Double Difference Earthquake Location(hypoDD) - Geophysics at ...

Locating Earthquakes Using HypoDD
Hongfeng Yang
October 25, 2013
1 Equations
Locating earthquake is a classic but still active research subject. Accuracy of location results depends
on several factors, such as the station coverage, the number of available phases, and the
velocity model used [Pavlis, 1986]. Relative location methods can effectively reduce the uncertainties
due to structure and thus give more reliable results [Pavlis, 1992; Waldhauser and Ellsworth,
2000].
The arrival time difference ∆t of two events at one station can be written as
∆t = − −→ p · ∆ −→ x + ∆o, (1)
where −→ p is the slowness vector, ∆ −→ x represents the separation of the events, and ∆o stands for the
origin time difference between the two events. Here we assume that the spatial separation between
the two events is much smaller than the event-station distance, so that the ray paths between the
sources and the station are similar and the effect of lateral structural variation on ∆t can be ignored
[Waldhauser and Ellsworth, 2000]. In other words, the −→ p is same.
If we have many observations, equation (1) can be expressed as
∆t i = − −→ p i · ∆ −→ x + ∆o
= −(p xi ∆x + p yi ∆y + p zi ∆z) + ∆o,
(2)
where i stands for the ith station. We re-write equation (2) into matrix format
⎛ ⎞ ⎛
⎞
∆t 1 −p x1 −p y1 −p z1 1 ⎛ ⎞
∆t 2
⎜ ...
⎟
⎝ ... ⎠
= −p x2 −p y2 −p z2 1
∆x
⎜ ... ... ... ...
⎜∆y
⎟
⎟ ⎝
⎝ ... ... ... ... ⎠ ∆z⎠ , (3)
∆o
∆t n −p xn −p yn −p zn 1
or
d = Gm. (4)
In most cases the number of observations n will be greater than the number of unknown parameters
(4 in this case) so that equation 3 is overdetermined. If the number of differential times is not too
large, we use the singular value decomposition to obtain the solution
m = V Λ −1 U T d, (5)
where U is the matrix of eigenvectors that spans the data space, V is the matrix of eigenvectors
that spans the model parameter space, Λ is a diagonal matrix whose elements are non-negative
singular values of G.
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2 HypoDD
HypoDD is a Fortran77 computer program package for relocating earthquakes with the doubledifference
algorithm of Waldhauser and Ellsworth [2000]. The latest version of HypoDD is version
1.3 and is available from http://www.ldeo.columbia.edu/˜felixw/hypoDD.html.
This tutorial follows instructions on http://geophysics.eas.gatech.edu/people/cwu/teaching/hypoDD/hypoDD.
2.1 Input files
hypoDD reads input parameters and files from the hypoDD.inp, which contains general information
such station, DD time, output file, and etc. It also includes a few important parameters that are
used during inversion.
1. OBSCC and OBSCT
OBSCC and OBSCT define the minimum number of links between event pair for cross correlation
differetial time (CC) and catalog differential time (CT). This parameters determine the
strength of links between event pair, which is crucial for relocation process. Below is quoted
from the hypoDD manual.
“To prevent an ill-conditioned system of DD-equations, hypoDD ensures connectedness between
events by grouping events into clusters, with each cluster having a chain of links from
any event to any other event in the cluster. The strength of this chain is defined by a minimum
number of observations per event pair that build up the chain (OBSCC, OBSCT for
cross-correlation and catalog data, respectively). Typical thresholds are similar to the number
of degrees of freedom for an event pair, which is 8 (3 spatial and 1 time for each event). If P-
and S-wave data is used, the threshold has to be higher to actually reach 8 stations per event
pair. Increasing the threshold might increase the stability of the solution, but it might also
split up a large cluster into subclusters. Lowering the threshold might include more events in
a single cluster, but might decrease the stability of the solution. A high threshold, however,
does not necessarily guarantee stable relocations, as the spread of the partial derivatives is
not taken into account directly.”
2. ISTART
This parameter tells hypoDD whether the intial location of events is from a centroid location,
or is taken from network sources (different initial locations).
3. ISOLV
For a small number of events, it is easy to contol the performance of hypoDD by SVD
(ISOLV=1). However, it is more difficulte to assess the reliability of the locations in the case
of large number of events (thousands or more) that can only be relocated in LSQR mode
(ISOLV=2).
4. NSET and Weighting
NSET is number of iterations during inversion. Specifications during each iteration can be
defined by values of weighting and re-weighting.
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2.2 Data format
3 Waveform Cross Correlation
Let f(t) and g(t) represent two time series, conventional non-normalized cross-correlation of the
two is
C(t) = f(t) ∗ g(t) =
∫ ∞
−∞
f(t + τ)g(τ)dτ. (6)
We obtain differetial times of each event pair at one station using waveform cross correlation (CC).
Following are the procedures to perform CC.
The tutorial dataset is located at “hyangteachingHYPODDtutsmalldata”.
1. Preparation of sac files
All waveform data are saved in sac format named as r, t, z, where r usually refers to east
component and t stands for north component. Note this is not required, but is recommended
in order to use my following processing scripts. All the sac files are saved under directory
named by conventional event ID (year, month, day, hour, minute, second).
2. P and S arrivals
All P and S arrival times need to be manually picked (usually for a small dataset), extracted
from catalog arrivals, or estimated from a velocity model. In this tutorial, the P and S arrivals
for all events are estimated from detection times based on template event waveforms. They
are saved in the sac header files as t 1 and t 2 , respectively. So we can simply extract them
using saclst and save them into event/p.arr and event/s.arr, respectively.
Then two arrival time files, sta.parr and sta.sarr, need to be built following the format of
eve/sta.z arr 1/0, where “1” indicates good waveform and “0” represents noisy one. It is
also important to check the waveform quality in this step before running waveform cross
correlations.
3. check waveform quality
There is a GUI interface written by Dr. Lupei Zhu at SLU to read in waveforms and to turn
it on or off using mouse. This program is named “mcc.tcl”. To use it, prepare arrival times
following the above format and type “mcc.tcl sta.parr” and check waveform quality. One can
also adjust arrival times of individual traces by dragging and moving them by mouse.
4. waveform cross correlation
Once you have the sta.parr and sta.sarr, you can perform waveform cross correlation using
“crscrl.pl”. Refer to README in the example dataset for the usage of the perl script.
5. Check cross correlation results
Use “chkCluster.tcl t 1 /t 2 sta.cc” to check the quality of waveform cross correlation. Adjust
the trace individually if necessary.
6. Obtain differential times
Use “cc2dt.pl” to obtain differential times after waveform cross correlation. Refer to “README”
in the sample dataset.
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References
Pavlis, G. L., Appraising earthquake hypocenter location errors: a complete practical approach for
single-event locations, Bull. Seismol. Soc. Am., 76, 1699–1717, 1986.
Pavlis, G. L., Appraising relative earthquake location errors, Bull. Seismol. Soc. Am., 82, 836–859,
1992.
Waldhauser, F., and W. L. Ellsworth, A double-difference earthquake location algorithm: Method
and application to the northern Hayward fault, California, Bull. Seismol. Soc. Am., 90, 1353–
1368, 2000.
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