Research Article

ASPHAA: A GIS-Based Algorithm toCalculate Cell Area on a Latitude-Longitude (Geographic) Regular Grid

Monia SantiniDepartment of Forest Environmentand ResourcesUniversity of TusciaViterbo, Italy

Andrea TaramelliISPRA – High Institute forEnvironmental ResearchRoma, Italy

Alessandro SorichettaDipartimento di Scienze della Terra“Ardito Desio”Università degli Studi di MilanoMilano, Italy

AbstractOne characteristic of a Geographic Information System (GIS) is that it addresses thenecessity to handle a large amount of data at multiple scales. Lands span over anarea greater than 15 million km2 all over the globe and information types are highlyvariable. In addition, multi-scale analyses involve both spatial and temporal inte-gration of datasets deriving from different sources. The currently worldwide usedsystem of latitude and longitude coordinates could avoid limitations in data use dueto biases and approximations. In this article a fast and reliable algorithm imple-mented in Arc Macro Language (AML) is presented to provide an automatic com-putation of the surface area of the cells in a regularly spaced longitude-latitude(geographic) grid at different resolutions. The approach is based on the well-knownapproximation of the spheroidal Earth’s surface to the authalic (i.e. equal-area)sphere. After verifying the algorithm’s strength by comparison with a numericalsolution for the reference spheroidal model, specific case studies are introduced toevaluate the differences when switching from geographic to projected coordinatesystems. This is done at different resolutions and using different formulations tocalculate cell areas. Even if the percentage differences are low, they become relevantwhen reported in absolute terms (hectares).tgis_1200 351..378

Address for correspondence: Andrea Taramelli, ISPRA Institute for Environmental Protection andResearch, via di Casalotti, 300, 0166, Roma, Italy. E-mail: andrea.taramelli@isprambiente.it

Transactions in GIS, 2010, 14(3): 351–377

© 2010 Blackwell Publishing Ltddoi: 10.1111/j.1467-9671.2010.01200.x

1 Introduction

Global lands span over an area greater than 15 million km2 and information typesare highly variable. A variety of protocols and conventions have been established inorder to deal with comprehensive aspects of the environment (Mitchell 2003). Anincreasing number of anthropogenic and biophysical phenomena occurring at multiplespatial and temporal scales, such as land cover/use transformations or climate changes,as well as their assessment for atmospheric studies or hazard-vulnerability-risk analy-ses, are treated as an integrated system since years (United Nations 1951, Gringorten1972, Henderson-Sellers et al. 1986, Kimerling et al. 1999, McCallum et al. 2006). Tothis regard the Geographic Information System (GIS) is very helpful as it bothaddresses the necessity to handle a large amount of data at multiple scales and allowsthe spatio-temporal integration of datasets deriving from different sources. Modeling,in particular, benefits from GIS for either assessment studies or future scenarios aboutenvironmental problems, useful in planning prevention, mitigation and adaptationstrategies.

As an example, the application of numerical prediction models to strong weatherphenomena such as dust and sand storms is considered of key importance in developingcontrol/mitigation measures (Engelstaedter et al. 2006, Washington et al. 2006, Bachet al. 2007) in regions far from the source areas. These and many others models, forexample the General or Regional Circulation Models (Pasqui et al. 2004, 2005, Fox-Rabinovitz et al. 2006) or the Dynamic Global Vegetation Models (DGVMs; e.g. Sitchet al. 2003, Crevoisier et al. 2007) run on latitude-longitude (geographic) grids overwhich the areal extent of model inputs and parameters needs to be carefully estimated toprovide more reliable results.

In other cases, principally for hazard/risk mapping and predictions at scales fromglobal to regional, it is reasonable to work on a geographic basis given the size ofinvolved areas (Dilley et al. 2005). An accurate computation of the cell area on ageographic grid-based map is necessary to appropriately assess the extent of criticalconditions up to detecting hazard/risk hot-spot distribution (Dilley et al. 2005, Arnoldet al. 2006).

All these studies have a common baseline: interpolating the raw observations, thepost-processed data and the model inputs and outputs into a shared grid-structure(Snyder 1926, Meshcheryakov 1962, Kimerling et al. 1999, Sanyal and Lu 2004,Yuan 2005). Using geographic coordinate systems, many area-based socio-economicand environmental indicators represented in grid format (e.g. population density,biomass consumption, crop yield, pollutant dispersal) can be evaluated for globalanalyses with higher accuracy. The issue of transforming the coordinate systemof a grid, whose reference change and planimetric projections are critical steps assource of errors and approximations, suggests that geographic coordinates should bepreferred to calculate area, even in place of equal-area projections (Yetman et al.2000, Small and Cohen 2004, Hay et al. 2006, McCallum et al. 2006, Elvidge et al.2007).

To this effort a global geographic grid is useful for partitioning the Earth into ageometrically regular set of cells (Tobler and Chen 1986, Bonham-Carter 1994, Bur-rough 1997, 2001), after selecting the most suitable Earth’s shape representation.Although the geoid (i.e. the irregular surface perpendicular at each point to the force ofgravity) is the solid better approximating the elevation contours of the globe, most GIS

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databases, used for cartographic-planimetric purposes, rely on ellipsoidal (regular)models of the Earth. In particular, such ellipsoids are simplified to oblate spheroids,whose two horizontal axes have the same length and are longer than the third verticalaxis (Thomas 1952). The reference surface of the spheroid, known as the “Datum”,varies to best fit the geoid at different Earth’s locations and it is characterized by a fixedinitial point (origin), the distance between the geoid and the Ellipsoid at such a point andthe initial azimuth on the surface.

A further simplification allows approximating the Earth’s spheroid to the relatedauthalic sphere, being a perfect sphere having the same surface area as the Earth’sspheroid (Kimerling et al. 1999). Each portion of the spheroid surface extending betweentwo meridians and two parallels maintains the surface area in its corresponding “pro-jected image” on the authalic sphere. The possibility to choose between spherical orspheroidal models introduces a key issue.

Latitude coordinates provided by Global Position Systems (GPS) and stored intogeographic databases are usually defined “geodetic” (or “geographic”), meaning that thelatitude angle from the equator is measured from the intersection point between theequatorial plane and the line crossing normally the spheroid surface at the consideredpoint. The so-called “geocentric” latitude is instead measured from the center of theEarth and is more suited for sphere-based approaches. For each geodetic latitude on thespheroid surface the authalic latitude on the simplified spherical surface can be com-puted. Hereafter the spheroid model will be indicated as SPHD while the related authalicsphere as ASPH.

The ASPH can be partitioned into a geographic regular grid (Figure 1a), partiallymatching the criteria reported in Kimerling et al. (1999), whose cells cover the entireglobe without overlapping, and with the same topology and shape. Each portion of theEarth’s surface can be represented by cells having the same angular dimensions along theNS and EW directions (i.e. spherical quadrangles or triangles where meridians convergeto the poles).

Equally-spaced latitude degree intervals of the regular geographic grid represent therows of the grid, while the equally-spaced longitude intervals represent the columns ofthe grid. In this representation the width of the longitude is proportional to the cosine ofthe latitude, while the length of a given angle of latitude remains constant (equal to theauthalic sphere radius multiplied by the cell dimensions in radians; Figure 1b). Thisrepresents a well-known issue to solve when using a latitude-longitude geographic grid,as grid cells decrease in area towards the poles.

Although it was widely demonstrated that such a geographic grid provides nearlyerror-free conditions for area estimation (Snyder 1926, Perillo et al. 1999), the compu-tational procedure to define the area of the grid cells is crucial, as the units of measureof cell size and cell surface area are not the same.

In this article a fast and reliable algorithm implemented in Arc Macro Language(AML; ©ESRI, Redlands, California) syntax is presented to provide the automaticcomputation of the cell area in a regularly spaced geographic grid at multiple spatialscales. Such an approach is based on the above mentioned approximation of spheroidEarth’s surface to the authalic sphere.

In the next sections the algorithm is described, tested and applied to key case studiesin order to highlight how areas estimated from grids in Projected Coordinate Systems(PCSs) suffer from biases in comparison with the ones computed from grids in Geo-graphic Coordinate Systems (GCSs), and that such deviations differ according to latitude,

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a

b

Figure 1 (a) Example of the partitioning of a sphere surface into a grid with a cell sizedimension equal to 10° along both the X (longitude) and the Y (latitude) dimensions.Where meridians converge to the poles, the cells are spherical triangles; and (b) on ageographic grid meridians converge toward the poles, whereas the parallel inter-distance remains constant (modified from http://members.shaw.ca/quadibloc/maps/mcy0105.htm)

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cell resolution and projection scheme. Finally the differences between ASPHAA and adifferent cell area computation approach are evaluated.

2 Algorithm Description

The surface area S of a generic cell of the aforementioned grid, comprised between l1 andl2 degrees of longitude and between j1 and j2 degrees of (authalic) latitude (i.e. the greencell in Figure 2) can be calculated as (Tobler and Chen 1986):

S R R d d R d d R=⎛

⎝⎜⎞

⎠⎟= = −(∫∫ ∫ ∫cos cosϕ λ ϕ ϕ ϕ λ λ λ

λ

λ

ϕ

ϕ

ϕ

ϕ

λ

λ

1

2

1

2

1

2

1

2

2 22 1)) ∫ cosϕ ϕ

ϕ

ϕ

d1

2

(1)

giving:

S R= −( ) −( )22 1 2 1λ λ ϕ ϕsin sin (2)

where

0 � l1 � l � l2 � 2p-p/2 � j1 � j � j2 � p/2R = the (authalic) radius of the spherej1, j2 and l1, l2 are expressed in radians (p radians = 180 degrees).

Figure 2 Example of a generic cell of the grid comprised between l1 and l2 degrees oflongitude and between j1 and j2 degrees of (authalic) latitude. l and j are genericcoordinates of a point belonging to the cell, whereas R is the (authalic) Earth’s radius(modified from http://badc.nerc.ac.uk/help/coordinates/cell-surf-area.html)

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Based on the above assumption it is possible to verify that all the cells belongingto the same row of the grid (i.e. comprised in the same interval of latitude) have thesame surface area: the three factors of Equation (2) (R2, (sin j2 - sin j1) and (l2 - l1),respectively) are constant. Thus, the grid area computation can be simplified as it onlyrequires the evaluation of the cell surface associated to a single column of the grid,being that such an estimate is valid for all the other columns. Within one grid column,the surface area for each cell can be calculated starting from its upper and lower(authalic) latitudes. The whole procedure is better described in the following throughan explicative example.

The example considers a grid with five rows and seven columns (Figure 3a), thevariable NROW indicates the row numbering on the grid (from top to bottom), NROWSis the total number of rows (i.e. 5), CS is the cell size (in radians), YLL is the (authalic)latitude coordinate of the lower left corner of the grid (in radians), and R is the (authalic)Earth’s radius in m. When using GIS databases, before applying Equation (2) it is often

a

b

Figure 3 (a) This figure represents an equal angle grid having the same X and Yspacing (CS) in angular dimensions. For each row, the coordinates of the bottom edgeare computed from the coordinates of the bottom of the grid (YLL) plus the value(NROWS - NROW) times the cellsize (CS). The coordinates of the top edge are com-puted by YLL + (NROWS - NROW + 1) * CS; and (b) the $$rowmap is a built-in grid inArcInfo that can be used to identify row locations of a grid cell. The $$rowmap gridassigns 0 to all the cells in the first (upper) row, 1 to the cells in the second row, andso forth

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necessary to convert the geodetic coordinates stored by the grid reference system intoauthalic coordinates.

The cell area for the lowest row of the grid (the dark grey one in Figure 3a),considering Equation (2), is:

R R CS YLL CS YLL⋅ ⋅ ⋅ + ⋅( )( ) − ( )[ ]sin sin1 (3)

while the cell area for the next row (the light grey one) of the grid, still from Equation (2),is:

R R CS YLL CS YLL CS⋅ ⋅ ⋅ + ⋅( )( ) − + ⋅( )( )[ ]sin sin2 1 (4)

Such a method is coded in an AML script (Appendix A1) in order to derive, from ageneric latitude-longitude grid supplied in GRID format (ESRI 1990), a grid storing ineach cell the area for that same cell in square linear units (m2). The procedure can bedivided into six steps as follows.

1. The pi-Greco (p) value is fixed to 3.14159265358979, the CS and the GeodeticLower Left (GLL) corner coordinates are derived from the grid metadata andconverted into radians and the total number of rows (NROWS) is stored

2. The native geodetic Datum related to the dataset is selected among 24 coordinatesystems. For each one, the semi-major axis (SMA) and the flattening (FL) of thespheroid are considered

3. From FL the eccentricity e is computed as

e FL FL= ⋅( ) − ( )2 2 (5)

4. A further grid named ROW is created, spatially matching the input grid and keeping,for each cell, the value of the row occupied by that same cell (counting the rows fromtop to bottom). This is achieved by computing

ROW = +$$rowmap 1 (6)

where $$rowmap is an ArcInfo built-in grid that can be used to identify the rownumbering, but as it starts from 0 (Figure 3b), one unit is added to the whole grid inorder to get the right numbering starting from 1.

5. Recalling that cells belonging to the same row also have the same surface area, theauthalic latitudes for the lower and upper bound of each row (j1 and j2, respectively)are computed from (Snyder 1926):

ϕ1 1= ( )arcsin q q (7)

ϕ2 2= ( )arcsin q q (8)

where

q ee e

ee

12

2

111 2

12

11

1

1

= −( )⋅− ⋅

−⎡⎣⎢

⎤⎦⎥⋅

− ⋅( )+ ⋅

sinsin

lnsins

ϕϕ

ϕg

g

g

iinϕg1( )⎡⎣⎢

⎤⎦⎥

(9)

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q ee e

ee

22 2

22

211 2

12

11

= −( )⋅− ⋅

−⎡⎣⎢

⎤⎦⎥⋅

− ⋅( )+ ⋅

sinsin

lnsins

ϕϕ

ϕg

g

g

iinϕg2( )⎡⎣⎢

⎤⎦⎥

(10)

q ee e

e

e= −( )⋅

( )− ⋅

−⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥⋅

− ⋅ ( )( )+

1 21

12

12

1

22

sin

sinln

sinπ

π

π

⋅⋅ ( )( )⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥sin

π2

(11)

where jg1 and jg2 are lower and upper geodetic latitudes, respectively. In the case ofthe lowest row of the grid, jg1 ≡ GLL.

6. From SMA and q the authalic radius R is computed as (Snyder 1926):

R SMAq= ⋅2

2(12)

Now equation (2) can be applied.

According to the ArcInfo specifications, the upper limit of the grid size to run thisalgorithm is assumed to be 4,000,000 of rows and 4,000,000 of columns. This wouldmean a grid working at ca. 0.324″ lon ¥ 0.162″ lat cell resolution (ca. few meters) for thewhole globe, which would be computationally a very hard challenge for global applica-tions. However if the angular spacing (resolution) of grid cells is so large that areascomputed in m2 overcome the 32-bit limit of representation allowed for grids in ArcInfo,the computation can be also made in km2 by converting the SMA values from m to km.

3 Algorithm Testing

Even if the reliability of the ASPH in approximating the SPHD surface areas is widelyassessed (e.g. Tobler et al. 1995, Qihe et al. 1999), the amount of biases in estimating thecell surface area as a function of the grid resolution and latitude did not receive greatattention. Before testing the ASPHAA, a comparison was made between its results and anumerical solution of the grid-cell area taking as reference the source SPHD. A Fortran90 routine for the computation of the cell area on a regular (geographic) grid partitioninga SPHD into cells is implemented. The aim of this procedure was: (1) defining theperformances of the algorithm for different cell resolutions and latitudes in order toassess its accuracy when used for different study areas and scales; and (2) as the algorithmis working on ArcInfo format, which is able to store grid values with a 32-bit (float)precision, the biases in case of large cells (i.e. billion of sq. meters) have to be taken intoaccount for uncertainty evaluation.

As it was already highlighted for the ASPH, also for the SPHD all the columns of theregular grid have the same surface area. There is also a perfect symmetry between upper(north) and lower (south) semi-SPHDs. For this reason and in order to meet computa-tional effort requirements, only a single longitude column lying from 0° to 90°N latitude(semi-lune) was considered (Figure 4a).

Whatever is the chosen angular grid resolution, such a numerical procedure divideseach SPHD’s cell into further equal angle cells. Aiming to a precision of 1 m2 for area

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computation, it was first acknowledged that a 1″ ¥ 1″ cell represents a spherical trape-zium from 0° to 89°59′59″N latitude and a spherical triangle from 89°59′59″ to90°00′00″ N latitude (the northern side is one point, i.e. the pole; Figure 4b). At similarscales (1″ ¥ 1″ is well representing a 30 m ¥ 30 m square at the equator) it is possibleconsidering a purely 2D cell for which the biases from the real Earth’s curvature can beneglected. Thus the chosen numerical solution uses as minimum partitioning unit 1″ ¥ 1″cells (arc-cells hereafter).

The routine reads the grid resolution, the lower left corner geodetic latitude andlongitude and, for each cell along the single grid column, it first calculates the area foreach arc-cell it contains and then for the cell itself. The algorithm’s pseudo-code isreported in Appendix A2.

Seven grid resolutions were selected: 0.1, 0.25, 0.5, 1, 2.5, 5 and 10°, whose cellscontain, respectively, 360 ¥ 360, 900 ¥ 900, 1,800 ¥ 1,800, 3,600 ¥ 3,600, 9,000 ¥9,000, 18,000 ¥ 18,000 and 36,000 ¥ 36,000 arc-cells.

For each resolution the total surfaces both of the “numeric” SPHD and the ASPHwere calculated by multiplying the semi-lune area by the appropriate factor (e.g. 360 ¥2 in the case of 1° of resolution, 720 ¥ 2 for 0.5°, etc.). Both calculations show a goodagreement (deviations less than 10-5–10-6%) from the SPHD area analytically calculatedas:

A SMASMI

SMISMA

= ⋅ + ( )

( )( )⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

⎧

⎨⎪

⎩⎪

⎫

⎬⎪

⎭⎪

⋅2 22

πsin arccos

ln11 + ( )( )⎡

⎣⎢⎤⎦⎥

( )⎧

⎨⎪

⎩⎪

⎫

⎬⎪

⎭⎪

sin arccosSMISMA

SMISMA

(13)

where SMA and SMI are semi-major and the semi-minor axes, respectively.Then, taking the SPHD model as reference, for each resolution the mean Absolute

Percent Deviation (APD) and Root Mean Square Error (RMSE) of ASPH were computed.Graphs in Figure 5 report exponential curves that fit the relationships (blue points) of thegrid resolution with APD (R2 = 0.989) and RMSE (R2 = 0.988). In the APD case it is

ba

Figure 4 (a) In grey an example of semi-lune on a sphere; and (b) example of sphericaltriangle

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noteworthy that when the angular cell spacing increases the biases among the two –ASPH-based and SPHD-based – algorithms decrease. For the RMSE, as expected, thebehavior is inverted: the minor percent deviation at coarser resolutions corresponds to amuch larger reference area. Absolute ASPHAA deviations appear raised when latitudeincreases (not showed here), this being more evident at finer resolutions, and suggestingthat resolution- and latitude-related correction factors could be introduced to improveASPHAA evaluations.

It is easy to extrapolate that in the case of a resolution finer that 0.1° the abovetrends are respected. A concrete example is reported in the Global Demography Projectmade for the year 1995 by Tobler et al. (1995). Starting from the grid storing the peopleliving in each cell at 5′ ¥ 5′ (ca. 0.083° ¥ 0.083°) of resolution, the population density gridwas calculated by dividing the abovementioned population count grid by the cell areagrid, and considering an authalic spherical surface with a radius of 6,731,007.178 m.This result was compared with the one derived dividing the same population count gridby the cell-area grid calculated with ASPHAA. The maximum difference consisted of oneinhabitant and occurred in 0.23% of cells. In Figure 5 the APD and RMSE for the sphere-and spheroid-based algorithms are reported (red points). When included in the fittingcurves previously computed for APD and RMSE, their R2 decreases to 0.896 and 0.927,respectively. It is interesting to note that authalic latitudes were not used in the case ofsphere-based computations of Tobler et al. (1995).

4 ASPHAA Applications

In this section several application examples are shown in order to assess when and howthe GIS-based ASPHAA could be useful within different research topics, overcoming thelimits of both computing grid cell area by other procedures and using databases inprojected coordinate systems. In particular, algorithm evaluation is given in two direc-tions: first, there is an assessment of its performance at different spatial scales (global,continental and regional) and second, the influence of area calculation procedures on the

Figure 5 In blue are reported the APD (left) and RMSE (right) of ASPHAA algorithm fromSPHD-based numerical solution of the cell area for different angular cell resolutions. Thered point represents the same evaluations made in the work of Tobler et al. (1995). Thefitted line refers only to the blue points

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distribution assessment of biophysical and socio-economic factors, considered bothseparately and coupled, and commonly used for risk analyses, is evaluated. The aim is tohighlight the impact on area calculation due to the data source type and/or aggregation.

4.1 Global Applications

4.1.1 Crop production

In the context of global change studies, modifications in land use pressures were recentlyrecognized to have a key role in greenhouse gas emission increase or reduction (McGuireet al. 2001, Albani et al. 2006). One of the main research topics is the tentative couplingbetween biophysical aspects of land use changes and their socio-economic drivers. Thishas been done, for example, by Ronneberger et al. (2009), who tried to integrate theglobal agricultural Kleines Land Use Model (KLUM; Ronneberger et al. 2005) and theGlobal Trade Analysis Project (GTAP) model (Hertel 1997). A useful dataset for similarevaluations was built by Monfreda et al. (2008), which combined the FAOSTAT (FAO2006a) and Agro-MAPS (FAO 2006b) databases with a number of national censuses andsurveys to create an extensive global dataset on crops for the year 2000. Each cell of theirgrid-based dataset contains the fractions of harvested area (hectares) and the yield(metric tons per harvested hectare) per year for 175 individual crops at a 5′ ¥ 5′ spatialresolution (http://www.geog.mcgill.ca/~nramankutty/Datasets/Datasets.html). In somecells a crop is harvested multiple times in a single year so that its surface is counted morethan once and its cell fractions can exceed one. While the crop harvested area could beuseful as an initial condition for most of land use change models, the crop yield attributeis probably most helpful for socio-economic evaluation driving land use demandsthrough market dynamics. Since such a spatial dataset was derived by merging censusstatistics from different sources and scales and implementing spatial analysis proceduresto interpolate data and to fill voids (Monfreda et al. 2008), a direct comparison betweenaggregate (administrative level) and spatialized statistics is not easy to accomplish.

The four globally most diffused crops, i.e. wheat, rice, maize and soybean (Monfredaet al. 2008), were chosen to compare ASPHAA-based areas with those resulting fromprojecting the grids with an equal-area global projection (Goode Homolosine, seeAppendix A3 for projection parameters), maintaining the source Datum (WGS 1984)and choosing a 10,000 m mesh size resolution, which is comparable with 5′ cell angulardimensions. On average, the projected grid-based area shows a bias (overestimation)compared to the ASPHAA reference area of about 0.2% (ca. 1,129,000 ha). The yield isaffected on the order of more than 3,800,000 tons being overestimated each year.Concentrating on the two most diffuse crops, i.e. wheat and rice, is it possible to observethat the former spans a much huger latitude interval (Figure 6a) than the latter(Figure 6b); this is the reason why it appears better when balancing the overestimationsand underestimations by ASPHAA, resulting in lower percent differences with censusdata.

Then, a test of the differences among areas (in decimal million of hectares) resultingfrom census databases and the ASPHAA algorithm is carried out using the aggregationof the 175 crops into 11 macro-categories as in Table 3 from Monfreda et al. (2008).Results (Figure 7) show that, on average, the ASPHAA algorithm underestimates the areafor 1%, and for each crop group the percentage increase with the total surface extentslightly following a logarithmic law (R2 = 0.68). In evaluating such differences one has to

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consider that biases can be due to the transformation of administrative unit boundarymaps from vector to grid and to the interpolation procedures that propagate the error ofmerging crops into macro-categories.

4.2 Risk Analysis

A further interesting research area, which integrates biophysical and human factors, isthe analysis of risk related to natural hazards. After the Indian Ocean Tsunami in 2004and the South-east Asia earthquakes and Central America hurricanes in 2005 (Arnoldet al. 2006), the Global Natural Disaster Risk Hotspots (GNDRH) project was launchedby the World Bank and Columbia University Center for International Earth Science

a

b

Figure 6 Map of harvested areas (m2) per year related to (a) wheat and (b) rice

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Information Network (CIESIN). Here a noticeable effort was made in order to producea comprehensive geodataset attempting to assess multi-hazard risk analysis at a global-scale and to identify the hot-spots, which are regions where the risk of natural disastersis particularly high or regions would impact other adjacent areas. Data are presented inthe form of gridded global maps with a 2.5′ ¥ 2.5′ spatial resolution (http://www.ldeo.columbia.edu/chrr/research/hotspots/coredata.html).

The risk level was evaluated by combining the natural hazard probability and thehistorical vulnerability for the considered area with the exposure of elements at risk(lives, properties, etc.). In particular the global risk assessment focused on two disasterconsequences, mortality and economic losses, for six major natural hazards: earth-quakes, volcanoes, landslides, floods, droughts and cyclones. In order to provide arelative representation of risk, absolute numbers (hazard frequency, mortality and eco-nomic losses) were converted into index values from 1 to 10 corresponding to deciles oftheir distribution. Finally a multi-hazard risk index was created by summing decilecategory values among all six natural hazards. This multi-hazard index reflects thenumber of hazards assumed relatively significant in a particular grid cell (Dilley et al.2005). Considering as attributes both the frequency of the six natural hazards and theirrisk in terms of mortality and economic losses, areas belonging to each attribute decileand for each type of hazard were calculated both with ASPHAA and after projectinggeographic grids into an equal-area world projection (i.e. Sinusoidal; see Appendix A3for projection parameters), maintaining the source Datum (i.e. WGS 1984) and choosinga 5,000 m mesh size resolution, comparable with 2.5′ cell angular dimensions. Percentdeviations of the projected- from the ASPHAA-based areas are reported in the graphs ofFigures 8a–c showing significant differences mainly in the evaluation of hazard conse-quences (economic losses and mortality), lower for floods, droughts and earthquakes andhigher for cyclones, landslides and volcanoes.

When considering multi-hazards indices, the difference in percent seemsto increase with the increase of the multi-hazard index itself for frequency and mor-tality, following an exponential law and having correlation coefficients equal to 0.78

Figure 7 Percent differences between areas (Mha) reported in Monfredaet al. (2008) and computed by ASPHAA after aggregating 175 crops into 11macro-categories

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a

b

c

Figure 8 Percent differences between areas computed using PCS and GCS for thedifferent risk deciles regarding hazard: (a) frequency; (b) mortality; and (c) economiclosses

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and 0.74, respectively. For economic losses, it is more difficult to detect a trend sincenumerous risk classes disappear when switching from geographic to a projected coor-dinate system.

4.3 Continental Applications

4.3.1 Dynamical global vegetation models

The first case at a continental scale considers a widely used Dynamical Global VegetationModel (DGVM), the Lund-Postdam-Jena (LPJ) scheme (Sitch et al. 2003). LPJ is aprocess-based model simulating the water and carbon exchanges between the biosphereand atmosphere by means of a given set of parameters and input variables. For each gridcell, vegetation is described in terms of the fractional coverage of 10 different PlantFunctional Types (PFTs) that are able to compete for space and resources. The PFTs mayin principle co-exist at any location, depending on plant competition and a set ofenvironmental constraints. Their relative proportion is determined by the competitionamong vegetation types with specific ecological strategies for dealing with temperature,water and light. Among numerous land classification systems, the MODerate resolutionImaging Spectroradiometer (MODIS) sensor data elaborations supplied by NASA (http://modis.gsfc.nasa.gov/data/dataprod/index.php), offer the land cover classification usefulto represent LPJ’s PFTs. As a proxy of a typical PFT partitioning reproduced through LPJsimulations, five maps of fractional cover (referring to Crop, C; Grassland and Wetlands,GW; Evergreen Broadleaved Forests, EBF; Evergreen Needleleaved Forests, ENF; andDeciduous Broadleaved Forests, DBF) were here used for Europe in a grid format and ata 0.25° ¥ 0.25° spatial resolution.

First for each pixel and then for the whole map, the total area (in m2) covered by thefive vegetation classes was computed, either using ASPHAA or the approach imple-mented in LPJ, which calculates the area for a pixel on a lat/lon grid as (http://www.pik-potsdam.de/research/cooperations/lpjweb):

S CS CS LATc= ⋅( )⋅ ⋅( )⋅ ( )111 000 111 000, , cos (14)

where LATc is the geodetic latitude of the cell center.Table 1 shows the differences in absolute area and percent between the two algo-

rithms. It is worth noting the cumulated underestimation of the LPJ approach for all the

Table 1 Absolute and percent differences between total surfaces computed by theprocedure implemented in LPJ and the ASPHAA algorithm

PFT ASPHAA (m2) LPJ’s algorithm (m2) Diff (km2) Diff (%)

C 4,819,400,523,776 4,788,854,980,608 -30,545.54 -0.6338DBF 1,329,023,811,584 1,319,833,960,448 -9,189.851 -0.69147EBF 65,319,903,232 64,964,202,496 -355.7007 -0.54455ENF 2,870,795,304,960 2,846,025,580,544 -24,769.72 -0.86282GW 1,428,607,336,448 1,420,176,785,408 -8,430.551 -0.59012TOT 10,513,146,880,000 10,439,855,509,504 -73,291.37 -0.69714

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considered PFTs. Though such differences are not higher than 1%, they are of the orderof 104–106 hectares in absolute terms.

Making a comparison between ASPHAA and the algorithm in LPJ at differentresolutions and at global scale as in Section 3, their percent differences remain around0.6%.

An interesting conclusion is that the LPJ method appears to switch from over- tounder-estimation when exceeding 15–20° latitude if compared with the SPHD andASPH calculations. This reinforces the hypothesis about the usefulness of introducinglatitude-based correction factors, especially to the most simplistic sphere-relatedapproaches.

4.3.2 The CORINE land cover dataset

The COordination foR Information oN the Environment (CORINE) project (http://www.eea.europa.eu/publications/COR0-landcover) provided first in 1990 and then in2000 the most detailed (at 1:100,000 scale) Pan-European land cover dataset, withaccuracy higher than 85% (Feranec et al. 2010). Working on a smaller coverage withrespect to global applications, the CORINE dataset for the year 2000 was here used totest the ASPHAA at finer resolutions. The dataset is available in a native vector format(http://www.eea.europa.eu/themes/landuse/clc-download) consisting of 100 km by100 km tiles having a Lambert Azimuthal Equal Area (LAEA) PCS (see Appendix A3 forprojection specification). All the tiles were first back-projected in GCS using the sameDatum (ETRS1989) as the LAEA, and they were then mosaiced into four longitudinalzones as shown in Figure 9.

Both projected and geographic versions of the vector maps for the four zones wererasterized, using a 500 m ¥ 500 m and a 0.005° ¥ 0.005° resolution for the former andthe latter, respectively. While for the projected grids the total land use surfaces werecomputed considering a cell area equal to 250,000 m2, to the four lat/lon grids theASPHAA was applied.

Table 2 shows the results in terms of percent differences (diff %) and absolutepercent differences (abs diff %) of aggregated land use areas calculated from PCS andGCS grids if compared with the reference surfaces calculated from source polygons.Either considering the whole European dataset or separately the four zones, theASPHAA-based area has a better correspondence with the reference area, in terms ofboth relative and absolute differences.

Results are influenced both by the accuracy of the source dataset and by thelevel of landscape fragmentation that causes biases during the vector to raster conver-sion, and for the ASPHAA-based areas also by the distance of the four zones from theorigin of the projection of the source dataset. The mean centers (i.e. the geographiccenters) for each land use polygon were first indentified, then their Euclidean distancesfrom the center of LAEA projection (lon = 10°E and lat = 52°N) were computed andaveraged when belonging to the same zone: as expected, in the case of absolute dif-ference for ASPHAA, the biases increase when increasing the averaged mean centerdistance.

4.4 Regional Application

The integration of grid-based biophysical parameters and high resolution demographicinformation at administrative level could be affected by the bias introduced through area

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Figure 9 Map showing the four longitudinal zones in which the CORINE dataset wasdivided. The black point represents the center of the LAEA projection (lon = 10°E; lat =52°N)

Table 2 Relative (diff %) and absolute (abs diff %) percent differences between totalgrid-based and vector-based (reference) surfaces

Region

Diff % Abs diff % Averaged Mean Centerdistance fromprojection origin (m)Projected ASPHAA Projected ASPHAA

Whole dataset 2.66 2.59 2.76 2.73 –Zone 1 8.44 8.02 8.50 8.32 1,421,053Zone 2 0.29 -0.03 0.58 0.53 618,126Zone 3 -1.63 -0.31 2.23 0.67 820,628Zone 4 -0.41 0.35 1.31 1.09 1,343,659

The averaged mean center distances represent the average distance of the mean centers for eachland use polygon across the whole domain and within each zone from the origin of projection (lon= 10°E, lat = 52°N)

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calculation in a regular projected grid at the regional (sub-continental) scale. As anexample of comparing and combining datasets from different sources and with differentdetail, a hurricane hazard/risk estimate in the Caribbean is here described for theHurricane Stan event affecting these regions on September 2005 (Pasch and Roberts2005). In this approach datasets about land use (http://bioval.jrc.ec.europa.eu/products/glc2000/products.php), rain (http://www.cpc.noaa.gov/products/janowiak/cmorph_description.html), wind speed (http://www.nhc.noaa.gov/pastall.shtml), topography(SRTM; http://seamless.usgs.gov/index.php) and bathymetry (GEBCO; http://www.gebco.net/) were processed and used for detecting areas vulnerable to storm surges,floods and high speed winds. Three multi-hazard severity classes (low, medium and high)were extracted by combining those representing the same severity classes referring to thesingle hazards, in order to assess the number of people affected by the considered event.

Results are shown in Table 3, where is it possible to see the differences in number ofpixels belonging to the three hazard severity classes, producing a total affected area (inkm2) diverging when using geographic or projected (UTM, see Appendix A3 for projec-tion parameters) coordinate systems, maintaining the source Datum (WGS 1984) andusing 0.0008333333° ¥ 0.0008333333° and 90 m ¥ 90 m resolutions, respectively.

The underestimation of the total affected area within the Caribbean by the projectedcoordinate system propagates the error to the final calculation of elements at risk in termsof affected population. Using population data at a 2.5′ ¥ 2.5′ angular resolution from theGridded Population of the World dataset (GPWv3; http://sedac.ciesin.columbia.edu/gpw/), the 1,418,634 affected people estimated using ASPHAA on the geographic grid isreduced to 1,010,210 affected people when projecting the grid using the UTM scheme forthe zone 17, maintaining the source Datum (i.e. WGS 1984) and choosing a 5,000 ¥5,000 m linear mesh size spacing. This high underestimation by the projected grid couldbe related to the fact that UTM projection is a conformal and not an equal-areaprojection.

5 Conclusions

With the advances of global position systems, remote sensing and GIS-computing tech-nologies, there is an increasing need for practical, reliable and fast tools for spatial dataanalysis and processing. Either different spatial scales and resolutions, or biophysical andsocio-economic data have to be considered in such evaluations. This allows a morein-depth analysis, through observation and modeling, of processes and phenomenaoccurring all over the globe, which are related to the dynamics of climate changes andeconomic development.

Table 3 Table shows the total area calculated with the two different coordinate systems(geographic and projected) within three classes of hazards related to Hurricane Stan

Hazard class Lat – Lon grid UTM projection

1 24,053,670,000 pixels 24,051,000,000 pixels2 8,026,730,000 pixels 7,993,510,000 pixels3 5,687,519,000 pixels 5,530,470,000 pixelsTotal affected area 37,767 km2 35,959 km2

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In this article, a well-developed algorithm, implemented into a GIS platform(ArcGIS9.x) and named ASPHAA, is proposed to calculate cell areas on gridded geo-datasets in geographic coordinate systems and assuming a spherical shape for the Earth.

Given several constraints due to byte representation for grid formats in ArcGIS andto the variety of resolutions that can be used under different research ambits, theanalytical algorithm was first compared with a numeric solution of the cell area in thecase of a more realistic spheroidal Earth’s model and considering seven different gridresolutions. Results confirm that:

1. the absolute percent difference does not exceed 0.01%;2. the root mean square error is less than 30 ha in the worst case (coarser resolution,

i.e. 10°); and3. there is an increase of biases at higher latitudes.

Then the algorithm was used at different spatial scales (from global to regional) andunder different topics (land use assessment and scenarios, natural hazard risk evaluation,vegetation modeling) highlighting several points as listed below.

Deviations from the true areas can be due first of all to vector-to-raster conversionperformed when working at large scales to meet computational requirements. Moreoverprojected grids, both using equal-area (e.g. Goode Homolosine, Sinusoidal, LAEA) orother schemes (e.g. the conformal UTM) can create biases that, even if less than 2% and4.5% at scales from global to regional, respectively, correspond to very large extents inabsolute terms, ranging from millions of hectares to hundred thousands of hectares atscales from global to regional. The increase of percent deviations can be due to severalreasons: aggregation from single to macro-categories in classifying data, the use ofnon-equal-area projections, the spread of the study area far from the projection point ofthe origin or the evaluations made by integrating data from different sources and withoriginally different spatial details. Also a recent study of Usery and Seong (2000) andYildirim and Kaya (2008) showed for example how, using projected coordinate systems,total areas for several attributes is affected by projection type, latitude and with gridresolution; on the contrary very high performances of ASPHAA at any latitude andresolutions are here demonstrated.

The comparison between ASPHAA and a simplified sphere-based algorithm withoutregard to authalic latitude conversion shows differences, whose sign is latitude-dependent, which need to be considered for evaluating the uncertainty in applicationswhere such algorithms are used.

Although this study suggests that several improvements can be introduced in thepresented algorithm, i.e. latitude- and resolution-based correction factors, others can beintuitively mentioned, as the accounting for a non-flat surface morphology of the cell,determining true areas certainly larger than from purely 2D estimations.

At this stage, ASPHAA proved to be an interesting tool to improve multi-thematicand integrated research efforts through performing a more reliable area computation inmulti-scale and multi-approach applications.

Appendix A1 – ASPHAA AML Code

/*january 28th, 2009/* command line: &r ASPHAA <grid name>

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&args grid&sv pi = 3.14159265358979&describe %grid%&sv NROWS = %grd$nrows%&sv YLL = %grd$ymin%*%pi%/180&sv CS = %grd$dx%*%pi%/180&ty ‘Available spheroids:’&ty ‘1 – AIRY 1830’&ty ‘2 – AUSTRALIAN 1965’&ty ‘3 – BESSEL 1841’&ty ‘4 – BESSEL 1841 NAMIBIA’&ty ‘5 – CLARKE 1866’&ty ‘6 – CLARKE 1880’&ty ‘7 – EVEREST 1830 INDIA’&ty ‘8 – EVEREST 1830 MALAYSIA’&ty ‘9 – EVEREST 1956 INDIA’&ty ‘10 – EVEREST 1964 MALAYSIA & SINGAPORE’&ty ‘11 – EVEREST 1969 MALAYSIA’&ty ‘12 – EVEREST PAKISTAN’&ty ‘13 – FISHER 1960’&ty ‘14 – GRS 80’&ty ‘15 – HELMERT 1906’&ty ‘16 – HOUGH 1906’&ty ‘17 – INDONESIAN 1974’&ty ‘18 – INTERNATIONAL 1924’&ty ‘19 – KRASSOVSKY 1940’&ty ‘20 – MODIFIED AIRY’&ty ‘21 – SOUTH AMERICAN 1969’&ty ‘22 – WGS 72’&ty ‘23 – WGS 84’&ty ‘24 – ETRS 1989’&sv spheroid:= [response ‘Choose the reference spheroid:’]&if %spheroid% = 1 &then

&do&sv SMA = 6377563.396&sv fl = 0.00334085067870327

&end&else &if %spheroid% = 2 &then

&do&sv SMA = 6378160.0&sv fl = 0.003352891899858333

&end&else &if %spheroid% = 3 &then

&do&sv SMA = 6377397.155&sv fl = 0.0033427731536659813

&end&else &if %spheroid% = 4 &then

&do&sv SMA = 6377483.865&sv fl = 0.003342773176894559

&end&else &if %spheroid% = 5 &then

&do&sv SMA = 6378206.4&sv fl = 0.0033900753039287908

&end

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&else &if %spheroid% = 6 &then&do&sv SMA = 6378249.145&sv fl = 0.003407561308111843

&end&else &if %spheroid% = 7 &then

&do&sv SMA = 6377276.345&sv fl = 0.0033244493186534527

&end&else &if %spheroid% = 8 &then

&do&sv SMA = 6377298.556&sv fl = 0.003324449343845343

&end&else &if %spheroid% = 9 &then

&do&sv SMA = 6377301.243&sv fl = 0.0033244493543833783

&end&else &if %spheroid% = 10 &then

&do&sv SMA = 6377304.063&sv fl = 0.003324449295589469

&end&else &if %spheroid% = 11 &then

&do&sv SMA = 6377295.664&sv fl = 0.0033244492833663726

&end&else &if %spheroid% = 12 &then

&do&sv SMA = 6377309.613&sv fl = 0.003324292418982392

&end&else &if %spheroid% = 13 &then

&do&sv SMA = 6378155.0&sv fl = 0.003352329944944847

&end&else &if %spheroid% = 14 &then

&do&sv SMA = 6378137.0&sv fl = 0.0033528106875095227

&end&else &if %spheroid% = 15 &then

&do&sv SMA = 6378200.0&sv fl = 0.0033523298109184524

&end&else &if %spheroid% = 16 &then

&do&sv SMA = 6378270.0&sv fl = 0.0033670034351006867

&end&else &if %spheroid% = 17 &then

&do

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&sv SMA = 6378160.0&sv fl = 0.0033529256086395256

&end&else &if %spheroid% = 18 &then

&do&sv SMA = 6378388.0&sv fl = 0.003367003387062615

&end&else &if %spheroid% = 19 &then

&do&sv SMA = 6378245.0&sv fl = 0.0033523298336767685

&end&else &if %spheroid% = 20 &then

&do&sv SMA = 6377340.189&sv fl = 0.0033408506318589907

&end&else &if %spheroid% = 21 &then

&do&sv SMA = 6378160.0&sv fl = 0.003352891899858333

&end&else &if %spheroid% = 22 &then

&do&sv SMA = 6378135.0&sv fl = 0.00335277945669078

&end&else &if %spheroid% = 23 &then

&do&sv SMA = 6378137.0&sv fl = 0.0033528106718309896

&end&else &if %spheroid% = 24 &then

&do&sv SMA = 6378137.0&sv fl = 0.0033528106811823171

&end&sv e [sqrt [calc 2*%fl% - %fl%*%fl%]]&sv e1 [calc 1 - %e%*%e%]&sv e2 [calc 1/[calc 2*%e%]]&sv ee [calc %e%*%e%]&sv sin90 [sin [calc %pi%/2]]&sv m1 [calc 1 - %e%*%sin90%]&sv m2 [calc 1 + %e%*%sin90%]&sv mln [log [calc %m1%/%m2%]]&sv sin90_2 [calc %sin90%*%sin90%]&sv q [calc %e1%*[calc [calc %sin90%/[calc 1 - [calc %ee%*%sin90_2%]]]- [calc %e2%*%mln%]]]&sv R2 [calc %SMA%*%SMA%*[calc %q%/2]]&sv cost [calc %R2%*%CS%]gridsetcell %grid%setwindow %grid% %grid%row = $$rowmap + 1lgl = ((%NROWS% - row)*%CS%) + %YLL%/*lgl = lower geodetic latitudeugl = ((%NROWS% + 1 - row)*%CS%) + %YLL%/*ugl = upper geodetic latitude

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q1 = %e1%*(sin(lgl)/(1 - %ee%*POW(sin(lgl),2)) - ((1/(2*%e%))*LN((1 -%e%*sin(lgl))/(1 + %e%*sin(lgl)))))q2 = %e1%*(sin(ugl)/(1 - %ee%*POW(sin(ugl),2)) - ((1/(2*%e%))*LN((1 -

%e%*sin(ugl))/(1 + %e%*sin(ugl)))))cell_area = (%cost%*abs((q2/%q%) - (q1/%q%)))

q&IF [exists row -grid] &THENKILL row&IF [exists lgl -grid] &THENKILL lgl&IF [exists ugl -grid] &THENKILL ugl&IF [exists q1 -grid] &THENKILL q1&IF [exists q2 -grid] &THENKILL q2

Appendix A2 – Numeric Solution for the Quadrangle Area on the Spheroid –Pseudo-code

For each cell of the gridCA = 0GLL = NROWS - NROW*CSGLLA = GLLGUL = (nrows-nrow+1)*CSGULA = GLL + 1/3,600ACA = 0For each arc-cellCLL = arctg (SMI2/SMA2)*tan(GLLA*pi/180.))If it is not the highest latitude cell and arc-cell ThenCUL = arctan (SMI2/SMA2)*tan(GULA* pi/180.))

ElseCUL = pi/2

End IfDCL = CUL - CLL

RLL = sqrt((((SMA2*cos(GLLA*pi/180.))2) + ((SMI2*sin(GLLA*pi/180.))2))/(((SMA*cos(GLLA*pi/180.))2) + ((SMI*dsin(GLLA*pi/180.))2)))RUL = sqrt((((SMA2*cos(GULA*pi/180.))2) + ((SMI2*sin(GULA*pi/180.))2))/(((SMA*cos(GULA*pi/180.))2) + ((SMI*dsin(GULA*pi/180.))2)))!Using the Law of cosines (Carnot’s theorem)Dlat = sqrt(RLL2 + RUL2 - 2*RLL*RUL*cos(DCL)DLlow = RLL*cos(CLL)*((CS/nac)*pi/180.)DLupp = RUL*cos(CUL)*((CS/nac)*pi/180.)

If it is not the highest latitude cell and arc-cell ThenACA= sqrt (Dlat 2-((DLupp - DLlow)/2)2)*(DLupp + DLlow)/2.

ElseACA = sqrt (Dlat 2-((DLupp - DLlow)/2)2)*DLlow)/2.

End IfCA = CA+ACAGLLA = GLL + i* (1/3,600)

If it is not the highest latitude cell and arc-cell ThenGULA = 90.

ElseGULA = GLLA + (1/3,600)

End If

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End ForCA = CA*NSL

End ForCA = Cell areaGLL = Geodetic Lower Left cell corner latitude (in degrees)GLLA = Geodetic Lower Left corner latitude lower Arc-cell in the cell(in degrees)GUL = Geodetic Upper Left cell corner latitude (in degrees)GULA = Geodetic Upper Left corner latitude lower Arc-cell in the cell(in degrees)ACA = Arc-Cell Area (m2)CLL = geoCentric Lower Left corner latitude (in radians)CUL = geoCentric Upper Left corner latitude (in radians)DCL = difference between geocentric Upper and Lower Left corner latitude(in radians)RLL = Spheroid radius (m) at CLLRUL = Spheroid radius (m) at CULDlat = measure (in meters) of the meridian arc between upper and lowerarc-cell cornerDLlow = measure (in meters) of the parallel arc between left and rightlower arc-cell cornersDLupp = measure (in meters) of the parallel arc between left and rightupper arc-cell cornersNac = number of arc-cells in a cell columni = order number of the arc-cells in the cell first column, counted frombottom to topNSL = number of columns with arc-cell horizontal resolution containedinto a semilune with cell resolution.

Appendix A3 – Projection Parameters

Projection: Goode_HomolosineFalse_Easting: 0.000000False_Northing: 0.000000Central_Meridian: 0.000000Linear Unit: Meter (1.000000)Geographic Coordinate System: GCS_WGS_1984Angular Unit: Degree (0.017453292519943299)Prime Meridian: Greenwich (0.000000000000000000)Datum: D_WGS_1984Spheroid: WGS_1984Semimajor Axis: 6378137.000000000000000000Semiminor Axis: 6356752.314245179300000000Inverse Flattening: 298.257223563000030000Projection: SinusoidalFalse_Easting: 0.000000False_Northing: 0.000000Central_Meridian: 0.000000Linear Unit: Meter (1.000000)Geographic Coordinate System: GCS_WGS_1984Angular Unit: Degree (0.017453292519943299)Prime Meridian: Greenwich (0.000000000000000000)Datum: D_WGS_1984Spheroid: WGS_1984Semimajor Axis: 6378137.000000000000000000Semiminor Axis: 6356752.314245179300000000

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Inverse Flattening: 298.257223563000030000Projection: Lambert_Azimuthal_Equal_AreaFalse_Easting: 4321000.000000False_Northing: 3210000.000000Central_Meridian: 10.000000Latitude_Of_Origin: 52.000000Linear Unit: Meter (1.000000)Geographic Coordinate System: GCS_ETRS_1989Angular Unit: Degree (0.017453292519943299)Prime Meridian: Greenwich (0.000000000000000000)Datum: D_ETRS_1989Spheroid: GRS_1980Semimajor Axis: 6378137.000000000000000000Semiminor Axis: 6356752.314140356100000000Inverse Flattening: 298.257222101000020000Projection: Transverse_MercatorFalse_Easting: 500000.000000False_Northing: 0.000000Central_Meridian: -75.000000Scale_Factor: 0.999600Latitude_Of_Origin: 0.000000Linear Unit: Meter (1.000000)Geographic Coordinate System: GCS_WGS_1984Angular Unit: Degree (0.017453292519943295)Prime Meridian: Greenwich (0.000000000000000000)Datum: D_WGS_1984Spheroid: WGS_1984Semimajor Axis: 6378137.000000000000000000Semiminor Axis: 6356752.314245179300000000Inverse Flattening: 298.257223563000030000

Acknowledgements

We thank Marco Traini for research assistance, and the referees for their critical reviewsthat helped to improve the manuscript. This work has been supported by the followinggrant: Fondazione Cassa di Risparmio di Foligno Award #1 Università di Perugia – Corsodi Laurea in Protezione Civile. When this study began, Andrea Taramelli was affiliatedwith the LDEO of Columbia University and with the Department of Earth Sciences of theUniversity of Perugia and Alessandro Sorichetta was affiliated with the Columbia Uni-versity’s Center for International Earth Science Information Network (CIESIN).

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