__________________________

`Corresponding author address: Phillip L. Spencer, National
Severe Storms Laboratory, 1313 Halley Circle, Norman, OK 73069,
E-mail:
spencer@nssl.noaa.gov`

Abstract

`For diagnostic purposes, the "traditional"
approach to estimating derivatives employs objective analysis to
provide a gridded field from the original observations, which are
typically not uniformly distributed in space. However, there exist
alternative methods involving derivative estimation via line integral
("triangle") techniques that do not involve a prior mapping of the
field onto a uniform grid. It has been suggested that these give
improved results. Empirical testing of the differences between wind
field derivative estimation using two different schemes is done with
prototypical examples of the techniques. Test results verify that the
line integral method indeed provides substantial improvements over
the traditional scheme. The magnitude of the improvement is shown to
depend on the degree of irregularity of the data distribution, as
expected. Although the particular prototype methods chosen have the
property that the line integral method truncates the amplitude of the
input field slightly more than the traditional scheme, the pattern of
the field is significantly better using the line integral technique
than with the traditional method. An unexpected result is that the
improvement by the line integral method over the traditional approach
does not diminish as the wavelength of the input field increases. It
is shown that this is a consequence of overfitting of the field to
the station observations, causing local discontinuities in the field
that produce errors in the gradient calculations, even in situations
where the distribution of data is uniform. Overall, the test results
make it abundantly clear that the traditional method is generally
inferior to derivative estimates via the line integral methodology.
`

There are at least two reasons for performing what has been described as "objective analysis" on observations: (1) it is a part of the process used for initialization of numerical simulation models, and (2) it is perceived as the beginning of diagnostic analysis of the observations. In what follows, our primary emphasis is on the methods used for diagnosis of observed data when those observations are obtained from sites distributed nonuniformly in space. That is, we are not particularly concerned herein with the use of objective analysis for the purpose of initializing numerical simulation models, although it should be clear that these topics are closely related. Modern methods for producing initial conditions as input to numerical simulations are complex, computationally intensive, and involve many more issues than we are prepared herein to consider (see Daley 1991; Chapters 6-12, for more information on initialization topics). Objective analysis for diagnostic purposes is in widespread use and typically is done without all of the complex manipulations that are involved in model initialization.

Furthermore, we are particularly concerned with the estimation of spatial derivatives from the observed data. As discussed in Doswell and Caracena (1988; hereafter DC88), spatial derivatives are often the most important quantities used in diagnostic calculations. Hence, obtaining the best spatial derivative estimates possible from a given set of observations can be critical to the diagnosis.

Given the widespread use of "canned" objective analysis systems (e.g.; the GEMPAK program ,which is based on Koch et al. [1983]), it is common practice to use objective analysis methods to produce a gridded data set from the observed variables. Using this gridded field, derivatives are then estimated by finite differencing methods. Hereafter, the two-step process of first producing a gridded field from the irregularly-distributed observations and then using finite differences to develop derivative estimates will be referred to as the "traditional" method. Obviously, there are many possible variations of such a scheme, employing various combinations of different objective analysis and finite differencing techniques. For reasons to be explained, we are not concerned with trying to replicate all possible such schemes. Rather, we will develop a prototype for traditional methods that we believe will serve to exemplify all the others.

Ceselski and Sapp (1975) presented an alternative approach, based on what seems to be the first introduction of the technique by Bellamy (1949). This method differs from the traditional method in that it estimates derivatives directly from the ungridded observations using line integrals. Schaefer and Doswell (1979) showed that such a scheme seems to produce improved derivative estimates. Later, in DC88, a theory was developed that explained the source of the improvements in the derivative estimates. The process of mapping irregularly-distributed observations onto a grid and then using finite differences to estimate derivatives is theoretically distinct from estimating derivatives directly from the nonuniform observations; the methods necessarily will give different answers. DC88 showed, at least in theory, that the traditional method introduces distortions into the derivative estimates that the line integral method does not. They also showed that in the exceptional case of uniformly-distributed observations, the results should theoretically be the same.

Since most meteorological observations are not uniformly distributed, the apparent improvement via line integral methods seems to be an important issue. The traditional method remains in widespread use for diagnostic purposes, in spite of some evidence that line integral methods give superior results. In fact, Caracena (1987) showed that the traditional approach was unnecessary; derivative estimation to any desired order of differentiation without grids is quite possible. Zamora et al. (1987) presented a simple technique to produce derivative estimates for the wind field using line integrals, called the "linear vector point function" (or LVPF) method. However, Davies-Jones (1993) demonstrated that all line integral methods are essentially the same when they assume a linear variation of the field between vertices of the triangles typically used for estimating the derivatives.

Our goal in this study is to provide some empirical tests that will provide quantitative information about the differences between derivative estimates calculated the traditional way versus those found using the line integral method (over triangles formed by the vertices of irregularly-distributed stations). In what follows, the latter will be referred to as the "triangle" method. Section 2 provides details of how we performed the analyses using the two contrasting techniques. The methodology will be applied to the problem of estimating first derivatives of the wind field, but can be extended easily to estimating higher order derivatives, or derivatives of any scalar field. In section 3, a description of the experiments will be provided, including the creation of analytically-specified "data" and the development of artificially-produced sampling arrays. Results of the calculations will be presented in section 4, followed by a discussion of those results in section 5.

** **

We have chosen to use the wind field as our testbed for empirical evaluation of the differences between the two different techniques. This is not necessarily the simplest choice, for the obvious reason that the wind is a vector field, not a scalar. There are four first-order derivatives in the horizontal wind field, instead of just two when considering derivatives of a scalar field. However, this allows us to explore the same issues presented in the simple tests performed by Schaefer and Doswell (1979) and, in some sense, can be considered a more challenging test of the methodology than a simple scalar can provide. A more detailed presentation of the test procedures is given in the next section.

Given that there are many different objective
analysis methods and a number of alternative finite differencing
schemes, there are numerous possible prototypes for the traditional
method. There even exist many possible variations on the triangle
method, including different ways of triangulating the station array
(e.g., see Steinacker et al. 2000), and different ways to put the
resulting fields of the derivative estimates onto a uniform grid for
display purposes.^{[2]} In order to keep the testing within practical bounds,
we have chosen some simple options, as will be developed in what
follows. Clearly, these simple options could be viewed as "strawmen"
that do not represent the "state of the art" in objective analysis.
We do not contend that our prototypes represent the best possible
schemes (however "best" might be defined) for derivative estimation.
Nevertheless, there are at least two reasons for making the choices
we have made.

First, we do not see any reason to believe that
simple versions of the schemes will necessarily demonstrate
significantly different behavior with respect to the issues that
concern us in this test. Higher order finite differencing or more
sophisticated objective analysis methods^{[3]} applied to the traditional scheme simply cannot obviate
the principles shown theoretically in DC88. They may alter the
*quantitative* outcome of the tests, but will not cause a
*qualitative* change in the results.

Second, the actual practice of objective analysis as applied to diagnoses based on observations (e.g., the widely-used scheme developed by Koch et al. [1983]) is not significantly different from what we have chosen. Arguably, the biggest issue is whether or not to use multiple passes of the traditional objective analysis method. We will demonstrate a potentially negative impact of multiple passes in what follows. We believe that our test is appropriate for helping those who employ the traditional scheme of derivative estimation to decide for themselves whether or not to consider switching to the line integral method for situations wherein derivatives are important to the diagnosis.

As discussed in DC88, the two schemes both
involve a mapping/smoothing operation and a derivative estimation
operation. However, they differ in the *order* in which these two
operations are performed. DC88 show that for irregularly-distributed
data, these operations do not commute; that is, the outcome
necessarily involves different answers. It also is shown in DC88 that
mapping/smoothing derivative estimates taken directly from the
observations is the "proper" order to perform the operations.

* *

a. The traditional scheme

Our prototype scheme uses a simple, single-pass distance-dependent weighted averaging scheme of the "Barnes" type (Barnes 1964). That is, it uses a Gaussian weighting of the form

where denotes the value of the analyzed field at the
Cartesian gridpoint (*x _{i}*,

where

in (2) is the Euclidean distance between the
data point and the grid point in question. The parameter
*l*
determines the characteristics of the analysis; this is the "shape
parameter" for the Gaussian weighting scheme. For the horizontal wind
field, the variables at the grid points will be the vector components
.

For finite differencing, we employ the simplest possible scheme (one that is widely used in diagnostic practice, however), the centered, 2nd-order scheme defined by

Observe that the differencing is done using the
*analyzed*
wind field, not the original data, which is characteristic of the
traditional scheme. Once all four spatial derivatives are computed,
the following quantities are determined from the finite difference
estimates of the wind vector component derivatives on the
grid:

where the analyzed divergence is , the analyzed vorticity is , the analyzed stretching deformation is , and the analyzed shearing deformation is . These are the four "kinematic" quantities that describe the linear (first order) characteristics of the wind field. They are more physically meaningful for wind field analysis than the four spatial derivatives; vorticity and divergence are commonly diagnosed from the wind field.

* *

b. The triangle scheme

For the line integral method, we begin with a Delauney triangulation of the station network, as described in Ripley (1981; pp. 39-41). In this application, any triangle created by this process with an included angle of less than 15º is omitted from consideration. Removing such narrow triangles from the analysis prevents potential problems with near-colinearity of the vertices. For each retained triangle, the four kinematic quantities of the wind field are computed using the LVPF method described in Zamora et al. (1987). This method assumes linearity between the vertices of the triangles and assigns the values to the triangle centroid. In order to plot the results, these quantities are treated as "data" located at the triangle centroids for an objective analysis using the same method as just given for the traditional scheme. Note that any objective analysis scheme using distance-dependent weighting will smooth the resulting analysis, to an extent that depends on the choice of the parameter(s) of the scheme (Stephens 1967). That is, in general, the schemes used for objective analysis behave as low-pass filters, with the amplitude and phase response of the scheme being determined by the choice of a weighting function and the parameters of the chosen scheme.

** **

Our goal is to determine quantitatively how well the two different prototypical methods estimate the derivatives of the wind field, as embodied in the four kinematic quantities. Since any test using real data necessarily means that we cannot know the "true" answer, we do not herein present any tests on real data. Rather, we follow the approach of Schaefer and Doswell (1979), using analytically-specified input fields, but extend the tests to explore the sensitivity of the differences between the schemes to certain variables.

The analysis grid is 161 X 161 grid
points, where D* _{g}* is the grid spacing (2.5 arbitrary units). The origin of
the (

* *

a. Analytic data field structure

Our wind component data are defined as
functions of (*x,y*) at any point using

where the wavelength, which is variable, is
given by *L*. The amplitude of the variation is 10 arbitrary units.
This field is simply a "checkerboard" of cyclonic and anticyclonic
vortices, so the analytic vorticity, which is given by

is also a "checkerboard" of positive and negative vorticity centers (Fig. 1). Note that the amplitude of the vorticity increases as the wavelength decreases, which mimics the behavior of real velocity fields. Although we are not considering it, the analytic stretching deformation is

The other two kinematic quantities, divergence and shearing deformation, are identically zero for the analytic wind field.

* *

b. Creation of artificial observing site arrays

As already noted, we wish to know
quantitatively the dependence of the improvements associated with the
triangle method on the degree of irregularity of the observation site
distribution, since Doswell and Caracena (1988) suggested that it
should increase as the irregularity increases. To accomplish this, we
employ the methodology developed by Doswell and Lasher-Trapp (1997),
whereby the location of each observation in an initially uniform
array is randomly perturbed by a distance no greater than
*D _{s}*, the
so-called

* *

c. Variables tested

The variables we considered in our empirical
testing were: (1) *D _{s}*, the scatter
distance that measures the degree of irregularity in the observing
site spacing, (2)

** **

a. Pattern distortion results

As a demonstration of the outcomes over what
could be considered a typical application, consider Fig. 3. The
wavelength of the input wave, *L* = 10D, is more or less in the
middle of the range identified as "marginally sampled" in DC88.
Although the alternating pattern of positive and negative vorticity
centers is captured by the traditional method, distortions of that
pattern are quite evident (Fig. 3a). Extrema within the field are noticeably shifted from
their true locations, with some being amplified relative to the
analytic values, and large gradients are created in places where the
true gradients should be relatively small. Close inspection reveals
that these bogus gradients occur in relative data voids (caused by
the irregularity of the data distribution), which agrees with the
findings of Barnes (1994) and Spencer et al. (1999). This general
finding also confirms those of Schaefer and Doswell (1979; see their
Fig. 6 and accompanying discussion).

On the other hand, the triangle method produces a pattern that is visually better than the traditional method (Fig. 3b). The gradients and locations of the extrema are analyzed much more faithfully with the triangle scheme, but it is noted that the peak values are somewhat reduced in comparison with those produced by the traditional method. Because the traditional method has amplified some of the peak derivative estimates beyond their true values, this general reduction in amplitude with respect to the traditional method is not necessarily a drawback of the triangle approach.

A similar analysis for *L* = 25D, is done to represent a
"well-sampled" wave (according to DC88), with only a modest
nonuniformity in the data array (*D _{s}* = 0.2D). This has a similar outcome (Fig. 4a, Fig. 4b) to that with

To complete this preliminary look at the
results, we repeat the analysis for *L* = 25D but with *D _{s}* = 0.8D again, a case of a highly irregular sample of a
relatively long wave. Figure 5 demonstrates the extreme distortion of
the field produced by the traditional method when compared to the
triangle method (Fig 5a), whereas for this situation, the triangle method has
underestimated the amplitude of the peaks but again reproduces the
pattern with considerable fidelity (Fig. 5b). Note the relationship between the peaks in the
estimated vorticity and the data voids in Fig. 5a, even for a wave
that is "well-sampled" by this data array.

* *

b. Sensitivity to parameters

As a way of summarizing the results of all the
parameter sensitivity studies, we compute the root mean square error
(*RMSE*) as
the difference between the analyzed and the true vorticity,
where

and where *N* is the number of
gridpoints over which the *RMSE* is computed. The
calculation of *RMSE* is done only over the interior part (i.e., the center
81 X 81 array) of the analysis grid, to avoid contamination of this
statistic by boundary errors in the analysis (see Pauley 1990). This
measure of the goodness of the fit has some limitations, but it is
considerably more meaningful than computing the *RMSE* at the
*data point*
locations (Glahn 1987).

Figure
6a shows that, for uniformly
distributed data, there is very little difference between the two
methods except for *l* <1.0D and for
wavelenths below those considered "marginally sampled." As discussed in Pauley and Wu (1990), there is very
little reason ever to use a shape parameter that small, since it
results in a serious overfitting of the data. Hence, it appears that
there is no obvious benefit to the triangle method for uniformly
distributed data, a result to be expected from the findings in DC88.
However, when the data are moderately nonuniform (Fig. 6b), using values of 1.0D* < l* < 1.5D (i.e., values of *l* recommended by Pauley
and Wu [1990]) results in some improvement by the triangle method for
wavelengths greater than about *L* = 11D. For a "saturated"
nonuniform distribution of data (Fig. 6c), the triangle method yields lower *RMSE* even when using a
relatively "tight" fit (*l* = 1.0D) for wavelengths greater
than about *L* = 7D, which is in the "marginally sampled" range. Again,
these results suggest that it is the *longer* wavelengths that
the greatest benefit to using the triangle method is found. This is
not the result expected in DC88, as noted.

* *

c. An issue tied to nonuniformity

Before we discuss the reasons for this,
however, it is useful to consider another aspect of the experiments,
discovered during the experimental design. For many purposes, authors
have used the average distance to the nearest neighbor (denoted here
as ) as a way
to characterize the "data spacing" of nonuniform arrays. Since Pauley
and Wu (1990), inter alia, have made some specific recommendations
about how to choose the shape parameter based on the average data
spacing, this is worthy of some consideration. Figure 7 shows that as the scattering approaches "saturation,"
the value of
gets smaller, owing to clustering. However, the total number of
observations remains the same, so the reduction in does not represent
an improvement in the data spacing, in terms of the fidelity with
which the data can represent an input wave of some wavelength
*L* >
. It would be
seriously misleading to believe that the sampling improves as its
nonuniformity increases! An improved representation of the data
spacing in nonuniform data arrays is that recommended by Koch et al.
(1983),

where *M* is the number of
observations and *A* is the area within which the data are distributed. For
large *M*,
this reduces to:

Thus, represents the data spacing that would result if the
data within the area *A* were uniformly distributed. Since we already knew the
initial, unperturbed data spacing in our experiments, this is the
parameter we used. Strictly speaking, even this is inappropriate,
since nonuniform sampling *increases* the effective
data spacing over that associated with a uniform distribution, as
discussed in DC88 and recently considered by Trapp and Doswell
(2000).

* *

d. Origins of the improvement over the traditional scheme at long wavelengths

In DC88, it is suggested that "... for wavelengths longer than 12Dx, the data can be considered continuous for many practical purposes, and finite difference derivative estimates will be accurate to within five percent ... " Our results do not show this. Instead it seems that even for very long waves, considerable distortion of the diagnosed derivative fields results from using the traditional method.

By doing a series of traditional analyses with
different values of *l* (not shown) during the initial mapping of the
(*u _{k},v_{k}*) data onto the
grid to derive the fields, it is possible to filter most of the
"noise" from the traditional estimates of the derivative fields
(e.g., the vorticity). The results of this experiment are summarized
in Fig.
8a, Fig. 8b, Fig. 8c, which shows that as the data become increasingly
nonuniform, the minimum

According to the theory developed in DC88, the separate processes of estimating the derivatives and mapping/smoothing the field onto a uniform grid only commute when the data array coincides with the analysis grid. In cases where the data are on a uniform grid, it is possible to estimate the derivatives first, using (3) directly on the data array, and then mapping/smoothing the estimates onto the analysis grid. This procedure, which can be carried out only in the case of a uniform data array, is called the "alternative" procedure, to distinguish it from the "traditional" and "triangle" schemes. In order to test this, we considered four separate experiments.

On occasion, studies involving the diagnosis of
derivatives using the traditional method employ a post-operative
filter to remove the resulting "noise" (as exemplified by Fig. 3a) in
the derivatives.^{[5]} We tested this by applying a simple five-point digital
filter, with a spectral response function shown in Fig. 9a, Fig. 9b, to the gridded derivative estimates of the traditional
scheme. A single pass of this filter through the derivative estimates
from the traditional method produced no apparent change in the
derivative field (not shown). There is a simple reason for this:
observe that our 161 x 161 analysis grid is considerably denser (by a
factor of eight) than the data array. This means that reduction of
the amplitude of 2D waves (i.e., on the data array, which would be 16 grid
lengths on the analysis grid) to negligible levels (say, to one
percent of their initial amplitude) would require many passes of this
digital filter. Reduction of the 2D wave amplitude in this
fashion to a low level necessarily results in a large loss of
amplitude that would extend across the whole spectrum. If a coarser
analysis grid than the one we used is chosen to try to get around
this problem, the consequence is an increased spatial truncation
error during the finite differencing step of the traditional scheme,
not an attractive option. Hence, it seems that adding a
post-operative filter to the traditional scheme does not appear
capable of producing results comparable to the triangle method and
can not account for the discrepancy we are trying to resolve.
Additional discussion of how the relative coarseness of the data and
analysis grids affects the resulting analyses can be found in
Appendix A.

The difference in the order of smoothing between the methods is not the direct source of the discrepancy, since the effect of asymmetric smoothing is negligible when the data are uniformly distributed (cf. Fig. 6). However, the difference in results between the methods is indeed related to way the smoothing is done, as we show below.

In our experiments, the derivative estimates
using the traditional method are particularly inferior at long
wavelengths when a relatively small shape parameter, *l*, is used to fit the
data relatively closely, even with uniform data spacing (cf. Fig.
6a). For large *l*, the differences between the methods are minor, but for
small *l*
(and, in particular, for 1.0D < *l* < 1.5D, where the objective analysis is being done properly),
the traditional method is somewhat inferior. This suggests that the
superiority of the triangle method over the traditional scheme might
be associated with a type of overfitting of the data.

We can illuminate this with the following
somewhat exaggerated experiment. For *L* = 25D, and *D _{s}* = 0.0D, we choose

** **

For most reasonable values of the shape
parameter, *l*, it is clear that the traditional method produces
markedly inferior estimates of the derivative fields when compared to
the triangle method. ^{[5]} Contrary to early expectations expressed in DC88, this
difference is *not* mitigated for features with well-sampled wavelengths.
On the contrary, it seems that it is precisely in the long input
wavelengths where the improvements obtained by using the triangle
method are greatest. Hence, if the goal of doing an objective
analysis is to diagnose derivatives with as much accuracy as
possible, then there seems little alternative to doing the derivative
estimates via the triangle (i.e., line integral) methods.

There does not seem to be any simple way to create derivative estimates via the traditional method that compare favorably to those derived by the triangle approach. We tried a heavier initial smoothing and a post-operative digital filter to attempt to improve on the results using the traditional scheme, but found that neither of these changes gave derivative estimates as good as those obtainable via the triangle scheme.

In the generally unrealistic situation of a uniform data array, our results confirm the theory developed in DC88. That is, the traditional and alternative schemes give virtually identical results when the data and analysis grids coincide. Interestingly, in this situation, the triangle scheme also gives similar but not quite identical results. By using line integrals to estimate the derivatives, the triangle method is doing something different, but we are not prepared to give a complete analysis of the process in comparison to the traditional approach.

We recognize that this outcome calls into question a rather substantial body of existing work involving derivative estimates, in view of the common use of the traditional method in diagnostic studies. We also realize that performing the line integral analysis involves more effort: a triangulation of the data points is needed, and line integrals need to be computed. Further, no common existing software can do the evaluations in this fashion. Until such software is made generally available, those wishing to do diagnosis involving derivatives will have to write their own software for the triangle approach. Nevertheless, the results of our experiments make it clear that using available software to estimate derivatives using the traditional method is done at some considerable peril to the accuracy of the calculations.

Generally speaking, objective analysis of meteorological data for diagnostic purposes is done with the tightest fit to the data that gives acceptable results, in order to preserve the amplitude of the input field. When doing this with the traditional method, unfortunately, the result is fields with distorted patterns, owing to the traditional method's inherent inability to reproduce extrema and gradients accurately when doing a close fit to the data. We note that multiple pass objective analysis schemes (i.e., those involving successive corrections, rather than just a single-pass mapping of the data from the irregular station distribution to a regular grid) generally will exaggerate this tendency for localized overfitting of the fields in the vicinity of the observation sites. Hence, it is not clear that multiple-pass objective analysis schemes can be given high recommendations if derivative estimates are the goal of the diagnosis.

The traditional method produces inferior results even in situations with uniform spacing and long input wavelengths if the analysis is done with a relatively tight fit to the input data. As the data distribution becomes increasingly nonuniform, the traditional method's inferiority to the triangle method is even more obvious. Given that virtually all routine observational networks (and even most experimental networks) have at least some degree of nonuniformity, it is not clear that using the traditional method should be the default choice for derivative estimation, in spite of its current widespread use in diagnostic studies. We hope that this set of experiments provides motivation for others to give serious consideration to changing their diagnostic derivative estimation methodology.

`Acknowledgments``. We appreciate helpful comments concerning an earlier
version of this manuscript from Mark Askelson and Jeff Trapp. We also
acknowledge several helpful discussions on this general topic with
Sonia Lasher-Trapp. Dave Watson provided us with considerable
assistance in developing the Delauney triangulation algorithm,
without which none of this would have been feasible. Finally, we
thank the two anonymous reviewers, whose comments helped us clarify
and improve the manuscript.`

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According to the theory developed in DC88, the separate processes of estimating the derivatives and mapping/smoothing the field onto a uniform grid only commute when the data array coincides with the analysis grid. In cases where the data are on a uniform grid, it is possible to estimate the derivatives first, using (3) directly on the data array, and then mapping/smoothing the estimates onto the analysis grid. This procedure, which can be carried out only in the case of a uniform data array, is called the "alternative" procedure, to distinguish it from the "traditional" and "triangle" schemes. In order to test this, we considered four separate experiments.

First, we considered the 21 x 21 data array unperturbed from its original uniform arrangement, analyzing the result onto the 161 x 161 analysis grid using l = 1.0D. The analytic field for L = 10 D (cf. Fig. 1) can be compared to the results shown in Fig. A1a, Fig. A1b, Fig. A1c. All three methods give good results with respect to the pattern. The resulting amplitudes vary among the three different schemes, but the traditional scheme retains the highest fraction of the amplitude.

Second, we left everything the same except for reducing the size of the analysis grid to the same size as the data array and aligning the data array with the analysis grid. Results (Fig. A2a, Fig. A2b, Fig. A2c) make it clear that the outcomes of the traditional and alternative schemes are virtually identical in this case, including the amplitudes. This seems to confirm the notion developed in DC88 that the processes of differentiation and smoothing do indeed commute when the data and analysis grids coincide. The triangle scheme is also very similar to the other two, but it retains slightly more of the amplitude.

Third, we repeated the experiment with the 21 x 21 data array unperturbed from its original uniform arrangement, analyzing the result onto the 161 x 161 analysis grid, but reduced the shape parameter to l = 0.5 D. That is, we deliberately "overfitted" the data. As can be seen in Fig. A3a, Fig. A3b, Fig. A3c, the resulting analyses show that the traditional method fails to produce a good estimated derivative field, even with uniformly distributed data. The alternative scheme is much improved, although it apparently is suffering from substantial truncation error, as would be expected with such a coarse grid. The triangle method exhibits a slightly reduced amplitude compared to the alternative scheme in this case.

Fourth, keeping l = 0.5 D, we repeated the experiment where the analysis grid and data array coincide exactly (Fig. A4a, Fig. A4b, Fig. A4c). In this case, the traditional scheme actually produces a better field pattern than with a fine analysis grid. As with the previous example, the traditional and alternative schemes give virtually identical results, and the triangle scheme is very similar, only this time it seems to retain slightly less amplitude than either of the other schemes.

In the generally unrealistic situation of a uniform data array, our results confirm the theory developed in DC88. That is, the traditional and alternative schemes give virtually identical results when the data and analysis grids coincide. Interestingly, in this situation, the triangle scheme also gives similar but not quite identical results. By using line integrals to estimate the derivatives, the triangle method is doing something different, but we are not prepared to give a complete analysis of the process in comparison to the traditional approach.

We now introduce random errors into the
analytic wind field (4) in order to assess their impact upon the
resulting traditional and triangle vorticity analyses. To do this,
each component of every wind "observation" within the data array is
assigned a different random error, generated by multiplying random
numbers (between ±1, drawn from a uniform distribution) by a
value RE_{max} , the maximum allowable random error. The random errors
are then added to the analytic wind components and the analyses are
performed. RE_{max} is varied from 0 to 10 m s^{-1}, the maximum wind
amplitude (see Eq. 4), for various combinations of the scatter
distance (*D _{s}*) and wavelength
(

Figure B1a suggests that for
RE_{max}
< 4 m s^{-1}, the traditional method is superior to the triangle
method for the given values of *D _{s}* and

For a more irregular data distribution
(*D _{s}* = 0.8D), the triangle method is
superior to the traditional method for RE

A contour field of the COPs for various
combinations of the scatter distance and wavelength is presented in
Fig.
B2. Interpretation of this figure is
best done by example. For *D _{s}* = 0.0D and

From Fig. B2, we make three primary
observations. First, for most wavelengths within and below the
"marginally-sampled" range (6D < *L* < 12D, as defined by DC88), the traditional method RMSE is
lower than the triangle method RMSE for all degrees of data
irregularity. Again, this is because the traditional method is better
able to represent the true amplitude of the waves. For irregular data
distributions (*D _{s}*¹ 0.0D), however, we have seen that despite having a larger
RMSE, the triangle method produces a superior pattern of the
vorticity field.

Second, for a perfectly regular data
distribution (*D _{s}* = 0.0D) and for wavelengths
exceeding the "marginally-sampled" range, the COPs are very nearly
zero, suggesting that the two methods are roughly equivalent. This
result is consistent with our previous findings (e.g., Fig.
6a).

Finally, for irregular data distributions --
those found in real-life observational networks -- and wavelengths
greater than those considered "marginally-sampled", the triangle
method is superior over a large range of RE_{max}. In fact, even for
only slightly irregular data distributions (0.1D < *D _{s}* < 0.4D), the triangle method generally is superior for values
of RE

Figure Captions

Figure
1. Analytic vorticity for
*L* =
10D; units are
arbitrary

Figure
2. Delauney triangulation of the
observing array perturbed using *D _{s}* = 0.8D. Triangles containing an included angle < 15º
have been removed and are not considered in any analysis.

Figure 3. Vorticity estimates using
(Fig.
3a) the traditional method, and
(Fig.
3b) the triangle method, for
*L *=
10D, *l* = 1.0D, and *D _{s}* = 0.8D.

Figure
4a, Figure 4b. As in Fig. 3, except for *L *= 25D and *D _{s}* = 0.2D.

Figure
5a. Figure 5b. As in Fig. 3, except for *L *= 25D.

Figure 6. Plot showing the *difference* (triangle -
traditional) of the *RMSE* for vorticity between the triangle method and the
traditional method, as a function of the shape parameter,
*l*, and
the wavelength, *L*, of the input analytic wave (as multiples of the data
grid spacing, D)
for (Fig.
6a) uniformly distributed data
[*D _{s}* = 0.0D], (Fig. 6b)

Figure
7. Box and whisker plot of the distance
to the nearest neighbor for each of the 441 data points, as a
function of *D _{s}*, the scatter
distance used to create nonuniform data distributions. The box and
whisker plots show the minimum, first quartile, third quartile, and
maximum values from the distribution associated with each scatter
distance value. The vertical bars depict the average value within the
distribution.

Figure 8. Contours of the logarithm of the
*RMSE*
associated with the traditional technique, as a function of shape
parameter, *l*, and the scatter distance, *D _{s}*, for an input wavelength,

Figure 9. (Fig. 9a) Schematic of the
simple 5-point smoother, with (Fig. 9b)
its associated response function as a function of the wavelength, in
multiples of the grid distance *D _{g}*.

Figure 10. Plot of the traditionally computed (solid line) and
true (dashed line) values along the row *j* = 81 in the analysis
grid for (Fig. 10a) the analyzed wind
component, ,
and (Fig. 10b) the finite difference
form of the derivative , using *L* = 25D,
*D _{s}* = 0.0D, and

Figure A1. Estimated vorticity field with uniform
(*D _{s}* = 0) 21 x 21 data array, 161 x 161 analysis
grid,

Figure A2. As in Fig. A1, except for a 21 x 21 analysis grid that coincides with the data array. [Fig. A2a , Fig. A2b, Fig. A2c]

Figure A3. As in Fig. A1, except for *l* = 0.5D.
[Fig. A3a,
Fig. A3b,
Fig. A3c ]

Figure A4. As in Fig. 11, except for a 21 x 21 analysis grid that coincides with the data array. [Fig. A4a , Fig. A4b, Fig. A4c]

Figure B1. RMSE for vorticity as a function of RE_{max},
the maximum random error, for (Fig.
B1a) *D _{s}* = 0.0D,

Figure B2. Contour plot of the
cross-over point (COP) for various values of the scatter constant
*D _{s}* and the wavelength