Remote Sensing of Environment 145 (2014) 131–144

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Remote Sensing of Environment

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Global sensitivity analysis of the spectral radiance of asoil–vegetation system

Alijafar Mousivand a,⁎, Massimo Menenti a, Ben Gorte a, Wout Verhoef b

a Department of Geoscience and Remote Sensing (GRS), Delft University of Technology (TUDelft), Stevinweg 1, 2628 CN Delft, The Netherlandsb Faculty of Geo-Information Science & Earth Observation (ITC), University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands

⁎ Corresponding author. Tel.: +31 15 2788338; fax: +E-mail addresses: A.Mousivand@tudelft.nl (A. Mousiva

(M. Menenti), B.G.H.Gorte@tudelft.nl (B. Gorte), w.verhoe

http://dx.doi.org/10.1016/j.rse.2014.01.0230034-4257/© 2014 Elsevier Inc. All rights reserved.

a b s t r a c t

a r t i c l e i n f o

Article history:Received 18 July 2013Received in revised form 24 January 2014Accepted 28 January 2014Available online 26 February 2014

Keywords:Global sensitivity analysisCoupled radiative transfer modelSLCMODTRANWinding stairs samplingSoil–vegetation systemTOA radianceImagine spectroscopy

We analyzed the sensitivity of Top-Of-Atmosphere (TOA) radiance and surface reflectance of a soil–vegetationsystem to input biophysical and biochemical parameters using the coupled Soil–Leaf-Canopy radiative transfermodel SLC and MODTRAN. We applied variance-based global sensitivity analysis for different atmospheric con-ditions and observation configurations. Among 23 input parameters, crown coverage, leaf area index, leaf incli-nation distribution function and soil moisture were found to be the most influential parameters driving theoutput variance of the radiance between 400 and 2500 nm with a few exceptions. Hapke's soil parameters andthe canopy layer dissociation factor were recognized to have marginal influence on the output radiance. It isalso found that a large portion of uncertainty in the output radiance is driven by the interaction effects amonginput parameters in the visible (~550 nm),whereas the red-near infrared (~670 nm), seems to have fewer inter-action effects. The effect of solar/view direction is found to be significant on TOA radiance sensitivity to the inputparameters. The results also confirmed that the sensitivities of surface reflectance are comparable to the TOA ra-diance sensitivities when the atmosphere is clear and visibility is high. Since coupled surface-atmosphere RTmodels can be computationally intensive, this work also introduces an improvement to the design and samplingof screening methods for efficient sensitivity analysis of computationally expensivemodels. The improvement isbased on three elements: a) generating sample points by Sobol's sequence generator; b) variational analysis inthe parameter space using the winding stairs method; c) use of mean and variance sensitivity measures. The re-sults with 1200model runs demonstrated high correlation (92%) with variance-based global sensitivity analysisusing 49,152 model runs, in determining the most influential and non-influential parameters.

© 2014 Elsevier Inc. All rights reserved.

1. Introduction

Terrestrial targets are characterized by heterogeneity at all scalesand the observed spectral radiance across the 0.4 μm–2.5 μmspectral region is determined by a complex combination of targetgeometry and bio-geophysical and chemical composition of targetelements (Jacquemoud & Baret, 1990; Verstraete, Pinty, & Myneni,1996). Extraction of information on observed targets from measure-ments of spectral radiance becomes significantly more efficient ifsome a priori knowledge is available about parameter and model un-certainties (Goel, 1988; Kimes, Knyazikhin, Privette, Abuelgasim, &Gao, 2000; Verstraete et al., 1996). This knowledge specifies whichtarget properties are the most significant determinants of spectralradiance observed in a specific spectral region. To this end, either exten-sive and detailed experimental data or a model of radiative transfer inthe soil–vegetation–atmosphere system can be used. A sufficiently

31 15 27 83711.nd), M.Menenti@tudelft.nlf@utwenteitc.nl (W. Verhoef).

realistic model makes exploration of a broad range of target and ob-servation conditions easier and effective through a systematic analy-sis of the sensitivity of modeled spectral radiance.

For given parameters, a sensitivity study quantifies how mucheach parameter contributes to the observed spectral radiance. Thisindicates wavelength dependency and identifies spectral ranges(bands) containing the most information about specific parameters.Identifying which target properties (parameters) control spectralradiance at a given wavelength and which ones have a negligible in-fluence on spectral radiance at the same wavelength is very usefulfor quantitative remote sensing. For instance, identification of themost influential parameters on existing measurements of radiancehelps to identify retrievable parameters (Bicheron & Leroy, 1999;Ceccato, Gobron, Flasse, Pinty, & Tarantola, 2002; Kimes et al.,2000; Mu, Yan, & Li, 2008). Furthermore, knowing the response ofspectral radiance to biophysical parameters helps to determine ap-propriate spectral band combinations for the estimation of parame-ters of interest (Goel & Thompson, 1985; Kimes et al., 2000). Ananalysis of the sensitivity of modeled spectral radiance to targetproperties can address the issues mentioned above.

132 A. Mousivand et al. / Remote Sensing of Environment 145 (2014) 131–144

There is a variety of methods to conduct Sensitivity Analysis (SA)and different aspects of SA should be considered. Consensus about anexact terminology does not seem to exist yet (Cacuci, Ionescu-Bujor, &Navon, 2005). We have distilled from the literature a conceptualscheme about aspects and methods of sensitivity analysis. We maydistinguish three aspects of SA briefly reviewed in the followingparagraphs.

1.1. Sampling of the parameter space

SA may be carried out by exploring the neighborhood of a singlelocation in the parameter space (local SA) or by considering a num-ber and distribution of samples sufficient to characterize reliablythe entire parameter space (global SA). Evaluation of sensitivity ata single location or at a limited number of locations can be useful toevaluate well-defined target and observation conditions. In general,a systematic exploration of the parameter space is required. Theideal solution therefore would be to employ a full factorial design,i.e. testing all possible combinations of parameter values, to explorethe entire parameter space. However, in the specific case of radiativetransfer in the Soil–Vegetation–Atmosphere system, full factorial de-sign is not practical. In some cases, an approximation of the full fac-torial design can be achieved with a reduced number of samples,using a fractional factorial design (Box, Hunter, & Hunter, 1978).When only a fraction of all possible combinations is considered, thedistribution of samples is important to select an appropriate and rep-resentative sample set. In fact, sample size and sampling scheme(distribution of the samples) determine howmuch reliable informationcan be extracted from SA (Campolongo, Cariboni, & Saltelli, 2007). Effi-cient sampling of the parameter space requires some a priori knowledgeof the probability distribution of parameter values to adapt the sam-pling of the parameter space to the likely parameter values. This ensuresthat the unlikely parameter values and unlikely combinations are notgiven equal weight to the more likely ones.

1.2. Variation of parameters

We may change the values of parameters “One (parameter) At aTime” (OAT) or we may change “All selected relevant Parameters Si-multaneously” (APS). The OAT approach has been the most frequentlyused approach in both remote sensing (Asner, Wessman, Bateson, &Privette, 2000; Bach & Verhoef, 2003; Barton & North, 2001; Bicheron& Leroy, 1999; Peres & DaCamara, 2004; Privette, Myneni, Tucker, &Emery, 1994; Wang, Qu, Hao, & Zhu, 2008; Yao, Liu, Liu, & Li, 2008)and other fields (see Saltelli et al., 2008) because of its simplicityand inexpensive computational time. The OAT approach in remotesensing basically measures the sensitivity to a specific parameterby calculating discrete partial derivatives of spectral reflectance/radiance with respect to the input parameters (target and observationconditions). The APS sensitivity analysis scans the entire parameterspace, for examplewith aMonte Carlo approach, to explore the possiblerange of variation for all parameters. In an APS exercise, all the param-eters are varied simultaneously which provides information on thepossible interactions among them as well. Although not extensive,the literature does contain a few studies that applied APS methods toquantify the sensitivity of output radiance/reflectance to input parame-ters of interest in remote sensing (Bacour, Jacquemoud, Tourbier,Dechambre, & Frangi, 2002; Ceccato et al., 2002; Morris, Kottas, Taddy,Furfaro, & Ganapol, 2008; Pingheng & Quan, 2011).

1.3. Sensitivity measures

Measures of sensitivity for local sampling of the parameter spaceare limited to deterministic approaches such as partial derivatives. Inthe more general case of the exploration of the entire parameterspace, statistics have to be defined and calculated to measure the

sensitivity over a large number of sample points and for all parametersconsidered (Cacuci et al., 2005).With this approach, the output varianceis decomposed into fractions of total variance which quantify thecontribution of one or more parameters to the output variance.Many statistical sensitivity measures are currently in use; well-known are correlation-based methods (Saltelli & Marivoet, 1990),regression-based (Rabitz & Aliş, 1999), variance-based sensitivity mea-sures (e.g., Fourier Amplitude Sensitivity Test, FAST (Cukier, Fortuin,Shuler, Petschek, & Schaibly, 1973), Sobol's method (Saltelli, 2002;Sobol, 1993) and more recently density-based methods (Borgonovo,2006). These methods present elaborate and accurate sensitivitymeasures in which simultaneous parameter variation and robust sta-tistical measures are combined. However, they are computationallyexpensive and complex because of the required large number ofmodel evaluations (Saltelli et al., 2008).

The computational burden of a coupled surface-atmosphere radi-ative transfer (RT) model run depends strongly on the calculationmode and the accuracy of the computation. As an example, thecoupled Soil–Leaf-Canopy RT model SLC (Verhoef & Bach, 2007)and MODTRAN (Berk et al., 1998, 1999) run may range from a fewseconds, in a very simple case, to several minutes running the atmo-spheric code in an accurate mode. This makes the use of sophisticatedmethods impractical. The challenge addressed here is how to design amethod to estimate sensitivities with an accuracy comparable to so-phisticated global methods but with much less computational load.Morris (1991) proposed the elementary effect method as a screeningmethod to identify a subset of non-influential inputs among a largenumber of parameters at a low computational cost. Though reducingthe computational load, screening methods still suffer from someproblems associated with the random sampling strategy which willbe disscussed in Section 3.2.1. Moreover, the atmosphere influencesTop-Of-Atmosphere (TOA) radiance considerably. In order to give anindication of the impact of the atmosphere on the sensitivity of dif-ferent spectral bands to the input parameters, we also investigatethe sensitivity of Bottom-Of-Atmosphere (BOA) reflectance to com-pare the results with the TOA radiance sensitivities.

This study aims at addressing four issues: (i) proposing an improve-ment of the design and sampling of screeningmethods for efficient sen-sitivity analysis of computationally expensive models (ii) using SA toidentify influential and non-influential parameters on TOA radiance,(iii) comparison between TOA radiance sensitivity and BOA reflectancesensitivity to input parameters, and; (iv) investigating the effect ofviewing and illumination geometry on TOA radiance sensitivity. Inorder to achieve these goals, we estimate sensitivity of TOA radianceand BOA reflectance to input parameters for a variety of sun-target-sensor observation geometries and three different atmospheric visibili-ties by both the proposed design and a variance-based method. Weevaluate the efficiency and accuracy of the proposed design againstthe variance-based method.

The remainder of the paper is organized as follows. In Section 2, weintroduce the approach used for the proposed design. Section 3.1 de-scribes the radiative transfermodels applied to simulate top and bottomof atmosphere radiance and reflectance. We discuss various aspects ofsensitivity analysis and themethodologies used to estimate sensitivitiesin detail in Section 3.2. Section 4 presents the results of the evaluation ofthe proposed design and the results of the application of sensitivityanalysis to identify influential and non-infleuntial parameters. The com-parison of TOA and BOA sensitivities and the effect of viewing and illu-mination geometry on TOA radiance sensitivity are also discussed in thissection. Finally, Section 5 concludes the paper with a summary.

2. Approach

The analysis of the sensitivity of TOA radiance andBOA reflectance totarget composition and architecture was carried out using the coupled

133A. Mousivand et al. / Remote Sensing of Environment 145 (2014) 131–144

SLC andMODTRAN radiative transfer models according to the approachdescribed here.

2.1. Sampling of the parameter space

We applied the Sobol sequence generator (Sobol, 1993) to selecta number (n) of baseline points in the parameter space. Sobol's se-quence generator distributes new sample points using the location ofprevious points. This maximizes the uniformity of the sampling and,unlike random sampling, prevents gaps or clusters to appear. Thesesample points are distributed within a k-dimensional space, where k isthe number of the input parameters.

2.2. Variation of parameters

The values of parameters at a baseline point are perturbed se-quentially one at a time (see Fig. 1a). Each step reaches a new pointin the parameter space that is different from the previous samplepoint in only one dimension (one input parameter). The procedurecontinues until all input parameters are perturbed once and endsup at a new sample point which is different from the starting point. Inthis way for each baseline point, k + 1 new points are constructed se-quentially to form a trajectory (i.e. a sequence of sample points) in the

Fig. 1. Examples of blocks inwinding stairs sampling (a) and radial design sampling (b) ina three dimensional hypercube.

parameter space. Having n baseline points, at the end, there are n trajec-tories comprising n(k + 1) sample points. The cost of the method (thenumber of required model runs) is then n(k + 1).

2.3. Measures of sensitivity

Along a trajectory, the ratio of the variation in the output radiancesof two successive sample points to the variation in the input parameteritself is calculated. This is the measure of the effect of the perturbedparameter along that segment of the trajectory. To compute sensitiv-ities of spectral radiance to a certain input parameter over the entiresample space, the mean value and standard deviation of these ratiosare calculated for each wavelength. Furthermore, a confidence levelof 95% has been set for each sensitivity index to ascertain that the esti-mated results are consistent. This has been done by bootstrapping anal-ysis. To evaluate the proposed design, we have compared it with thevariance-based method. The comparisons are done on the basis ofboth direct comparison of spectral sensitivity and of parameter ranking.In the latter case, sensitivity results are transformed into ranks in orderof importance and the correlation coefficients of the ranks, estimated byKendall's tau approach (Abdi, 2007), are compared.

3. Methods and data

3.1. Top-Of-Atmosphere (TOA) radiance

TOA radiance is defined as a combination of surface reflectance, at-mospheric effects and target's surroundings. This combination can bemodeled effectively using the four-stream radiative transfer theory pro-posed by Verhoef and Bach (2003, 2007, 2012).

LTOA ¼ Eso cosθsπ

ρso þτss rsd þ τsdrdd

1−rddρddτdo þ τssrsoτoo þ

τsd þ τssrsdρdd

1−rddρddrdoτoo

� �

ð1Þ

The definitions are given in Table 1; the overbars indicate spatialfiltering over the surroundings of the target to account for the adja-cency effect. The four successive summands on the right-hand sidecorrespond to the contributions related to (i) target-independent at-mospheric path radiance, (ii) adjacency effects, (iii) directly reflectedsolar illumination from the target, and (iv) directly reflected sky illumi-nation on the target. The four-stream approach requires four surfacereflectance factors plus seven atmospheric coefficients to simulateTOA radiance (Table 1). The surface reflectance components are calcu-lated by the SLC radiative transfer model and the atmospheric parame-ters are calculated by the MODTRAN atmospheric model. We haveneglected the adjacency effect here since we are calculating TOA radi-ance for a single target. Surface bidirectional reflectance simulated bythe SLC has also been used for the BOA reflectance sensitivity analysis.

3.1.1. Soil–Leaf-Canopy (SLC) RT modelThe integrated Soil–Leaf-Canopy RT model SLC (Verhoef & Bach,

2007) was used to calculate Top-Of-Canopy (TOC) reflectance. SLCsimulates TOC reflectance spectra over the spectral range between400 and 2500 nm for arbitrary sun-target-sensor geometries. Themodel comprises structural and optical properties of the canopy,leaf and soil and the observation and illumination geometry. Infact, SLC is a four-stream extended version of the SAIL RT model(Verhoef, 1984) which uses four fluxes, including direct solar flux,diffuse downward flux, diffuse upward flux and directly observedflux (radiance). The model generates as outputs four reflectance fac-tors of the canopy plus two canopy absorptances (Verhoef & Bach,2007). The four canopy reflectances are rso, rdo, rsd, and rdd inTable 1 and the two canopy absorptances are canopy absorptancefor direct solar incident flux and canopy absorptance for hemispher-ical diffuse irradiance.

Table 1Variables modeled by SLC and MODTRAN used in the study.

Variable Description RT model

LTOA TOA radiance –

rso Surface bidirectional reflectance factor SLCrdo Surface hemispherical–directional reflectance for diffuse incidence SLCrsd Directional–hemispherical reflectance over surroundings SLCrdd Bi-hemispherical reflectance over surroundings SLCρso TOA atmospheric bidirectional reflectance MODTRANρdd BOA spherical albedo of the atmosphere MODTRANτss Direct atmospheric transmittance from the sun to the ground MODTRANτoo Direct atmospheric transmittance from the ground to the sensor MODTRANτsd Diffuse atmospheric transmittance from the sun to the ground MODTRANτdo Diffuse atmospheric transmittance from the ground to the sensor MODTRANEs

o Extraterrestrial solar irradiance on a plane perpendicular to the sunrays –

θs Local solar zenith angle –

134 A. Mousivand et al. / Remote Sensing of Environment 145 (2014) 131–144

Unlike its predecessors, SLC can treat the soil background as ananisotropic reflector. To simulate soil reflectance and its variabilitywith moisture, a non-Lambertian extended version of Hapke's soilBRDF model (Hapke, 1981) containing the hotspot and soil moistureeffects has been incorporated. The PROSPECT leaf reflectance model(Jacquemoud & Baret, 1990) is applied to compute transmittanceand reflectance of green and brown leaves, in a version that can con-sider brown pigments (Verhoef & Bach, 2003). The 4SAIL2 model(Verhoef & Bach, 2007) is employed to simulate canopy reflectanceas a two-layer structure canopy reflectance model where a crownclumping effect (as in forests) is taken into account. The clumping ef-fect is described by two additional parameters; vertical crown cover-age (Cv) and tree shape-factor (Zeta). The latter is equal to the ratioof crown diameter to crown height. Furthermore, two parametersare used in 4SAIL2 to characterize the mixture of green and brownleaves in the canopy, namely the fraction of brown leaf area (fb) and adissociation factor (D). When all green leaves are in the top layer andall brown leaves are laid down in the bottom layer, the dissociation fac-tor is equal to 1. In case of dissociation equal to zero, all leaves are ho-mogeneously mixed over the two layers.

The 4SAIL2model accuracy has been verified by comparison againstother models in the RAMI project (Widlowski et al., 2007) and withsatellite and ground measurements (Verhoef & Bach, 2007). In theRAMI-3, the 4SAIL2 is shown to be in good agreement with the en-semble of all other models with less than 4% difference in BRF simu-lations. Verhoef and Bach (2007) have demostrated that the SLCmodel fit CHRIS-PROBA data of the Upper Rhine Valley in Germany fordifferent types of vegetations and bare soil with an average RMS errorof 1.6% for nadir viewing, 2.3% for ±36° and 3.3% for oblique obser-vations with ±55° between measured and simulated reflectances.SLC requires 26 input parameters (Table 2) where in this study thesimulations are done for a range of sun-target-sensor geometries.For more details, the reader is referred to Verhoef and Bach (2007).

3.1.2. MODTRAN (MODerate resolution TRANsmittance and radiance code)MODTRAN is a state-of-the-art code developed initially from the

LOW resolution TRANsmittance 7 (LOWTRAN 7) model. It modelsscattering and absorption effects in the atmosphere with high accu-racy and high spectral resolution up to 1 cm−1. The scaled DISORT(DIScrete Ordinate Radiative Transfer) algorithm is integrated for ac-curate computations of atmospheric multiple scattering and the cor-related k algorithm is applied for regions with significant absorptioneffects (Berk et al., 1998). All these features make MODTRAN a reli-able candidate for realistic simulation and analysis of remote sensingmeasurements. MODTRAN is well-suited for both at-sensor radiancesimulation (Guanter, Segl, & Kaufmann, 2009; Laurent, Verhoef, Clevers,& Schaepman, 2011; Schläpfer & Nieke, 2005; Verhoef & Bach, 2007,2012) and for atmospheric correction (Adler et al., 1999; Schläpfer &Richter, 2002) from the UV to thermal infrared wavelengths. In the cur-rent study, theMODTRAN codewas used to construct a look-up table of

six atmospheric coefficients in Eq. (1) for a number of different atmo-spheric scenarios.

3.2. Sensitivity analysis

This section presents the theoretical basis and the knowledgenecessary for the reader to understand different aspects of sensitivityanalysis. Following our procedure outlined in the introduction, we dis-cuss different aspects of the two methods for sensitivity analysis; thevariance-based method and the proposed improvement. Each subsectionis devoted to a specific aspect of sensitivity analysis and relevant meth-odologies are described in detail.

3.2.1. Sampling of the parameter spaceRandom sampling, used in the Morris screening method, is simple

and perhaps the most well-known amongst sampling strategies. Al-though widely used, assuming uncorrelated sample points mightlead to clusters and gaps in the distribution of samples and to poorlydistributed samples across the parameter space. This is more likelywhen the number of sample points is small relative to the numberof input variables. Stratified sampling partly copes with this problembut still distributes samples randomly over subspaces. In fact, thereexists a relatively high discrepancy between sample sets generatedby random sequences in high dimensional spaces (Saltelli et al.,2008). The discrepancy of a sequence is low if the number of points inthe sequence falling in an arbitrary set B is almost proportional to themeasure of the set B. Low-discrepancy sequences offer a more effectiveway to sample the parameter space uniformly while workingwith highdimensional spaces. Sobol sampling is a low-discrepancy sequence thatgenerates a uniform distribution in probability space. This method gen-erates a new sample point with a priori knowledge of where the previ-ous ones are located. In the first place, it samples from a multivariateuniform distribution, then goes back and fills in gaps left unsampledin repeated loops (Press, Teukolsky, Vetterling, & Flannery, 2007).The sequence substantially improves the sampling in a multivariateparameter space. Among different sampling strategies, Sobol sam-pling has been recognized as one of the most appropriate samplingschemes to distribute samples over the parameter space while workingwith hypercube spaces (Saltelli et al., 2008). To apply Sobol sampling,the actual parameter values are scaled to a uniform distribution in(0,1). This is characterized by the cumulative distribution function(CDF). If function F is a cumulative distribution function with inversefunction G, then U = F(X) and X = G(U) relates a random variable Xwith CDF F to a random variable U distributed uniformly on (0,1).

3.2.2. Variation of parametersTo explore the parameter space, two parameter variation techniques

have been employed. Radial design sampling andwinding stairs samplingare applied with the variance-based method and the proposed designrespectively. These strategies are explained in detail below.

Table 2Input parameters of the SLC model.

Parameter Interpretation Unit min max

Hapke_b Hapke phase function parameter b [−] 0.5 1Hapke_c Hapke phase function parameter c [−] 0.3 1Hapke_h Hapke hot spot width parameter [−] 0.05 0.8Hapke_B0 Hapke hot spot magnitude parameter [−] 0.05 0.8SM Soil moisture percentage [−] 0.05 0.6N_g Mesophyll Structural parameter in Prospect, green leaves [−] 1 2.5Cab_g Chlorophyll content in Prospect, green leaves μg/cm2 10 80Cw_g Water content in Prospect, green leaves g/cm2 0.001 0.1Cdm_g Dry matter content in Prospect, green leaves g/cm2 0.001 0.01Cs_g Senescence factor Prospect, green leaves [−] 0.001 0.9N_b Mesophyll Structural parameter in Prospect, brown leaves [−] 1 3Cab_b Chlorophyll content in Prospect, brown leaves μg/cm2 20 90Cw_b Water content in Prospect, brown leaves g/cm2 0.001 0.1Cdm_b Dry matter content in Prospect, brown leaves g/cm2 0.001 0.01Cs_b Senescence factor Prospect, brown leaves [−] 0.001 0.9LIDFa LIDF parameter a, which controls the average leaf slope [−] −1 1LIDFb LIDF parameter b, which controls the distribution's bimodality [−] −1 1Diss Layer dissociation factor [−] 0.1 0.9Zeta Tree shape factor [−] 0.1 0.9Hot Hot spot size parameter [−] 0.001 0.1fb Fraction brown leaf area [−] 0 1Cv Vertical crown cover percentage [−] 0 1LAI Total (green + brown) leaf area index m2m2 0 6SZA Sun zenith angle degree – –

VZA Viewing zenith angle degree – –

RAA Relative azimuth angle degree – –

135A. Mousivand et al. / Remote Sensing of Environment 145 (2014) 131–144

3.2.2.1. Radial design sampling. Assume A1 = {a11, a12, a13, …, a1k} andB1 = {b11, b12, b13, …, b1k} are two synthetic canopy sample pointsgenerated by the Sobol sequence generator, where the first subscriptdenotes sample point number and the second subscript denotesinput parameter number in a k-dimensional space. In radial designsampling, a new sample is drawn by moving a step in one directionwithin the parameter space each time. For instance, the first newsample point is generated by moving a step from the original point(A1) towards the second point (B1) in the direction of the first pa-rameter, in which all the entries are from A1 but the first of whichis from B1 and the new sample point is A1

B ¼ b11; a12; a13;…; a1kf g.Similarly, the second new sample point is obtained by replacing thesecond parameter value of A1 by the corresponding parametervalue of B1 so that AB

2 = {a11, b12, a13, …, a1k} and so on. This processcontinues until all parameters are varied once and forms a block. Inother words, the radial design begins with an original point andmoves away from the point in one direction each step, thenreturning back to the original point and moves in a different direc-tion until all possible directions (the number of inputs) areaccounted for. Hence, the kth sample point is AB

k = {a11, a12,a13, …, b1k}. In such a design all the samples are symmetric to thefirst sample, which is characterized by a radial pattern (Fig. 1b).Using radial design, for each sample point a block is constructed bya sample point from matrix A and k samples of AB (Fig. 2). Sensitivityindices are computed from n such blocks, each of size (k+1). Conse-quently, the total cost of the method is n(k + 1).

3.2.2.2. Winding stairs sampling. The winding stairs sampling technique(Jansen, 1999; Saltelli et al., 2010) draws new sample points bymovinga step in a new direction from the previous sample point (unlike theradial design that all the samples are a step away from the originalpoint). Given the two samples A1 and B1, the first new point is gener-ated by moving a step away from A1 in the direction of the first inputparameter so that BA1 = {b11, a12, a13, …, a1k}. Likewise, the secondpoint is drawn by moving a step from the previous generated sampleB1

A in the direction of the second input parameter giving BA2 = {b11,

b12, a13, …, a1k}, and so on. Therefore, the kth sample points is BAk ={b11, b12, b13, …, b1k} that is the same as sample point B1. Windingstairs sampling begins with a sample point and moves away from

that point each step in a new direction until all input parametersare perturbed once and ends up at a new point. New samples aredrawn in a fixed cyclic order that constructs k new points aftereach cycle. Since the sample points differ in a single input parametervalue, the design would ultimately have a stair-like form (Fig. 1a).With winding stairs, for each sample point a block is built from ma-trix A and k samples of BA (Fig. 2). Sensitivity indices will be comput-ed by n such blocks. Hence, the total cost of the method is n(k + 1).The winding stairs method makes a more balanced use of elementsfrom matrices A and B, while a radial sampling block is mostly popu-lated by elements from matrix A. In both designs, sampling blockscontain the same number of total-order sensitivity effects per eachparameter which maintains the balance for each block internally(Saltelli et al., 2010). This helps to increase the number of blocks ar-bitrarily without worrying about the balance of the design. Overall,the winding stairs method has been shown to provide better esti-mates of sensitivity indices for small sample sizes (Chan, Saltelli, &Tarantola, 2000).

3.2.3. Measures of sensitivityAn effective and robust sensitivity analysis method requires con-

sideration of a set of sensitivity indices. These indices are a measureof the first-order effect (also known as the main effect), second-order effects (the mutual interaction effect among parameters),higher order and the total-effect measures (the first-order effectplus all higher-order effects). With k input parameters, the entirenumber of first and higher-order effects would be 2k − 1 indices.Nevertheless, it is often desirable to compute only two sensitivity in-dices; first-order and total effect for each input parameter in a typicalsensitivity analysis (Saltelli et al., 2010). These indices are addressedin the next section.

3.2.3.1. Sensitivity indices. Variance-based analysis is based on the de-composition of the total variance of the model output into compo-nents related to the different parameters and their interactions.The analysis relies on (i) the Sobol decomposition of a function,and (ii) the law of total variance (or variance decomposition formula)(Chen, Jin, & Sudjianto, 2004). Given an RT model LTOA = f(X), whereLTOA is the simulated radiance (we shall, for brevity, write L) and X =

Fig. 2. An illustration of matrices A,B,AB and BA for winding stairs sampling (BA) and radial design sampling (AB).

136 A. Mousivand et al. / Remote Sensing of Environment 145 (2014) 131–144

(x1,x2,…xk) is a vector of k input parameters, X is assumed to contain in-dependent components and have a joint density which is uniform overΩk, the k-dimensional unit hypercube, Ωk = (X|0 ≤ xi ≤ 1; i = 1,…, k).The Sobol decomposition of the function f(X) is given by (Sobol, 1993)

f Xð Þ ¼ f 0 þXki¼1

f i xið Þ þX

1≤ ib j≤k

f ij xi; xj

� �þ

X1≤ ib jbm≤k

f ijm xi; xj; xm� �

þ…þ f12…k

ð2Þ

where i, j, m∈ {1,…, k} and f0 = E(L) is the expected value of f(X) overall parameters:

f 0 ¼ Ε Lð Þ ¼Z

Ωk

f xð Þdx ð3Þ

This expansion is a type of ANOVA decomposition. fi(xi) is the first-order function for parameter xi and fij(xi,xj) indexes the second-order in-teraction of the parameters xi and xj. An input parameter has interac-tions with other parameters if its impact on the output depends onthe value of the other parameters. All the terms in Eq. (2) are orthogonalwith zero mean. Therefore, by averaging out all the parameters but ximinus the constant f0, one can obtain a decomposition onlywith respectto parameter xi as follows;

f i xið Þ ¼ Ε Ljxið Þ−Ε Lð Þ ¼Z

Ωk−1

f xð Þd x∼ijxið Þ− f 0 ð4Þ

where x~ i means all the input parameters except xi. Similarly, a decom-position with respect to parameters xi and xj is

f ij xi; xj

� �¼

Z

Ωk−2

f xð Þd x∼ijjxij� �

− f i xið Þ− f j x j

� �− f 0 ð5Þ

Therefore, the total variance Var(L) can be written as the sum ofthe partial variances by squaring and integrating Eq. (2) and, ac-cording to the orthogonality property of the decomposition termsin Eq. (2) (Chen et al., 2004),

Var Lð Þ ¼Xki¼1

Vi þX

1≤ ib j≤k

Vij þX

1≤ ib jbm≤k

Vijm xi; x j; xm� �

þ…þ V12…k ð6Þ

where Vi ¼ Varxi Ex�iLjxið Þ

� �is the contribution of xi to Var(L), Vij ¼

Varxij Ex�ijLjxij

� �� �is the contribution of the interaction of xi and xj

to Var(L), and so on up to V12 … k , which is the contribution of the in-teraction of all the parameters to Var(L). Eq. (6) decomposes the totalvariance into distinct components related to each parameter and

interactions. The measure of importance of a parameter can thenbe expressed as the expected reduction in the total variance, giventhat parameter, i.e., E(Var(L) − Var(L|xi)), which according to the lawof variance is equal to Vi in Eq. (6). Likewise, Vij is the expected reduc-tion in the total variance, given two parameters, xi and xj. Variance-based sensitivity indices are estimated as ratios between the partial var-iances and the total variance.

Si ¼ Vi=Var Lð Þ ð7Þ

Sij ¼ Vij=Var Lð Þ ð8Þ

Si is the main effect of parameter xi on the total variance and Sij isthe second-order effect of xi. The main effect indicates the relativeimportance of the parameter on themodel output. This effect is addi-tive on the model output and independent of the value of the otherinput parameters (Monod, Naud, & Makowski, 2006). A measure ofthe total-order effect (STi) can be interpreted as the effect of each pa-rameter on the output variance while all interactions with other pa-rameters are taken into account. This denotes the main effect of aparameter and all other higher-order effects.

STi ¼ Ex�iVarx�i

Ljx�ið Þ� �

=Var Lð Þ ¼ 1� ðV�i=Var Lð ÞÞ ð9Þ

Assuming three parameters, the total effect index of a parameter asreported by Saltelli et al. (2010) is represented by a sequence of sensi-tivities;

ST1 ¼ S1 þ S12 þ S13 þ S123 ð10Þ

An ideal sensitivity method must be able to take into account theinteraction among different input parameters; in particular for non-linear models. This demands quantification of all effects and calcula-tion of all indices defined above with a sufficient number of samples.The computation of the sensitivity indices of the applied methods isdiscussed in the following sections.

3.2.3.2. Sensitivity indices of the variance-based method. Among globalstatistics analysis methods, variance-based methods are often re-ferred to as accurate global methods in the literature (Saltelli et al.,2010). For this study, the variance-based approach, as described inSaltelli et al. (2010), was evaluated as a reference to measure firstand total sensitivity indices. The method implements the radial designsampling technique described in Section 3.2.2.1. The blocks generatedby the radial designmethod are used to construct conditional variancesand the total variance of the model. Further, sensitivity indices are de-rived from such variances. The first-order sensitivity index is given by

Si ¼1n

Xnj¼1

f Bð Þ j f AiB

� �j− f Að Þ j

� �

Var Lð Þ ð11Þ

137A. Mousivand et al. / Remote Sensing of Environment 145 (2014) 131–144

Here, A and B are the two independent sampling matrices, AiB is the

design matrix constructed by the radial design sampling in Section3.2.2.1 (Fig. 2), where all columns are from A but the ith column isfrom B and f(.) denotes the evaluation of the RTmodel at sample points(output radiance). Likewise, the total indices were estimated withJansen's formula (Jansen, 1999), which has been shown to usually per-form best among available formulas (Saltelli et al., 2010).

STi ¼12n

Xnj¼1

f Að Þ j− f AiB

� �j

� �2

Var Lð Þ ð12Þ

3.2.3.3. Sensitivity indices in the proposed design. The proposed methodcombines the Sobol sampling scheme with winding stairs parametervariation (see Section 3.2.2.2). The Sobol scheme efficiently distributesthe sample points over the entire parameter space and avoids gapsand clusters. This gives a fairly good exploration of the parameterspace even with a limited number of sample points. The ratio betweenthe variation in the output radiance of two successive sample pointsand the variation in the input parameter itself is calculated. Therefore,along a trajectory, k ratios are calculated, one for each parameter. Hav-ingn such ratios for each parameter, one can calculate statistics to assesssensitivity. We have used themean of the absolute value of all these ra-tios for each parameter as the total-order effect of that parameter.

S ¼

Xnt¼1

f iþ1; tð Þ x1;…; xkð Þ− f i;tð Þ x1;…; xkð Þdxi

��������

nð13Þ

where t is trajectory number, n stands for the number of trajectories, dxiis the variation in parameter i and f(i,t) denotes the radiance of samplepoint i. Furthermore, the standard deviation of the ratios for each pa-rameter has been calculated to represent the degree of nonlinearity ofthe given parameter.

3.3. Bootstrap confidence intervals

A sensitivity study is incomplete without an understanding of theaccuracy of the estimated indices. High uncertainty might emerge inthe estimation of sensitivity indices due to the nature of samplingstrategy used in sensitivity analysis (Archer, Saltelli, & Sobol,1997). The bootstrapping approach (Efron & Tibshirani, 1993) wasapplied with blocks (i.e. blocks of sample points) to estimate confi-dence bounds on the sensitivity indices. In this approach, the gener-ated samples are resampled with replacement many times to createa number of bootstrap replicate data sets and form a distribution ofsensitivity indices. The percentile method (Efron & Tibshirani,1993) is then utilized to construct bootstrap confidence intervals inwhich for a 95% confidence interval, the lower and upper 2.5% per-centiles of the generated distribution of the bootstrap are used. It isalso worth noting that the bootstrap resampling comes at no furthercost, because model evaluation of all blocks is already done.

3.4. Evaluating parameter ranking

Since the ultimate goal is to identify influential and non-influentialparameters, i.e. ranking the input parameters based on their impor-tance, a rank transformation is done by simply replacing the sensitivityscores of all methods with their respective ranks and then assessing theKendall correlation between differentmethods on the ranks rather thanthe corresponding sensitivity values. Hence, if the number of parame-ters is k, then the largest score among the sensitivity values is assignedrank 1, the next largest value is assigned rank 2, etc. The Kendall corre-lation coefficient (also known as Kendall's tau (τ) coefficient) is a non-

parametric test to measure the strength of the relationship betweentwo ranked samples (Abdi, 2007). τ varies between−1 and 1 and yieldsa positive correlation if the measured ranks of both methods increasetogether, whilst mismatching between two ranks results in a negativecorrelation. Having two sets of ranks each containing k parameters,there are k(k − 1)/2 possible comparisons of pairs (Xi,Yi) and (Xj,Yj).Kendall's tau is given by

τ ¼ NC−ND12k k−1ð Þ

ð14Þ

where NC is the number of concordant pairs and ND stands for the num-ber of discordant pairs. This test provides a consistent measure of thesimilarity between ranked parameters. This measure reflects well theimportance of agreement on the top ranks.

3.5. Setting up the sensitivity experiments

All input parameters and their ranges of variation were selectedidentically for all the sensitivity experiments. A set of biophysical pa-rameter valueswas derived from the literature and fieldmeasurementswith the goal of covering a wide range of vegetation conditions. This setwas used to define the range of variability of each parameter (Table 2).The sensitivity analysis was conducted with a variety of different sun-target-sensor geometries. During each experiment, two Solar ZenithAngles (SZA) were set at 30° and 60° corresponding to high and lowsun elevations, and View Zenith Angle(VZA) was allowed to take onfive values of 0°, 15°, 30°, 45° and 60° and Relative Azimuth Angle(RAA) two values of 30° and 150°. Thus, there were 20 sets of sun-target-sensor geometries. Furthermore, this test was repeated forthree different atmospheric profiles with visibility values at 10, 23and 50 km, representing a very hazy, moderate and clear sky respec-tively, to quantify the effect of solar/view geometry and the atmo-sphere on the sensitivity results.

The acquisitions were simulated with the observation configurationlisted in Table 2. The spectral sampling of the simulations was adjustedto 1 nm, i.e. 2101 contiguous spectral bands were taken between 400and 2500 nm. These spectral bands were then convolved to matchthe Hyperion sensor characteristics. In order to mimic actual Hyperi-on images, several spectral bands were removed due to water ab-sorption, intersection bands and very low signal so that eventually196 spectral bands were used in the 400–2400 nm region for furtheranalysis. Therefore, the spectral bands corresponding to regions be-tween 1350 and 1420 nm, 1800 and 1930 nm, and above 2400 nmwere removed and are not included in the graphs.

4. Results and discussion

4.1. Evaluation of the proposed design

The performance of the proposed design was compared to thevariance-based method based upon the ranking of the most influen-tial and non-influential input parameters in order of importance at agiven wavelength. A high correlation value (see Eq. (14)) betweentwo methods indicates high degree of similarity in the ranking ofthe parameters.

While applying Sobol's sequence, additional uniformity can bereached by increasing the sample size by taking n-data pointswhere n= 2a and a is an integer (Sobol, 1967). Based on this recom-mendation, n was set to 211 resulting in 2048*(23 + 1) = 49,152synthetic canopies to calculate thefirst-order and total-order sensitivityindices in the variance-based approach. Preliminary analysis on thenumber of sample points showed that the assessed sensitivity indiceswith this sample size are highly correlated to the ones with larger

138 A. Mousivand et al. / Remote Sensing of Environment 145 (2014) 131–144

sample sizes. Therefore, the sample size is deemed adequate to give re-liable sensitivity indices. As stated in Section 3.3, since bootstrappinganalysis comes at no extra cost, the results were resampled 10,000times (with replacement) to have reliable estimates.

In order to select a proper number of samples for the proposeddesign, we investigated its convergence behavior in terms of samplesize. Six different number of trajectories at 10, 20, 50, 70, 100 and200, corresponding to 240, 480, 1200, 1680, 2400 and 4800 samplepoints respectively, were taken to inspect whether an increasingnumbers of trajectories would effectively improve the performanceof the method. The results revealed that increasing the number oftrajectories improves the correlation between the proposed designand the varaiance-based method to some extent. However, the pro-posed design tends to converge toward an almost stable average cor-relation of 92% with the variance-based method and no significantdifference is found by increasing the number of trajectories beyound50. This suggests that a good level of agreement may be reached withthis number of trajectories. Therefore, the number of 50 trajectorieswas selected for further analysis in the study, hence the total cost ofthe method is 50*(23 + 1) = 1200 simulations.

Fig. 3 illustrates two pairwise comparisons between the proposeddesign and the variance-based method in terms of parameter ranking.Two solar/view geometries are plotted, a nadir view with low solar ze-nith angle and very high view and solar zenith angles. It is shown thatthe method is in good agreement with the variance-based method forthe entire spectrum. However, the results indicate that the correlationis slightly higher from the visible to the near-infrared (~1400 nm) re-gion. An explanation for this is the fact that Kendall's τ is more sensitiveto agreement in the top ranks than in the lower ranks. Therefore, overthe visible to near-infrared region where there are a group of dominantparameters with distinguished sensitivity values that attain top ranks,both methods can equally rank the parameters and a higher correlationis to be expected, whereas for the rest of the spectrum there are moreparameters with similar low sensitivity values which potentially leadsto less similarity in parameter ranking. The correlation coefficients, av-eraged over the entire spectrum, are 0.92 and 0.93 for the low andhigh zenith angles, respectively. In general, for all the solar/view geom-etries and different atmospheric profiles applied in this study, themeancorrelation coefficients were 0.91 to 0.93. The results demonstrate thatthe proposed design is reasonably accurate in parameter rankingwhile applying only 1200 model runs versus 49,152 model runs withthe variance-based method.

Fig. 3. Kendall tau correlation coefficient between the proposed design (1200model runs)and the variance-basedmethod (49,152model runs). The blue line represents solar zenithangle of 60°, viewing zenith of 60° and relative azimuth angle of 150°; the green line rep-resents solar zenith angle of 30°, nadir viewing angle and relative azimuth angle of 150°.

4.2. Influential and non-influential parameters

Reliable quantitative information can be retrieved on a parame-ter, if that parameter is strongly related to the output radiance(Kimes et al., 2000; Verstraete et al., 1996). This implies that a pa-rameter is retrievable if the sensitivity of output radiance to that pa-rameter is significant. In other words, a high sensitivity value meansthat the parameter is responsible for a significant portion of the outputvariance of the radiance which leads to a greater chance of retrieval ofthat parameter. In an ideal case, the necessary and sufficient conditionfor an input parameter to be non-influential is that the main effectand the interaction effects with other parameters are zero, whichshows that the contribution of the parameter on the output varianceis zero. This condition allows assigning an arbitrary value to that param-eter within its domain. In practice, input parameters with little effectcan be treated as non-influential if their effect is very small comparedto the other parameters for a given wavelength. For instance, chloro-phyll strongly influences the visible spectrum by absorption whileshowing negligible effect in the shortwave infrared region.

To assess the actual influence of a given parameter, both the maineffect and the interaction effects among parameters have to be takeninto account. To show the importance of interaction effects amongstinput parameters, we have plotted in Fig. 4 the sum of the first-ordereffects and total-order effects for a given solar viewing geometry.The blue line represents the sum of all the total-order effects, thered line indicates the sum of the first-order effects and the greenarea is the interaction effects among parameters. It should be notedthat the sum of first-order effects over all input parameters cannotexceed one, and does not equal one unless all the interaction termsare zero. By contrast, the sum of total-order effects over all input pa-rameters is never less than one, and equals one only if all interactionterms are zero, because this sum includes every first-order effectonce and every interaction term multiple times.

Fig. 4 depicts that the sum of the first-order and total-order effects isalmost symmetric with a few exceptions due to some round-off error. Itis shown that the sum of the interaction effects among the input param-eters reaches amaximum in the visible, and in particular near the greenregion (~550 nm) corresponding to a peak in the reflectance of vegeta-tion. This means that a large portion of uncertainty in the output radi-ance is driven by interaction among the factors in this region. Whilethe red-near infrared (~670 nm), corresponding to maximum vegeta-tion absorption, seems to have the least impact on the interaction effectswhere approximately 85% of the output variance in TOA radiance is ex-plained by the main effects. The possible conclusion is that the greenchannel is very sensitive to a number of parameters that are correlated

Fig. 4. Sum of the first-order effects, interaction effects and total-order effects over all theparameters for nadir view angle and solar zenith angle of 30°.

139A. Mousivand et al. / Remote Sensing of Environment 145 (2014) 131–144

and cause considerable interaction effects. But the red-near infrared(~670 nm) is only sensitive to a few dominant parameters (such asLAI, Cv and SM) and therefore it includes only the interaction effectsof them.

Since the total-order effect of a parameter includes the first-orderplus all the interaction effects, we have used total-order effect toidentify the dominant and negligible input parameters. The total-order effects estimated by both methods are given in Figs. 5 and 6 fora solar zenith angle of 30°, nadir viewing zenith and relative azimuthangle of 150°. Fig. 5 illustrates the most influential parameters and theinput parameters with the lowest sensitivity values are shown inFig. 6. Direct comparisons between sensitivity values estimated byboth methods show that for some parameters the proposed design un-derestimates sensitivities. This happens if the sample space is small andthe model is nonlinear. However, in general, there is good agreementbetween both methods for selecting influential and non-influential pa-rameters at any given wavelength.

4.3. Physically-based understanding of the radiative processes that governthe simulations

The radiation in the range from 400 to 700 nm is the photosyn-thetically active radiation for plants. Spectral features in this regionare primarily controlled by leaf pigments and in particular chloro-phyll concentration. Therefore, spectral radiance is most sensitiveto chlorophyll content, senescence factor and other relevant param-eters such as the fraction of brown leaf area (fb) and dissociation fac-tor (Diss). The effect of chlorophyll content and senescence factordisappears beyond approximately 700 nm and 1000 nm respectivelydue to the fact that chlorophyll and other leaf pigments are transparentto infrared radiation. When the subscripts for green (g) and brown(b) leaves are not given, wemean both leaf types. Leafmesophyll struc-ture (N) parameter and dry matter content (Cdm), on the other hand,influence the near-infrared spectra depending on the relative thicknessof the mesophyll. In general, the sensitivity of the observed radiance to

Fig. 5. Total-order sensitivity effect of the most influential parameters for solar zeni

N and Cdm is relatively small, especially for drymatter contentwhich isless than 3% at its highest value. Cdm and N parameters also affect theshort-wave infrared region. Water content (equivalent water thickness(Cw)) has negligible impact on the visible and the near-infrared butinfluences the shortwave infrared considerably because of radiationabsorption by leaf water content. Water content is responsible formore than 10% of the output variance between 1200–1300 nm and1500–1700 nm (Fig. 5). Leaf area index (LAI) and vertical crown cov-erage (Cv) are very influential parameters affecting the output radi-ance over almost all the spectral bands. This is explained by the factthat LAI and Cv are the indicators of the presence and density of veg-etation and determine radiation interception by the canopy. Even asmall variation in the value of these two parameters causes somechange in the observed radiance. Nonetheless, the highest sensitivityto LAI is seen in the red-near infrared where the maximum change inthe absorption of radiation from vegetation occurs. The impact of LAIand Cv remain almost stable from 1400 nm onwards, while the mostdramatic declines are seen around 550 nm and 700 nm because ofthe sharp peaks in the influence of chlorophyll and LIDFa parameters.Similarly, an increase in the influence of water content causes a de-crease in the impact of LAI and Cv. Leaf inclination distribution functionparameter a (LIDFa) controls the average leaf slope, which in turn con-trols radiation interception by the leaves and therefore is an importantparameter contributing to the output radiance, but the parameter b thatcontrols the distribution's bimodality is shown to be non-influentialover almost the entire spectrum. The influence of LIDFa goes up around700 nm and remains steady until around 1300 nmwhen there begins adecline. This is due to the high infrared reflectivity of leaves which iscontrolled by LIDFa parameter. Soil moisture affects the observed radi-ance at all wavelengths but its influence is larger for the shortwave in-frared spectra. All the Hapke's soil parameters (except Hapke's phasefunction parameter b) and hot-spot effect show very small effect on al-most all the spectral bands (Fig. 6). Soil parameters have a secondary ef-fect (background effect) on the reflectance/radiance of a canopy whichis generally low, especially with the increase of vegetation cover. There

th angle of 30°, nadir viewing zenith angle and relative azimuth angle of 150°.

Table 3Total-order sensitivity effect of the five simulated Hyperion spectral bands.

Band 17 Band 33 Band 55 Band 141 Band 210

N_g 0.0505 0.0039 0.0054 0.0088 0.0046Cab_g 0.1611 0.0173 0.0000 0.0000 0.0000Cw_g 0.0000 0.0000 0.0010 0.1371 0.0572Cdm_g 0.0002 0.0000 0.0090 0.0016 0.0033Cs_g 0.0656 0.0002 0.0026 0.0000 0.0000N_b 0.0210 0.0017 0.0035 0.0040 0.0022Cab_b 0.0687 0.0072 0.0000 0.0000 0.0000Cw_b 0.0000 0.0000 0.0006 0.0724 0.0299Cdm_b 0.0001 0.0000 0.0055 0.0008 0.0014Cs_b 0.0316 0.0001 0.0017 0.0000 0.0000LIDFa 0.1216 0.0137 0.1775 0.0354 0.0206LIDFb 0.0037 0.0007 0.0034 0.0019 0.0009Diss 0.0059 0.0005 0.0001 0.0026 0.0012Zeta 0.0052 0.0045 0.0112 0.0100 0.0067Hot 0.0016 0.0003 0.0021 0.0012 0.0005fb 0.0885 0.0067 0.0061 0.0485 0.0194Cv 0.1651 0.4494 0.5374 0.2520 0.3368LAI 0.0898 0.3198 0.1935 0.2154 0.2466SM 0.1014 0.1410 0.0324 0.1795 0.2403Hapke_b 0.0112 0.0167 0.0035 0.0107 0.0099Hapke_c 0.0027 0.0061 0.0013 0.0068 0.0069Hapke_h 0.0012 0.0027 0.0006 0.0030 0.0031Hapke_B0 0.0033 0.0075 0.0016 0.0084 0.0086

Fig. 6. Total-order sensitivity effect of the input parameters with the lowest sensitivity values for solar zenith angle of 30°, nadir viewing zenith and relative azimuth angle of 150°.

140 A. Mousivand et al. / Remote Sensing of Environment 145 (2014) 131–144

is another reason, however, that might explain the low effect of thethree Hapke's parameters: Hapke_B0 and Hapke_h parameters areonly responsible for the soil's hot spot effect and with Hapke_c param-eter, they only have an influence on the bi-directional reflectance factor(rso in Table 1). This implies that the effect of these parameters is mar-ginal on the output radiance since in the four-stream approach, imple-mented in this study, four different reflectance factors are considered.

4.4. Detailed interpretations of the sensitivities of five Hyperion spectralbands

Wehave selected five Hyperion spectral bands (the vertical red linesin Figs. 5 and 6) as examples for more detailed interpretations of allthe input parameters. The selected spectral bands are centered at519, 681, 905, 1558 and 2244 nm in accordance to Hyperion spectralbands 17, 33, 55, 141 and 210 respectively. The results for the fivespectral bands are summarized in Table 3 and Fig. 7. Table 3 liststhe sensitivity of each spectral band to all input parameters. Fig. 7also provides the total-order sensitivity index of these five bandswith a comparison to the proposed design.

Parameters with zero value in Table 3 (and Fig. 7) have simply noinfluence on the related spectral band. For instance, band 17 andband 33 are not sensitive to Cw and Cdm. This is because these twobands are located in the photosynthetic pigments region where varia-tions in the amount of water content or cell structure do not impactthe output radiance. On the other hand, band 141 is the best suitedband to study Cw since it is located in a very sensitive region to varia-tions in canopy water content. Aside from the ideal case, i.e. zero sensi-tivity value, one may decide to define a parameter as negligibleaccording to the accuracy required. Measure of this accuracy can be de-finedwith reference to a fixed number of decimal places, i.e. the numberof digits following the decimal point. This can be interpreted as approx-imate with a precision equal to the number of zero-valued digits onecan assume the parameter in question as negligible without affectingtoomuch the output radiance. For instance in band210,with a precisionof two decimal digits, wemay label N, Cab, Cdm, Cs, Diss, LIDFb, Hot andHapke-h as parameters with negligible effect. Likewise, Cw, Cdm, Hotand Hapke_h are parameters for which band 17 is insensitive with

three decimal digits precision. If a parameter is labeled as negligiblefor a spectral band, it means that the band cannot be used for the re-trieval of that parameter.

This analysis is useful in the context of parameter retrieval problemsto decide which parameters can be kept constant during retrieval. Afurther analysis of the sensitivity of each spectral band to input pa-rameters can be of help in identifying which spectral band is moresensitive to a certain parameter. This, in turn, is translated into find-ing the optimal band selection in a parameter retrieval problem. Asdiscussed before, based on the sensitivity concept, the spectralbands that present the highest sensitivity to a given parameter aremore likely suitable for retrieval purposes than those with verysmall sensitivity values. This can be best demonstrated by employing

Fig. 7. Total-order sensitivity effect of the five simulated Hyperion spectral bands.

141A. Mousivand et al. / Remote Sensing of Environment 145 (2014) 131–144

an inversion algorithm, in which the results extracted from an optimalband selection are validatedusing ground truthmeasurements.However,this is beyond the scope of this paper and will be addressed in futurestudies.

4.5. TOA radiance sensitivity vs BOA reflectance sensitivity

We have also performed the sensitivity analysis for BOA reflectanceto investigate the impact of the atmosphere on the sensitivity valuesof input parameters. The atmosphere has a complicated influence onthe observed radiance in the optical solar spectral range. This influ-ence is specifically significant on visible and the infrared radiation.Two major effects of the atmosphere are scattering and absorption.Atmospheric absorption is primarily due to gaseous and aerosols ab-sorption properties. The scattering effect is caused by molecules andaerosols (Kaufman & Tanré, 1996). We have used three different vis-ibilities, as an indicator of aerosol load, at 10 km, 23 km and 50 kmcharacterizing hazy to clear atmospheres. A rural aerosol type anda mid-latitude summer atmosphere were selected and the watervapor content was assumed to be 2 g/cm. Fig. 8 compares the total-order sensitivity of BOA reflectance and two TOA radiance sensitivi-ties for atmospheric visbilities of 10 km and 23 km. For the sake ofclarity, we only plot the comparisons of some dominant parameters.Strong atmospheric absorption bands are omitted from the graphsand only BOA sensitivities are shown for these bands.

The measured sensitivities of BOA spectral reflectance are similarto the TOA radiance sensitivities with a few exceptions when the at-mosphere is clear and visibility is high (Fig. 8). When visibility de-creases there are some significant differences between the measuredsensitivities for BOA and TOA. It is not surprising that the most remark-able differences are seen in the visible spectra for dominant parametersdriving the uncertanity in the output. It is well known that atmosphericscattering is more significant for visible wavelengths where the

scatteringmodifies the propagation direction and causes gain on the ra-diation reaching the sensor. The hazier the atmosphere becomes, thelarger the influence it has on the sensitivity to the input parameters.This difference is larger for parameters relevant to leaf pigments suchas Cab and fb in the visible region. The results, as shown in Fig. 8, indi-cate the influence ofmost parameters is less affected by the atmospherefor the shortwave infrared wavelengths. A possible explanation for thisis that scattering continuously decreases from visible towards the infra-red and consequently has less impact on the sensitivities. It is also evi-dent from this figure that a unique pattern cannot be introduced sothat visibility decreases or increases the influence of a parameter simi-larly over the entire spectrum. This is due to the fact that a decrease inthe sensitivity to a certain parameter causes an increase in the othersensitivities. These findings suggest that in general the sensitivity ofTOA radiance to the input parameters in the case of clear atmosphereis similar to BOA reflectance sensitivity excluding the strong atmo-spheric absorption wavelengths.

4.6. Effect of viewing and illumination geometry on TOA radiance sensitivity

Canopy reflectance is anisotropic and exhibits variation in sensitivityvalues as a function of illumination and viewing geometries. The SLC isable to model anisotropic reflectance which makes it possible to inves-tigate the impact of geometry on the sensitivity. The in-depth analysis ofthe geometry effect on the sensitivity and finding the optimal geometryconfigurations for parameter retrieval is out of the scope of the currentpaper and will be discussed in future studies. Here, we have plotted thespectral sensitivity for four different viewing/illumination directions inFig. 9 to give an impression about the influence of the sun-target-sensor geometry on the sensitivity of TOA radiance to two dominant pa-rameters; Cv and LAI. The impact of anisotropy is wavelength depen-dent. As shown in Fig. 9, in general, for both parameters the observeddifference between solar/view directions is not significant in the visible,

Fig. 8. Total-order sensitivity of BOA reflectance and two TOA radiance sensitivities for atmospheric visibilities of 10 km and 23 km.

Fig. 9. Comparison of the total-order sensitivity of TOA radiance to CV (top) and LAI(bottom) for four viewing/illumination directions.

142 A. Mousivand et al. / Remote Sensing of Environment 145 (2014) 131–144

while it is larger in the NIR and SWIR wavelengths. From 800 nm on-wards, the difference in sensitivity for nadir viewwith SZA of 30° andSZA of 60° with a large VZA of 60° reaches more than 10% and 5% forCv and LAI respectively. The sensitivity in the spectral range from1400 nm to 1800 nm is very high at SZA of 30° with nadir viewwhile from 700 nm to 1400 nm the lowest sensitivity is observed forthis geometry. A possible explanation for this might be that increasingor decreasing the sensitivity value to one parameter affects the othersensitivities, for example, in the spectral region from 2000 nm onwardsan increase in the sensitivity at the low solar/view angles to Cv causes adecrease in the sensitivity for the same geometry to LAI.

5. Summary and conclusions

We assessed the sensitivity of TOA radiance and BOA reflectanceof a soil–vegetation system to the input biophysical and biochemicalparameters using the coupled Soil–Leaf-Canopy radiative transfermodel SLC and MODTRAN. An improvement in the design and sam-pling of screening methods was proposed based on a Sobol samplingstrategy and winding stairs parameter variation. The results demon-strated that the proposed improvement is efficient in identifyingdominant and non-dominant input parameters compared to thevariance-based method with modest computational demands (only1200 model runs against 49,152 model runs in the variance-basedmethod). In general, therefore, it seems that the proposed designcan be used instead of complex global methods when the objective isto identify influential and non-influential parameters among many

143A. Mousivand et al. / Remote Sensing of Environment 145 (2014) 131–144

other parameters or with computationally expensive methods. In par-ticular it can be used where it is necessary to have fast and appropriatesensitivity estimates of a coupled surface-atmosphere radiative transfermodel at low computational cost.

Among 23 input parameters, crown coverage (Cv), leaf area index(LAI), leaf inclination distribution function (LIDFa) and soil moisture(SM) were found to be the most influential parameters driving theoutput variance of the radiance from 400 nm to 2500 nm with afew exceptions. Chlorophyll content (Cab) and water content (Cw)were also identified as influential parameters for the visible and short-wave infrared radiation respectively. On the other hand, Hapke's soilparameters (h, B0 and c), dissociation factor and LIDFbwere recognizedto havemarginal influence on the output radiance. It is also found that alarge portion of uncertainty in the output radiance is driven by the in-teraction effects among input parameters in the visible (~550 nm),whereas the red-near infrared (~670 nm), seems to be less related tothe interaction effects. The results of this research support the idea offinding retrievable parameters from existingmeasurements of radianceusing sensitivity analysis.

This study has also found that generally TOA radiance sensitivi-ties to dominant biophysical parameters are similar to BOA reflec-tance sensitivities provided that the visibility of the atmosphere ishigh (e.g. more than 23 km). This suggests that both TOA radianceimages and atmospherically corrected reflectance images can beused to retrieve influential biophysical parameters of interest. However,when the visibility decreases there are some significant differencesbetween the measured sensitivities for BOA and TOA, in particular forthe visible part of the spectrum. It was also shown that the impact ofviewing and illumination geometry iswavelength dependent and influ-ences NIR and SWIR wavelengths significantly, while the observed dif-ference between different solar/view directions was not considerablein the visible.

The proposed method requires two independent sample sets,though recent studies suggest that all first-order effects can be ob-tained even from a simple Monte Carlo run (Plischke, Borgonovo, &Smith, 2013), which would decrease computational cost noticeably.This can be considered as a further research step in developing sen-sitivity analysis methods for radiative transfer models if only first-order effects are demanded.

Acknowledgments

The authors are grateful to anonymous reviewers for their construc-tive andhelpful comments,which considerably helped in improving themanuscript.

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