energy and moment are two independent measures of the strength of a seismic event. Their physical meaning and how they are calculated was described in a previous blog post. Analysis of the relationship between the energy and moment of events can provide insight into seismic sources. For example, blasts or ore pass noise, falsely processed as real events, tend to have distinct zones on an energy-moment chart. In general, events with higher-than-average energy are associated with high relative stress. Energy index is a parameter used to estimate effective stress. To calculate energy index, the mean energy-moment relationship must be defined. Energy index is the log difference in energy from the mean energy-moment relationship. When comparing energy index in different software or from separate sites, it is important to note that if the energy-moment relationship is not the same, the energy index will not be consistent. The most common method of fitting a linear relationship between two variables is known as least squares regression (LSR). This method essentially minimizes the vertical (Y-axis) difference between the data points and the line of best fit. For the energy-moment case, this would be minimizing the energy difference. LSR is designed for cases where the independent variable (X-axis) is known perfectly (zero error) and the error is only associated with the dependent variable (Y-axis). This is not suitable for the energy-moment case as there is uncertainty in both the energy and moment parameters. The uncertainty in moment and the uncertainty in energy are also generally not the same scale. There are several linear regression methods that account for uncertainty in both parameters. Orthogonal regression minimizes the perpendicular difference between the data and best fit line, assuming a constant ratio between the X and Y variances. There is also a method known as weighted least squares which does not assume a correlation between the uncertainty of the two variables. A less complicated approach is to use the quantile-quantile (QQ) plot of the data. This plots the smallest energy against the smallest moment, the second smallest energy against the second smallest moment, etc. This approach has the effect of normalizing the different scale variances of each parameter. The ordinary LSR method can then be applied to the QQ data to obtain an accurate line of best fit. This is equivalent to the orthogonal regression method. The figure below shows the difference between the QQ fit and the least squares fit of the energy-moment data at the Tasmania mine. The QQ fit is a better match to the highest point density zones. The poor LSR fit is because the variance in energy is higher than the variance in moment. The distribution of the energy and moment departure indices is plotted below. Both distributions are slightly asymmetrical, likely due to the various seismic mechanisms superimposed. The wider variance in the energy index introduces a bias that has the effect of making a shallower least squares fit as it tries to minimise the vertical departure in the centre of the chart. If you are doing your own energy index calculations, or using different software, you should be aware of the method used to define the energy-moment relationship and the linear constants used. The least squares approach or chi-square regression should be avoided. The QQ based method is the approach used in mXrap and you can find the fitted linear equation in the footer of the energy-moment chart. You might see different parameters for “global” energy index and “local” energy index. This distinction comes from the different energy-moment relationships used. The global EI relationship is based on all events that pass the quality filter. The local EI relationship is based on events that pass the current base filter (volumetric, parameter ranges etc.).

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Plane based analysis

The Grid Based Analysis now has a plane based function. The user can select a plane (defined in the plane editor window) and display cumulative or average parameters on that plane. Like the grid, the average and cumulative parameters are calculated differently. For the average parameters, the plane points are treated identically to the grid – the events around each point are found and the average parameter is calculated (the plane is essentially a 2D grid). The cumulative parameters are calculated on a ‘splatter’ basis; where events within a certain distance of the plane are projected onto the plane and their impact is distributed among the plane points within their source radius. Plane based analysis can be used to try to evaluate the change in seismic parameters along a fault plane, or simply to view changes in parameters easily as a slice. The plane can be dynamically overridden, allowing the user to ‘sweep’ through their mine. Several transparency options are available, including transparency based on a single value, based on the value of the parameter of interest or based on the number of events in the area (analysis quality).

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Seismic source parameters – quick guide

As mentioned in the last blog post, energy and moment are independently calculated based on the displacement and velocity spectra of the recorded waveforms. Another spectral parameter is the corner frequency. The figure on the left shows the corner frequency (f0) on theoretical displacement, velocity and acceleration spectra. The calculation of corner frequency relies on fitting a reliable source model to the observed spectra. Many commonly used source parameters are derived from Energy, Moment and Corner Frequency. Below is a quick guide to these parameters, illustrated with an Energy-Moment chart that has events coloured by the relevant parameter. The corner frequency is indicative of the dimensions of the source (source radius in the case of a circular fault). This is a physical relationship easily demonstrated. In the linked video, you can see and hear the decrease in frequency as the length of the ruler is increased. Another example is the change in frequency resulting from changing the length of vibrating guitar strings. For the same physical reasons, larger seismic events tend to have lower frequencies. The radius of the seismic source is calculated from f0. In theory, the source volume can be calculated based on the Moment and source radius. In practice however, “Apparent” volume is more commonly used to approximate the source volume. The source volume is proportional to the cube of source radius, therefore any errors in the source radius parameter (or corner frequency) are amplified. The method of calculating Apparent Volume is more stable, based on Energy and Moment.

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Seismic energy and moment

You know that energy and moment are parameters to describe seismic events. But what exactly is their physical meaning for a seismic event source and how are they calculated? Moment and energy are both separate (but related) measures of the strength of a seismic event. A similar example is a car engine, the performance is described with two separate (but related) measures: power (hp or kW) and torque (Nm). In a simplified piston and crankshaft arrangement, the torque is the twisting force exerted by the force of the piston on the lever arm (crankshaft). Power relates to the rate at which work is done and how fast the torque is applied (torque x RPM). So, moment, energy and power are all related measures of the system performance. You might have heard that energy and moment are independent source parameters. This is to distinguish them from derived parameters (parameters that are calculated from energy, moment, corner frequency etc). They are independently calculated but they are not unrelated to one another. Moment is related to the displacement (strain) of the source. Energy is related to the speed at which the displacement happens. In general, higher stress conditions lead to higher rates of displacement and therefore higher energy relative to moment. What does Moment physically mean? So, you know that moment is a force applied to a lever arm. You might be wondering, where is the lever arm for a seismic source? In the context of seismic sources, moment is a force couple. Two equal and opposite forces, with a notional distance between them, forms a definite moment. Let’s look at a force couple applied to a small crack. The images below are displacement results from a simple Phase2 model of a small horizontal and vertical crack. The arrows indicate the direction of the displacement. Notice that the displacement pattern is essentially the same for both force couples. This is why, when you do an inversion from the observed waves trying to model the source, the solution comes down to a double-couple. There is no way to distinguish between the two possible solutions. The displacement field caused by a dislocation on a plane is fundamentally equivalent to that produced by a double-couple. For a homogeneous and isotropic medium, the moment of a seismic event caused by the shear fracture on a plane is: M = G x D x A Where, G = Shear stiffness of the rock D = Average displacement A = Area of slip How is Moment calculated? We are rarely in a position to be able to measure the area of slip or the amount of displacement. In practice, moment is calculated from seismic waveforms, usually in the far-field (outside the source volume). The Brune model is used to relate the characteristics of the seismic source to the characteristics of the recorded waveform. The model is based on a circular disc (penny) shaped dislocation surface where a tangential stress drop is applied instantaneously, resulting in a shear wave propogating perpendicular to the fault surface. To compute Moment, a Fourier transform is required to convert the displacement waveform from the time domain to the frequency domain. The frequency content is also referred to as the spectrum of the signal. Moment is proportional to the spectral level (Ω0); the plateau of the displacement spectrum at lower frequencies. The spectra for each sensor must be corrected for geometric attenuation and decay and the Brune model must be fitted to the signal. Moment can then be computed as: M0 = 4πρV3Ω0R Where: ρ = rock density V = the sonic velocity in rock R = the distance to the source In theory, the Brune model is only applicable to the S-wave but in practice, the same method is used for the P-wave. The final Moment for a seismic event is the average of the S-wave and P-wave moment. M0 = (Mp + Ms)/2 What does Energy physically mean? While seismic moment is a better description of the intensity of a seismic event within the near-field, seismic energy is a better description of the potential damage outside the source volume. The energy source parameter does not represent the total work done during the event, rather the energy that is radiated away from the source. The elastic energy radiated by a seismic event is only a fraction of the total work done by the source. How is Energy calculated? Similarly to moment, energy is calculated in the frequency domain, except energy uses the velocity spectrum rather than displacement. The radiated energy is proportional to the velocity-squared spectrum integrated across the full frequency domain. The total energy for a seismic event is the sum of the P-wave and S-wave energy. E = Ep + Es Conclusions The calculations for seismic energy and moment are complex and there are several assumptions and sources of error such as: Error associated with integrating recorded wave to displacement in the time domain Assumptions associated with the Brune source model Error associated with fitting the Brune model to the displacement and velocity spectra, including when bandwidth limitations of seismic systems result in a poorly constrained fit Error associated with the calculation of source location (R)

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