ICES Journal of Marine Science, 58: 1184–1194. 2001
doi:10.1006/jmsc.2001.1106, available online at http://www.idealibrary.com on
Patchy distribution fields: sampling distance unit and
reconstruction adequacy
I. Kalikhman
Kalikhman, I. 2001. Patchy distribution fields: sampling distance unit and reconstruction adequacy. – ICES Journal of Marine Science, 58: 1184–1194.
A mathematical model (Kriging) was used to examine the effects of choosing various
units of sampling distance on the adequacy of reconstructing patchy distribution fields.
The model simulates fish or plankton patches (or gaps) of different shapes and spatial
orientations, and an acoustic survey by parallel transects along which a unit of
sampling distance is set. Conformity of the reconstructed fields to those originally
generated was evaluated by calculating their correlations. If a priori information on
patch orientation is available, the optimal ratios of the distance between transects and
the sampling distance unit to the corresponding autocorrelation radius for the field
have been determined, ensuring the required match between the reconstructed field and
its original. The experiments with a certain unit of sampling distance confirm that
a posteriori determination of patch orientation allows reconstruction of the best field
attainable on the basis of the data from the survey conducted. In cases of field
movement, the criterion for choosing a survey direction is based on the relationship
between the dimension of moving patches in the direction of movement and that of the
surveyed area; the criterion remains valid when a unit of sampling distance is set along
transects.
2001 International Council for the Exploration of the Sea
Keywords: acoustic survey design, distance between transects, sampling distance unit,
patchy distribution field, anisotropy, autocorrelation radius, reconstruction, conformity, coefficient of determination.
Received 30 July 2000; accepted 1 June 2001; published electronically 22 August 2001.
I. Kalikhman: Yigal Allon Kinneret Limnological Laboratory, Israel Oceanographic
& Limnological Research Ltd, Tel-Shikmona, PO Box 8030, Haifa 31080, Israel;
tel.: 972 4 851 5202; fax: 972 4 851 1911; e-mail: [email protected]
Introduction
One major goal of any acoustic survey is to reconstruct
the distribution field studied. In this case, the survey
design is considered optimal if it ensures the required
conformity of a reconstructed field to the actual one
with minimal expenditures (Thompson, 1992). It is
assumed that the size and structure of the actual distribution field under natural conditions should be known.
However, in a real situation, the parameters of distribution fields are never known. Therefore, the mathematical simulation method was used to optimize survey
design and improve the algorithm of data analysis; fish
and plankton locations in Lake Kinneret were used as
prototypes in simulating distribution fields (Kalikhman
& Ostrovsky, 1997).
The major aspects of the acoustic survey design are
the distance between transects and the unit of sampling
distance. The latter is the length of a survey transect
1054–3139/01/061184+11 $35.00/0
along which the acoustic measurements are averaged to
receive one sample. The sampling distance unit may be
as short as 0.1 mile to distinguish dense schools, or as
long as ten miles in the case of species widely distributed
over large areas of the ocean; usually, the unit of
sampling distance may be within the range one to five
miles (MacLennan & Simmonds, 1992). However, with
the beginning of the use of acoustic surveys for ecological purposes (e.g. Rose & Leggett, 1990), a more preferable rationale for selecting the sampling distance unit
is required. Therefore, the mathematical simulation
method was used again. The difference between the
present mathematical model and that described in the
above mentioned study by Kalikhman & Ostrovsky
(1997) consists of the fact that the unit of sampling
distance is set along transects.
Since the distribution of fish and plankton is usually
patchy (Steele, 1976), the sampling distance unit should
be chosen on the basis of statistical characteristics of a
2001 International Council for the Exploration of the Sea
Patchy distribution fields
patchy distribution field. The main objective of the
present study is to determine criteria for choosing the
unit of sampling distance allowing one to obtain a
realistic image of an actual patchy field. A further
objective consists of comparing these criteria with those
for choosing the distance between transects. For that
reason, regarding patchy distribution fields, surveys
were simulated exactly on the same fields as those
studied in Kalikhman & Ostrovsky (1997); the fields
consisting of gaps were added to test the applicability of
the algorithm to this kind of distribution (e.g. Greene
et al., 1994; Genin et al., 1994).1
Mathematical model design
The mathematical model is based on the approach
described by Kizner et al. (1983) and Zaripov et al.
(1983). The surveyed area is a rectangular array of
numbers making up a square matrix of 50 lines and 50
columns. Each of these 2500 numbers represents the
value of a field in an elementary area (node). To simulate
a distribution field, patches (or gaps) are generated and
placed into the surveyed area. In the outside patches the
field variable is zero, while outside gaps it is the constant
background. Inside a patch (gap) the variable increases
(decreases) from the outer border towards the centre
according to a law obtained by analyzing data from real
acoustic microsurveys (Williamson, 1980). Patches
(gaps) may be circular (isotropic or non-directional) or
elliptical (anisotropic or directional) with different sizes
and spatial orientation; separate or overlapping (partially or fully), forming larger patches (gaps) with more
complicated shapes. Patches (gaps) are either immovable
(static) or movable (dynamic); moving patches (gaps)
have a realistic shape of a comet (Yudanov, 1971).
Random fluctuations can be superimposed on the
simulated patchy field.
A resulting (constructed) patchy distribution field is
then sampled by simulation of an acoustic survey. We
distinguish the survey path and the general direction of
the survey. Transects are disposed perpendicularly to the
general survey direction. Thus, the entire path of the
survey consists of parallel transects and connecting
tracks perpendicular to transects. The distances between
parallel transects are regular and equal to the length of
connecting tracks (D). As mentioned above, a unit of
sampling distance (d) is set along transects. This is the
length of a transect along which the values of a signal
power are integrated and averaged. The field values
sampled along transects over each sampling distance are
assumed to be measured without error and are used to
1
Gaps are a special case of patchiness, corresponding to areas
(or volumes) of habitable space in which organisms are noticeably reduced in abundance relative to background levels
(Greene et al., 1994).
1185
reconstruct the field. The reconstructed field again
consists of the full set of nodes of the array representing
the surveyed area. The corresponding values of the
reconstructed field and that originally generated are
then compared for all the nodes. Their conformity is
evaluated by the coefficient of determination which
represents the square of the standard Pearson
correlation coefficient (r) (Croxton et al., 1968).
The autocorrelation radius (R) is used as a parameter
of a field in choosing the distance between transects of a
survey (Gandin, 1965). When the axes are located
horizontally (or vertically), the autocorrelation radius is
determined from a plot of the autocorrelation function
as the distance from the coordinate centre to the point
where the lag is zero. This distance is equal to the lag
unit multiplied by the lag. When rotating the axes to a
certain angle, each lag unit increases. If the angle is such
that the axis does not cross at least three points, the
angle is replaced by another one, close to the initial one,
where the axis crosses the required amount of points. An
important property of the autocorrelation function consists of the fact that it is sensitive only to variable
deviations while constant levels have no effect (Stull,
1988). This enables us to use the autocorrelation function when seeking the autocorrelation radii of distribution fields that include gaps. We suggest using the same
characteristics of the field for choosing the unit of
sampling distance. Throughout the paper, R, D, and
d are given in proportion to the size of the square
representing a surveyed area.
Mathematical experiments
Further on, the reconstruction of the distribution fields
will be taken into consideration. The gridding methods
use weighted interpolation algorithms. The Kriging
method widely used in reconstruction of the distribution
fields was tested. It is assumed that this algorithm has an
underlying variogram. The search option controls the
remoteness of points in which gridding operations have
no effect (Cressie, 1991).
Static fields
The gridding method is chosen to enable the maximal
correlation between the reconstructed field and that
originally generated to attain. Preliminary experiments
indicate that in most cases, the maximal correlation is
attained when using Kriging with the linear variogram
model. In all experiments, this method is used to exclude
the effect of the gridding methods on relationships of
interest. If an original field is anisotropic, the correlation
between the reconstructed field and the originally generated one depends upon the orientation of the ellipse for
1186
I. Kalikhman
the data weighting and search. Preliminary results of
experiments indicate that the maximal correlation is
attained if, in reconstructing the field, the major axis of
this ellipse is adjusted to the direction of patch elongation. When designing a real survey, a priori information
on patch orientation may or may not be available.
Use of a priori information on patch orientation and
autocorrelation radii for the field.
In carrying out this series of simulation experiments, it
was assumed that a priori information was available
either on the field isotropy, or on the anisotropic patch
orientation (in surveying fields along shelf edge or in
frontal zones it is often observed that patches are
extended along the edge or front and not in any other
direction). Thus, in reconstructing the fields, the major
axis of the ellipse for the data weighting and search was
adjusted to the direction of the patch elongation.
The adequacy of field reconstruction depends on two
factors. The first factor is the ratio of the distance
between transects to the autocorrelation radius for the
field in the direction of the survey (D/Rs). According to
the results presented in the study by Kalikhman &
Ostrovsky (1997) and used throughout this section, a
patchy field can be reconstructed properly (r2 >0.70)
if the ratio of the distance between transects to the
autocorrelation radius in the survey direction is
D/Rs <1.0–1.5 (later on, the values (r2)t =0.70 and
(D/R)t =1.5 will be named as the threshold ones). If this
condition is met, the adequacy of reconstruction may
depend on the second factor: the ratio of the sampling
distance unit to the autocorrelation radius for the field in
the direction perpendicular to the survey direction
(d/Rp). Let us consider this partial dependence under the
following conditions: for each of the 12 distribution
fields, 12 replicated simulations with various units of
sampling distance are conducted (the minimal distance is
d=1/50, then d=1/16, d=1/12, d=1/11 and so on up to
d=1/3); the total number of mathematical experiments
in this series is 144.
For each simulated isotropic field (Figure 1, first
panel), the autocorrelation radius is equal to R=0.15
(left), R=0.09 (middle), or R=0.16 (right). For these
fields, the condition of adequate reconstruction is met if
the distance between transects does not exceed the
following threshold value: Dt =0.23 (left), Dt =0.14
(middle), or Dt =0.24 (right). The simulations conducted indicate that the survey with the distance
between transects close to the threshold (second panel:
D=0.22, left; D=0.14, middle; D=0.24, right) results
in the coefficient of determination being higher
than the threshold value (r2 =0.83, left; r2 =0.77, middle;
r2 =0.85, right).
In the case of the left field, the increase of the unit of
sampling distance from d=1/50 to d=1/7 results in
lowering the coefficient of determination from r2 =0.83
(second panel) to r2 =0.74 (third panel). For the middle
field, the increase of the unit of sampling distance from
d=1/50 to d=1/12 results in lowering the coefficient of
determination from r2 =0.77 (second panel) to r2 =0.74
(third panel). For the right field, the increase of the unit
of sampling distance from d=1/50 to d=1/7 results in
lowering the coefficient of determination from r2 =0.85
(second panel) to r2 =0.81 (third panel). Thus, the experiments conducted confirm the fact that in cases of
isotropic fields, with a constant ratio of the distance
between transects to the autocorrelation radius in the
survey direction (D/Rs), the adequacy of reconstruction
of a field depends on the ratio of the sampling distance
unit to the autocorrelation radius in the direction
perpendicular to the survey direction (d/Rp).
In the case of the left field, the surveys carried out at
D=0.22 and d=1/7 in the two perpendicular directions
give similar results of r2 =0.74 (third panel) and r2 =0.69
(fourth panel); for the middle field, the surveys carried
out at D=0.14 and d=1/12 in the two perpendicular
directions give similar results of r2 =0.74 (third panel)
and r2 =0.71 (fourth panel); for the right field, the
surveys carried out at D=0.24 and d=1/7 in the two
perpendicular directions also give similar results of
r2 =0.81 (third panel) and r2 =0.74 (fourth panel). Thus,
for an isotropic field and a given distance between
transects and sampling distance unit, the adequacy of
reconstruction of a distribution field is independent of
the direction of the survey performed.
For each simulated anisotropic field (Figure 2, first
panel), the autocorrelation radii in directions of abscissa
and ordinate axes are equal to Rx =0.27 and Ry =0.13
(left), Rx =0.06 and Ry =0.23 (middle), or Rx =0.13 and
Ry =0.28 (right). Optimal direction for the survey of the
left field is the abscissa, while for the middle and right
ones it is the ordinate (Kalikhman & Ostrovsky, 1997);
the condition of adequate reconstruction is met if the
distance between transects does not exceed the following
threshold value: Dt =0.41 (left), Dt =0.35 (middle), or
Dt =0.42 (right). The simulations conducted indicate
that the survey with the distance between transects close
to the threshold (second panel: D=0.40, left; D=0.32,
middle; D=0.40, right) results in the coefficient of determination being higher than the threshold value (r2 =0.92,
left; r2 =0.83, middle; r2 =0.90, right).
In the case of the left field, the increase of the unit of
sampling distance from d=1/50 to d=1/8 results in
lowering the coefficient of determination from r2 =0.92
(second panel) to r2 =0.86 (third panel). For the middle
field, the increase of the unit of sampling distance from
d=1/50 to d=1/8 results in lowering the coefficient of
determination from r2 =0.83 (second panel) to r2 =0.66
(third panel). For the right field, the increase of the unit
of sampling distance from d=1/50 to d=1/6 results in
lowering the coefficient of determination from r2 =0.90
Figure 1. First panel: simulated patchy isotropic fields. Second to fourth panels: paths of simulated surveys and the reconstructed
fields (third and fourth panels show the unit of sampling distance set along the transects; the surveys shown in these panels are
directed perpendicularly to each other). Right column represents the gaps.
Figure 2. First panel: simulated patchy isotropic fields. Second to fourth panels: paths of simulated surveys conducted in the
directions of the major or minor axes of autocorrelation ellipses, and distribution fields as reconstructed using a priori information
on patch orientation (third and fourth panels show the unit of sampling distance set along the transects). Right column represents
the gaps.
Patchy distribution fields
(second panel) to r2 =0.86 (third panel). Thus, the
mathematical experiments confirm that in the case of
anisotropic fields, similar to that of isotropic ones, with
a constant ratio of the distance between transects to the
autocorrelation radius in the survey direction (D/Rs),
the adequacy of field reconstruction depends on the
ratio of the sampling distance unit to the autocorrelation
radius in the direction perpendicular to the survey
direction (d/Rp).
In the case of the left field, a survey carried out at
D=0.40 and d=1/8 in the direction of patch elongation
gives almost the same results in terms of original field
adequacy (r2 =0.86, third panel) as a survey at D=0.12
and d=1/4 conducted in a perpendicular direction
(r2 =0.90, fourth panel). For the middle field, a survey
carried out at D=0.32 and d=1/8 in the direction of
patch elongation permits us to reconstruct the original
field with almost the same adequacy (r2 =0.66, third
panel) as a survey conducted at D=0.10 and d=1/3 in a
perpendicular direction (r2 =0.69, fourth panel). Finally,
for the right field, a survey carried out at D=0.40 and
d=1/6 in the direction of patch elongation gives even
better results (r2 =0.86, third panel) than a survey conducted at D=0.14 and d=1/3 in a perpendicular direction (r2 =0.81, fourth panel). Thus, the surveys in the
direction of patch elongation with a larger distance
between transects and smaller unit of sampling distance
permit us to reconstruct the original field with about the
same adequacy as those in a perpendicular direction
with a smaller distance between transects and a larger
unit of sampling distance.
For each simulated rotated anisotropic fields (an
example of rotation for 30 is given in Figure 3, first
panel), the autocorrelation radii in directions of abscissa
and ordinate axes are equal to Rx =0.25 and Ry =0.18
(left), Rx =0.11 and Ry =0.20 (middle), or Rx =0.18 and
Ry =0.27 (right). The condition of adequate reconstruction is met if the distance between transects do not
exceed the following threshold value: Dt =0.40 (left),
Dt =0.30 (middle), or Dt =41 (right). The simulations
conducted indicate that the survey with the distance
between transects close to the threshold (second panel:
D=0.40, left; D=0.30, middle; D=0.24, right) results in
the coefficient of determination being higher than the
threshold value (r2 =0.85, left; r2 =0.79, middle; r2 =0.94,
right).
The increase in the angle between the survey direction
and that of patch elongation makes us set a smaller
distance between transects so that we may achieve the
same adequacy of reconstructing an original patchy
distribution field. At the same time, a larger unit of
sampling distance can be set. For example, in the case of
the left field, the survey carried out at D=0.40 and
d=1/7 with the angle between the survey direction and
that of patch elongation equal to 30 gives even better
results in terms of original field adequacy (R2 =0.77,
1189
third panel) than the survey conducted at D=0.28 and
d=1/6 with the angle between the survey direction and
that of patch elongation equal to 60 (r2 =0.74, fourth
panel). For the middle field, the survey carried out at
D=0.30 and d=1/12 with the angle between the survey
direction and that of patch elongation equal to 30
permits us to reconstruct the original field with the same
adequacy (r2 =0.61, third panel) as the survey conducted
at D=0.14 and d=1/7 with the angle between the survey
direction and that of patch elongation equal to 60
(r2 =0.61, fourth panel). Finally, for the right field, the
survey carried out at D=0.24 and d=1/7 with the angle
between the survey direction and that of patch elongation equal to 30 gives almost the same results in terms
of original field adequacy (r2 =0.88, third panel) as the
survey conducted at D=0.16 and d=1/5 with the angle
between the survey direction and that of patch elongation equal to 60 (r2 =0.85, fourth panel). This is
explained, ultimately, by the relationship of autocorrelation radii located in two reciprocally perpendicular
directions (the autocorrelation radius located at a
smaller angle to the major axis of autocorrelation ellipse
is always larger than that situated at a larger angle).
The results of the simulated surveys with various
ratios of sampling distance units to autocorrelation
radius are considered below (Figure 4; it should be
remembered that these dependences are obtained under
the assumption that D/Rs <1.5). The existence of a
distinct relationship fitting the generalized data set (r2 vs.
d/Rp) confirms the possibility of using the autocorrelation radius in the direction perpendicular to the survey
direction as a parameter of a field in choosing the unit of
sampling distance. From Figure 4 it can be seen that
with the d/Rp ratio increasing to 1.0–1.5, the coefficient
of determination changes only slightly and remains
comparatively high, while with further increase, this
coefficient falls considerably. For this reason, we suggest
choosing the unit of sampling distance to be less than
(1.0–1.5)Rp. Thus, when designing a real survey, if a
priori information is available on the autocorrelation
radius for a field in any direction, the unit of sampling
distance in this direction can be chosen. This unit
ensures the required match between the field reconstructed on the basis of the data from the survey
designed and that really existing in the water body,
although unknown.
A posteriori test for anisotropy.
In the absence of a priori information on patch orientation, a survey is usually conducted in an arbitrary
direction and the distribution field is reconstructed
assuming the area for the data weighting and search is
circular (isotropic). However, the mathematical experiments show that such a hypothesis may lead to considerable distortions in reconstructing anisotropic fields
Figure 3. First panel: simulated patchy isotropic fields. Second to fourth panels: paths of simulated surveys conducted in arbitrary
directions, and distribution fields as reconstructed using a priori information on patch orientation (third and fourth panels show
the unit of sampling distance set along the transects). Right column represents the gaps.
Patchy distribution fields
1191
(R=0.27 or R=0.24, depending on the transect chosen).
The field reconstructed with this type of orientation of
the ellipse for the data weighting and search is the
optimal one, attainable on the basis of the data from the
survey conducted. If the rotation of this ellipse does not
indicate any substantial change in the autocorrelation
radius for the reconstructed field, the latter is assumed to
be isotropic. Thus, the algorithm remains valid when a
unit of sampling distance is set along transects and can
be used to determine a posteriori the direction of patch
elongation.
Dynamic fields
Figure 4. Coefficient of determination between the reconstructed field and that originally generated as a function of the
d/Rp ratio. Filled symbols correspond to surveys of patchy
distribution fields; empty symbols, to those of gappy fields.
Colours correspond to the results of the simulated surveys of
the fields shown in Figures 1–3; dark blue, Figure 1, left and
right; red, Figure 1, middle; crimson, Figure 2, left and right;
light blue, Figure 2, middle; yellow, Figure 3, left and right;
green, Figure 3, middle (surveys corresponding to the empty
red, light blue and green circles are not shown in Figures 1–3).
The coefficient of determination regarding isotropic fields is
obtained by averaging the results of the surveys carried out in
the directions of abscissa and ordinate axes. The solid line
represents the non-linear regression ensuring the minimal sum
of squared residuals (the ends of the straight lines indicate the
bend of non-linear regression). Dashed lines are the borders of
the confidence interval with the probability p=0.99.
(Kalikhman & Ostrovsky, 1997). As shown in the same
paper, better results are obtained if the field is reconstructed as anisotropic, with the major axis of the ellipse
for the data weighting and search adjusted to the
direction of the patch elongation.
Let us test this algorithm under the assumption that a
unit of sampling distance is set along transects. An
original anisotropic distribution field is presented in
Figure 5 (first panel, first position). A survey with the
distance between transects D=0.40 and the unit of
sampling distance d=1/7, and the field reconstructed as
an isotropic one, are given in the second position of
Figure 5. Let us consider the same survey but with the
field reconstructed as an anisotropic one (Figure 5,
second panel). The rotation of the ellipse for the data
weighting and search is presented in the first to third
positions. When the major axis of the ellipse for the data
weighting and search coincides with the direction of
patch elongation (third position), the maximal autocorrelation radius for the reconstructed field is obtained
Let us consider the same conditions as those taken in
the study by Kalikhman & Ostrovsky (1997): i.e. all
patches were moved in the same direction uniformly and
rectilinearly; simulated surveys were undertaken in the
opposite, same and perpendicular directions relative to
that of patch movement; the speed of patch movement
was 0.7, or 0.5, or 0.3 of that of the survey. Survey
design was examined regarding the conditions widely
met in practice (McAllister, 1998): i.e. the field was
reconstructed disregarding the information on the patch
movement and compared with the average original field.
Let us test the criterion for choosing a survey direction in the case when the unit of sampling distance is set
along transects. In Figure 6 (first panel), the following
conditions are, in particular, considered: the survey is
carried out in the opposite direction relative to that of
the patch movement; the speed of patch movement is
equal to 0.7 of the survey; the distance between transects
is D=0.14; the unit of sampling distance is d=1/7. The
average original field is shown in the second panel, first
position. The situation that could be described by the
Doppler effect occurs when the survey and the patches
move in the opposite directions (second position); the
Doppler effect is also observed when the survey and
the patches move in the same direction (third position).
The survey in the opposite direction leads to the
maximal coefficient of determination between the reconstructed field and the average original one (r2 =0.64).
The survey in the perpendicular direction results in the
minimal coefficient of determination (r2 =0.49).
In general, as shown by the simulations, if the extension of patches in the direction of movement exceeds
that of the surveyed area, a survey in the same direction
may result in missing patches, and therefore produces
poorer results as compared to that in the opposite
direction (Figure 6). In contrast, if the extension of
patches in the direction of movement is smaller than that
of the surveyed area, a survey in the opposite direction
may result in missing patches, and so gives poorer results
compared to that in the same direction. Mathematical
experiments with lower speeds of patch movement
confirm the regularity observed. Thus, the algorithms
1192
I. Kalikhman
Figure 5. First panel, first position: simulated patchy rotated anisotropic field (earlier considered in Figure 3, left). Reconstructed
as a result of the survey with the distance between transects D=0.40 and the unit of sampling distance d=1/7 when no a priori
information on patch orientation is available: first panel, second position, as an isotropic field (the area for the data weighting and
search is circular); second panel, as an anisotropic field (the angle between the major axis of the ellipse for the data weighting and
search and the direction of patch elongation is indicated). Transects used to estimate the autocorrelation functions for the
reconstructed field are shown with straight crimson lines.
remain valid for cases where a unit of sampling distance
is set along transects.
Conclusions
Survey design and the algorithm of data analysis for
distribution fields consisting of gaps are exactly the same
as they are for patchy distribution fields. A priori
information on patch orientation and autocorrelation
radii for the field in two reciprocally perpendicular
directions allows optimization of the survey design and
the algorithm of data analysis. To some extent it is
possible to compensate the change in adequacy of reconstructing a patchy distribution field by an increase of the
distance between transects and a corresponding decrease
of the sampling distance unit, and vice versa. It is
practically expedient to carry out a survey with a larger
distance between transects and a smaller unit of sampling distance. A field can be reconstructed properly
(r2 >0.70) if D/Rs <1.0–1.5 and d/Rp <1.0–1.5, where Rs
and Rp are the autocorrelation radii for the field in
the direction of the survey and in the perpendicular
direction.
If a priori information on the field is not available, the
direction of patch elongation should be determined
when reconstructing the field, i.e. a posteriori. Testing
the algorithm with the assumption that the unit of
sampling distance is set along transects confirms its
correctness for this case. The direction of the major axis
of the ellipse for the data weighting and search, for
which the autocorrelation radius for the reconstructed
field is maximal, indicates the direction of patch elongation. If the rotation of this ellipse does not indicate
any substantial changes in the autocorrelation radius
for the reconstructed field, the latter is assumed to be
isotropic.
With respect to the field movement, the test of the
criterion for choosing a survey direction confirms its
applicability for the case when a unit of sampling
distance is set along transects. The criterion is based on
the comparison of the dimension of moving patches in
Patchy distribution fields
1193
Figure 6. A moving field (earlier presented in Figure 2, left, as an immovable one) and the path of the simulated survey carried out
in the opposite direction relative to that of the patch movement. The speed of patch movement is 0.7 of that of the survey; the
distance between transects D=0.14 and the unit of sampling distance d=1/7. First panel, sequence of positions of the field and the
survey. Second panel, average original field (first position); the fields reconstructed from the results of the surveys in the opposite,
same and perpendicular directions (second to fourth positions). Smaller arrows indicate the direction of the patch movement, larger
ones, the directions of the surveys.
the direction of movement and that of the surveyed area.
If the extension of moving patches exceeds that of a
surveyed area, a survey in the opposite direction gives
better results; in contrast, if the extension of moving
patches is smaller than that of a surveyed area, it is
reasonable to carry out a survey in the same direction.
Acknowledgements
I wish to thank the reviewer Dr Robert Kieser and the
anonymous reviewer for the attentive reading of my
paper and constructive criticism which allowed me to
improve the manuscript substantially. I am grateful to
the Israeli Ministry of Absorption, the Israeli Ministry
of Science and Technology, the Rich Foundation, AID
and BSF, whose financial support made this study
possible.
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