1. Moment tensor inversion using observations of unknown amplification
Dr Rosalia Daví
Dr Vaclav Vavryčuk
Institute of Geophysics, Academy of Sciences, Praha, Czech Republic

2. We propose a method for calibrating seismic networks in order to retrieve accurate moment tensors based on a joint inversion of large datasets of earthquakes for moment tensors and for amplifications of stations of the network.
The method is suitable for detecting technical problems at the stations (reverse polarities, incorrect orientation and amplification of sensors), for quantify local site effects and unify seismic networks.
Synthetic tests and numerical modeling (mapping the station distribution, the velocity model, the hypocenter locations of West-Bohemia).
Application in West-Bohemia (retrieval of highly accurate moment tensors).

3. METHODOLOGY
System of equations of the standard moment tensor inversion of amplitudes for one event
where
G is the Nx6 matrix of the Green’s function amplitudes
is spatial derivatives of the Green’s tensor calculated for the ith station.
u is the N-vector of the displacement amplitudes observed at N stations
m is the 6-vector of moment tensor components

4. IT IS IMPORTANT TO HAVE HIGH VARIABILITY OF FOCAL MECHANISMS
If we incorporate into the inversion one uncalibrated station with index i = N+1 and unknown station amplification C(N+1)
Combining the equations we obtain:
IT IS IMPORTANT TO HAVE HIGH VARIABILITY OF FOCAL MECHANISMS

5. Numerical modeling
Generation of datasets of events with synthetic focal mechanisms.
Calculation of synthetic amplitudes.
Contamination of the amplitudes with random noise.
Multiplication of the noisy amplitudes by synthetic station amplifications.
Application of the procedure to calibrate the network and to retrieve the moment noise.

6. Partial network calibration
3 different datasets consisting of 10, 50 and 200 events with 2 station configurations defined as the sparse and dense configurations.
The synthetic P-wave amplitudes were contaminated with uniformly distributed random noise of 3 levels: up to 10%, 25% and 50% of the noise-free amplitude.

7. Events
Fixed stations
Noise level [%]
Δsparse
Δdense 10
0.072
0.041 1 25
0.164
0.085 50
0.260
0.169
0.031
0.017
0.071
0.045
0.113
0.063
0.009
200
0.024
0.047
0.040
0.034
0.029 5
0.148
0.125
0.012
0.013
0.053
0.049
0.008
0.007
0.022
0.018
0.039
0.028 -
0.056
0.105
0.023
0.042
0.005
0.016
The inversion for the station amplifications is performed repeatedly 100 times for different random noise.
The mean and standard deviations of the retrieved station amplifications are calculated.
The inversion code perform better with a high number of stations with known amplifications, a low level of noise and a high number of jointly analyzed events.

8. Sensitivity to the station location
The stations located in the proximity to the nodal lines display higher standard deviations compared to the other stations.
If the condition of a variety of focal mechanisms is satisfied, the inversion yields accurate results independently of the station locations.

9. Complete network calibration
Amplification of the 1st station is fixed
The other amplifications are calculated
All amplifications are normalized
N sets of N retrieved amplifications
Repeating for all stations
Improved initial guess of station amplifications
Amplification of the 2nd station is fixed
Amplification of the Nth station is fixed
Averaging of N amplifications for each station
Simple initial guess of station amplifications
Simple initial guess
Calculation of the 1st station amplification
The other amplifications are fixed
Normalization of the N retrieved amplifications
New set of station amplifications
Improved initial guess
Starting set of station amplifications
Difference in two successive
sets of amplifications is small
Calculation of the 2nd station amplification
Calculation of the Nth station amplification
Repeating for all stations
Final set of station amplifications
Yes No
New iteration
Complete network calibration
The complete network calibration can adjust station amplifications by including the local site effects at all stations.
The simplest way to calibrate the complete network (of N stations), is to perform the calibration in iterations.

10. Convergence
(a)
(b)
Simple initial guess
Improved initial guess
1) We assume a simple initial guess when the starting values of all station amplifications equal 1.
2) We estimate the starting values using the improved initial guess.
The iteration process with the improved initial guess (b) converges much faster.

11. Accuracy
True station amplifications
Retrieved station amplifications
(a)
(b)
KAC
The retrieved amplifications are slightly biased from the true amplifications (noise in the data).
The lowest accuracy is achieved for station KAC (in the intersection of the nodal lines).
Difference between the true and retrieved amplifications in percent.

12. The RMS values are higher for the uncalibrated network.
The RMS values decrease for corrected moment tensors.
Data set
DC [%]
CLVD [%]
ISO [%]
abs(CLVD) [%]
abs(ISO) [%]
RMS
Uncorrected moment tensors
Noise-free data
77.6
-17.2
-4.7
17.7
4.7
0.248
Noisy data
77.5
-17.1
-4.6
17.8
4.8
0.266
Corrected moment tensors
100.0
0.0
0.000
92.5
-0.7
0.1
5.6
1.8
0.115
Uncorrected moment tensors
DC component
Non-DC components
RMS
Corrected moment tensors
The focal mechanisms (left-hand plots) are better clustered for uncalibrated moment tensors (artificial).
The uncorrected moment tensors display significant false negative CLVD and ISO components.
The corrected moment tensors display smaller CLVD and ISO components and form a cluster centered around the origin of coordinates.

13. WEST BOHEMIA REGION
Czech Republic
Germany
22 short-period seismic stations (yellow triangles).
Epicentres of the 2008 swarm (red circles) .
Depth of 7.6 to 10.8 km.
(from Vavryčuk, 2011)
200 micro-earthquakes: min 20 stations, high signal-to-noise ratio and highly accurate hypocenter locations.

14. (a)
Simple initial guess
(b)
Improved initial guess
Convergence
Iterations with the improved initial guess converge faster.
Good convergence of the iterations and a reasonable accuracy of the station amplifications (the variability of the focal mechanisms is higher for observed data).

15. Relative amplification
Accuracy
Station
Station name
Relative amplification
Abs. error
Rel. error [%] 1
BUBD
1.01
0.02
2.0 2
HOPD
1.24
0.03
2.2 3
HRC
0.62
3.5 4
HRED
1.16
0.06
5.6 5
KAC
0.85
0.05
6.0 6
KOC
0.65
0.01 7
KOPD
0.70 8
KRC
0.88
1.6 9
KVC
1.12
2.7 10
LAC
1.06
0.04
3.3 11
LBC
0.82
1.0 12
LOUD
1.45
1.3 13
NKC
1.00
1.5 14
NKCN
1.4 15
PLED
0.79 16
POC
1.40 17
POLD
1.04
1.9 18
SKC
1.02
2.3 19
SNED
0.80
1.1 20
TRC
1.90 21
VAC
0.84
0.9 22
ZHC
0.78
2.1
The accuracy of the amplification corrections was estimated using a jack-knife test (the iterative procedure was run 50 times on subsets of 100 randomly selected events).
The achieved accuracy of the station amplifications is ~ 4% for the majority of stations.
The only exceptions are stations KAC and HRED with accuracy of 6.0% and 5.6%.

16. Accuracy KAC station (unfavorable position).
HRED
TRC
KAC station (unfavorable position).
HRED station (rather high noise level).
No station with a reversed polarity.
TRC station (incorrect calibration of the sensor, incorrect value of the gain factor or an anomalous medium response).
The scatter of the amplification corrections is high: the values range from 0.62 to 1.45.

17. The DC part of the moment tensors is rather stable.
Uncorrected moment tensors
DC component
Non-DC components
RMS
Corrected moment tensors
The DC part of the moment tensors is rather stable.
The non-DC form a more compact cluster for corrected MT.
CLVD and ISO components are less compressive.
Dataset
DC [%]
CLVD [%]
ISO [%]
abs(CLVD) [%]
abs(ISO) [%]
RMS
Uncorrected moment tensors
76.3
-16.3
-2.3
19.0
4.6
0.191
Corrected moment tensors
83.8
-9.7
-1.5
12.5
3.7
0.112
The average value of the CLVD changed from -16.3% to -9.7%.
The average value of the RMS is reduced from 0.19 to

18. Conclusions
We propose a method for calibrating seismic networks in order to retrieve accurate moment tensors based on a joint inversion of large datasets of earthquakes for moment tensors and for amplifications of stations of the network.
The inversion works better for dense networks (good focal sphere coverage, high variety of focal mechanisms and large datasets).
It detects: reverse polarities, incorrect orientation and amplification of sensors, anomalous local site effects at stations.
The tests (for accuracy and stability of the method) show that the moment tensors is retrieved with higher accuracy.

19. The inversion was applied to calibrate the WEBNET network (dataset of 200 micro-earthquakes that occurred in 2008).
The moment tensors retrieved using amplitudes of properly calibrated stations of the WEBNET network display lower RMS than the original moment tensors.
The focal mechanisms are not changed but the non-DC components of moment tensors changed significantly (one origins of spurious non-DC components might be linked to inaccurate amplifications).
The method is can be used for data gathered: (1) in laboratory experiments, (2) in boreholes or in mines (calibration and orientation are frequently unknown), or (3) in field experiments (networks are inhomogeneous).

20. Thank you for your attention!