The HybridMT is an all-in-one example module. It contains the basic functions for tremors processing, namely: locator (see localization: HypoDD), calculation of the seismic moment tensor (MT), and calculation of the Mw magnitude. The module requires optimization modules for counting locations. Provides SWIP5 with the following functionality:

• Location estimation,
• Calculation of the seismic moment tensor and at the same time the magnitude Mw,
• Drawing and displaying seismic moment tensors in the form of "beach balls".

Description of the method of the seismic moment tensor estimation

MT counting is based on the hybridmt-v1.2.2 program by Grzegorz Kwiatek (Wiejacz 1992, Kwiatek i Martinez-Garzon 2016). The table below shows the equivalent components of the seismic moment tensor:

The mechanism is calculated from the amplitudes of the first inputs of the P waves or the first inputs of the P and S waves. The seismic process is assumed to be rectangular, which allows the determination of the full Green's function and the time course of the source function to be omitted. According to Aki i Richardsa (2002), Fitch i in. (1980) oraz De Natale i in. (1987), the recorded displacement for the P wave is

while for wave S it is

where ρ is the average density of the medium, r is the source-receiver distance, α and β are the average velocities of the P and S waves at the source,  T^P and T^S are the travel times of the P and S waves, M is the seismic moment tensor, l is the direction of the wave in the receiver , γ is the direction of the wave at the source, and ̂P_γ and ̂P_I are the transverse directions at the source and receiver. The moment tensor is obtained by solving a set of N equations per U, where N is the number of stations that recorded the event. Six independent components of the moment tensor require a minimum of six equations, but the more the better. The system of equations is then treated as redundant and solved using the method of least squares (L2 norm), where the z-cost function is the sum of the squares of the residuals. The uncertainty of the MT estimate can be calculated using the normalized root mean square (RMS) error between the theoretical and estimated amplitudes (Stierle i in., 2014a, 2014b)

The full tensor is calculated first. Then the trace null (TN) condition is imposed on the solution and the cost function is minimized again. This gives us a solution to the focal mechanism without changing the volume in the source. When the zero trace and zero determinant conditions are set, the solution is constrained to a double couple (DC). The entire solution moment can also be decomposed into the components isotropic (ISO), compensated linear vector dipole (CLVD) and parts of the shear mechanism (DC).

There are two ways to determine MT:

The old method is based directly on the work of Wiejacza (1992) and the hybridmt-v1.2.2 program. It calculates MT only from the vertical components, determining the amplitude of the P wave according to the formula

A = A_z / cos(i)

where i is the wave incidence angle, while Az is the amplitude determined from the vertical component of the signal. This method is applicable to mine systems, which often consist of only one component sensor. In the case of surface systems, the assumption of a too flat ascent account due to the large distance and inaccurate model leads to overestimation of the amplitudes.

The new method makes it possible to calculate MT from both P and SV waves. It uses three signal components. It rotates the signal from the ZNE system to LQT, then for the P wave the amplitude is determined on the L component and for the SV wave on the Q component.

Counting moment tensors instruction

Before counting the MT, you should have the phases marked, the event located and the wave angles estimated. The calculation of the seismic moment tensor is called from the menu with the command PluginsFull MT. Then a window as shown in Fig. 26 appears.

Rys. 26  Okno Hybrid Moment Tensor

You can count MT for different locations (1). However, they must contain information about the angles for at least 6 waves. Locations that do not contain the angle of exit of the wave from the focus are ignored. If the wave ascent angle and back-azimuth are defined for peaks, they need not be defined for the focus. In the old method, you can choose whether the amplitude of the vertical component of the wave is corrected for the ascent angle - Apply incid (4). For P waves, in the seismogram window, the following are drawn for the Z components: velocity and displacement seismograms. The pair of seismograms indicated by the mouse is enlarged. The beginning and the end of the wave are marked on the velocity seismogram. The end of the wave is marked by right-clicking. The start of the wave defaults to the time of the selected phase, but can be changed by right-clicking while holding down the Ctrl key. After selecting a wave, the initial and final amplitude dashes are displayed on the displacement seismogram. This value is calculated using the polynomial integration method and may differ slightly from the displacement seismogram. When pressing Compute, full MT, trace null MT and double couple MT are counted and the corresponding "beach balls" are displayed (6). If you click on one of them, it will be displayed enlarged above.

• Estimation of the MT using the first inputs of the P wave
• Estimation of the MT using the first inputs of the P and SV waves
• Drawing of "beach balls"