A Discrete Global Grid for 

Photointerpretation

Joseph M. McCollum, Jamie K. Cochran, and 

Anita K. Rose

United States

Department of

Agriculture

Forest Service

Southern 

Research Station

Research Paper 

SRS–44
Authors:

Joseph M. McCollum, IT Specialist, Jamie K. Cochran, Supervisory 

Forester, and Anita K. Rose, Ecologist, Forest Inventory and Analysis 

Southern Research Station, Forest Service, U.S. Department of 

Agriculture, Knoxville, TN 37919.

July 2008

Southern Research Station

200 W.T. Weaver Blvd.

Asheville, NC 28804
For P1 classification, photointerpreters have historically 

counted dots or clusters of dots on unrectified photos to 

obtain area estimates. Photointerpreters have called each dot 

forested land, nonforested land, census water, or noncensus 

water. Typically 25 dots or clusters of dots have been done 

per plot. This method has several weaknesses. At times, 

dots have not been across the entire landscape but have been 

concentrated near plots. Second, photointerpreters have 

written their results on paper; and these results have been 

transcribed by keypunching, so transcription errors have 

been possible. In addition, dots printed on the photograph 

could obscure the landscape and transparent overlays are not 

necessarily reproducible.

The objective of this study was to create an improved 

method for forest area estimation. The new method was to 

make use of a grid that conforms with the current P2 grid, 

with dot counts distributed evenly across the landscape, 

and was to be capable of being made operational quickly. 

The improved methodology we have developed includes 

a program to record the photointerpreters’ call, to allow 

photointerpreters to see what is under the dot, and to allow a 

photointerpreter to check another’s results. 

Background

Figure 1 gives an idea of how global grids are developed. 

The base map data was obtained from the Global Resource 

Information Database of the United Nations Environment 

Programme (Global Resource Information Database-Geneva 

1992) and projected onto an icosahedron via the gnomonic 

projection by the authors. Note that each face of the 

icosahedron has a different aspect. 

An icosahedron is a solid having 20 faces that are shown in 

the figure as large equilateral triangles. Faces are outlined 

with wide white lines. Each triangle may be divided into 

nine smaller triangles, shown with thin black lines. Sets 

of six such triangles may be grouped to form a “mother 

hexagon” shown in medium black lines. This method also 

produces pentagons in 12 sets of 5 triangles. 

Snyder (1992) showed how to project the earth onto any of 

the platonic solids, but specifically onto an icosahedron.

Abstract

The Forest Inventory and Analysis (FIA) Program of the Forest Service, 

U.S. Department of Agriculture, collects its data in three phases. The first 

phase is collection of photointerpretation data or dot counts, the second 

phase is field collection of FIA plot data, and the third phase is collection 

of Forest Health Monitoring data. This paper describes the development of 

the Phase 2 (P2) and Phase 3 (P3) grids and discusses the creation of a new 

grid for Phase 1 (P1), complete with an efficient indexing scheme, which is 

essentially equivalent to the historical dot count grids. The P2 grid consists 

of one field site per approximately 6,000 acres. To create the new P1 grid, 

we decomposed the P2 grid by a factor of 27 to obtain new proposed 

P1 photointerpretation cells of about 220 acres. The new grid can be used 

for initial photointerpretation points to determine area estimates 

forested land.

Keywords: FIA, Forest Health Monitoring, global grids, hexagon, Phase 1, 

Phase 2, Phase 3, photointerpretation.

Introduction

The Forest Inventory and Analysis (FIA) Program of 

the Forest Service, U.S. Department of Agriculture, is 

responsible for inventorying forested land in the United 

States and its possessions. A three-phase sampling 

procedure is used in conducting inventories. Within the 

Southern Research Station, forest area estimation Phase 

1 (P1) has historically been accomplished by classifying 

aerial photographs on a forest/nonforest basis. In Phase 

2 (P2), a subsample of P1 points is visited to confirm the 

classification and estimate tree and stand-level attributes. In 

Phase 3 (P3), a subset of the P2 plots is visited to evaluate 

forest health. The chief end product is a report of the forest 

conditions in each State (Conner and others 2004). The 

results are often reported by survey unit, an FIA-defined 

group of contiguous counties within a State that usually 

have similar ecological characteristics. 

The hexagonal grid system FIA currently uses for sampling 

originated within the Environmental Protection Agency’s 

Environmental Monitoring and Assessment Program 

(Overton and others 1990). It was adopted by the Forest 

Health Monitoring (FHM) Program but geometrically 

translated, and was subsequently adopted by FIA. 

A Discrete Global Grid for 

Photointerpretation

Joseph M. McCollum, Jamie K. Cochran, 

and Anita K. Rose
2

Any hexagon may be decomposed into T = h2 + hk + k2

smaller hexagons, where h and k are any integers. Without 

loss of generality, it is possible to set h ≥ k ≥ 0 by reorienting 

axes. In virology, T is the “triangulation” number (Caspar 

and Klug 1962). Any product of triangulation numbers is 

also a triangulation number. In turn, the smaller hexagons 

may themselves be decomposed into smaller and smaller 

hexagons. The value for one iteration of T is the aperture, 

and the number of iterations of T is the resolution (Sahr and 

others 2003). A constant aperture size is preferred when 

defining the resolution, but mixed aperture sizes are possible 

in global grids. Common values of T include T = 3 (if h = 1, 

k = 1), which results in a hexagon consisting of one whole 

hexagon and six third-hexagons, as shown in figure 2A, and 

T = 4 (if h = 2, k = 0), which results in a hexagon consisting 

of one whole hexagon and six half-hexagons, as shown in 

figure 2B. 

When the FIA and FHM grids were distributed to FIA units 

around the country, they were distributed as polygon and 

point coverages. When projected into the cone-based Albers 

Equal Area projection, the cell size is indeed 5,937.2 acres, 

with a range of < 1 acre (Brand and others 2000). When 

projected into the plane-based Lambert Azimuthal Equal 

Area (LAEA) projection, the cell size is about 5,931 acres, 

with a range of about 40 acres, depending on the choice of 

projection point. When projected into the cylinder-based 

Cylindrical Equal Area projection, the cell size is again 

Today the projection is called the Icosahedral Snyder Equal 

Area (ISEA) projection (Carr and others 1997). It is not 

supported in widely used spatial software packages. 

The surface area (adapted from Pearson 1990) of an 

ellipsoid is:

where

RA = the authalic radius of the ellipsoid

a = the length of the semi-major axis of the ellipsoid

e = the eccentricity 

i = an index variable

The authalic radius of an ellipsoid is the radius of a 

sphere with the same surface area as the given ellipsoid. 

Equation (1) leads us to an estimate of 5.10 x 1014 m2 for 

the surface area of the earth, using either the Clarke 1866 

ellipsoid (a = 6378206.4 m, e = 0.0823) or the Geodetic 

Reference System 1980 ellipsoid (a = 6378137 m, e = 

0.0818), and consequently 1.70 x 1013 m2

 for the surface 

area of a mother hexagon. See ESRI (1991) for values of a

and e for the Clarke 1866 and Geodetic Reference System 

1980 ellipsoids. 

Figure 1—Earth projected on an icosahedron.

Figure 1—Earth projected on an icosahedron.

∑

∞

=

+

+ = = ⋅ − ⋅ 0

2

2 1

2 2 2 1 4 4 (1 ) ii

i

i

A S πR πa e e (1)
3

about 5,931 acres, with a slightly smaller range of about 

32 acres. 

For sufficiently densified arcs, all equal area projections 

should yield the same result. For the P2 and P3 grids, 

hexagons are defined as sets of six points. The paths that 

connect one point to the next are implied. Thus, spatial 

software packages infer a straight line in the current 

projection. If paths from one point to the next were defined 

rigorously, each equal area projection would yield the 

same result, although shape would be distorted from one 

projection to the next. 

Methods

Sahr and others (2003) list five design choices for 

constructing a global grid. The authors could not determine 

the design choices used in constructing the original FIA 

grid but desired a grid that would conform with it. First, the 

authors confirm that an icosahedron was used to construct 

the original grid. Second, the authors could not determine 

the orientation of the icosahedron. Third, they observe that if 

a mother hexagon is decomposed by a factor of 3, 11 times, 

and a factor of 4 once, the result is 2.40 x 108 m2 (5,930 

acres), the approximate size of a P2 hexagon. Fourth, they 

believe that the original projection method was ISEA, and 

since that was not available, that LAEA is the next best 

choice. Fifth, the authors have developed their own indexing 

system for the P2 grid (and consequently the P1 grid). 

While use of the cone-based Albers projection results in 

lower variation in the size of a hexagon, the authors found 

that the plane-based LAEA projection preserves the shape 

of the hexagons better. Under the Albers projection, segment 

length and interior angles had a higher variation than they 

did under the LAEA projection. A cursory analysis of 

the FIA grids showed that the standard deviation of the 

interior angles of the Albers grid is about 1o; the standard 

deviation of the interior angles of the LAEA grid is about 

0.1o. Moreover, White and others (1992) found that LAEA 

preserves geometry better than Albers.

Consequently, the authors started with the P2 grid and 

decomposed that one. Remember that cell sizes are not 

exactly equal under any supported projection, so it does not 

work to start with a hexagon of desired size and tessellate it 

over the plane. Beyond some distance, this grid would not 

conform with the P2 grid.

The authors thought about consistency with the past and 

considered a 25-fold decomposition (h = 5, k = 0), as shown 

in figure 3. 

Constructing this decomposition requires not just collection 

of hexagon centers and vertices but also calculation of 

several intermediate points. Note that there is a point that 

is 3/5 the distance from the center to each of the vertices. 

Note also that for each segment of the hexagon, there are 

two additional points that are 2/5 and 4/5 the distance from 

the center to each of the segment midpoints. Note also that 

there are two points on each segment that are 1/5 and 4/5 the 

distance from one vertex to the next. Thus, there are a total 

of 19 points on the interior of the hexagon, and 12 other 

points are shared with one other hexagon (they count as onehalf points), for a total of 25. 

Figure 2—Simple decompositions of a hexagon.

(A) (B)

Figure 3—Twenty-five fold decomposition of a hexagon.
4

is the r2-axis, coincident with the x-axis, and the p2-axis, 

running from south-southwest to north-northeast. 

If the cells of Resolution i are numbered in rows and paths 

as (ri,pi), then one way of relating Resolution 2 to Resolution 

1 is to convert (ri,pi) to rectangular coordinates, as shown in 

equations (2) through (5). 

where

x = the x-coordinate (east) of a point

y = the y-coordinate (north) of a point

q = the circumscribed radius of a Resolution 1 hexagon

pi

 = the path number of Resolution i

ri = the row number of Resolution i

Set x’s and y’s in equations (2) through (5) equal to each 

other to get a system of two equations, which is solved for r2

and p2, yielding:

Thus, it is possible to calculate the row and path of 

a Resolution 2 hexagon from its parent Resolution 1 

hexagon’s row and path. The reader can verify that for 

cells sharing centers, such as the Resolution 1 (1,1) and the 

Resolution 2 (0,3), that indeed r2 = r1 – p1 and p2 = r1 + 2p1. 

An adjustment must be made for cells not sharing centers.

 The program calculates the azimuth (starting from north) 

between the current point on the Resolution 2 grid and the 

corresponding Resolution 1 center. With this example, if the 

azimuth is near 30o, then add 1 to the path. If the azimuth 

is near 90o, then add 1 to the row. If the azimuth is near 

150o, then add 1 to the row and subtract 1 from the path. 

If the azimuth is near 210o, subtract 1 from the path. If the 

azimuth is near 270o, then subtract 1 from the row. Finally, 

if the azimuth is near 330o, then subtract 1 from the row and 

add 1 to the path. In the program the authors have written 

for this exercise, they define “near” as ± 30o, although in 

The only information that is required to construct a threefold decomposition are the hexagon centers and the vertices, 

and performing a Thiessen expansion on those points. To 

simplify programming tasks, the authors opted for a 27-fold 

decomposition of the P1 grid. 

Analysis of the grid is facilitated if cells are neatly 

numbered in rows and paths. Also, it is less work in the long 

run if the cells are numbered along the way rather than at the 

end of all iterations. Thus, a method of numbering the cells 

must be found. 

The method actually used by the authors is the simple one 

described by White and others (1992). It is shown in figure 

4. Ideally, one should start with the mother hexagon and 

number it as (0,0) for instance and derive new coordinates 

for each hexagon as it is created. Since this option was not 

available to the authors, they first painstakingly numbered 

the P2 cells in rows and paths. This process involved an 

iterative approach of guessing where centers would be, then 

finding which cells had no proposed center or more than 

one center. For the purposes of producing the P1 grid, the 

P2 grid is Resolution 1. The reader is also asked to imagine 

three sets of axes in figure 4. One set is the traditional eastwest x-axis and the north-south y-axis. Another set is the 

r1

-axis, running from southwest to northeast, and the p1-axis, 

which is coincident with the y-axis. The final set of axes 

Figure 4—Resolution 1 (top coordinate) to Resolution 2 (bottom 

coordinate).

2 1

3 x = ⋅ q ⋅r

2 1 1

3 y = 2 ⋅ q ⋅r1 + 3 ⋅ q ⋅ p13 y = ⋅ q ⋅r + 3 ⋅ q ⋅ p

2 2

1

2 x = q ⋅r + ⋅ q ⋅ p 2 2

1

2 x = q ⋅r + ⋅ q ⋅ p

2 2

1 y = ⋅ 3 ⋅ q ⋅ p

(2)

(3)

(4)

(5)

2 1 p1 r = r −

2 1 2 p1 p = r +

(6)

(7)
5

reality, it is likely no more than a few degrees under the 

Albers projection and much less than that under the LAEA 

projection. However, the authors notice that the deviation 

can grow much larger near the edge of the grid; thus when 

processing a State or survey unit, the authors included a 

large buffer of about 0.5o of latitude and longitude around 

the area of interest. 

Going from Resolution 2 to Resolution 3 is a little different. 

In figure 5, the Resolution 1 grid is shown in gray lines, 

the Resolution 2 grid is shown in thin black lines, and the 

Resolution 3 grid is shown in heavy lines and bold text.

In Resolution 3, the r3-axis is coincident with the r1-axis, 

and the p3

-axis is coincident with the p1-axis. Rows of 

hexagons are vertical columns, and paths of hexagons go 

from southwest to northeast. A method similar to the one 

used for figure 4 shows the equations to be:

Again the reader can verify these equations hold true for 

the cells that share centers, e.g., that the Resolution 3 

coordinates (0, -3) can be generated from the Resolution 2 

coordinates (1, -2). The angular adjustments are 30o out of 

phase from the previous resolution—occurring at 0o, 60o, 

120o, 180o, 240o, and 300o instead of 30o, 90o, 150o, 210o, 

270o, and 330o. 

Going from Resolution 3 to Resolution 4 is exactly like 

going from Resolution 1 to Resolution 2. At this point the 

authors stopped. However, some of FIA’s partners have 

expressed interest in a more intense photointerpretation. 

This method could be easily adapted to higher resolutions 

of aperture 3. However, the authors do not see a way of 

generalizing the method to any arbitrary aperture size. 

Rather, a strategy would have to be developed for each 

aperture size. The authors have shown the strategy for 

T = 25, for example.

Results and Discussion

The authors have produced P1 grids for each of the Southern 

States. The grids are seamless along State lines and survey 

unit lines even for Texas, which was processed one survey 

unit at a time. 

The authors’ vision is that photointerpreters can use a digital 

orthophotograph or other digital image along with the grids 

the authors have created and a computer program to record 

the results. 

In figure 6 we see an illustration of all three phases of 

FIA: the smallest cells, shown with fine black lines, are P1 

photointerpretation cells. There are 27 P1 hexagons per 

P2 hexagon—19 whole hexagons plus 12 half-hexagons 

plus 6 third-hexagons. The medium-sized cells, shown in 

heavy dark gray lines, are P2 FIA cells. A cell of any phase 

belongs to the State where its center is. The large cells 

shown in heavy gray lines are P3 cells. P3 cells come in 

two types: one shown in figure 6A, called the C16, consists 

of 13 whole P2 cells and one-half of 6 other P2 cells, for a 

total of 16. This configuration was used in States that did 

not have FHM before 1999. The other, which is called the 

C9, is shown in figure 6B and consists of seven whole P2 

cells and one-third of six other P2 cells for a total of nine. 

This configuration was used in States where FHM data were 

collected FHM before 1999. See McCollum and Cochran 

(2005) for details. All P2 cells have a FIA plot, and every 

C16 cell has a FHM plot, but only 5/9 of the C9

 cells contain 

a FHM plot. This strategy results in a FHM to FIA ratio of 

1-to-16.2; note that 1/9 x 5/9 = 5/81 = 1/16.2. 

Figure 5—Resolution 2 (bottom coordinate) to Resolution 3 (top 

coordinate).

3 2 2 r = 2r + p

3 2 2 p = p − r

(8)

(9)
6

If the horizontal line in figure 6 is a State boundary, then 

the gray areas in figure 6 represent P1 cells that will not 

necessarily “belong” to corresponding P2 cells unless cells 

are traded between States, cells in partial hexes are ignored, 

or a supplemental grid assigning P1 cells to P2 cells is 

created. The same problem arises if remote sensing is used 

and pixels are used instead of dots. 

If the horizontal line in figure 6 is a county boundary, 

other questions arise. Some analysts feel that cells in one 

county should not influence the data in another county. 

Others feel that all P2 plots should carry the same number 

of pixels if remote sensing is used, or the same number of 

photointerpretation dots if dot counting methods are used.

The authors have found a method to estimate forest area 

that satisfies a number of goals: the grid completely covers 

the landscape and maximizes the distance from one dot to 

the next for a fixed number of dots; placement of dots is 

repeatable, easily lends itself to spatial analysis, and does 

not require the intense investment in remote sensing, in

training of analysts, or acquisition of imagery or software.

LITERATURE CITED

Brand, G.; Nelson, M.D.; Wendt, D.G.; Nimerfro, K.M. 2000. The 

hexagon/panel system for selecting FIA plots under an annual inventory. 

In: McRoberts, Ronald E.; Reams, Gregory A.; Van Deusen, Paul C., eds. 

Proceedings of the first annual forest inventory and analysis symposium. 

Gen. Tech. Rep. NC–213. St. Paul, MN: U.S. Department of Agriculture 

Forest Service, North Central Research Station: 8–13.

Carr, D.; Kahn, R.; Sahr, K.; Olsen, T. 1997. ISEA discrete global grids. 

Statistical Computing and Statistical Graphics Newsletter. 8(2/3): 31–39.

Caspar, D.; Klug, A. 1962. Physical principles in the construction of 

regular viruses. Cold Spring Harbor Symposium on Quantitative Biology. 

27: 1–24. 

Conner, R.C.; Adams, T.; Butler, B. [and others]. 2004. The state of 

South Carolina’s forests, 2001. Resour. Bull. SRS–96. Asheville, NC: U.S. 

Department of Agriculture Forest Service, Southern Research Station. 67 p.

ESRI. 1991. Map projections and coordinate management. Redlands, CA: 

Environmental Systems Research Institute, Inc. 86 p.

Figure 6—All three phases of FIA.

(A) (B)
7

Pearson, F., II. 1990. Map projections: theory and applications. Boca 

Raton, FL: CRC Press, Inc. 372 p.

Sahr, K.; White, D.; Kimerling, A.J. 2003. Geodesic discrete global 

grid systems. Cartography and Geographic Information Science. 30(2): 

121–134.

Snyder, J.P. 1992. An equal-area map projection for polyhedral globes. 

Cartographica. 29(1): 10–21.

White, D.; Kimerling, A.J.; Overton, W.S. 1992. Cartographic and 

geometric components of a global sampling design for environmental 

monitoring. Cartography and Geographic Information Systems. 19(1): 5–22

Global Resource Information Database-Geneva, ed. 1992. World 

Databank II. Arc/Info export file. www.grida.no/db/gis/prod/html/toc.htm. 

[Date accessed: October 7, 2004].

McCollum, J.; Cochran, J. 2005. Forest Inventory and Analysis and 

Forest Health Monitoring: piecing the quilt. In: McRoberts, Ronald E.; 

Reams, Gregory, A.; Van Deusen, Paul C., eds. Proceedings of the fifth 

annual forest inventory and analysis symposium. Gen. Tech. Rep. WO–69. 

Washington, DC: U.S Department of Agriculture Forest Service: 182–189.

Overton, W.S.; White, D.; Stevens, D.S. 1990. Design report for EMAP 

environmental monitoring and assessment program. EPA/600/3-91/053. 

Washington, DC: U.S. Environmental Protection Agency, Environmental 

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McCollum, Joseph M.; Cochran, Jamie K.; Rose, Anita K. 2008. Photointerpretation 

on a discrete global grid. Res. Pap. SRS–44. Asheville, NC: U.S. Department of 

Agriculture Forest Service, Southern Research Station. 7 p.

The Forest Inventory and Analysis (FIA) Program of the Forest Service, U.S. 

Department of Agriculture, collects its data in three phases. The first phase is collection 

of photointerpretation data or dot counts, the second phase is field collection of FIA 

plot data, and the third phase is collection of Forest Health Monitoring data. This paper 

describes the development of the Phase 2 (P2) and Phase 3 (P3) grids and discusses the 

creation of a new grid for Phase 1 (P1), complete with an efficient indexing scheme, 

which is essentially equivalent to the historical dot count grids. The P2 grid consists of 

one field site per approximately 6,000 acres. To create the new P1 grid, we decomposed 

the P2 grid by a factor of 27 to obtain new proposed P1 photointerpretation cells of 

about 220 acres. The new grid can be used for initial photointerpretation points to 

determine area estimates for forested land.

Keywords: FIA, Forest Health Monitoring, global grids, hexagon, Phase 1, Phase 2, 

Phase 3, photointerpretation.
The Forest Service, United States Department of 

Agriculture (USDA), is dedicated to the principle 

of multiple use management of the Nation’s forest 

resources for sustained yields of wood, water, forage, 

wildlife, and recreation. Through forestry research, cooperation 

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National Forests and National Grasslands, it strives—as directed 

by Congress—to provide increasingly greater service to a 

growing Nation.

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