Exploring the Geostatistical Method for Estimating the Signal-To-Noise Ratio of Images

AbstractThe signal-to-noise ratio (SNR) has been estimated forremotely sensed imagery using several image-based methodssuch as the homogeneous area (HA) and geostatistical (GS)methods. For certain procedures such as regression, analternative SNR (SNRvar), the ratio of the variance in thesignal to the variance in the noise, is potentially moreinformative and useful. In this paper, the GS method wasmodified to estimate the SNRvar, referred to as the SNRvar(GS).Specifically, the sill variance c of the fitted variogram modelwas used to estimate the variance of the signal componentand the nugget variance c0 of the fitted model was used toestimate the variance of the noise. The assumptions requiredin this estimation are presented. The SNRvar(GS) was estimatedusing the modified GS method for six different land-coversand a range of wavelengths to explore its properties. TheSNR*var(GS) was found to vary as a function of both wave-length and land-cover. The SNR*var(GS) represents a usefulstatistic that should be estimated and presented for differentland-cover types and even per-pixel using a local movingwindow kernel.

IntroductionRemote sensing of properties distributed spatially within thephysical environment involves measurement with uncer-tainty. As a result, a pixel z(x0) (of spectral response z atlocation x0) in a remotely sensed image can be viewed asbeing comprised of the true or underlying pixel value u(x0)plus some measurement error e(x0):

z(x0) = u(x0) + e(x0). (1)

The term e(x0) usually comprises systematic es(x0) andrandom er(x0) errors such that

e(x0) = es(x0) + er(x0). (2)

Measurement error can reduce the accuracy of predic-tion of both continuous (e.g., nitrogen content of vegetation)and categorical (e.g., land-cover) variables from remotely

Exploring the Geostatistical Method for Estimating the Signal-to-Noise

Ratio of ImagesP.M. Atkinson, I.M. Sargent, G.M. Foody, and J. Williams

sensed imagery (e.g., Lucht and Lewis, 2000; Valor et al.,2002; Foody et al., 2004). Thus, it is important that remotesensing researchers and practitioners are aware of suchuncertainty, particularly where sensors are new or little dataare available on sensor performance. However, quantitativeassessment of accuracy is often missing from the imageprocessing chain associated with remotely sensed data(Schott, 1997).

A problem which has generated interest within theremote sensing community is how best to quantify the errore(x) given only a remotely sensed image z(x) in a givenwaveband. It is generally not possible to estimate the errore(x0) directly for pixel location x0 given only the observedvalue z(x0). If the error were known, the true value wouldalso be known and the error could be removed. Rather,researchers have sought to estimate the expectation of theabsolute error E[|e(x)|] over all x, that is, to estimate theerror using a statistical model. Systematic error can often beassociated with a known source within the remote sensorsystem, and so often it can be removed (Wrigley et al., 1984;Nichol and Vohora, 2004). Thus, most attention has focusedon estimating over all x the expectation of the absoluterandom error, referred to here as random noise or just noise(Curran and Dungan, 1989; Roger, 1996; Smith and Curran,1996; Schowengerdt, 1997; Corner et al., 2003).

Random noise at a single location x0 can be estimatedby measuring repeatedly the variable of interest z(x0) andusing the mean mz(x0) of the observed values to estimate thetrue value u(x0) (i.e., assuming es(x0) = 0) and the standarddeviation sz(x0) to estimate the random noise z(x0) (Dugginet al., 1985). However, for remotely sensed images, repeatedmeasurement at a single location is impractical (for a giventime and date). Fortunately, several methods exist forestimating noise from a single remotely sensed image basedon the image data alone (Smith and Curran, 1999), and someof these are described below.

It is important to interpret an estimate of random noisesz for a given remotely sensed image z(xi),i 1,2, . . . ,n(where n is the number of pixels) in relation to the true orunderlying image u(xi) (Equation 1). Traditionally, thesignal, estimated by the mean spectral response mz, has beenexpressed relative to the noise, estimated by the samplestandard deviation, sz using a signal-to-noise ratio (SNR)

PHOTOGRAMMETRIC ENGINEER ING & REMOTE SENS ING J u l y 2007 841

P.M. Atkinson and G.M. Foody are with the School ofGeography, University of Southampton, Highfield,Southampton SO17 1BJ, UK ([email protected]).

I.M. Sargent is with the Ordnance Survey, Romsey Road,Maybush, Southampton SO16 4GU, UK.

J. Williams is with QinetiQ, St Andrews Road, Malvern,Worcestershire WR14 3PS, UK.

Photogrammetric Engineering & Remote Sensing Vol. 73, No. 7, July 2007, pp. 841850.

0099-1112/07/73070841/$3.00/0 2007 American Society for Photogrammetry

and Remote Sensing

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(Lo, 1986; Curran and Dungan, 1989; Lillesand and Kiefer,2000) as

(3)

Several image-based methods of estimating image noiseand image SNR have been developed. Smith and Curran (1999)describe five approaches that estimate image noise by mini-mizing the component of the local standard deviation sz(W(x))due to underlying variation, where W(x) is a local kernel and represents a specific wave band. These are the Homoge-neous Area (HA) (Duggin et al., 1985), Nearly HomogeneousArea (NHA) (Boardman and Goetz, 1991), Geostatistical (GS)(Curran and Dungan, 1989), Homogeneous Block (HB) (Gao,1993), and Multiple Waveband (MW) (Roger and Arnold, 1996)methods. In each of the five approaches the SNR is estimatedby comparing some estimate of the mean mz (representing thesignal) to the standard deviation sz (representing image noise),as seen in Equation 3. Two of these five approaches are nowdescribed briefly: the HA method and the GS method.

The HA Method for Estimating the SNRIn the simple HA method a small window W(x0) (hereafterabbreviated to W) is defined within which the underlyingvariation in the image is expected to be homogeneous (thesame everywhere), or as close to this as possible. The signalis estimated by the local mean mz(W) within W and thenoise is estimated with the local standard deviation sz(W).The SNR is estimated by

(4)

To apply this method in practice it is necessary to selecta window within which the variance of the underlyingimage (i.e., the variance due to the signal devoid of noise) isexactly zero or as close to zero as possible.

The GS Method for Estimating the SNRIn comparison to the HA method, the GS method is prefer-able because it is not necessary to select locally homoge-neous areas. It centers on estimation and modeling of thevariogram (strictly the semi-variogram), a function represent-ing spatial dependence of measured values (Matheron, 1965and 1971). Spatial dependence is the tendency for proximatepixels to take more similar values than more distant ones.The model fitted to the variogram is often composed ofseveral components: at least one structured componentcombined with a nugget component (see below). In the GSmethod, the mean estimates the signal whereas the squareroot of the variance of the nugget component of the vari-ogram estimates the noise. The advantage of this approachover the HA approach is that the presence of spatiallystructured underlying variation is not detrimental to theestimation of image noise. In fact, the presence of spatialstructure is helpful in separating the underlying variationfrom the noise as part of the process of modelling thevariogram (Curran and Dungan, 1989; Chappell et al., 2001).

Often in remote sensing, regression is used to predictthe property of interest at the ground using remotely sensedimage data. For procedures such as regression, theresearcher is interested in the deleterious effect of noiserelative to variation in the signal (as opposed to the signalitself). For this reason researchers have proposed alternativeSNR measures. Of particular interest is the SNRvar defined by

(5)SNR*var VARIANCESIGNALVARIANCENOISE

s2us2e

SNR* mlz(W)slz(W)

.

SNR* SIGNALNOISE

mzsz

.

The SNRvar has been applied commonly to remotelysensed images (e.g., Schowengerdt, 1997). The informationcontained in the SNRvar is fundamentally different to that inthe SNRmean (from this point onwards the SNR of Equations 3and 4 is referred to as the SNRmean to distinguish it clearlyfrom the SNRvar). While the denominators are basically thesame (one is the square of the other), the numerators differ.In particular, in the SNRmean the numerator is the meanspectral response (often within a local window), while inthe SNRvar the numerator is the variance of the spectralresponse within a local window.

The SNRvar contains useful information where the objec-tive is to use the information in the image to predict someproperty of interest using a model. An example would bewhere the objective is to regress the image variable (e.g.,reflectance) on some ground variable (e.g., leaf area index(LAI)) with which it is moderately correlated (commonly thecase in remote sensing). Then the SNRvar conveys the ratio ofthe variation that may be correlated with the ground variableto the variation that is not. The point is that the extent towhich the underlying spatial variation (and, therefore, theinformation resolved by an image) is affected by image noisemay be of great interest to remote sensing researchers wishingto evaluate the viability of a particular investigation beforeactually executing it. Therefore, the SNRvar may be useful incircumstances where the SNRmean is not.

In this paper, we propose using a modification of the GSmethod to estimate a SNRvar(GS) statistic in which the signaland noise components are estimated using the variogram.The new SNRvar(GS) is estimated using the modified GSmethod and evaluated in a range of circumstances. Atkinsonet al. (2005) found that the SNR*mean varied with land-covertype and this variation had important implications for useand interpretation of the statistic. That analysis is extendedhere to the SNR*var(GS). In the next section, the GS method isdescribed and adapted to allow estimation of the SNRvar(GS).

The Geostatistical MethodGeostatistics is a set of techniques for the analysis of spatialdata (Journel and Huijbregts, 1978; Isaaks and Srivastava,1989; Goovaerts, 1997). Central to geostatistics is thevariogram, as described in the introduction. The variogramis itself best viewed as a parameter of the random functionmodel, described below.

The RF modelContinuous random variables (RVs) Z(x0) are characterizedfully by the cumulative distribution function (CDF) whichgives the probability that the RV Z(x0) at location x0 is lessthan or equal to a given threshold z:

F(x0;z) = Prob{Z(x0) z} z (6)

A Random Function (RF) is the spatial equivalent of aRV in which the inter-dependence between any two pointlocations may be expressed as a function of separatingdistance (Matheron, 1965 and 1971; Journel and Huijbregts,1978; Isaaks and Srivastava, 1989; Goovaerts, 1997). Arealization of a RF is termed a Regionalized Variable (RV). In geostatistics, spatial data (e.g., a remotely sensed image)are modeled as a RV (Goovaerts, 1997).

The spatial law of the RF Z(x) is given by the n-variateor n-point cdf:

F(x1,...,xn; z1,...,zn) = Prob{Z(x1)z1,...,Z(xn)zn} z (7)

defined for any choice of n and any location x. The entirespatial law is not required for most applications. Rather, theanalysis is usually restricted to CDFs involving at most two

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locations at a time. Since repeated observations are usuallynot available for any fixed location x0, the analysis isrestricted further to fixed vectors of distance and directionknown as lags, h. Each pair of locations {x, x h} forwhich data are available are then treated as repeats of h.This strategy amounts to the decision to adopt a stationaryRF model. Stationary, in this context, means that the vari-ogram parameter is constant across space. Such a decisionhas implications for the implementation of the modified GSmethod to estimate SNRvar(GS).

When the expected value E{Z(x)} exists and is independ-ent of x (invariant within a region V) we have

E[Z(x)Z(x + h)] = 0. (8)

If the variogram exists and depends only on h then

(9)

and the RF is said to be intrinsically stationary. The vari-ogram may exist given intrinsic stationarity only, but thecovariance function and autocorrelation function (whichimply second-order stationarity) may not exist.

Variogram Estimation and ModelingFor continuous variables, the sample semivariance isdefined as half the average squared difference betweenvalues separated by a given lag h. The sample variogrammay be estimated using:

(10)

where P(h) is the number of paired comparisons at aspecific lag h and z(x) represents an observation or realiza-tion of Z(x) at location x. Further, the variogram may beobtained either as an average representing all directions(omnidirectional) or for several different orientations wherevariation is anisotropic (i.e., depends on orientation). In asingle remotely sensed image many observations areavailable allowing greater confidence in the estimatedvariogram and increasing the number of orientations inwhich it can be estimated.

The estimated variogram obtained from Equation 11 is aset of semivariances at a discrete set of lags only. To allowstatistical prediction, it is customary to fit a continuousmathematical model to the sample variogram. The modelfitted must be such as to ensure that linear combinations ofthe RF cannot have negative variances. This property isreferred to as conditional negative semi-definiteness (CNSD).It is common practice to select a model from several whichare known a priori to be CNSD (McBratney and Webster,1986; Webster and Oliver, 1990). Two such models are givenbelow, both of which are used in this paper.

1. The nugget effect model:

(11)

where c0 is the nugget variance; and2. The spherical model:

(12)

where c is the sill variance or structured component and a is the range or non-linear parameter. In simple terms, thefitting procedure is as follows. The user chooses a type of

g(h) c 1.5

ha

0.5 ha 3

1

if h a

otherwise

g(h) c0 01 if h 0

otherwise

g(h) 1

2P(h)

P(h)

a1[z(xa) z(xa h)]2

g(h) 12

E {Z(x) Z(xh)}2

variogram model and decides whether an anisotropic modelis required (based on directional variograms). A model isthen fitted to the sample variogram usually by some auto-matic process such as weighted least squares (e.g., Cressie,1985 and 1991), that is, by minimizing:

(13)

where WSS is the weighted sum of squares, and (hk) is aset of weights at k 1,2, . . . K lags hk (Goovaerts, 1997).

Interpreting the Modeled VariogramConsider the coefficients of a fitted spherical model. The sillc of the spherical model provides information on theamount of variation present in V. More precisely, the sillestimates the a priori variance D2(v,) of Z (that is thevariance obtained on a support v within an infinitely sizedregion). The dispersion or sample variance D2(v,V) of Z (thatis, the variance obtained on a support v within a region V)is also predictable from the variogram through numericalintegration. The range a of the model provides informationon the scale of spatial variation.

The variogram models described may be used singly orin a positive linear combination (Goovaerts, 1997). Thenugget model is often fitted together with a structuredcomponent (e.g., spherical model) to represent a discontinu-ity at the origin. The nugget variance c0 (the sill of thenugget component) represents unresolved variation thatexists at a micro-scale (at an interval smaller than thesmallest distances of separation in the sampling frame) andmeasurement error, but it can also arise from uncertaintyboth in estimating the variogram and in fitting the model atshort lags (Atkinson, 1997). Where the amount of micro-scale variation is believed to be small (e.g., because observa-tions are abutting or overlapping, as for remotely sensedimagery), then the nugget variance can be assumed to bedue primarily to measurement error. This assumptionunderlies the Curran and Dungan (1989) GS method forestimating the SNR.

The Modified GS MethodThe two key arguments supporting the GS method are asfollows. First, noise is not correlated with the signal. Thisimplies that the variogram of the image z(h) is simply thesum of the variograms for the underlying variation u(h) andthe noise e(h):

z(h) = u(h) + e(h). (14)

Second, since imagery is already convolved by thepoint-spread function (PSF) of the sensor, proximate pixelsare likely to be highly autocorrelated. This means that in thelimit, that is, as h approaches zero, the contribution fromthe signal to the variance will approach zero.

(15)

Estimating SNRvarHere we present a new and simple estimator of the SNRvar.The SNRvar may be estimated by the ratio of the two vari-ogram model coefficients, c (representing the variance in theunderlying image, and potentially the sum of more than oneindividual sill component) and c0 (representing the variancedue to noise):

(16)SNR*var(GS) cc0

.

limit(h : 0)

gz (h) s2e.

WSS K

k1(hk)[g(hk) g(hk)]2


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It is implicit in such an approach that a second-orderstationary RF model must be adopted (i.e., so that the covari-ance exists) and a bounded model (with a clearly defined sill,e.g., the spherical model) must be fitted to the variogram ofthe stationary RF (Myers, 1989; Goovaerts, 1997).

In the analysis that follows, the SNRvar(GS) is estimatedfor six different land-cover types and for a range of differentwavebands to evaluate the behavior of the statistic. First, thefield site and data used are introduced briefly.

Field Site and DataA Compact Airborne Spectrographic Imager (CASI) image wasacquired over Falmouth, Cornwall in July 1999 by the UK

Environment Agency. The image was acquired in EnhancedSpectral Mode with 72 spectral bands of approximately 8.3nm width that were contiguous across the spectrum from400 nm to 900 nm. The increased integration time requiredfor these narrow wavebands results in pixels that aremarginally longer than they are wide. However, the imagewas resampled so that the pixels represented an area ofapproximately 4 m by 4 m.

The scene covered by the CASI image included a rangeof land-cover types (i.e., varying crop types). Six areascontaining relatively homogeneous areas of specific land-cover types were chosen. The spatial and spectral data forpixels contained wholly within these areas were extracted.The six regions are shown in Figure 1.


Figure 1. CASI sub-images, with a spatial resolution of approximately4 m, representing areas of (a) barley, (b) cauliflower, (c) corn, (d)grassland, (e) potato, and (f) woodland. (Atkinson et al., 2005; usedwith permission from International Journal of Remote Sensing,http://www.informaworld.com).

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Analysis

Example 1Two (of the available six) land-cover types, grassland andwoodland, were selected to illustrate the SNR*var(GS). Theremaining four land-cover types exhibited variation that was

similar to that for grassland, and so they were omitted toreduce redundancy in the presentation. Omnidirectionalvariograms for the two land-cover types, and for a sample ofeight wavebands (wavebands 1, 11, 21, 31, 41, 51, 61, and71), are shown in Figure 2. Variograms were estimated andmodeled for all wavebands, but only the sample of eight is


Figure 2. Variograms of (a, c, e, g, i, k, m, and o) woodland and (b, d, f, h, j, l,n, and p) grassland for wavebands (a and b) 1, (c and d) 11, (e and f) 21, (gand h) 31, (i and j) 41, (k and l) 51, (m and n) 61, and (o and p) 71 sampledfrom the complete set of 72 wavebands. The models shown were fitted usingweighted least squares approximation. Note that to allow the models to beshown fully, the scale on the abscissa for woodland reaches a maximum of 40 pixels (160 m), whereas for grassland the maximum is 14 pixels (56 m).The scales of the ordinates on all plots differ for the same reason. (Atkinson et al., 2005; used with permission from International Journal of Remote Sensing,http://www.informaworld.com).

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shown, again to reduce redundancy. All variograms wereestimated and fitted with models using weighted leastsquares approximation using the GSTAT software (Pebesmaand Wesseling, 1998). The wavelength ranges for thesewavebands are 401.6 nm to 501.3 nm (14 slightly overlap-ping bands with a bandwidth of 8.2 nm) and 500.1 nm to917.7 nm (58 slightly overlapping bands with a bandwidthof 8.4 nm). The variograms for the grassland and woodlandclasses were all fitted best (in the WSS sense) with a spheri-cal or nested spherical model combined with a nuggetcomponent.

The SNRvar was estimated using the modified GSmethod (Equation 16). The sill variance, the nugget vari-ance and the SNRvar are all plotted against wavelength foreach land-cover type in Figure 3 (note the different scaleson the ordinates of the plots). Three observations can bemade. First, the sill variance is highly dependent onwavelength and land-cover type. This is not surprising: asdescribed in the Geostatistical Method Section, the overallsill (c1, . . . ,cn and c0) represents the a priori varianceD2(v,) of the imagery per land-cover. This result simplymeans that variance in remotely sensed imagery is a


Figure 2. (Continued) Variograms of woodland and grassland for designatedwavebands. (Atkinson et al., 2005; used with permission from InternationalJournal of Remote Sensing, http://www.informaworld.com).

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function of wavelength and land-cover type. Geostatisticaldescriptions of texture have been devised (e.g., Carr, 1996)to exploit such variation to help to discriminate betweenland-cover types. Second, the nugget variance is alsodependent on wavelength and land-cover type, but doesnot follow exactly the same pattern as the sill variance.Third, as a consequence of the above relations, theSNR*var(GS) is also dependent on wavelength and land-covertype. Variation in the SNR with wavelength is well knownand understood. Variation of SNR*var(GS) with land-cover isnot surprising, but it is interesting. It means that thepredictive ability of models such as regression and classifi-cation models may also depend on variation in the ratio ofsignal variance-to-noise variance with land-cover type andwavelength. Thus, the SNR*var(GS) statistic might be used toevaluate predictive models prior to their application.

Example 2To investigate the dependence of SNR*var(GS) on land-cover typefurther, we conducted a second study using a single waveband.

A single, near-infrared waveband (880.1 to 888.5 nm)was extracted for each of the six land-cover types. Thisband was chosen using feature selection (the band whichproduced the greatest Mahalanobis distance between the

two spectrally closest bands) (Mather, 2004). Omnidirec-tional variograms were estimated and modeled as before.The nested spherical model combined with a nuggetcomponent provided the best (WSS) fit in each case exceptfor the variogram for the land-cover cauliflower for which asingle spherical component (plus nugget) was satisfactory(Figure 4). The woodland variogram is different to theothers in that (a) it has a markedly different scale on theordinate, and (b) a sizeable short-range structure with arange of about four pixels (or 16 m) is included, which ismost likely related to the size of the tree crowns.

The SNRvar(GS) was estimated using the modified GSmethod. The sill variance c, the nugget variance c0 and theSNR*var(GS) are all plotted against land-cover type in Figure 5(again, note the different scales on the ordinates of theplots). The differences in the estimates of c and c0 and theSNR*var(GS) between woodland and the other land-covers areimmediately apparent. This may be a function of the largenear-infrared reflectance for woodland. However, notice thatthe differences in the signal variance c are much greaterthan the differences in the noise variance c0. Examiningonly those classes representing crops or grasses there stillexist important differences in c, c0 and the SNR*var(GS)between land-cover types. The differences in SNR*var(GS)

Figure 3. The (a and b) sill variance, (c and d) nugget variance, and(e and f) SNR*var(GS) plotted against wavelength for both (a, c, and e)woodland and (b, d, and f) grassland. (Atkinson et al., 2005; usedwith permission from International Journal of Remote Sensing,http://www.informaworld.com).

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between all classes are large: the smallest value is less than3 (potato) and the largest value is greater than 80 (wood-land). This means that if the objective were to predict somebiophysical (e.g., leaf area index, biomass) or biochemical(e.g., nitrogen content) property from the imagery, then, apriori, the predictions for the woodland class would beexpected to be less affected by noise.

DiscussionWe now summarise the findings of this paper, make somesuggestions for improving the GS method for estimating theSNRvar(GS) and make some observations about the variogramrange coefficient.

Variation in SNR*var(GS) with Land-coverThe SNR*var(GS) varied with wavelength and land-cover type.Variation with land-cover type arises primarily becausevariance in the signal is a function of land-cover (e.g., it iswell known that different land-covers have different textures).However, variation in the SNR*var(GS) arises to some extent as afunction of the dependence of noise on land-cover type as

discussed by Atkinson et al. (2005). Specifically, componentsof the noise may be due to atmospheric effects that interferewith the signal such that the noise is partially a function ofthe signal.

Variation in SNR*var(GS) with land-cover, as demonstratedhere, means that it is important that local variation inSNR*var(GS) is presented. The SNRvar(GS) should at least beestimated and presented for each land-cover class. Preferably,the SNRvar(GS) should be estimated locally and the resultingestimates displayed as images (Atkinson et al., 2005).

Improving the GS MethodAn advantage of the GS method over other methods ofestimating the SNRvar is that it can be estimated by land-cover class, or locally per-pixel using a local kernel ormoving window leading to an image of SNR*var(GS) statistics.The suggestion that the SNR*var(GS) might be presented as animage has implications for how the GS method might beimproved. For example, it would be possible to estimate adistribution of noise estimates. This distribution could beused to refine the noise value used in estimating SNRvar(GS).For example, the smallest or near-smallest (e.g., 5-percentile)


Figure 4. Variograms estimated from the sub-images representing areas of (a)barley, (b) cauliflower, (c) corn, (d) grassland, (e) potato, and (f) woodland.The models shown were fitted using weighted least squares approximation.(Atkinson et al., 2005; used with permission from International Journal ofRemote Sensing, http://www.informaworld.com).

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value of SNR*var(GS) could be used to provide a cautious orconservative estimate of image SNRvar(GS) or one that wasleast contaminated by signal. Alternatively, the distributionof SNR*var(GS) estimates might be useful to investigators. Forexample, the distribution of values could be used to provideinformation on the uncertainty inherent in image-basedestimation of SNRvar(GS).

The variograms estimated in this paper were all omnidi-rectional, meaning that isotropy was assumed. While in thepresent case we checked for anisotropy, isotropy is notguaranteed generally. Anisotropy may be useful given thepresent objective: where the variogram varies with orientationdifferent estimates of c0 may arise. Then, it may be possibleto utilize such differences to obtain a more accurate estimateof noise than that provided by modeling the omnidirectionalvariogram. For example, given that the variogram models foreach orientation are of the same basic type it may be prefer-able to select the smallest c0, rather than the average, on thebasis that this reduces the possibility of underlying variationcontributing to the estimate.

Another property that may be useful for estimatingnoise is the correlation between adjacent bands. This

correlation should be close to one for much hyperspectraldata. However, the noise may be uncorrelated even betweenadjacent bands, depending on its source. Where this is thecase, the cross-variogram (representing the cross spatialdependence or cross-correlation between two variables)should yield useful information on noise. The cross-vari-ogram is due to underlying variation alone and contains nocomponent due to noise. This occurs where the noise termsare independent, and thus do not contribute to the cross-variogram (e.g., van der Meer and Bakker, 1997). Thus, forhyperspectral imagery, the cross-variogram represents apotentially useful alternative to the variogram.

Variogram Model RangesInterestingly, the variograms estimated from the woodlandclass produced a slightly larger total range (maximum of thetwo ranges of the nested variogram models) than did thevariograms of grassland (Figure 2). The full set of modelranges for each of the two land-cover types is shown inFigure 6. For grassland the range was relatively constantwith wavelength. However, for woodland there was a smallincrease in the range at around the position of the red-edge(Curran, 1994). Such variation with wavelength has beenobserved by others (Chavez, 1992; Atkinson and Emery,1999) and has implications for techniques for processinghyperspectral imagery (e.g., Atkinson and Emery (1999)found that strategies for sampling reflectance needed to bewavelength-dependent).

ConclusionThe SNR*var(GS) represents a useful image-based statistic thatmay be interpreted as the ratio of the component of variancethat may be correlated with (or useful in predicting) avariable of interest to the component of variance that is not


Figure 5. The (a) sill variance, (b)nugget variance and (c) SNR*var(GS)plotted as bar charts againstbarley, cauliflower, corn, grass-land, potato, and woodland.

Figure 6. The range plottedagainst wavelength for both (a) woodland, and (b) grassland.

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correlated with that variable (i.e., noise). Such information isof obvious utility in remote sensing where the objective defacto is to predict ground properties from images that are afunction of those ground properties. The GS method pro-vides a simple means of estimating this ratio. However, careshould be exercised when presenting this ratio. This paperhas shown that the SNR*var(GS) varies as a function of bothwavelength (as expected) and land-cover class. Therefore,the SNRvar(GS) should at least be estimated and presented perland-cover class and preferably as an image of local values.It would be interesting to extend this analysis to include aquantitative interpretation of the effects of the SNR*var(GS) onregression and classification models.

AcknowledgmentsThe authors thank the Environment Agencys NationalCentre for Environmental Data and Surveillance for con-tributing data. The research was performed with the aid offunding derived from the UK Ministry of Defence Corporateand Applied Research Programmes.

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(Received 5 August 2005; accepted 28 November 2005; revised 31January 2006)


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Exploring the Geostatistical Method for Estimating the Signal-To-Noise Ratio of Images