Illustrator glitch?
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- fredddddd
I have CS5.5.
When I try to make a box that is .3in square, it makes it to 0.3056 in x 0.2917 in????
Why?
- nato0
May have Snap to Pixel Grid turned on?
- fredddddd0
^ I don't think so, where is that? I checked and only have snap to point and snap to grid under "view" and both are turned off.
- nato0
You set it in the dialog box when you make a new document.
- fredddddd0
It's on print setting, so I didn't check that box.
- antagonlsta0
that's odd indeed.
- ernexbcn0
To compensate earth's gravity and the magnetic poles fluctuations, this happens at the floating point level in the CPU.
- ORAZAL0
Back azimuth and slowness anomalies observed at seismic arrays can be used to constrain local and distant
structural and propagation effects in the Earth. Observations of large systematic deviations in both azimuth and
slowness measured for several P phases (i.e., Pg, Pn, P, PKP) recorded at several IMS arrays show a characteristic
sinusoidal pattern when plotted as a function of theoretical back azimuth. These deviations are often interpreted as
the affect of the wavefield being systematically bent by refraction from a dipping velocity structure beneath the
array, most likely a dipping Mohorovičić discontinuity (Moho).
We develop a model-based technique that simultaneously fits back azimuth and slowness observations with a
ray-based prediction that incorporates a dipping layer defined by its strike and dip. Because the azimuth and
slowness deviations both vary as a function of true azimuth, fitting both residuals jointly will give a more consistent
calibration for the array. The technique is used to fit over 9900 observations at CMAR from a global distribution of
well-located seismic events.
Under the assumption that the dipping layer is the Moho with mantle velocity 8.04 km/sec and crustal velocity
6.2 km/sec, we estimate that Moho strike and dip under the CMAR array are 192.6° and 18.3°, respectively. When
the trend of the Moho is removed from the back azimuth and slowness residuals, both the sinuous trend and
variations with predicted slowness are mitigated. While a dipping interface model does not account for all of the
discrepancy between observed and predicted back azimuth and slowness anomalies, and additional calibration
whether empirical or model-based should be pursued, this technique is a good first step in the calibration procedure
for arrays exhibiting sinusoidal residual trends.
2012 Monitoring Research Review: Ground-Based Nuclear Explosion Monitoring Technologies 232OBJECTIVES
The goal of this project is to test the robustness of a ray-based prediction of azimuth and slowness anomalies to fit a
dipping Moho geometry beneath an array. To date the modeling of such anomalies has been simplified to removing
a sinusoidal-fit function from the back azimuth and slowness residuals separately. This approach does not properly
account for the effect of variable slowness at a given back azimuth. Here we present a more complete model
approach that simultaneously fits back azimuth and slowness observations with a ray-based prediction that
incorporates a dipping Moho. Because the azimuth and slowness deviations both vary as a function of true azimuth,
fitting both residuals jointly will give a more consistent calibration for the array.
RESEARCH ACCOMPLISHED
Seismic array data provide an advantage over single station data for monitoring purposes because they allow for the
additional measurement of the vector velocity of an incident wave front. Specifically, the back azimuth and
slowness may be determined as the seismic wavefronts sweep across the array of sensors. The back azimuth and
slowness parameters are commonly used to distinguish between different seismic phases, associate arrivals to an
event, and can even provide needed constraints on event location when an event is recorded at just one or two
stations. Since their inception, seismic arrays have been used for both monitoring as well as numerous studies of
Earth structure (see Rost and Thomas [2002] for an excellent review). However, to use arrays effectively they need
to be calibrated to account for local structural effects (e.g.,Ringdal and Husebye, 1982; Ram and Yadav, 1984;
Koch and Kradolfer, 1997; Tibuleac and Herrin, 2001; Bondar et al. 1999; Lindquist et al., 2007; Tibuleac and
Stoujkova, 2009).
For Earth models with spherically symmetric velocity structure, the great-circle path from the event to the station is
used to predict back azimuth, and theoretical ray parameter is used to predict slowness. Empirically, back azimuth
and slowness measurements are often observed to deviate significantly from theoretical predictions, and most
notably they can show a sinusoidal azimuthal dependence. Such a pattern suggests that the ray paths have been
systematically bent by refraction from a dipping velocity structure beneath the array, and the Moho is a likely
candidate (Figure 1). The functional relationship between the Moho strike φ and dip δ angles and the sinusoidal
pattern of back azimuth and slowness residuals was derived by Niazi (1966) and subsequently used in several array
calibration studies (e.g., Otsuka, 1966a, 1966b; Greenfield and Sheppard, 1969; Havskov and Kanesewich, 1978;
Koch and Kraundoffer, 1997; Tibuleac and Herrin, 1997; Schweitzer, 2001; Lindquist et al., 2007; Tibuleac and
Stroujkova, 2009).
Figure 1. Geometry of the incident and refracted wavefronts with respect to a dipping interface with a
specified orientation (defined by φ , δ). Note that the illustration is simplified to 2 dimensions,
whereas the actual problem is 3 dimensional. l is the directional vector of the incident beam
(determined by theoretical back azimuth and slowness), v is the directional vector of the refracted
beam, n is the normal vector to the dipping-Moho plane (using φ and δ), θ1 is the angle of incidence,
and θ2 is the angle of refraction. We use the iasp91 P-wave velocities, 8.04 and 6.2 km/sec
respectively, for v1 and v2, below and above the Moho.
2012 Monitoring Research Review: Ground-Based Nuclear Explosion Monitoring Technologies 233The discrepancies (hereafter called errors) of back azimuth and slowness can be attributed to deviations in the
assumption of spherically symmetric velocity structure (model error) and measurement error. In this paper we
address model error, even though measurement error is also significant. Studies based on the approach of Niazi
(1966) address back azimuth and slowness model errors. For array stations with uncommonly large errors, back
azimuth and slowness errors often exhibit a discernable trend when plotted as a function of theoretical back azimuth.
Empirical approaches may correct for these trends if the calibration data set provides outstanding geographic
coverage with minimal measurement error. However, a model-based approach may better exploit the data trend by
allowing trend extrapolation for data sets with limited geographic coverage. Using a model-based first step in the
calibration procedure will also improve subsequent empirical calibration because back azimuth and slowness
residuals (following a model-based correction) will better adhere to the zero-mean assumption that is implicit in
empirical calibration methods such as kriging.
To date the modeling has been simplified to removing a sinusoidal-fit function from the back azimuth and slowness
residuals. This approach does not properly account for the effect of variable slowness at a given back azimuth. Here
we present a more complete model approach to a dipping layer correction that simultaneously fits back azimuth and
slowness observations with a ray-based prediction that incorporates a dipping layer defined by its strike and dip. In
this method the free parameters would be the strike φ and dip δ angles of the Moho plane in contrast to the phase
and amplitude of cosine waves that are fit to back azimuth and slowness residuals separately. Because the azimuth
and slowness deviations both vary as a function of true azimuth, fitting both residuals jointly will give a more
consistent calibration.
We examine the effect of a dipping Moho discontinuity on the angle of incidence and the observed azimuth of a
seismic wave, and what follows is an overview of the geometry and derivation of the equations governing the
azimuthal dependence of both observed azimuth and slowness parameters, some examples of the ray-based forward
predictions, and finally some examples of fitting data.
2012 Monitoring Research Review: Ground-Based Nuclear Explosion Monitoring Technologies 234Azimuth and Slowness Measurements
We use back azimuth and slowness measurements made from P phases (i.e., Pg, Pn, P, PKP) recorded at several
IMS arrays between the years 1995 and 2009. These measurements are provided by the USNDC and are produced
from well-located events with epicenter error of 25 km or less (GT25) and mb > 4.5. The back azimuth and slowness
are estimated using a standard frequency-wavenumber algorithm (e.g., Capon, 1969; Kvaerna and Doornbos, 1986).
We include only those measurements having uncertainties of less than 15° for back azimuth and 1.5 sec/° for
slowness, and we further restrict the slowness to be less than 13.5 sec/° as such horizontally travelling waves with
high slowness cause instabilities in the optimization code. The global distribution of events provides good
back-azimuthal coverage at several arrays allowing us to discern patterns in the azimuth and slowness residuals.
The back azimuth and slowness measurements are next compared with theoretical predictions made using the iasp91
velocity model and residuals are formed. Examples of these residuals are shown in Figure 2 for the CMAR (Chiang
Mai, Thailand) and ILAR (Eilelson, Alaska) arrays. Both the back azimuth and slowness residuals show the
characteristic sinusoidal shape indicative of a dipping layer beneath the array. Note the effect of slowness (incoming
ray inclination) on the magnitude of the azimuth deflection. The effect of the dipping interface is strongest for rays
that approach the interface with a small slowness (i.e., at a steep angle).
We observe this pattern in the residual distribution at a number of IMS arrays and report results here for only
CMAR and ILAR as there are calibration studies for both of these arrays, by Tibuleac and Stroujkova (2009) and
Lindquist et al. (2007), which use the Niazi (1966) approach to determine Moho dip and strike and thus provide a
direct comparison with our results.
Figure 2. Back azimuth and slowness residuals plotted against true back azimuth for stations CMAR (left)
and ILAR (right). Color indicates theoretical slowness of the incident wavefront (in sec/°) based on
the iasp91 velocity model and essentially represents the epicentral range. The sinuous trend of the
residuals and the affect of slowness on the azimuth residual are both consistent with a dipping
interface beneath the array.
2012 Monitoring Research Review: Ground-Based Nuclear Explosion Monitoring Technologies 235Dipping Layer Determination
Niazi (1966) attributed systematic back azimuth and slowness deviations to dipping geologic interfaces under the
array, and he derived equations to compute these deviations for a dipping interface. Both back azimuth and slowness
residuals have an asymmetric cosine shape and their values should be close to zero for rays coming along strike and
maximum (absolute value) for rays coming up- or down-dip. This is indicated by the 360° periodicity and 90° phase
shift between slowness and back azimuth anomalies (Figure 2). The extent of the azimuth bias varies with the angle
of incidence, dip of the Moho, and velocity ratio. The pattern of azimuth perturbation can constrain the dip if the
velocity contrast is known, or vice versa, but the pattern is not unique when both dip angle and velocity contrast are
varied. Typically one determines the strike from a cosine fit to azimuth residuals then uses that strike to find the dip
angle from a cosine fit to the slowness residuals.
The strike is found from the crossover points of the back azimuth residuals and the dip is found from the amplitude
of the slowness residuals. The strike of the dipping Moho is the point with the largest azimuth residual following the
first zero crossing of the cosine fitting curve. According to Niazi (1966), if the azimuths are read clockwise the
direction of the dip is given by the point of transition of the back azimuth residuals from negative to positive values.
Both Tibuleac and Stroujkova (2009) and Lindquist et al. (2007) used this approach to determine the Moho
orientation beneath CMAR and ILAR respectively.
As the refraction of the seismic wavefront follows Snell’s law, the effect of a dipping interface on an incident
seismic ray may be more directly computed using vector arithmetic and Snell’s law
(http://en.wikipedia.org/wiki/Sn... allowing us to solve for the Moho strike φ and dip δ angles by
simultaneously fitting both the back azimuth and slowness residuals using:
ݒ ൌ ቀమ
భ
ቁ ݈ ቀమ
భ
Θଵܱܵܥ
Θଶܱܵܥ
ቁ ݊ [1]
and
Θଵܱܵܥ
ൌ ݊ · ሺ݈
ሻ
,
Θଶܱܵܥ
ൌ ට1 െ ቀమ
భ
ቁ ଶ ሺ 1 െ ܥܱܵଶ
Θଵ
ሻ
where l is the directional vector of the incident ray (determined by theoretical back azimuth and slowness), v is the
directional vector of the refracted ray, n is the normal vector to the dipping-Moho plane (using φ and δ), θ1
= angle of
incidence, and θ2
= angle of refraction. We use the iasp91 P-wave velocities below and above the Moho, 8.04 and
6.2 km/sec respectively, for v1 and v2.
We have written a MATLABTM code based on equation [1] to invert back azimuth and slowness residual data for the
strike and dip of an interface using a multi-variable optimization technique. The forward model may be written in
the generalized form:
ߙ∆ሺ
ߤ∆ ,
ሻ ൌ ݂ ሺ ߶, ߜ, ߙ
ߤ ,
ଵ ݒ ,
ଶ ݒ ,
ሻ [2]
ߙ∆ Where
ߤ∆ and
are differential back azimuth and slowness residuals for the i
th data point (i.e., the measured
data). The function f describes how an incoming plane wave with an expected back azimuth of ߙ
and slowness ߤ
,
based on the known source receiver configuration, interacts with an oriented plane with specified velocities above
and below the boundary ( ݒ ଵ
and ݒ ଶ
respectively). The goal is therefore to find the optimal orientation of the plane
that satisfies equation [2] (i.e., to determine φ and δ).
We chose to utilize a pre-existing algorithm from the MATLAB Optimization ToolboxTM called lsqcurvefit which is
a nonlinear curve-fitting routing that is designed to solve multivariant systems in a least squares sense. In our
specific case, the lsqcurvefit algorithm searches for the strike and dip of the plane by minimizing:
ߙ ,ߜ ,߶ ሼ ݂ ሺ ∑
ߤ ,
ଵ ݒ ,
ଶ ݒ ,
ሻ െ ሺ∆ߙ
ߤ∆ ,
ሻሽ
ୀଵ
2
[3]
2012 Monitoring Research Review: Ground-Based Nuclear Explosion Monitoring Technologies 236where the summation is over all n slowness/azimuth residual data points. The MATLABTM function utilizes the
trust-region-reflective optimization algorithm, based on the interior-reflective Newton method developed in
Coleman and Li (1994, 1996), to simultaneously search for the optimal strike and dip of the proposed orientation of
a plane defining the Moho.
Results
We applied this optimization technique to determine the best-fitting strike and dip of a Moho interface beneath the
CMAR array and compare the results using the same data set with Niazi’s technique. The azimuth residuals for 9903
P phases recorded at CMAR are shown in Figure 3 (top, left) and are fit by an asymmetric cosine curve of the form
1.04 - 14.2*cos(Z – 188.4°) where Z is the true back azimuth computed from iasp91. The curve fit is done using the
fminsearch function in MATLABTM, and as described by Niazi (1966) strike of the dipping Moho is the point with
the largest azimuth residual following the first zero crossing of the cosine fitting curve, which in this case is 188.4°.
The lower left plot in Figure 3 shows the azimuth residuals distribution after the cosine function has been removed.
Using our optimization method we find that the best fitting Moho at CMAR has strike 192.6° and dip 18.3°. The
predicted azimuth residuals from this Moho model are shown in Figure 3 (top, right), and when these values are
removed from the observed values the resulting distribution is shown in Figure 3 (lower, right). The major sinuous
trend in the residuals is removed, and the correlation between error and theoretical slowness is also mitigated. This
shows the importance of using a dipping interface model, which simultaneously fits azimuth and slowness data, in
contrast to simply fitting the azimuth trend with a sinusoid. First, a simple sinusoidal curve fit does not account for
the strong dependence on slowness. Second, the effect of a dipping interface is not a pure sinusoid. The deviation
from a sinusoid becomes stronger as dip angle and slowness increase.
Figure 3. Corrected azimuth residuals at CMAR using the two methods. (Left, top) Azimuth residuals at
CMAR with the cosine fit shown by the black line which has the form 1.04 – 14.2* cos(Z- 188.4) and
yields a Moho strike estimate of 188.4°. The fit is then subtracted from the residuals to give the
corrected distribution (Left, bottom). (Right, top) Azimuth residuals predicted for CMAR from a
dipping Moho model, with strike 192.6° and dip 18.3°, obtained from our optimization code which
inverts both the azimuth and slowness data for CMAR in Figure 2 simultaneously. (Right, bottom)
Azimuth residuals after fitting a dipping interface; both the sinuous trend and variations with
predicted slowness are mitigated. See Table 1 for goodness-of-fit statistics.
2012 Monitoring Research Review: Ground-Based Nuclear Explosion Monitoring Technologies 237Figure 4. Corrected slowness residuals at CMAR using the two methods. (Left, top) Slowness residuals for
CMAR with the cosine fit shown by the black line which has the form 0.27 + 1.2*cos(Z+70.4) and
yields a Moho dip estimate of dip 13.1° or 16.8°. The fit is then subtracted from the residuals to give
the corrected distribution (Left, bottom). (Right, top) Slowness residuals predicted for CMAR from
a dipping Moho model, with strike 192.6° and dip 18.3°, obtained from our optimization.
(Right, bottom) Slowness residuals after fitting a dipping interface; both the sinuous trend and
variations with predicted slowness are mitigated. See Table 2 for goodness-of-fit statistics.
Slowness residuals for CMAR are shown in Figure 4 (top, left). Slowness is not as strongly affected by incoming ray
angle as azimuth for the CMAR case, but the change in slowness can become strongly dependent on incident angle
if the interface dips steeply. Using the Niazi method, the Moho dip is found by forward prediction using the
strike=188.4° and the median incident angle (determined from the median slowness) for this data set. We find a
Moho dip of 13.1° or 16.8° depending on whether we minimize the fit to the azimuth residuals or the slowness
residuals respectively. Since this method does not account for the effects of varying incidence angle (or slowness), it
cannot provide corrections for the full suite of observed azimuths and slownesses as they are coupled. Our
optimization method shows significant variance and SMAD reduction (SMADR) over the Niazi method for both
azimuth and slowness although it is particularly good for azimuth as one may expect due to the strong dependency
on incident angle (see Table 1 and Table 2).
The predicted slowness residuals obtained from our optimization code are shown in Figure 4 (top, right), and the
slowness residuals after removing the predicted values are shown in Figure 4 (bottom, right) . There is a dependence
of the slowness residual on theoretical slowness (incoming ray inclination), but this effect is not as great as the effect
on azimuth residual (Figure 3). This observation is reflected in the statistics listed in Table 1 and Table 2 where the
SMAR values show our optimization method is significantly better for azimuth and moderately better for slowness.
Tibuleac and Stroujkova (2009) also estimated the parameters of a dipping Moho under CMAR (strike=197.7° or
213.5° and dip=18.4° or 15.5° using the forward-modeling method of Niazi; the results are for two different
methods) using the forward-modeling method of Niazi (1966). Our resulting Moho strike and dip estimates are in
very good agreement with these previously published values.
2012 Monitoring Research Review: Ground-Based Nuclear Explosion Monitoring Technologies 238Table 1. Summary Statistics for Azimuth
Residuals at CMAR
Observed Cosine fit Optimization
Corrected Corrected
Mean (°) 0.988 0.987 -0.745
Median (°) 0.649 2.720 -0.842
rms (°) 0.102 0.082 0.062
SMAD (°) 12.795 12.46 4.924
Variance (°) 121.38 78.92 44.82
SMADR 2.57% 61.51%
Variance
Reduction
34.98% 63.07%
Table 2. Summary Statistics for Slowness
Residuals at CMAR
Observed Cosine fit Optimization
Corrected Corrected
Mean (sec/°) -0.492 -0.492 0.100
Median (sec/°) -0.646 -0.801 -0.016
rms (sec/°) 0.012 0.009 0.008
SMAD (sec/°) 1.406 0.991 0.692
Variance(sec/° 1.618 0.876 0.813
SMADR 29.49% 50.82%
Variance
Reduction
45.84% 49.74%
CONCLUSIONS AND RECOMMENDATIONS
We have developed a new method for modeling the strike and dip of the Moho beneath a seismic array by
simultaneously fitting the sinusoidal pattern of azimuth and slowness deviations from a one-dimensional velocity
model. The new method is tested at CMAR for over 9900 observations, and we determine the Moho beneath this
array to have a strike angle of 192.6° and a dip angle of 18.3° which are in good agreement with previous estimates
made by Tibuleac and Stroujkova (2009).When the dipping Moho predictions are removed from the azimuth and
slowness residuals, both the sinuous trend and variations with predicted slowness are mitigated. While a single
dipping interface model does not account for all of the discrepancy between observed and predicted back azimuth
and slowness anomalies, this technique is a good first step in the calibration procedure for arrays exhibiting
sinusoidal residual trends. Additional modeling with multiple dipping interfaces included within the outlined
optimization scheme should be pursued.
ACKNOWLEDGEMENTS
Thanks to Ileana for helpful discussions and making available her MATLABTM script to fit the Moho dip angle
using the Niazi method. We are also grateful to Stan Ruppert for helpful comments on this paper.
2012 Monitoring Research Review: Ground-Based Nuclear Explosion Monitoring Technologies 239REFERENCES
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Monitoring System Network, Bull. Seismol. Soc. Am. 89, 989–1003.
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Minimization Subject to Bounds, Math. Program., 67 (2), 189-224, 1994.
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2012 Monitoring Research Review: Ground-Based Nuclear Explosion Monitoring Technologies 240
- bulletfactory0
dogear.
- doesnotexist0
nah, it's in the transform palette.
- ernexbcn0
We are getting somewhere here, I'm running some calculations.
- Mr_Right0
Have you checked the turbo encabulator?
[youtube]PL91A01BD0F8E561C0[/you...
