Checkerboard Test for 3-D Tomography Model of Washington
Cacadia Margin - 20 km cell dimensions in X, Y, and Z.
Introduction:
Checkerboard sensitivity tests are useful to assess the ability of
tomographic inversion to resolve structural details in the earth. In
tomographic inversion, lack of an explicit inverse
operator in the computational formulation makes it
difficult to estimate parameter uncertainties and resolution for the
inverted model. The checkerboard test is a useful alternative
which gives a general picture of the resolving power of tomographic
inversion. It can be applied to any inversion procedure
without knowledge of the internal operation of the inversion.
The idea is to superimpose a small perturbation (typically a 3-D or 2-D
regular "checkerboard" or grid pattern) signal onto the
tomographically inverted structure model, compute synthetic travel
times (data) to all observing stations from all sources used in the
original
tomographic inversion, and then invert the synthetic travel and arrival
times in the same manner as the actual data. Random
errors may added to the synthetic data to simulate random errors in the
actual observations. The ability of the tomographic method to
quantitatively recover
the perturbed model is then an estimate of the sensitivity of the
original inversion of real data to recover similar details in the real
earth.
There are
obviously many factors that influence the results of a
checkerboard test. These include such things as: the amplitude of the
perturbation signal, whether the perturbation signal is a smooth
variation such as a 2-D or 3-D sinusoid or a "square" wave (or other
wave form), the size of the grid (or the spatial wavenumber) of the
perturbation signal, whether variation is allowed in depth as well as
in latitude and longitude, and whether the spatial wavenumber of the
perturbation is the same in all three dimensions. In the following
discussion, we outline the scheme used here to compute the
checkerboard test, and also briefly discuss aspects of the results.
Description of Method:
The form of the velocity
structure models is a 3-D grid with velocity values (actually slowness;
conversion between velocity and slowness is handled in the code)
specified at points in the grid corresponding to points in the earth.
For purposes of discussion, the final inverted model is called
MODELA. First, synthetic data are generated using MODELA and all
of the sources and receivers used to generate MODELA. These synthetic
data are then "inverted" as if they were the actual data, using the
same procedure as with the original inversion, and the resulting model
is called: MODEL0. Note that MODEL0 will be close to, but not
exactly equal to, MODELA because we used the actual data with MODELA,
but used slightly different synthetic data to generate MODEL0. MODEL0
will be used as a reference model. Note also that MODEL0 is generated
using synthetic data with no errors, so the fit to the data should be
nearly perfect.
Next, add the checkerboard "perturbations" to MODEL0, and generate
theoretical travel times using this perturbed model as before. The
checkerboard pattern is a sinusoid in 3-D with specified wavelength set
independently for each of the 3 spatial directions. For our examples,
we set the spatial wavelength to be 40 km in all three directions,
resulting in cubic "cells" 20 km on each side of alternating sign. The
actual values added to MODEL0 are sinusoidal perturbations
up to maximum 10% of the velocity value at each grid node. Note that
since the
perturbations are computed as percentages of the velocity values, the
actual pattern added is not uniform over the model. To the
synthetically generated data, normally distributed standard
error with zero mean and 0.05 s standard deviation are added to
simulate random
error in the real data.
Using MODEL0 as the starting model, the synthetic data are inverted
to stability using the same parameters as used in inversion of
the actual data. The resultant model is called: MODEL1. The difference
between MODEL0 and MODEL1 (MODEL1-MODEL0), scaled to the
range -.1 to .1 using the reference velocity in MODEL0, is the output
of the checkerboard test. These values are plotted in percentage (-100%
to 100%).
If the recovery of the perturbation is perfect,
the result should be an exact sinusoidal pattern in 3-D with maximum
values in the center of each cell. With no recovery of the
perturbations (null resolution), the values in the center of each cell
are zero. However, the following caveat applies -- since the
plots in this map view are horizontal slices through the model, if a
slice falls on a zero value or node of the perturbation sinusoid, the
plot should show zero (white cell values) even if resolution is
perfect. The origin of the perturbation sinusoid is actually at -11 km
(11 km above sea level),
the starting position of the velocity model in depth. With checkerboard
cells of 20 km, the first node in depth is at 9 km, the second node
in depth at 29 km, and so on. A careful look at the 3-D checkerboard
maps will show low amplitude values at these depths even when the
resolving power of the inversion is high.