IEEE Signal Processing - March 2018 - 109

polarity of amplitudes with the inverted moment tensor before
Velocity estimation
stacking. The block W [i] corresponding to nonzeros in x gives
An accurate velocity model is crucial for the correct location of
the information about the locations of events. Using a convenmicroseismic events. Unlike applications such as radar, where
tional least squares approach as in (12) for this submatrix W,
wave propagation occurs in a homogeneous medium (free
space), elastic models of the earth are needed to capture the
we obtain an estimate of the source mechanism. In a monitorsignificant variations in velocities of both P- and S-waves that
ing region containing a large number of grid points, the size
result in large differences in ray propagation.
of the dictionary W can be prohibitively huge. In addition, the
Both surface and downhole monitoring arrays require vecoherency between spatially close submatrices is usually quite
locity model calibrations. Surface arrays often use only a single
high due to the continuity of the wave propagation, which also
velocity model if the array is dense and has a large aperture, as
makes accurate location resolution very challenging.
explained in the "Introduction" section. It is well known that
The time-reversal method (RTM) is an effective tool to
the increased coverage of the receiver array results in greatimage the location and source mechanism of microseismic
er robustness in the location of the microseismic events, i.e.,
events. Energy emitted by microseismic events and transmitted
reduced sensitivity to the velocity model.
through the media in the form of wave
Although surface microseismic monitoring
propagation can be refocused back to the
in seismology, it is
arrays with wide aperture result in robust
source. Mathematically, the time-reversal
well known that, for
horizontal positioning, they tend to have
concept is simply to solve the wave equahomogeneous media, the
large vertical uncertainty in the estimated
tion backward in time using the received
moveout curve across a
event locations [2], [6].
wave field at the sensor array [16]. For a
linear array is a hyperbola.
Downhole arrays require a velocity modgiven velocity model of reasonable quality
el that models both P- and S-wave propaand a sufficient array aperture, the extrapogation, if not more complicated anisotropic models. Also,
lated wavefield will concentrate to a focus point, which can
downhole microseismic velocity modeling is particularly
be considered as an approximation of the source location. By
challenging as usually both depth and horizontal position are
examining the energy radiation pattern at this focus point, the
extremely sensitive to the velocity model. A majority of curfocal mechanism can be estimated as well.
rent injection projects, especially hydraulic fracturing of shale,
Clearly, the wavefield extrapolation I (x, t) in a 3-D velocity
are done in relatively simple flat-layered sedimentary basins.
model gives a 4-D data volume, 3-D in space and 1-D in time,
Therefore, the usual starting velocity model is a layered model.
to be searched for the focus
Downhole microseismic models usually use the sonic logs, i.e.,
-1
r
measurement of P- and S-wave velocities along a borehole or
I (x, t) = F ' / D (x ri, ~) G (x ri, x, ~) 1,
(13)
i
vertical seismic profiling, to build an initial isotropic velocity
model. Very often, the velocity model is then adjusted to locate
perforation shots to match their correct known locations [29].
where D (x ri, ~) is the data received on sensor ri at frequency
r is the complex conjugate of the approximated Green's
The velocity model adjustment can be realized in a number of
~, G
ways, but it usually involves fitting S-P-wave traveltime diffunction, and F -1 is simply the inverse Fourier transform. To
ferences. The S-P-wave traveltime differences that cannot be
obtain the source location and origin time, we need to find t 0
explained by an isotropic model are then fitted by a transverse
when I (x, t) has its maximum amplitude or exhibits a reasonisotropy (TI) with vertical axis of symmetry velocity model [4]
ably focused image for the seismic events in the 4-D cube. This
or a more complex velocity model [3].
method is called arithmetic-mean RTM (AmRTM). One of the
Downhole calibration further requires learning the oriadvantages of RTM-based methods is that we do not need to
entation of 3C sensors as polarization measurements will be
pick the wave arrivals. Usually, we solve only one wave equaused to locate the microseismic events. Orientation of the
tion to simulate the wave propagation using all of the traces as
monitoring geophones often is determined from back aziinput data, which implements the summation in (13) implicitly.
muths of P-waves generated by the perforation shots located
In general, the source signature of a microseismic event is
in the treatment well, assuming isotropy and lateral homogefar from an impulse, so the focus point in AmRTM is usually a
neity of the medium between wells. Therefore, the orientation
smeared region rather than a distinct bright spot. This increasof the geophones in the monitoring well is determined relative
es the difficulty and accuracy of the detection and localization.
to the position of the treatment well at depths corresponding
To this end, different imaging conditions have been proposed,
to the perforations. The orientation (in a geographical coore.g., geometric-mean RTM (GmRTM) [27]:
dinate system) of the monitoring array and of the observed
microseismic event hypocenters can, however, be obtained
1 (x) = / % W ri (x, t),
(14)
t i
only from the positions of the receivers and the perforations.
Therefore, any error in the positioning of the monitoring array
-1
r
or uncertainty in the locations of the perforations is directly
The
GmRTM
can
be
where W ri = F {D (x ri, ~) G (x ri, x, ~)}.
propagated into errors when estimating positions in the fracregarded as a cross correlation, and we collapse the time axis
ture system.
for the ease of localization.
IEEE Signal Processing Magazine

March 2018

109

Table of Contents for the Digital Edition of IEEE Signal Processing - March 2018