IEEE Signal Processing - March 2018 - 106

Number of Traces

unaligned CCFs. From the relative time delays, updated first
breaks are calculated.
4) Align each CCF, and sum the aligned CCFs to produce a
stacked CCF.
5) Modify the unstacked and unaligned CCFs by cross correlating with the stacked CCF.
6) Terminate when the sum of squares error is acceptable; otherwise, go to step 3.
The aforementioned steps are carried out for S-arrivals and then
for P-arrivals (after muting the S-arrivals). An example is shown
in Figure 8.
A different strategy for time picking is to admit that some fraction of the picks will be false and then use a classifier to find the
true picks for the event. Such a classifier needs to learn the pattern of a seismic event from all arrival-time picks and apply a rule
to cluster the picks into two groups: true event and false picks.
The noisy picks are discarded, so the classifier performs a type
of denoising, although, strictly speaking, this is not denoising of
the signal. Since the true first-arrival times of any isolated seismic event result in a predictable moveout curve on a monitoring
receiver array, a parametric model for valid moveouts can be used
to build a classifier for true picks of an actual seismic event.
In seismology, it is well known that, for homogeneous media,
the moveout curve across a linear array is a hyperbola. To utilize
this prior knowledge, a random sample consensus (RANSAC)based [17], [30] arrival-time event clustering method has been
developed to classify the events and nonevents in the picks. In a
RANSAC-based method, two steps, i.e., hypothesize and test, are
implemented iteratively as follows:
■ Hypothesize: A minimal sample subset (MinSet, denoted as
X kM) is randomly selected from the data set, and the unique
model parameters (i k) are computed for this chosen MinSet
for kth iteration.

5
0

500

1,000
1,500
2,000
Number of Samples
(a)

2,500

P-Arrival
S-Arrival

10
5
0

500

1,000
1,500
2,000
Number of Samples
(b)

2,500

Figure 8. (a) Noisy traces [the original waveform is the same as in Figure 5(a)], where SNR = -2.53 dB and (b) picked P- and S-arrivals together
with original traces. (Figure used with permission from [22].)
106

a b/2 d/2 x
P (x, y, t; i) = 6x y t@ >b/2 c e/2H >yH
d/2 e/2 f t
x
+ 6g h i@ >yH + j = 0.
t

(8)

The RANSAC framework finds the parameter vector i =
[a, f, i] T in a ten-dimensional space. Since each hypothesis
and test step is numerically inexpensive, we can run this alternating scheme for a large number of iterations so that a good
minimal subset (containing all event picks) can be picked with
high probability.
The method has been tested on a data set of 50 s collected by
the Long Beach nodal dense array in Southern California, which
contains 5,200 sensors [40], [41]. Figure 9(a) shows the top view
of the dense sensor array with the event location marked in red.
In (b), the 3-D view is shown with all arrival time picks indicated
by using open circles. Using the RANSAC-based classification
approach, the true picks are successfully selected from all of the
picks, and the optimal hyperbolic surface is delineated using a
red mesh.

Source mechanism-based detection, localization,
and moment tensor estimation

Test: All elements in the entire data set (X D) are checked to
determine which ones can be labeled as inliers, i.e., consistent with the hypothesized model in the sense that the distance from the model's moveout curve is less than some
prescribed value (d) . The set of all such inliers is called a
consensus set (ConSet, denoted as X kC) .
For a surface array, the proposed method must try to fit
an underlying hyperbolic surface from the set of time picks
{(x n, y n, t n)} for n = 1, f, N, where x n and y n indicate the spatial locations of the corresponding sensor. The surface can be
defined using a quadratic equation in (x, y, t) that takes the following general form:
■

As discussed in the "Introduction" section, microseismic monitoring from large arrays using migration-based detection and
location techniques is subject to selecting a threshold that
results in a constant rate of false positives, i.e., detection of
random noise interpreted as real events. This threshold, which
depends on SNR, is commonly applied to the stacked trace to
identify a true event, assuming a consistent SNR across the
array. However, an abnormal value of SNR in a stacked trace
may result from a sudden change of noise on a few receivers,
e.g., when a car drives by or when the well head of a monitoring borehole is hit. To differentiate between such nonstationary noise and a true signal, we can measure the coherency
of the signal across the array along the expected moveout. A
noise burst in part of the monitoring array would show low
coherency across the array while a true signal should show
high coherency. Thus semblance, which is a measure of coherency, is used as an additional differentiator between true signals and false positives.

IEEE Signal Processing Magazine

March 2018

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