Is there always extra bandwidth in non-uniform spatial sampling?
Ralf Ferber* and Massimiliano Vassallo, WesternGeco London Technology Center;
Jon-Fredrik Hopperstad and Ali Özbek, Schlumberger Cambridge Research
Summary
We address the question of whether or not there always is a
“bandwidth benefit” in non-uniform spatial sampling of
geophysical data. Answering this question is, for example,
important in the context of random sampling of seismic
data, as it recently has been shown that there can be such a
benefit under certain assumptions on the spectral structure
of the data. Assume that a fixed number of sensors are
placed either uniformly (i.e., on a regular grid) or nonuniformly (either randomly distributed or following any
suitable non-uniform sampling scheme). The bandwidth
supported by uniform sampling is that of the Nyquist
wavenumbers corresponding to the sampling distances on
the regular grid. The bandwidth supported by the nonuniform sampling we propose here refers to the maximum
bandwidth of data that could be reconstructed by a linear
operator at arbitrary sampling locations within the survey
area without unacceptably high reconstruction error.
Without making further assumptions on the spectral
structure of the data, i.e., especially without assuming
sparseness of the data spectrum, we will argue that we see
no such bandwidth benefit in non-uniform sampling in the
examples we have investigated.
Introduction
Chen and Allebach (1987) set out to find a sampling
location set among a candidate group of such sets that is
optimal for data reconstruction over a given class of bandlimited broadband signals. If the bandwidth is given, the
sampling locations are usually designed to be uniform with
the sampling location distances such that the corresponding
Nyquist wavenumbers are greater than the assumed
bandwidth. However, in many applications it is impossible
to place sensors at regular locations. The theory of
compressive sampling even requires randomly distributed
sensors (Candès et al., 2004; Hennenfent and Herrmann,
2008; Moldoveanu, 2010). We consider here a reverse
problem; that of finding the maximum bandwidth
supported by a given set of non-uniform sampling
locations. We consider that a set of sampling locations
supports broadband signals of a certain bandwidth if the
data can be reconstructed at non-sampling points within an
acceptable accuracy. Hence, it makes sense to look for the
maximum bandwidth supported by the set of sampling
locations. To eliminate edge effects, the area of data
reconstruction could be restricted to an inner part of the set
of data sampling locations. To test whether a certain
bandwidth is supported by the sampling locations, we
suggest proceeding as follows: firstly, compute the
maximum of the minimum mean-square interpolation error
for a range of reconstruction locations within the selected
subarea of the point set. Secondly, if this maximum is
lower than the acceptable reconstruction error, flag the
bandwidth as being supported by the set of sampling
locations. If this test is done in an exhaustive search over
bandwidth, the bandwidth domain can be split into a region
that the set of sampling locations supports and a
complementary region that it fails to support. The approach
developed here is valid for any dimensionality of the
sampling problem, while a typical example would be twodimensional spatial sampling of geophysical data. All
examples we give to illustrate the approach will be spatially
two-dimensional for ease of visualization.
Reconstruction of data from non-uniformly spaced
samples
We wish to reconstruct the signal from non-uniformly
spaced sampling locations by a linear “convolution-like”
process; the signal at non-sampling locations will be
estimated by a weighted summation of existing samples in
the direct neighborhood of the reconstruction location. The
weights, of course, must be selected such that the
reconstruction error is minimal. Optimal reconstruction of
one-dimensional signals by a weighted summation of data
sampled non-uniformly is given by Yen’s 4th theorem
(1956). Chen and Allebach showed an extension of Yen’s
approach to two-dimensional sampling, which can be easily
extended to any number of dimensions. The reconstruction
method described by Yen’s 4th theorem is optimal for
spatially broadband signals (Özbek et al., 2010); the
assumed signal bandwidth is a critical parameter. In all the
work mentioned above, the signal bandwidth was assumed
known and fixed, such that, depending on the sampling
locations, a reconstruction might be possible or not. For
infinitely long one-dimensional data, it is well known that
the maximum bandwidth supported by the sampling
locations can be estimated as the Nyquist wavenumber
corresponding to the average sampling distances (Beutler,
1974; Jerri, 1977). Moore and Ferber (2008) added a
bandwidth optimization strategy to the reconstruction of ndimensional non-uniformly sampled seismic data as a way
of finding the maximum bandwidth supported by the
sampling locations. In this paper, we first show that the
maximum bandwidth supported by non-uniform sampling
locations is not uniquely defined. In fact, the wavenumber
domain can be split into a region around the zero
wavenumber supported by the sampling locations, and a
complementary region that is not supported. We also show
that, for broadband data, there is no significant bandwidth
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Extra bandwidth in non-uniform sampling
gain in non-uniform sampling compared to uniform
sampling.
Minimum mean-square error interpolation operator
As stated above, we wish to reconstruct data at nonsampling locations using a weighted summation over data
in the direct vicinity of the non-sampling location. The
weights w ( w1 , w2 , , wN )T form a reconstruction operator
applied to the data d
(d , d2 , , d N )T , with the data
represented by a function d ( x ) of an M-dimensional
sampling location vector. The reconstruction by means of
weighted summation can be mathematically expressed as
dˆ wT d . We select here the minimum mean-square error
reconstruction operator (Chen and Allebach, 1987; Moore
and Ferber, 2008), that depends on the sampling locations,
the interpolation location, and the assumed data bandwidth
(hence used as the bandwidth parameter of the Yen-4
operator), but not specifically on the details of the data or
its wavenumber spectrum. The reconstruction operator can
be computed from the matrix/vector equation
w(k )
S 1 (k )r (k ),
with k as the bandwidth parameter, whose components are
the maximum wavenumbers in each of the M dimensions
of sampling. The sampling location matrix that must be
inverted to get the reconstruction operator is
S (k )
M
si , j , with si , j
m 1
sin 2 km ( xi ,m
2 km ( xi ,m
x j ,m )
x j ,m )
.
Similarly, the interpolation vector, r , also a product of
sinc functions, depends on the differences between
sampling locations and the output location for data
reconstruction, and is given by
M sin 2 k ( x
ym )
m
i ,m
r (k ) ( r1 ,..., rN )T , with ri
.
2 km ( xi ,m ym )
m 1
The minimum mean-square error (MMSE) can now be
computed as: MMSE 1
N
wi ri .
i 1
Examples
As a first example, we show here a bandwidth support
spectrum for a simple uniform sampling scheme, consisting
of 25 sampling locations on a uniform sampling grid of
12.5 by 12.5 m. For a range of x- and y- wavenumbers, we
compute the maximum value of the MMSE of the
corresponding reconstruction operator, the maximum being
taken over the central survey area. These maxima as
function of the x- and y-wavenumbers are displayed colorcoded in Figure 1B. The wavenumbers reside in the range
of zero to 0.1 1/m, for both coordinate axes of the sampling
scheme. Only positive wavenumbers, i.e. one quadrant of
Figure 1: Bandwidth support spectrum (B) for 25 sensors on a uniform 12.5- by 12.5- m grid. The lower wavenumbers below 0.03 1/m are well
supported, with a transition region before the non-supported wavenumber (greater than 0.04 1/m) appear. The left side of Figure (A)
shows a non-quadratic bandwidth box and how it relates to a single entry in the bandwidth support spectrum.
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Extra bandwidth in non-uniform sampling
the full bandwidth support wavenumber spectrum, are
displayed due to the inherent symmetry. The corresponding
maximum MMSE values are displayed on a logarithmic
color-coded scale, such that deep blue colors denote very
small reconstruction errors, while red colors denote large
errors. For comparison, we indicate the Nyquist
wavenumbers of the infinite regular grid (i.e., kx = ky =
0.04 /m) by a blue dot. Figure 1A clearly shows that the
wavenumber domain splits into two regions: one region of
low wavenumbers, supported by the non-uniform sampling
scheme, and the other region with larger wavenumbers that
are not supported. There is also a transition zone between
these regions. The bandwidth support spectrum should be
interpreted as follows: assume a bandwidth of the data
characterized by two wavenumbers in spatial x- and ydirections, for example kx = 0.02 1/m and ky = 0.04 1/m,
i.e., one assumes that the data resides in the wavenumber
box highlighted in green color in Figure 1A. For this
bandwidth box there is one entry in the bandwidth support
spectrum highlighted by the blue dot indicated by the green
arrow. This particular scheme of 25 sensors on a 12.5- by
12.5-m grid does not support this bandwidth box well, as it
is in the transition zone from good to poor support. The
Nyquist wavenumbers of the 12.5- by 12.5-m uniform grid,
i.e. kx = ky = 0.04 1/m, are also on the edge of the
bandwidth support area due to the fact that only 25 sensor
locations are available for data reconstruction. The size of
the transition zone will shrink with a larger number of
sensors, which would allow use of a more accurate
interpolation operator.
As a second example we show a simple non-uniform
sampling scheme, also consisting of 25 sampling locations
derived from the uniform sampling grid of 12.5- by 12.5-m
used above by random variations around the uniform
sampling locations. For the same range of x- and ywavenumbers we again compute the maximum value of the
MMSE of the corresponding reconstruction operator, the
maximum being taken over the central reconstruction
locations (inside the red rectangle in Figure 2A). These
maxima as a function of the x- and y- wavenumber are
again displayed color-coded in Figure 2B. For comparison
we indicate again the Nyquist wavenumbers of the
underlying regular grid (i.e. kx = ky = 0.04 1/m) by a blue
dot. The diagram depicted in Figure 2B clearly shows that
the wavenumber domain again splits into two regions: one
region with low wavenumbers, clearly supported by the
non-uniform sampling scheme, and the other region with
larger wavenumbers that are not supported. There is also a
transition zone between these regions. The bandwidth
support region is larger than that of the uniform sampling
scheme used above. Higher wavenumbers in one direction,
ky = 0.08 for example, can be supported if the wavenumber
of the other direction is reduced, kx = 0.01 for example,
see yellow dot in Figure 2B, such that non-uniform
Figure 2: Bandwidth support spectrum (Figure 2B) for a non-uniform sampling scheme (sensor locations depicted in Figure 2A), created by
random variations around the uniform grid used in the first example. The maximum mean square error for the linear reconstruction
operators is computed over the inner part of the survey area inside the red rectangle in Figure 2A.
© 2011 SEG
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59
Extra bandwidth in non-uniform sampling
sampling could be regarded as being more versatile than
the uniform sampling schemes. With this type of nonuniform sampling the wavenumber transition zone seems to
follow a curve in which the product of the corresponding
wavenumbers is the product of the wavenumbers of the
underlying uniform grid (kx k y 0.042 m 2 ). The
bandwidth corresponding to the Nyquist wavenumbers of
the underlying uniform grid however is not better supported
by this non-uniform sampling scheme, as it still sits on the
edge of the transition zone from good to poor bandwidth
support.
As a third example, we look at another non-uniform
sampling scheme for 25 sensors placed in the same survey
area as before, but this time using Hammersley points
(Tien-Tsin et al., 1997), depicted in Figure 3A, to define
the sensor locations. These points form a well-known lowdiscrepancy sequence and have been used, for example, for
quasi-Monte Carlo integration. The bandwidth support
spectrum, Figure 3B, now is more anisotropic and
somewhat larger than that of the random sampling scheme
above, but fundamentally similar, and again not showing a
bandwidth benefit as compared to uniform sampling (blue
dot in Figure 3B).
Conclusions
For spatially broadband signals, i.e., without assuming
anything else about the signals other than bandlimitation,
we presented a technique to calculate the maximum
bandwidths that a non-uniform sampling scheme supports.
We introduced the so-called bandwidth support spectrum as
a function that contains the maximum minimum mean
square reconstruction error of the corresponding optimum
linear reconstruction operator, the Yen-4 operator, as a
function of those limiting wavenumbers. The bandwidth
support spectrum splits the wavenumber plane into three
distinct areas; those low-wavenumbers well supported by
the sampling scheme and those higher wavenumbers
clearly not supported, with a transition zone between. We
showed that non-uniform sampling schemes can be more
versatile than uniform schemes, as they can support
wavenumbers that a particular uniform scheme with the
same number of sensor locations cannot support. However,
we also showed, that under the restriction of an identical
number of sensor locations in the survey area, the nonuniform sampling schemes show no bandwidth gain
beyond that of the Nyquist wavenumbers associated with
the uniform sampling scheme.
Figure 3: Bandwidth support spectrum (Figure 3B) for a non-uniform sampling scheme using Hammersley points (sensor locations depicted in
Figure 3A), created by random variations around the uniform grid used in the first example. The maximum mean square error for the
linear reconstruction operators is computed over the inner part of the survey area inside the red rectangle in Fig. 3A.
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