Hengl 06 Finding the Right Pixel Size

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Computers & Geosciences 32 (2006) 1283–1298

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Finding the right pixel size$

Tomislav Hengl�

European Commission, Directorate General JRC, Institute for Environment and Sustainability, Soil and Waste Unit, TP 280,

Via E. Fermi 1, I-21020 Ispra (VA), Italy

Received 6 May 2005; received in revised form 23 November 2005; accepted 24 November 2005

Abstract

This paper discusses empirical and analytical rules to select a suitable grid resolution for output maps and based on the

inherent properties of the input data. The choice of grid resolution was related with the cartographic and statistical

concepts: scale, computer processing power, positional accuracy, size of delineations, inspection density, spatial

autocorrelation structure and complexity of terrain. These were further related with the concepts from the general statistics

and information theory such as Nyquist frequency concept from signal processing and equations to estimate the

probability density function. Selection of grid resolution was demonstrated using four datasets: (1) GPS positioning data—

the grid resolution was related to the area of circle described by the error radius, (2) map of agricultural plots—the grid

resolution was related to the size of smallest and narrowest plots, (3) point dataset from soil mapping—the grid resolution

was related to the inspection density, nugget variation and range of spatial autocorrelation and (4) contour map used for

production of digital elevation model—the grid resolution was related with the spacing between the contour lines i.e.

complexity of terrain. It was concluded that no ideal grid resolution exists, but rather a range of suitable resolutions. One

should at least try to avoid using resolutions that do not comply with the effective scale or inherent properties of the input

dataset. Three standard grid resolutions for output maps were finally recommended: (a) the coarsest legible grid

resolution—this is the largest resolution that we should use in order to respect the scale of work and properties of a dataset;

(b) the finest legible grid resolution—this is the smallest grid resolution that represents 95% of spatial objects or

topography; and (c) recommended grid resolution—a compromise between the two. Objective procedures to derive the

true optimal grid resolution that maximizes the predictive capabilities or information content of a map are further

discussed. This methodology can now be integrated within a GIS package to help inexperienced users select a suitable grid

resolution without doing extensive data preprocessing.

r 2005 Elsevier Ltd. All rights reserved.

Keywords: Grid resolution; Scale; Inspection density; Point pattern analysis; Variogram; Terrain complexity

e front matter r 2005 Elsevier Ltd. All rights reserved

geo.2005.11.008

structions to derive a suitable grid resolution

p://hengl.pfos.hr/PIXEL/

332 785535; fax: +39 0332 786394.

ess: [email protected].

1. Introduction

A grid cell, popularly known as pixel, is thefundamental spatial entity in a raster-based GIS(Gatrell, 1991; DeMers, 2001). Although there ispractically no difference between pixel and grid cell,geoinformation scientists like to emphasize thatpixel is a technology and grid is a model (De By,

.

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ARTICLE IN PRESST. Hengl / Computers & Geosciences 32 (2006) 1283–12981284

2001). A grid means ideal properties—orthogonalmatrix, fixed resolution, which a raster image doesnot necessarily has to fit. For example, an aerialphoto first needs to be ortho-rectified and thenresampled to a regular grid to (approximately) fitthe grid model (Rossiter and Hengl, 2002).

Grid cell can be also related (but should not beconfused) with the support size, which is typically afixed area or volume of the land that is beingsampled. Support size can be increased by usingcomposite samples or by averaging point-sampledvalues belonging to the same blocks of land. Ingeostatistics, one can also control the support size ofthe output models by averaging multiple predictionsper regular blocks of land, which is known as‘‘block kriging’’ (Heuvelink and Pebesma, 1999).This means that we can sample at point locations,then make predictions for blocks of 10� 10m. Thelatter often confuses GIS users because we canproduce predictions at regular point locations(point kriging) and then display them using a rastermap, but we can also make predictions for blocks ofland (block kriging) and display them using thesame raster model (Bishop et al., 2001). Thisdistinction is especially important for the validationof the spatial prediction models because it canlead to serious misconceptions—validating a point

BASE MAPS

scale S

BASE MAPS

scale S+FINER GRID

Disaggregation

BASE MAPS

scale S-

COARSER GRID

Aggregation

LARGE SCALES

SMALL SCALES

DOWNSCALIN

UPSCALING

Fig. 1. Upscaling and downscaling in a grid-based GIS. S indicates

McBratney (1998).

model (support size of few centimeters) at 1 kmsupport or vice versa can be quite discouraging(Stein et al., 2001).

Although the raster structure has a number ofserious disadvantages such as of under- and over-sampling in different parts of the study area andlarge data storage requirements, it will remain themost popular format for spatial modelling in thecoming years (DeMers, 2001). What makes itespecially attractive is that most of the technicalcharacteristics are controlled by a single measure:grid resolution, expressed as ground resolution inmeters. The enlargement of grid resolution leads toaggregation or upscaling and decrease of gridresolution leads to disaggregation or downscaling.As grid becomes coarser, the overall informationcontent in the map will progressively decrease andvice versa (McBratney, 1998; Kuo et al., 1999; Steinet al., 2001). In cartography, coarser grid resolu-tions are connected with smaller scales and largerstudy areas, and finer grid resolutions are connectedwith larger scales and smaller study areas. Theformer definition often confuses non-cartographersbecause bigger pixel means smaller scale, whichusually means larger study area (Fig. 1). Note inFig. 1, that both aggregation and disaggregationcan be done before or after geo-computation. If the

MODEL

scale S

DERIVED MAPS

scale S

MODEL

scale S+

DERIVED MAPS

scale S+

MODEL

scale S-

DERIVED MAPS

scale S-

NO

YES

G

scale: S� are smaller scales and Sþ are larger scales. Based on

ARTICLE IN PRESST. Hengl / Computers & Geosciences 32 (2006) 1283–1298 1285

model is linear, the two routes should yield the sameresults (Heuvelink and Pebesma, 1999); if not, therecan be serious differences. Aggregation is fairlyuseful procedure to reduce the small scale variationand get better idea about the general pattern (Steinet al., 2001). In contrast, if our objective is to locateextremes (hot spots), then aggregation is somethingwe should avoid.

Many researchers investigated the effects of gridresolution on the accuracy of their models. Applica-tions range from mapping of soil properties (Flor-insky and Kuryakova, 2000), modelling of surfacerunoff (Kuo et al., 1999; Molnr and Julien, 2000), seacurrents (Davies et al., 2000) or modelling ofmeteorological data (McQueen et al., 1995; Nodaand Niino, 2003). In the case of terrain data,aggregation of grid resolutions will seriously deterio-rate accuracy of terrain parameters (Weihua andMontgomery, 1994; Dietrich et al., 1995; Thompsonet al., 2001). Wilson et al. (2000) demonstrated theimpact the grid resolution makes on hydrologicalanalysis: as the grid resolution increased from 30 to200m the flow-path length and specific catchmentarea maps changed drastically. The grid resolutioncan also be crucial for the accuracy of the simulationmodel such as surface runoff or erosion models(Sanchez Rojas, 2002). Kienzle (2004) gives asystematic overview of effects of various gridresolutions on the reliability of terrain parameterssuggesting finer grid resolutions from 5–20m. Lianget al. (2004), on the other hand, observed impacts ofdifferent spatial resolutions on modelling surfacerunoff and concluded that the resolution needs to beimproved only to a critical level after which themodel will not necessarily perform better. Weihuaand Montgomery (1994) also got a substantialimprovement with 10m grid resolution over 30 and90m data, but 2 or 4m data gave only marginalimprovements. Florinsky and Kuryakova (2000)focused specifically on the importance of gridresolution of terrain parameters on the efficiency ofspatial prediction of soil variables. They plottedcorrelation coefficients versus different grid resolu-tions and looked for the grid size with most powerfulprediction efficiency. Bishop et al. (2001) suggesteduse of the Shannon’s information criterion to selectthe optimal block size for block kriging. All theseexperiments clearly prove two things: (1) gridresolution plays an important role for the efficiencyof the mapping and (2) its selection can be optimized,to a certain level, to satisfy both processingcapabilities and representation of spatial variability.

Although much has been published on the effectof grid resolution on the accuracy of spatialmodelling, choice of grid resolution is seldom basedon the inherent spatial variability of the input data(Vieux and Needham, 1993; Bishop et al., 2001). Infact, in most GIS projects, grid resolution is selectedwithout any scientific justification. In the ESRI’spackage ArcGIS, for example, the default outputcell size is suggested by the system using some trivialrule: in the case the point data is being interpolatedin Spatial Analyst, the system will take the shortestside of the study area and divide it by 250 toestimate the cell size (ESRI, 2002). Obviously, suchpragmatic rules do not have a sound scientificbackground.

This motivated me to produce methodologicalguides to select a suitable grid resolution for outputmaps based on the inherent properties of the inputdata. I tried to relate the choice of grid resolution tomeasurable cartographic and statistical conceptssuch as: scale, processing power, positional accu-racy, inspection density, spatial dependence struc-ture and complexity of terrain. I will firstrecommend some general rules of thumb to selectthe grid resolution and then demonstrate how toselect a legible grid resolution given the realdatasets.

2. Methods

2.1. Grid resolution and cartographic concepts

Although we live in a digital era where we donot necessarily work with hard copy maps,spatial resolution and extent are still stronglyrelated with the traditional cartographic concepts(Quattrochi and Goodchild, 1997; Goodchild,2001). For example, in traditional soil carto-graphy the scale of an existing map is commonlyassessed by estimating either the maximum locationaccuracy (MLA) or average size area (ASA)of the polygons on the ground (Rossiter, 2003).These cartographic definitions can also be used toestimate the suitable grid resolution for a givenmapping scale. As a rule of thumb, Rossiter(2003) suggests that four grid cells can beconsidered equivalent to the minimum legibledelineation (MLD), which is the smallest size areathat we map. According to the definition of Vink(1975) the MLD is 0:25 cm2 on the map, sothe suitable grid resolution can estimated based on


the scale number ðSNÞ:

pp

ffiffiffiffiffiffiffiffiffiffiffiffiMLD

4

r¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiSN2 � 0:000025

p2

¼ SN � 0:0025 (1)

where p is the grid (pixel) size and MLD is theminimum legible delineation area on the ground inm2. This means that for a 1:50K scale, MLD is6.25 ha and suitable grid resolution is 125m, whichseems fairly coarse. Somewhat larger grid resolu-tions from 0.5 to 3mm on the map have been alsorecommended by Valenzuela and Baumgardner(1990).

The grid resolution can also be related to theMLA, which commonly ranges from 0.25mm tomaximum of 0.1mm on the map (Vink, 1975). Thisgives the smallest legible resolutions

pXSN �MLA ¼ SN � 0:00025 ð0:0001Þ (2)

So for 1:50K scale, the smallest legible gridresolution is 12.5m (5m). Resolutions finer than5m truly do not make sense as it will be hard tovisualize or print them at this scale of work.Following notions in microscopy, and the Nyquistfrequency concept from signal processing (Shannon,1949), which states that the original signal can bereconstructed if sampling frequency is two times theoriginal frequency, McBratney et al. (2003, Table 1),suggested that there should be at least 2� 2 pixelsto represent smallest rounded objects of interest andat least two pixels to represent the width ofelongated objects. The smallest objects are typicallyof size 1� 1mm on the map, so that the gridresolution can be determined using the p ¼ 0:5mmrule (Fig. 2a). The former can be used as the

0.00

025

0.00

05

0.0025

Sca

le n

umbe

r

Grid resolution (m)(a)

200K

150K

50 100 150 200

100K

50K

Fig. 2. Popular cartographic rules to select grid resolution: (a) relationsh

between grid resolution and density of observation points for soil map

universal rule of thumb to relate scale with gridresolution.

2.2. Grid resolution and computer processing power

The grid resolution can also be related with the sizeof area and processing power of our computer.Although we might insist on using the finest gridresolution possible, the calculation time will increaseexponentially (cubically) with the total number ofpixels in a map (McBratney, 1998). This means thatgrid resolution needs to match capabilities of ourcomputer and time given to complete a GIS project.Following the popular Moore’s law, Lagacherie andMcBratney (2005) discussed the relationship betweenthe grid resolution and processing capabilities ofstandard desktop PC’s and discovered the followingrough relationship between the log of the image sizeand the current year (Fig. 3a):

log10ðmÞ ¼ 0:14 � ðY � 1955Þ (3)

where log10ðmÞ is the logarithm of the image size inpixels and Y is the current year. According to thisformula, the standard image size in year 2005 wouldbe about 107 pixels, which makes an image of 3000�3000 pixels.

The grid resolution can also be estimated bydividing the size of the study area by the number ofpixels that computer can handle. For example, if thesize of area is about 100 000 km2 and computer canhandle 107 pixels, we should probably work withresolutions of 100 or more meters. Based on thisprinciple, we can also derive the coarsest globalstandard resolution. This is the resolution of an

Grid resolution (m)50 100 150 200

4 / cm2

2.5 / cm2

1 / cm2Obs

erva

tions

/ km

2

(b)

20

15

10

5

ip between grid resolution and cartographic scale; (b) relationship

ping applications.

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Imag

e si

ze (

pixe

ls)

Year Standard grid resolution (m)(a)

1012

1010

108

106

104

102

1980 5002000 10002020 15002004 2000

Year

(b)

2040

2030

2020

2010

2000

Fig. 3. (a) Growth of standard image size in pixels also follows Moore’s Law. (b) By year 2040, images of almost all Earth should be

available in resolutions of 25m or better. Note inflection point at year 2026.

T. Hengl / Computers & Geosciences 32 (2006) 1283–1298 1287

image covering the whole Globe. The surface ofEarth is about 5:10� 108 km2 (Yoder, 1995), whichmeans that the coarsest global standard resolutionin the year 2005 is about 7 km. If the Eq. (3) iscorrect, the coarsest global standard resolution inyear 2040 will be about 25m (Fig. 3b). At that timethe computers will be so powerful that they will beable to handle images of million by million pixels!These are of course rather simple models and realfigures might differ from application to application.

2.3. Grid resolution and GPS positioning

If the scale of a project is unknown or non-standard, we can assess it by analyzing the mappingmethodology. For example, the choice of the gridresolution can be related with the positionalaccuracy of our field positioning method. This isespecially important for integration of GPS withremote sensing images and aerial photos where weshould be certain that our GPS reading will (mostprobably) fall inside the right pixel. To ensure this,we first need to estimate the confidence radius of thepositioning method using some control points. Theconfidence radius is the radius of a circle where weexpect the most ðP ¼ 95%Þ of the points to appear(Arnaud and Flori, 1998). It is normally evaluatedusing the error vector ðrEÞ, which is as a differencebetween the measured ðX GPS;Y GPSÞ and the truelocation of the control point ðX T ;Y T Þ:

rE ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðX GPS � X T Þ

2þ ðY GPS � Y T Þ

2

q(4)

As a rule of thumb, one should select the gridresolution so that the area of circle described by theerror radius is equal or smaller than the area of pixel(Fig. 4b):

p2Xr2E � p (5)

where r2E is the average error radius and p is the gridresolution. The recommended grid resolution is theone where most of the points (95%) would fallwithin the pixel

p ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2EðP¼95%Þ � p

q� 1:8 � rEðP¼95%Þ (6)

where rEðP¼95%Þ is the 95% probability error radiusor confidence radius derived from the cumulativedistribution of the error vectors from a set ofmeasurements (Fig. 4a). Garmin (www.garmin.com), for example, claims that their handheldreceivers (selective availability turned off) achievehorizontal error of not more than 15m in 95% oftime. This would be then compatible with a gridresolution of 27m.

2.4. Grid resolution and remote sensing systems

As with raster-based GIS, grid resolution controlsmany aspects of remote sensing systems used formapping. Characteristics of objects are scale depen-dent and their total number, area, average size orperimeter of objects being mapped will differ fordifferent grid resolution (Lillesand and Kiefer, 2000,pp. 598–603). In fact, one will never be able toabsolutely determine how many islands or lakes are

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grid resolution

confidenceradius

error vector

(a) (b)

Fig. 4. Some important concepts for evaluation of positioning methods: (a) distribution of error vectors follows log-normal distribution;

(b) confidence radius where most of measurements fall should not have larger area than grid node.

T. Hengl / Computers & Geosciences 32 (2006) 1283–12981288

in the world and what is their perimeter. Althoughwe cannot actually change the grid resolution inremote sensing images, we should at least be certainthat the images are adequate for our mappingapplication. We can do that by inspecting the size of

smallest spatial objects that are being mapped andthen select the appropriate grid resolution, i.e.remote imaging system. Again, there should be atleast four pixels to represent smallest objects and atleast two pixels to represent the narrowest objects,which can be expressed mathematically as

pp

ffiffiffiffiffiffiffiffiffiffiffiaMLDp

4if So3

wMLD

2if S43

8>><>>:

(7)

where aMLD is the area of the smallest objects, wMLD

is the width of the narrowest objects and S is theshape complexity index derived as the perimeter toboundary ratio:

S ¼P

2 � r � p; r ¼

ffiffiffia

p

r(8)

where P is the perimeter of polygon, a is the area ofpolygon and r is the radius of circle with the samesurface area (Hole, 1953). Arbitrary value of 3 isused to differentiate between compact and narrow/long polygons. This means that we can first samplesize of spatial objects (e.g. crop plots, water bodies,forest patches, roads) in part of the study area andthen decide on the suitable sensor. To be morecertain, one can plot the histogram with cumulative

distributions and derive the 5% probability area, i.e.5% probability width of the reference objects(Garbrecht and Martz, 1994). For example, if theobjective of our mapping project are agriculturalplots, and if the smallest plots are about 1 ha, thegrid resolution should be at least 50m, which meansthat we should use Landsat or similar imagery. Ifthe smallest plots are about 0.05 ha (corresponds tothe scale 1:5000), we will need images with resolu-tions of 10m and finer.

In addition to the size of smallest objects, one alsoneeds to take into account how contrasting is anobject of interest when compared with its surround-ings. The local contrast between adjacent objects canbe assessed by statistically comparing the reflectancevalues between the target object ðrT Þ and surround-ing object ðrSÞ. Obviously, the higher the difference,the less strict we have to be about the gridresolution. For example, consider the centerlineroad markers of size of 20 cm. Such markers will bevisible even on the 60 cm resolution QuickBirdimages because of high contrast with the surround-ing objects. The impact of local contrast in the caseof aggregation is illustrated in Fig. 5b: if thecontrast is fairly low, the resampling will totallydiminish point or line object so they cannot bedistinguished from the surroundings.

2.5. Grid resolution and point samples

In many mapping projects, a map is made out ofthe point samples collected in the field and then used

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Fig. 5. Impact of local contrast on visibility of objects for point, line and polygon features: (a) high contrast between target object and

surrounding ðrT ¼ 0:4%; rS ¼ 99:6%Þ; and (b) low contrast between target object and surrounding ðrT ¼ 78:4%; rS ¼ 94:1%Þ. Fine

resolution images have been resampled to coarser grids using bilinear resampling.


to make predictions. To be consistent, everymapping project should have approximately anequal density of samples per area, also calledinspection density. Obviously, the denser the ob-servation points, the larger the scale of mapping. Acartographic rule, used for example in soil mapping,is that there should be at least one (ideally four)observation per 1 cm2 of the map (Avery, 1987,Table 1). This principle can be used to estimate theeffective scale of a data set consisting of sampledpoints only. For example, 10 observations per km2

corresponds to the scale of about 1:50K. The sameprinciple can be also expressed mathematically

SN ¼

ffiffiffiffiffiffiffiffiffiffi4 �

A

N

r� 102 . . .SN ¼

ffiffiffiffiffiA

N

r� 102 (9)

where A is the surface of the study area in m2 and N

is the total number of observations. Rememberfrom Eq. (1) that the scale number can be used toestimate the grid resolution. If we take the inter-mediate number of 2.5 observations per cm2 andcombine it with the p ¼ 0:5mm on the map rule ofthumb, with a bit of reduction, we finally get asimple formula

p ¼ 0:0791 �

ffiffiffiffiffiA

N

r(10)

So, for example, if we deal with 100 samples and thesize of the area is 10 km2, a recommended grid

resolution would be 25m (see also Fig. 2b), whichcan also be expressed as 160 pixels per point sample.

Grid resolution can also be related to geometry ofpoint patterns, i.e. distance between the sampledpoints (Boots and Getis, 1988). Following, again,the Nyquist frequency concept (Shannon, 1949) thegrid resolution should be at most half the averagespacing between the closest point pairs

pphij

2(11)

where hij is the average distance between two closestpoint pairs also called mean shortest distance. In thecase of regular point samples, the formula simplifiesto

p ¼ 0:5 �

ffiffiffiffiffiA

N

r(12)

or 4 pixels per point sample. Note that thedifference between the factors in Eqs. (10) and(12) is fairly big. You should keep in mind thatEq. (12) is valid only if we are dealing with(spatially) absolutely regular point samples. If thepoint samples show more random or clustereddistribution of points, we need to be somewhatmore strict about the grid resolution. If we aredealing with random point samples, then theaverage spacing between the closest point pairs isapproximately half the spacing between closestpoint pairs in regular point samples (Fig. 6b). This

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(a) (b)

Distance (m)

Pro

babi

lity

of fi

ndin

gon

e po

int n

eigh

bour

1.00

0.75

0.50

0.25

(c)

regular

point sampling

random

point sampling

Fig. 6. Selection of grid resolution in relation to different point

sample patterns: (a) in situation of regular sampling, coarsest grid

resolution is half average spacing to closest point pair; (b) in

situation of random sampling, coarsest grid resolution should be

about two times finer; (c) probability of finding one point in

neighborhood for regular and point samples.


is because random sampling has equal probabilityof producing totally clustered and totally regularsamples (Fig. 6c). See also the supplementarymaterials (http://hengl.pfos.hr/PIXEL/) for furtherargumentation. The factor then modifies to

p ¼ 0:25 �

ffiffiffiffiffiA

N

r(13)

We can be even more strict and look for a gridresolution where 495% of closest point pairs donot fall into the same pixel. This can be done usingsome well known point pattern analysis algorithmssuch as the one described in Boots and Getis (1988)and Rowlingson and Diggle (1993). We first need toderive the probability of finding the first point pairat different distances and for a given data set. Wecan then select the 5% probability smallest distance(see Fig. 6c)

pXhijðP¼5%Þ (14)

In addition, we can inspect the spatial correlation

structure of the point data set and use thisinformation to select the grid resolution. Here thekey is to estimate the range of spatial dependence.Obviously, a variable that is spatially auto-corre-lated at shorter distances would require finer gridresolution and vice versa. Selection of grid spacingcan be related with estimation of the optimal binsize that can estimate the probability densityfunction, used for example in statistics to displayhistograms (Izenman, 1991). This gives a rathersimple formula

p ¼ hR �m�13 (15)

After we have determined the range of spatialdependence (hR), we need to count the number ofpoint pairs (m) within that range and then derive asuitable grid resolution that can be used to representthe spatial dependence structure.

Another useful thing of estimating the spatialdependence structure is that we get an estimate ofthe nugget variance, i.e. the variation that happensat zero distances. This is a part of variation that wewould typically like to remove or diminish, whichcan be achieved by using for example block kriging(Heuvelink and Pebesma, 1999; Bishop et al., 2001).Because the nugget variance refers to microscalevariation (in fact, infinitely small distances), anyblock size with positive area would average outshort-range variation. To be on the safe side, we canincrease the size of the block ðBÞ depending on theamount of nugget variation. A satisfactory blocksize can be estimated by comparing the derivedestimation variance with the original variation inthe data, so-called normalized estimation error

sE% ¼sE

sz

(16)

where sE is the estimation error (precision) and sz isthe total variation in the spatial data. This ratio isequivalent to the coefficient of multiple determina-tion ðR2Þ used in regression analysis to show howmuch of the variation has been explained by amodel. If sE%p40% this means that the predictionmodel has explained 85% or more of the variationin the data. In contrary, when the sE% gets close to100%, this means that the prediction model iscompletely weak ðR2 ’ 0Þ. In practice, we can runblock-kriging predictions for different block sizesand than select the block size for which thenormalized estimation error is good enough tomap the whole area (e.g. sE% ¼ 40%). Similarly,

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19

21

25

27

23

z (m

)

p = 2.5 m

p = 1.2 m

p = 0.5 m

x (m)

x (m)

x (m)

x (m)

19

19

19

21

21

21

25

25

25

27

27

27

23

23

23

0

0

0

5

5

5

10

10

10

15

15

15

0 5 10 15

Fig. 7. Schematic example showing effect of grid resolution on

representation of topography: too coarse grid resolution ðp ¼ 2:5mÞwill misrepresent topography; whereas finer grid resolution ðp ¼

0:5mÞ will be more effective in representing all peaks and channels.


there is no gain in using larger block sizes if there isno nugget variation or if already 95% of variation isexplained by the prediction model ðsE% ¼ 23%;R2 ¼ 0:95Þ.

2.6. Grid resolution and terrain analysis

The key problem when selecting a grid resolutionfor terrain analysis is that there can be a significantdifference between the surface elevation on acoarser grid versus the actual topography, meaningthat some peaks and channels might disappear in araster DEM. In general, an increase in the detail inthe DEM will also mean more accurate terrainparameters. This increase, however, depends on thegeneral variability of the landscape. For example, agenerally simple and smooth landscape might notneed a fine resolution DEM. Even more so, if thegrid resolution is too fine it might introduce localartefacts or slow down computation of terrainparameters. Obviously, we need a grid resolutionthat optimally reflects the variability of the elevationsurface and is able to represent the majority ofgeomorphic features (Borkowski and Meier, 1994;Kienzle, 2004).

A suitable grid resolution can be derived for givensampled elevations (e.g. contours) and based on thecomplexity of (sampled) terrain. Imagine a one-dimensional topography with a number of inflectionpoints (Fig. 7). We can again connect the problemof the grid resolution with the Nyquist–Shannonsampling theorem (Shannon, 1949; Tempfli, 1999).In our one-dimensional example above, terrain isthe signal and its frequency is determined by thedensity of inflection points. Hence, the grid resolu-tion should be at least half the average spacingbetween the inflection points:

ppl

2 � nðdzÞ(17)

where l is the length of a transect and nðdzÞ is thenumber of inflection points observed. We can alsobe more strict and look for the 5% probabilityspacing between the inflection points. In theexample in Fig. 7 there are 20 inflection pointsand average spacing between them is 0.8m. Hence,a grid resolution of at least 0.4m is recommended.Cumulative distribution of distances between theinflection points shows that 5% threshold of thesmallest spacing between the inflection points is0.2m. Hence, grid resolution of at least 0.4m and atmost 0.2m is recommended.


In the 2D case, the suitable grid resolution can beestimated from the total length of contours. Herethe contours present mapped changes of fixedelevations. We do not actually have a map ofinflection points, but can approximate them usingthe contour map. The suitable grid resolution isthen

p ¼A

2 �P

l(18)

where A is the total size of the study area andP

l isthe total cumulative length of all digitized contours.A more strict approach is to evaluate the density ofcontour lines in an area and derive the 5%probability smallest width of contours.

3. Case studies

3.1. Example 1: GPS positioning

In this example, I will first demonstrate how toselect a grid resolution based on the evaluation ofthe GPS positioning method. In this case, 100positioning fixes were recorded using the single-fixGPS positioning method (Arnaud and Flori, 1998)at the control point with a known location. Thefluctuation of the GPS readings can be seen inFig. 8a. The errors ranged from 0.7 to 23.9m,average error was 8.5m with a standard distributionof 5.2m. The error vectors seem to follow the

GPS single-fix

True location of the point

freq

uenc

y

20

10

0

0(a) (b)

20 m

Fig. 8. Selecting grid resolution based on confidence radius of positio

location of point; (b) histogram of error vectors, average error vector

log-normal distribution (Fig. 8b). The theoreticallog-normal distribution gave the 95% probabilityradius of 19.1m, while the experimental distributionshows a somewhat higher value (20.4). Eq. (6) givesus a suitable grid resolution of 34.4m. If this gridresolution is selected, most (95%) of GPS fixes willfall within the right pixels. This number for examplecorresponds to the resolution of the Landsatimagery. A more accurate positioning methodwould be needed to locate points within finer gridresolutions. For example, August et al. (1994)showed that using the averaging of multiple GPSfixes, the 95% radius of a standard GPS method canbe decreased to up to 2.5 times by averaging 300replicates. If we would like to use a GPS positioningwith grid resolutions of about 15m, then we wouldneed to use GPS positioning with averaging (5minper point). Higher positional accuracy (5–20 times)can be achieved by using differential correction,which can improve accuracy to less than threemeters on average. Such accuracy would becompatible with grid resolutions within the range2–10m (SPOT or IKONOS imagery).

3.2. Example 2: monitoring agricultural plots

I will now demonstrate how to select a remotesensing sensor based on the size of agricultural plots(Fig. 9a). The polygon map consists of 121 polygonsin total. The smallest polygon is 0.005 ha, the

95% probabilityconfidence radius

average error

experimentaldistribution

theoreticaldistribution

20 m100

ning method: (a) 100 single-fix GPS measurements around true

and 95% probability confidence radius.

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(a)0 200 m

(b)

polygon size

grid resolution

freq

uenc

yC

umul

ativ

e pr

obab

ility

24

6ha

12

0.50

0.25

0.75

1.00

30

0

0

0 50 100 150 200 m

50% probability

5% probability

Fig. 9. Selection of grid resolution based on average size of objects observed: (a) agricultural plots; (b) distribution of surfaces for compact

plots ðSo3Þ and related grid resolution.


biggest is 6.903 ha, average size of polygons is0.824 ha with standard deviation of 1.005 ha. Thepolygons were then separated into two groupsaccording to the shape complexity index. In thiscase only six polygons classified for narrow poly-gons ðS43Þ. For each of these, an average widthhas been estimated by taking regular measurements(10 per polygon). A histogram of areas of compactpolygons and a cumulative theoretical log-normaldistribution can be seen in Fig. 9b.

I further on derived the 5% inverse cumulativedistribution value assuming the log-normal distri-bution. I got 0.046 ha, which means that the pixelsize should be about 20m. The coarsest legible gridresolution for this data set ðP ¼ 50%Þ would be70m ðA ¼ 0:5 haÞ. If resolutions coarser than 70mare used to monitor agriculture for this area, then inmore than 50% of the areas there will be less thanfour pixels per polygon. Note that in this case weare not using the true smallest polygon size but asomewhat higher figure (0.046 ha) because thesmallest value (0.005 ha) is not representative.Further inspection of the widths showed that theaverage width of the narrow polygons is about 16m,which gives a somewhat more strict grid resolutionof about 8m. However, the narrow polygonsoccupy only 0.9% percent of the total study area,

so we do not have to be as strict. Finally, I wouldrecommend that satellite imagery in a range from 10to 70m can be used to monitor agriculture in thisstudy area.

3.3. Example 3: point data interpolation

In this example, I will use the Wesepe point datapreviously used in numerous soil mapping applica-tions (De Gruijter et al., 1997). The dataset consistof 552 profile observations where various soilvariables have been described. The target variableis the membership value to enk earth soil type toanalyze the spatial dependence structure. The valuesrange from 0 to 1, with an average of 0.232 and astandard deviation of 0.322. The total size of thearea is 12:1 km2, which gives a sampling density ofabout 45 observations per km2, which correspondsto the scale of 1:25K, i.e. grid resolution of 12m(Eq. (10)). If we inspect the spreading of the points,we see that the average spacing between the closestpoint pairs is about 120m, which is fairly close tothe regular point sampling (for this data set—148m). The cumulative distribution showed that95% of points are at distances of 5 and more meters.This means that the legible grid resolutions arebetween 5 and 150m (Eq. (12)).


Automated fitting of the variogram using a globalmodel further gave a nugget parameter ðC0Þ of0.042, a sill ðC0 þ C1Þ of 0.097 and a rangeparameter ðRÞ of 175.2m, which means that thevariable is correlated up to the distance of about525m ðhRÞ. There are 11 807 point pairs within thisrange, which finally means that the optimal lag/gridsize would be about 23m. The pattern analysis ofthe point data set further shows that there is clearregularity in the point geometry: most of thedistances are grouped at 180m. The final inter-polated map in resolution of 10m can be seen inFig. 10d. If we apply block-kriging, the nuggetvariation, which seems to be significant in this caseðC0 ¼ 0:042Þ, would be diminished and the modelwould explain 71% of variation at the support sizeof 10m and 78% at the support size of 60m. Note

Distance (m)

Pro

babi

lity

1.00

0.75

0.50

0.25

2000 m0(a)

(b)

(d)

200 400 600(c)

800

Fig. 10. Selection of grid resolution based on point pattern analysis: (a)

point in neighborhood and graph of distances to closest point;

(http://www.gstat.org), (d) interpolated map using ordinary kriging at

that although further increasing of the block sizewould results in somewhat better normalizedestimation error, on the other hand, we wouldunnecessarily loose a level of detail by increasing theblock size above 120m because more than 50%point pairs would be within the same blocks. Alsonote that this sampling density corresponds to thescale of 1:25K, which means that the support sizeshould not exceed 25m (Fig. 11).

3.4. Example 4: contour data for terrain modelling

In this case study, I will demonstrate how a gridresolution can be selected from a map of contours,i.e. a dataset consisting of lines digitized from atopo-map. The study area is described in detail inHengl et al. (2004). Contour lines were extracted

0.0

0.1

0.2

0.3

0.4

0.5

Distance (m)400 800 1200

Sem

ivar

ianc

e

1600

0.120

0.090

0.060

0.030

Nugget = 0.042

R=175.2 m

C0+C1 = 0.097

exponential model

219

13701667

3390 4194

5751

51096432

5854

6554

63275315

3167

a set of 552 soil profile observations; (b) probability of finding one

(c) variogram and parameters fitted automatically in gstat

grid resolution of 10m.

http://www.gstat.org

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0.00

0.10

0.20

0.30

0.40

0.50

0.000

0.020

0.040

0.060

0.080

0.100

B = 0 B = 10 B = 60

R2 = 31% R2 = 71% R2 = 78%

Fig. 11. Kriging predictions (above) and the kriging estimation error (below) for different block sizes. Increasing the block size (support)

typically leads to lower estimation error and total reduction of the nugget variation. Compare with the punctual estimation ðB ¼ 0Þ on the

left.


from the 1:50K topo-map (Fig. 12a), with thecontour interval of 10m and supplementary 5mcontours in areas of low relief. The total area is13:69 km2 and elevations range from 80 to 240m.There were 127.6 km of contour lines in total, whichmeans that the average spacing between thecontours is 107m. The grid resolution should beat least 53.5m to present the most of the mappedchanges in relief. I then derived the distance fromthe contours map using the 5m grid and displayedthe histogram of the distances (Fig. 12b) to derivethe 5% probability distance. Absolutely shortestdistance between the contours is 7m, and the 5%probability distance is 12.0m. Finally, I canconclude that the legible resolution for this dataset is within the range 12.0–53.5m. Finer resolutionsthan 12m are unnecessary for the given complexityof terrain. Note that selection of the most suitablegrid resolution based on the contour maps is scaledependant. For the contour lines digitized from the1:5K topo maps (Fig. 12c), the average spacingbetween the lines is 26.6m and the 5% probability

distance is 1.6m. This means that, at 1:5K scale, therecommended resolutions are between 1.6 and13.3m.

4. Discussion and conclusions

There are three important concepts brought outin this paper that need to be further emphasized.First, principles from the general statistics andinformation theory, such as Nyquist frequencyconcept from signal processing and equations toestimate the probability density function, can beclosely related to selection of the grid resolution.Second, there are three standard grid resolutionsthat can be derived for each input data:

�
Coarsest legible grid resolution—this is the largestresolution that we should use given a specificscale, positional accuracy, size of objects or/andcomplexity of terrain. Using resolutions coarsestthan the coarsest legible resolution means that weare either not respecting the scale of work,

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Fig. 12. Selection of grid resolution based on complexity of terrain: (a) and (c) contours from 1:50 and 1:5K topographic maps, (b) and

(d) histograms of distances between contours for two scales.


positional accuracy, inspection density, size ofobjects being mapped or the complexity ofterrain.
� Finest legible grid resolution—this is the smallestgrid resolution that represents the most (95%) ofspatial objects or topography. This is the finestmeaningful resolution which corresponds to theconcept of the maximum location accuracy.Resolutions finer than the finest legible resolutionare probably just waste of memory. � Recommended grid resolution—this is a compro-mising resolution, usually set as the intermediatenumber between the coarsest and finest resolu-tions.
Third, the choice of the grid resolution needs to beconsidered in relation to a number of inherentproperties of a dataset or study area. The para-

meters and summary equations to derive coarsestand finest grid resolutions discussed in this paperare given in Table 1. These formulas can now beintegrated within a GIS package to help inexper-ienced surveyors derive grid maps without doingextensive data preprocessing. A simple calculatorcan be found via the website listed at the beginningof the paper.

From all discussed statistical approaches todetermine the true optimal pixel size, two seems tobe most promising. The first is to select the pixel sizethat yields best predictive properties. For example,if we wish to use terrain parameters to predict soilproperties over entire study area, we can test terrainparameters derived at various resolutions and thenselect the one that offers the highest correlationcoefficient with the target variable (Florinsky andKuryakova, 2000). A problem of this approach is

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Table 1

Summary equations to select grid resolution: SN is scale factor, rE is positioning error, rE is average positioning error, a is average size of

delineations, aMLD is area of the minimum legible delineation, wMLD is width of narrowest legible delineation, A is surface of study area, N

is number of sampled points in study area, hij is spacing between closest point pairs, hij is average spacing between closest points, hR is

range of spatial dependence, m is number of point pairs within range of spatial dependence, and l is total length of contours

Aspect Coarsest legible resolution Finest legible resolution Recommended compromise

Working scale pSN � 0:0025 XSN � 0:0001 ¼ SN � 0:0005GPS positioning error p1:8 � rEðP¼99%Þ XrE �

ffiffiffipp

¼ 1:8 � rEðP¼95%Þ

Size of reference objects pffiffiap

4X

ffiffiffiffiffiffiffiffiffiffiwMLDp

2¼

ffiffiffiffiffiffiffiffiffiaMLDp

4

Inspection density p0:1 �ffiffiffiAN

qX0:05 �

ffiffiffiAN

q¼ 0:0791 �

ffiffiffiAN

qDistance between points p hij

2XhijðP¼5%Þ ¼ 0:25ð0:5Þ �

ffiffiffiAN

qSpatial dependence structure p hR

2XhijðP¼5%Þ ¼ hR �m

�13

Complexity of terrain p APl

XwMLD

2 ¼ A

2�P

l


that it can be fairly time consuming to test allpossible combinations of different grid resolutions.In addition, the prediction power versus gridresolution graph might give a set of different peaksfor different target variables, so that we cannotselect a single ‘optimal’ grid resolution. Finally, thisgrid resolution is then valid only for this study areaand its effects might be different outside the area.The second analytical approach to select theresolution is to derive information content fordifferent block size values and then select the onethat offers the richest amount of information perunit area (Bishop et al., 2001). In this case, only asingle combination of the prediction model andblock size should yield the highest informationcontent. Such principles still need to be refined andtested in various case study to see if they really leadto optimal grid resolutions with maximum informa-tion content for a given level of detail.

Selection of the right pixel size will remain anissue that is relative to application type and projectobjectives. Moreover, standard grid resolutions(20–200m in most cases) with which we worktoday, will soon shift toward finer and finer, whichmeans that we need to consider grid resolution intime context also. No absolute ideal pixel size exist,that is for sure. One should at least try to avoidusing resolutions that do not comply with theinherent properties of the input datasets.

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