ASTM D6982-09(2016)
(Practice)Standard Practice for Detecting Hot Spots Using Point-Net (Grid) Search Patterns
Standard Practice for Detecting Hot Spots Using Point-Net (Grid) Search Patterns
SIGNIFICANCE AND USE
4.1 Search sampling strategies have found wide utility in geologic exploration where drilling is required to detect subsurface mineral deposit, such as when drilling for oil and gas. Using such strategies to search for buried wastes and subsurface contaminants, including volatile organic compounds, is a logical extension of these strategies.
4.2 Systematic sampling strategies are often the most cost-effective method for searching for hot spots.
4.3 This practice may be used to determine the risk of missing a hot spot of specified size and shape given a specified sampling pattern and sampling density.
4.4 This practice may be used to determine the smallest hot spot that can be detected with a specified probability and given sampling density.
4.5 This practice may be used to select the optimum grid sampling strategy (that is, sampling pattern and density) for a specified risk of not detecting a hot spot.
4.6 By using the algorithms given in this practice, one can balance the cost of sampling versus the risk of missing a hot spot.
4.7 Search sampling patterns may also be used to optimize the locations of additional ground water monitoring wells or vadose zone monitoring devices.
SCOPE
1.1 This practice provides equations and nomographs, and a reference to a computer program, for calculating probabilities of detecting hot spots (that is, localized areas of soil or groundwater contamination) using point-net (that is, grid) search patterns. Hot spots, more generally referred to as targets, are presumed to be invisible on the ground surface. Hot spots may include former surface impoundments and waste disposal pits, as well as contaminant plumes in ground water or the vadose zone.
1.2 For purposes of calculating detection probabilities, hot spots or buried contaminants are presumed to be elliptically shaped when projected vertically to the ground surface, and search patterns are square, rectangular, or rhombic. Assumptions about the size and shape of suspected hot spots are the primary limitations of this practice, and must be judged by historical information. A further limitation is that hot spot boundaries are usually not clear and distinct.
1.3 In general, this practice should not be used in lieu of surface geophysical methods for detecting buried objects, including underground utilities, where such buried objects can be detected by these methods (see Guide D6429).
1.4 Search sampling would normally be conducted during preliminary investigations of hazardous waste sites or hazardous waste management facilities (see Guide D5730). Sampling may be conducted by drilling or by direct-push methods. In contrast, guidance on sampling for the purpose of making statistical inferences about population characteristics (for example, contaminant concentrations) can be found in Guide D6311.
1.5 This standard does not purport to address all of the safety concerns, if any, associated with its use. It is the responsibility of the user of this standard to establish appropriate safety and health practices and determine the applicability of regulatory limitations prior to use.
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Designation: D6982 − 09 (Reapproved 2016)
Standard Practice for
Detecting Hot Spots Using Point-Net (Grid) Search Patterns
This standard is issued under the fixed designation D6982; the number immediately following the designation indicates the year of
original adoption or, in the case of revision, the year of last revision.Anumber in parentheses indicates the year of last reapproval.A
superscript epsilon (´) indicates an editorial change since the last revision or reapproval.
1. Scope 2. Referenced Documents
1.1 Thispracticeprovidesequationsandnomographs,anda 2.1 ASTM Standards:
D5730Guide for Site Characterization for Environmental
reference to a computer program, for calculating probabilities
of detecting hot spots (that is, localized areas of soil or Purposes With Emphasis on Soil, Rock, the Vadose Zone
and Groundwater (Withdrawn 2013)
groundwater contamination) using point-net (that is, grid)
searchpatterns.Hotspots,moregenerallyreferredtoastargets, D6051Guide for Composite Sampling and Field Subsam-
pling for Environmental Waste Management Activities
are presumed to be invisible on the ground surface. Hot spots
may include former surface impoundments and waste disposal D6311Guide for Generation of Environmental Data Related
toWaste ManagementActivities: Selection and Optimiza-
pits, as well as contaminant plumes in ground water or the
vadose zone. tion of Sampling Design
D6429Guide for Selecting Surface Geophysical Methods
1.2 For purposes of calculating detection probabilities, hot
spots or buried contaminants are presumed to be elliptically
3. Terminology
shaped when projected vertically to the ground surface, and
3.1 Definitions:
search patterns are square, rectangular, or rhombic. Assump-
tions about the size and shape of suspected hot spots are the 3.1.1 hot spot—a localized area of soil or groundwater
contamination.
primary limitations of this practice, and must be judged by
historical information. A further limitation is that hot spot 3.1.1.1 Discussion—A hot spot may be considered as a
boundaries are usually not clear and distinct. discretevolumeofburiedwasteorcontaminatedsoilwherethe
concentration of a contaminant of interest exceeds some
1.3 In general, this practice should not be used in lieu of
prespecified threshold value. Although hot spots are more
surface geophysical methods for detecting buried objects,
likely to have variable sizes and shapes and not have clear and
including underground utilities, where such buried objects can
distinct boundaries, ellipitically shaped hot spots or targets
be detected by these methods (see Guide D6429).
with well defined edges are assumed for the purposes of
1.4 Search sampling would normally be conducted during
calculating detection probabilities. The assumption that hot
preliminary investigations of hazardous waste sites or hazard-
spots have elliptical shapes is not inconsistent with known
ous waste management facilities (see Guide D5730). Sampling
historical patterns of contaminant distribution.
may be conducted by drilling or by direct-push methods. In
3.1.2 sampling density—the number of soil borings (that is,
contrast, guidance on sampling for the purpose of making
sampling points) per unit area.
statistical inferences about population characteristics (for
3.1.3 semi-major axis, a—one-half the length of the long
example, contaminant concentrations) can be found in Guide
axis of an ellipse. For a circle, this distance is simply the
D6311.
radius.
1.5 This standard does not purport to address all of the
3.1.4 semi-minor axis, b—one-half the length of the short
safety concerns, if any, associated with its use. It is the
axis of an ellipse.
responsibility of the user of this standard to establish appro-
priate safety and health practices and determine the applica- 3.1.5 target—the object or “hot spot” that is being searched
bility of regulatory limitations prior to use. for.
1 2
This practice is under the jurisdiction of ASTM Committee D34 on Waste For referenced ASTM standards, visit the ASTM website, www.astm.org, or
Management and is the direct responsibility of Subcommittee D34.01.01 on contactASTM Customer Service at service@astm.org. ForAnnual Book ofASTM
Planning for Sampling. Standards volume information, refer to the standard’s Document Summary page on
Current edition approved May 1, 2016. Published May 2016. Originally the ASTM website.
approved in 2003. Last previous edition approved in 2009 as D6982–09. DOI: The last approved version of this historical standard is referenced on
10.1520/D6982-16. www.astm.org.
Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States
D6982 − 09 (2016)
3.1.6 threshold concentration—the concentration of a con-
taminant above which a hot spot is considered to be detected.
3.1.7 unit cell—the smallest area into which a grid can be
divided so that these areas have the same shape, size and
orientation. For a triangular grid, the unit cell is a 60°/120°
rhombuscomprisedoftwoequilateraltriangleswithacommon
side.
3.2 Symbols: a= lengthofthesemi-majoraxisofanellipse
b= length of the semi-minor axis of an ellipse
A = area of target or hot spot. For an ellipse, A = πab.
T T
A = search area
S
S= the“shape”ofanellipticaltarget(thatis,theratioofthe
length of the semi-minor axis to the length of the semi-major
axis of an ellipse, b/a)
G= the distance between nearest grid nodes of a unit cell
Q= the ratio of the length of the long side of a rectangular
grid cell to the length of the short side
A = the area of the unit cell. For a square, A = G . For a
C sq
rectangle A = Q·G . For a 60°/120° rhombus, A =[(√3)/
re rh FIG. 1 Projection of Boundaries of Subsurface Contamination to
2]G . The inverse of A is the sampling density the Ground Surface
C
β= the probability of not detecting a hot spot
P(hit)= probability of detection (that is, 1 − β)
5.3 Thesearchpatterniseitherasquare,arectangular,oran
4. Significance and Use
equilateral triangular grid. Borings are made at the intersec-
4.1 Search sampling strategies have found wide utility in
tions of grid lines (that is, nodes) (Fig. 2).
geologic exploration where drilling is required to detect
5.4 Borings or direct-push devices are directed downward
subsurface mineral deposit, such as when drilling for oil and
vertically and the detection of the target is unambiguous. Such
gas. Using such strategies to search for buried wastes and
an assumption presumes that the full length of a boring would
subsurface contaminants, including volatile organic
be subject to analysis as contiguous intervals of the boring. If
compounds, is a logical extension of these strategies.
sampling intervals are discontinuous, then contamination
4.2 Systematic sampling strategies are often the most cost-
might be missed if it occurred between sampled intervals. If
effective method for searching for hot spots.
sampling intervals are too long, then a hot spot may not be
detected because of dilution of a hot spot with less contami-
4.3 This practice may be used to determine the risk of
nated portions of the sampled interval. The criteria for detec-
missing a hot spot of specified size and shape given a specified
tion of contaminants may be prespecified threshold concentra-
sampling pattern and sampling density.
tions (for example, screening levels) that would trigger further
4.4 This practice may be used to determine the smallest hot
investigation of sites or facilities.
spot that can be detected with a specified probability and given
5.5 The area of the borehole or direct-push device is
sampling density.
infinitely small compared to the target area. The algorithms
4.5 This practice may be used to select the optimum grid
used in this practice assume that boreholes or direct-push
sampling strategy (that is, sampling pattern and density) for a
devices have no area, but rather are vertical lines projected
specified risk of not detecting a hot spot.
downward from grid nodes.
4.6 By using the algorithms given in this practice, one can
6. Preliminary Considerations
balance the cost of sampling versus the risk of missing a hot
spot.
6.1 Before designing a hot spot detection strategy, a pre-
liminary investigation of the area containing possible hot spots
4.7 Search sampling patterns may also be used to optimize
or targets should be conducted. From historical records, physi-
the locations of additional ground water monitoring wells or
cal layout of buildings and equipment, known transportation
vadose zone monitoring devices.
pathways, landscape features, and eyewitness accounts, one
5. Assumptions
may be able to identify areas with a high probability of
5.1 One or more targets or hot spots exist and are equally subsurface contamination.Areas with different expected prob-
likely to occur in any part of the search area. abilities of detection of a hot spot or other target should be
clearly mapped.
5.2 When projected vertically upward to a level ground
surface, the target appears as an ellipse or a circle (Fig. 1).The 6.2 Within areas of relatively uniform expected probability
probablesizeandshapeofahotspotcanonlybeguessedfrom of hot spot or target detection, sampling grids of prespecified
past site or facility records, known layout of the site or facility, grid spacing G and type (for example, square, rectangular, or
and personal knowledge. triangular) may be overlain. Areas with smaller hot spots
D6982 − 09 (2016)
FIG. 2 Grid Patterns for Detecting Hot Spots. Borings are Made at the Grid Nodes
should have correspondingly higher sampling densities com- target can only be hit once and the probability P of detecting
paredtoareaswithlargehotspots.However,areaswithgreater the hot spot is simply equal to the ratio of the area of the target
hazard from missing a hot spot should also have correspond- A to the area of the unit cell A (that is, P = A /A ).
T C T C
inglyhighersamplingdensitiesthanareaswithalesserhazard.
7.2 Case 2—If the longest dimension of an elliptical target
Ideally, the starting point for each grid and its orientation
is greater than the grid spacing (that is, 2a > G), then the target
should be randomly determined.
may be hit more than once. In this case, algorithms developed
6.3 When searching for hot spots, threshold concentrations by Singer and Wickman (1) employing affine transformations
for detection may be established by a regulatory authority. and programmed in FORTRAN by Singer (2) are required to
Whether or not a threshold concentration is exceeded will calculate the exact probability of detecting the target. This
depend upon the physical distribution of the contaminant, the program is limited to ellipses having a shape S between 0.05
volumeofthesamplingdevice,thesamplingintervalsselected, and 1.0 and the ratio a/G between 0.05 and 1.0. Singer’s
and the sensitivity of the analysis. If contamination occurs in a algorithms have been adapted by J. R. Davidson (3) to the
discrete layer, then the probability of detecting a hot spot will personal computer (PC) running under the MS DOS operating
decrease with increasing volume of material sampled in a bore system. Supporting documentation for this program,
hole or if the sampling interval exceeds the depth of the ELIPGRID-PC, is available from Oak Ridge National Labora-
discrete hot spot layer. The analytically determined contami- tory (4, 5).
nant concentration may then be less than the threshold concen-
7.3 Randomly Oriented Elliptical Target—The probability
tration because of the dilution of the hot spot layer with
ofdetectingatarget,P(hit),ofaspecifiedsizeashapeSandfor
uncontaminated layers of soil or waste. Further, a hot spot
a specified grid G spacing can be obtained from nomographs
confined to a discrete layer may be missed entirely by not
showninFigs.3and4forsquareandequilateraltriangulargrid
sampling that layer. For this reason, continuous sampling is
sampling patterns, respectively. Data for these nomographs
recommended.
weregeneratedusingtheELIPGRID-PCprogram.Tousethese
6.4 Detection of contaminant levels in samples above graphs, first calculate the ratio a/G. Then draw a vertical line
thresholdconcentrationsmaytriggermoredetailedsamplingto from the point represented by the ratio a/G on the x-axis of the
better define the spatial extent of hot spots or buried contami- graph to the curve representing the prespecified shape of the
nation. Again, a grid sampling strategy will be the most ellipse. Then draw a horizontal line to the y-axis. For shapes
efficient. other than those shown on the graphs, one must interpolate
7. Determining Hot Spot Detection Probabilities
7.1 Case I—Ifthelongestdimensionofanellipticaltargetis
The boldface numbers in parentheses refer to the list of references at the end of
less than or equal to the grid spacing (that is, 2a ≤ G), then the this standard.
D6982 − 09 (2016)
FIG. 3 Nomograph Relating the Probability of Detecting a Single Hot Spot to the Ratio a/G for Selected Shapes (b/a)
Using a Square Grid with Grid Spacing G.
FIG. 4 Nomograph Relating the Probability of Detecting a Single Hot Spot to the Ratio a/G for Selected Shapes (b/a)
Using a Triangular Grid with Grid Spacing G.
between curves with closest values of S. The value on the grid spacing to detect an elliptical target of shape at a
y-axisrepresentstheprobabilityofatleastonehitofthetarget. prespecified probability of detection. In this case, draw a
Using these same graphs, one can also determine the required horizontal line from the prespecified probability of a hit to the
D6982 − 09 (2016)
curve representing the prespecified shape of the ellipse. Then orientation is close to 30°, 90°, or 150° whereas a square grid
draw a vertical line down to the x-axis. From the ratio a/G at is more efficient when the angle of orientation is between 25°
the point of intersection with the x-axis, one can determine the and 65° or between 115° and 155°. Between 25° and 35°,
minimum required grid spacing. Similarly, one can also deter- efficiencies are nearly the same. These orientations minimize
mine the smallest sized hot spot of a given shape that can be the probabilities of hitting the same target more than once
detected for a given grid spacing and probability of detection which would result in less efficient sampling.
by calculating a from the ratio a/G and grid spacing G. 8.3.2 Rhombic Grid versus Square and Equilateral Trian-
Alternatively, one can use the computer program ELIPGRID- gular Grids—A rhombus is a parallelogram having opposite
PC. sides equal in length. A rhombus is also a square if the inside
anglesare90°.Twoe
...
This document is not an ASTM standard and is intended only to provide the user of an ASTM standard an indication of what changes have been made to the previous version. Because
it may not be technically possible to adequately depict all changes accurately, ASTM recommends that users consult prior editions as appropriate. In all cases only the current version
of the standard as published by ASTM is to be considered the official document.
Designation: D6982 − 09 D6982 − 09 (Reapproved 2016)
Standard Practice for
Detecting Hot Spots Using Point-Net (Grid) Search Patterns
This standard is issued under the fixed designation D6982; the number immediately following the designation indicates the year of
original adoption or, in the case of revision, the year of last revision. A number in parentheses indicates the year of last reapproval. A
superscript epsilon (´) indicates an editorial change since the last revision or reapproval.
1. Scope
1.1 This practice provides equations and nomographs, and a reference to a computer program, for calculating probabilities of
detecting hot spots (that is, localized areas of soil or groundwater contamination) using point-net (that is, grid) search patterns. Hot
spots, more generally referred to as targets, are presumed to be invisible on the ground surface. Hot spots may include former
surface impoundments and waste disposal pits, as well as contaminant plumes in ground water or the vadose zone.
1.2 For purposes of calculating detection probabilities, hot spots or buried contaminants are presumed to be elliptically shaped
when projected vertically to the ground surface, and search patterns are square, rectangular, or rhombic. Assumptions about the
size and shape of suspected hot spots are the primary limitations of this practice, and must be judged by historical information.
A further limitation is that hot spot boundaries are usually not clear and distinct.
1.3 In general, this practice should not be used in lieu of surface geophysical methods for detecting buried objects, including
underground utilities, where such buried objects can be detected by these methods (see Guide D6429).
1.4 Search sampling would normally be conducted during preliminary investigations of hazardous waste sites or hazardous
waste management facilities (see Guide D5730). Sampling may be conducted by drilling or by direct-push methods. In contrast,
guidance on sampling for the purpose of making statistical inferences about population characteristics (for example, contaminant
concentrations) can be found in Guide D6311.
1.5 This standard does not purport to address all of the safety concerns, if any, associated with its use. It is the responsibility
of the user of this standard to establish appropriate safety and health practices and determine the applicability of regulatory
limitations prior to use.
2. Referenced Documents
2.1 ASTM Standards:
D5730 Guide for Site Characterization for Environmental Purposes With Emphasis on Soil, Rock, the Vadose Zone and
Groundwater (Withdrawn 2013)
D6051 Guide for Composite Sampling and Field Subsampling for Environmental Waste Management Activities
D6311 Guide for Generation of Environmental Data Related to Waste Management Activities: Selection and Optimization of
Sampling Design
D6429 Guide for Selecting Surface Geophysical Methods
3. Terminology
3.1 Definitions:
3.1.1 hot spot—a localized area of soil or groundwater contamination.
This practice is under the jurisdiction of ASTM Committee D34 on Waste Management and is the direct responsibility of Subcommittee D34.01.01 on Planning for
Sampling.
Current edition approved Nov. 15, 2009May 1, 2016. Published December 2009May 2016. Originally approved in 2003. Last previous edition approved in 20032009 as
D6982D6982 – 09.–03. DOI: 10.1520/D6982-09.10.1520/D6982-16.
For referenced ASTM standards, visit the ASTM website, www.astm.org, or contact ASTM Customer Service at service@astm.org. For Annual Book of ASTM Standards
volume information, refer to the standard’s Document Summary page on the ASTM website.
The last approved version of this historical standard is referenced on www.astm.org.
3.1.1.1 Discussion—
A hot spot may be considered as a discrete volume of buried waste or contaminated soil where the concentration of a contaminant
of interest exceeds some prespecified threshold value. Although hot spots are more likely to have variable sizes and shapes and
Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States
D6982 − 09 (2016)
not have clear and distinct boundaries, ellipitically shaped hot spots or targets with well defined edges are assumed for the purposes
of calculating detection probabilities. The assumption that hot spots have elliptical shapes is not inconsistent with known historical
patterns of contaminant distribution.
3.1.2 sampling density—the number of soil borings (that is, sampling points) per unit area.
3.1.3 semi-major axis, a—one-half the length of the long axis of an ellipse. For a circle, this distance is simply the radius.
3.1.4 semi-minor axis, b—one-half the length of the short axis of an ellipse.
3.1.5 target—the object or “hot spot” that is being searched for.
3.1.6 threshold concentration—the concentration of a contaminant above which a hot spot is considered to be detected.
3.1.7 unit cell—the smallest area into which a grid can be divided so that these areas have the same shape, size and orientation.
For a triangular grid, the unit cell is a 60°/120° rhombus comprised of two equilateral triangles with a common side.
3.2 Symbols: a = length of the semi-major axis of an ellipse
b = length of the semi-minor axis of an ellipse
A = area of target or hot spot. For an ellipse, A = πab.
T T
A = search area
S
S = the “shape” of an elliptical target (that is, the ratio of the length of the semi-minor axis to the length of the semi-major axis
of an ellipse, b/a)
G = the distance between nearest grid nodes of a unit cell
Q = the ratio of the length of the long side of a rectangular grid cell to the length of the short side
2 2 2
A = the area of the unit cell. For a square, A = G . For a rectangle A = Q·G . For a 60°/120° rhombus, A = [(√3)/2]G .
C sq re rh
The inverse of A is the sampling density
C
β = the probability of not detecting a hot spot
P(hit) = probability of detection (that is, 1 − β)
4. Significance and Use
4.1 Search sampling strategies have found wide utility in geologic exploration where drilling is required to detect subsurface
mineral deposit, such as when drilling for oil and gas. Using such strategies to search for buried wastes and subsurface
contaminants, including volatile organic compounds, is a logical extension of these strategies.
4.2 Systematic sampling strategies are often the most cost-effective method for searching for hot spots.
4.3 This practice may be used to determine the risk of missing a hot spot of specified size and shape given a specified sampling
pattern and sampling density.
4.4 This practice may be used to determine the smallest hot spot that can be detected with a specified probability and given
sampling density.
4.5 This practice may be used to select the optimum grid sampling strategy (that is, sampling pattern and density) for a specified
risk of not detecting a hot spot.
4.6 By using the algorithms given in this practice, one can balance the cost of sampling versus the risk of missing a hot spot.
4.7 Search sampling patterns may also be used to optimize the locations of additional ground water monitoring wells or vadose
zone monitoring devices.
5. Assumptions
5.1 One or more targets or hot spots exist and are equally likely to occur in any part of the search area.
5.2 When projected vertically upward to a level ground surface, the target appears as an ellipse or a circle (Fig. 1). The probable
size and shape of a hot spot can only be guessed from past site or facility records, known layout of the site or facility, and personal
knowledge.
5.3 The search pattern is either a square, a rectangular, or an equilateral triangular grid. Borings are made at the intersections
of grid lines (that is, nodes) (Fig. 2).
5.4 Borings or direct-push devices are directed downward vertically and the detection of the target is unambiguous. Such an
assumption presumes that the full length of a boring would be subject to analysis as contiguous intervals of the boring. If sampling
intervals are discontinuous, then contamination might be missed if it occurred between sampled intervals. If sampling intervals are
too long, then a hot spot may not be detected because of dilution of a hot spot with less contaminated portions of the sampled
interval. The criteria for detection of contaminants may be prespecified threshold concentrations (for example, screening levels)
that would trigger further investigation of sites or facilities.
D6982 − 09 (2016)
FIG. 1 Projection of Boundaries of Subsurface Contamination to the Ground Surface
5.5 The area of the borehole or direct-push device is infinitely small compared to the target area. The algorithms used in this
practice assume that boreholes or direct-push devices have no area, but rather are vertical lines projected downward from grid
nodes.
6. Preliminary Considerations
6.1 Before designing a hot spot detection strategy, a preliminary investigation of the area containing possible hot spots or targets
should be conducted. From historical records, physical layout of buildings and equipment, known transportation pathways,
landscape features, and eyewitness accounts, one may be able to identify areas with a high probability of subsurface contamination.
Areas with different expected probabilities of detection of a hot spot or other target should be clearly mapped.
6.2 Within areas of relatively uniform expected probability of hot spot or target detection, sampling grids of prespecified grid
spacing G and type (for example, square, rectangular, or triangular) may be overlain. Areas with smaller hot spots should have
correspondingly higher sampling densities compared to areas with large hot spots. However, areas with greater hazard from
missing a hot spot should also have correspondingly higher sampling densities than areas with a lesser hazard. Ideally, the starting
point for each grid and its orientation should be randomly determined.
6.3 When searching for hot spots, threshold concentrations for detection may be established by a regulatory authority. Whether
or not a threshold concentration is exceeded will depend upon the physical distribution of the contaminant, the volume of the
sampling device, the sampling intervals selected, and the sensitivity of the analysis. If contamination occurs in a discrete layer, then
the probability of detecting a hot spot will decrease with increasing volume of material sampled in a bore hole or if the sampling
interval exceeds the depth of the discrete hot spot layer. The analytically determined contaminant concentration may then be less
than the threshold concentration because of the dilution of the hot spot layer with uncontaminated layers of soil or waste. Further,
a hot spot confined to a discrete layer may be missed entirely by not sampling that layer. For this reason, continuous sampling is
recommended.
6.4 Detection of contaminant levels in samples above threshold concentrations may trigger more detailed sampling to better
define the spatial extent of hot spots or buried contamination. Again, a grid sampling strategy will be the most efficient.
7. Determining Hot Spot Detection Probabilities
7.1 Case I—If the longest dimension of an elliptical target is less than or equal to the grid spacing (that is, 2a ≤ G), then the
target can only be hit once and the probability P of detecting the hot spot is simply equal to the ratio of the area of the target A
T
to the area of the unit cell A (that is, P = A /A ).
C T C
7.2 Case 2—If the longest dimension of an elliptical target is greater than the grid spacing (that is, 2a > G), then the target may
be hit more than once. In this case, algorithms developed by Singer and Wickman (1) employing affine transformations and
programmed in FORTRAN by Singer (2) are required to calculate the exact probability of detecting the target. This program is
limited to ellipses having a shape S between 0.05 and 1.0 and the ratio a/G between 0.05 and 1.0. Singer’s algorithms have been
The boldface numbers in parentheses refer to the list of references at the end of this standard.
D6982 − 09 (2016)
FIG. 2 Grid Patterns for Detecting Hot Spots. Borings are Made at the Grid Nodes
adapted by J. R. Davidson (3) to the personal computer (PC) running under the MS DOS operating system. Supporting
documentation for this program, ELIPGRID-PC, is available from Oak Ridge National Laboratory (4, 5).
7.3 Randomly Oriented Elliptical Target—The probability of detecting a target, P(hit), of a specified size a shape S and for a
specified grid G spacing can be obtained from nomographs shown in Figs. 3 and 4 for square and equilateral triangular grid
sampling patterns, respectively. Data for these nomographs were generated using the ELIPGRID-PC program. To use these graphs,
first calculate the ratio a/G. Then draw a vertical line from the point represented by the ratio a/G on the x-axis of the graph to the
curve representing the prespecified shape of the ellipse. Then draw a horizontal line to the y-axis. For shapes other than those
shown on the graphs, one must interpolate between curves with closest values of S. The value on the y-axis represents the
probability of at least one hit of the target. Using these same graphs, one can also determine the required grid spacing to detect
an elliptical target of shape at a prespecified probability of detection. In this case, draw a horizontal line from the prespecified
probability of a hit to the curve representing the prespecified shape of the ellipse. Then draw a vertical line down to the x-axis.
From the ratio a/G at the point of intersection with the x-axis, one can determine the minimum required grid spacing. Similarly,
one can also determine the smallest sized hot spot of a given shape that can be detected for a given grid spacing and probability
of detection by calculating a from the ratio a/G and grid spacing G. Alternatively, one can use the computer program
ELIPGRID-PC.
7.4 Oriented Elliptical Target—If the orientation of the elliptical target with respect to the grid lines is specified, then the
probability of detecting the target must be determined using the computer program ELIPGRID-PC.
8. Comparing the Relative Efficiencies of Search Patterns
8.1 The efficiency of a search pattern is measured as the probability that a target (for example, hot spot) will be hit at least once.
Given the same sampling density, a sampli
...
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