scaling and uncertainty in the relationship between the

E LS EV I E R

Scaling and Uncertainty in the Relationship between the NDVI and Land Surface Biophysical Variables: An Analysis Using a Scene Simulation Model and Data from FIFE

M. A. Friedl,* F. W. Davis,* J. Michaelsen,* and M. A. Moritz*

Biophysical inversion of remotely sensed data is con- strained by the complexity of the remote sensing process. Variations in sensor response associated with solar and sensor geometries, surface directional reflectance, topography, atmospheric absorption and scattering, and sensor electrical-optical engineering interact in complex man- ners that are difficult to deconvolve and quantify in individual images or in time series of images. We have developed a model of the remote sensing process to allow systematic examination of these factors. The model is composed of three main components, including a ground scene model, an atmospheric model, and a sensor model, and may be used to simulate imagery produced by instruments such as the Landsat Thematic Mapper and the Advanced Very High Resolution Radiometer. Using this model, we examine the effect of subpixel variance in leaf area index (LAI) on relationships among LAI, the fraction of absorbed photosynthetically active radiation (FPAR), and the normalized difference vegetation index (NDVI). To do this, we use data from the first ISLSCP Field Experiment (FIFE) to parameterize ground scene properties within the model. Our results demonstrate interactions between sensor spatial resolution and spatial autocorrelation in ground scenes that produce a variety of effects in the relationship between both LAI and FPAR and NDVI. Specifically, sensor regularization, nonlinearity in

*Center for Remote Sensing, and Department of Geography, Boston University

*Institute for ComputatiLonal Earth System Science and Depart- ment of Geography, University of California, Santa Barbara

Address correspondence to Mark. A. Friedl, Dept. of Geography, Boston Univ., 675 Commonwealth Ave., Boston, MA 02215.

Received 13 February 1995; revised 21 June 1995.

REMOTE SENS. ENVIRON. 54:233-246 (1995) ©Elsevier Science Inc., 199!5 655 Avenue of the Americas.. New York, NY 10010

the relationship between LAI and NDVI, and scahng the NDVI all influence the range, variance, and uncertainty associated with estimates of LAI and FPAR inverted from simulated ND VI data. These results have important implications for parameterization of land surface process models using biophysical variables such as LAI and FPAR estimated from remotely sensed data.

INTRODUCTION

Sophisticated strategies have evolved over the last two decades to extract land surface biophysical information from remotely sensed data (e.g., Smith, 1983; Strahler et al., 1986; God, 1987; Asrar et al., 1989; Dozier, 1989; Dubayah, 1992; Sellers et al., 1992a; Myneni et al., 1993). As part of this process, researchers have developed models to simulate interactions among electromag- netic radiation, land surfaces, the atmosphere, and remote sensing devices that have produced significant advances in our understanding of the complex nature of remotely sensed imagery (Suits, 1972; Verhoef, 1984; Li and Strahler, 1985; Tanr~ et al., 1990; Kerekes and Landgrebe, 1989b; Hall et al., 1992). These studies have shown that well-defined functional relationships exist between remotely sensed data and key land surface biophysical parameters such as leaf area index (LAI) and the fraction of absorbed photosynthetically active radiation (FPAR) (e.g., Asrar et al., 1989; 1992; Goward and Huemmrich, 1992; Sellers et al., 1992a). Opera- tional remote sensing of these quantities, however, is complicated by a variety of effects including atmospheric path radiance and absorption, surface direc-

0034-4257 ! 95 / $9.50 SSDI 0034-4257(95)00156-U

234 Friedl et al.

tional reflectance, subpixel heterogeneity in land cover, and topography, among others (Duggin, 1985; Duggin and Robinove, 1990; Myneni and Asrar, 1994; Dubayah, 1992).

The First ISLSCP Field Experiment (FIFE) was designed to investigate the use of satellite remote sensing for retrieving land surface climate parameters. Re- suits obtained from FIFE have shown that many variables may be accurately estimated from satellite data including incident solar radiation (Frouin and Gautier, 1990); net radiation (Starks et al., 1990), soil moisture (Wang et al., 1992), and LAI and biomass (Asrar et al., 1989; Friedl et al., 1994). Indeed, one of the key results from FIFE has been to show the viability of many models to study critical land surface properties and processes (see Hall et al., 1992), and, by extension, to use these data as input to biophysical models of global change processes such as biogeochemical cycling, or mass and energy exchange between the land surface and the atmosphere (Sellers et al., 1992b; Running et al, 1989; Ruimy et al., 1994; Potter et al., 1993).

Investigations from FIFE have identified a number of scientific issues that must be resolved in order for satellite-based land surface climatology to become oper- ational. Effects of sensor calibration (Markham et al., 1992), atmospheric scattering and absorption (Hal- thorne and Markham, 1992), topography (Dubayah, 1992), and scene illumination geometry and nonlam- bertian reflectance (Deering et al., 1992) introduce po- tentially large uncertainties into radiance-based estimates of surface biophysical properties. Furthermore, linking point-based process models (e.g., Potter et al., 1993) to area-based measurements from sensors may create severe scale dependencies in model outputs due to the effects of subpixel land surface heterogeneity. At present it is unknown how errors in radiometry, model formulation or specification, and scaling propagate through the entire modeling framework, but the amount of system-dependent noise is likely to be considerable (Davis et al., 1991).

The general objectives of our work here are to developed and test a modeling framework that allows examination and quantification of the sources of error in satellite-based estimates of land surface variables. In this article, we present a detailed description of a physically based scene simulation model developed for this purpose, and use this model to study scaling processes in the relationship between the normalized difference vegetation index (NDVI) and land surface biophysical variables. To do this, we use data from FIFE to simulate Landsat Thematic Mapper (TM) imagery under varying ground scene conditions, and use these simulated data to examine the effect of subpixel spatial autocorrelation in ground scene properties on scaling

and uncertainty in the relationship between the NDVI and both LAI and FPAR.

BACKGROUND

Scene Simulation Modeling Several remote sensing simulation models have been developed over the past two decades. Most of these models have considered specific components of the remote sensing data acquisition process. Some studies have modeled the performance characteristics of electrical-optical sensors (Peters, 1982) emphasizing the effect of sensor properties on important sensor diagnostics such as the signal-to-noise ratio. Other studies have used similar approaches, but have coupled sensor simulation models with atmospheric radiactive transfer models to study the effects of sensor parameters on in-flight sensor performance (e.g., Kidd and Wolfe, 1976; Schueler and Thorne, 1982; Isaacs and Vogelmann, 1988).

More recent models have followed the conceptual framework proposed by Strahler et al. (1986), where the image acquisition process is treated as having three subcomponents: a scene component, an atmospheric component, and a sensor component. These models have employed two-dimensional reflectance arrays to simulate land surface boundary conditions, thereby explicitly including some of the spatial effects inherent in the sampling and regularization process intrinsic to the remote sensing data acquisition process (Reeves et al., 1987; Kerekes and Landgrebe, 1989b). Such models have been used to examine sensor performance under different scene and atmospheric conditions, focusing primarily on the impact of sensor properties and atmospheric characteristics on the accuracy of land cover maps produced by automated classification routines (e.g., Kerekes and Landgrebe, 1989b).

A different approach has been to develop analytical models of specific components of the remote sensing image acquisition process. Such models allow a more theoretical analysis of processes such as image regularization (Jupp et al., 1988; 1989), or the combined effects of atmospheric path radiance and absorption, and sensor electrical-optical engineering on image statistical properties and classification accuracies. For example, Moba- serri et al. (1980), and Kerekes and Landgrebe (1991) present parametric models to estimate statistical classification algorithm accuracies under varying scene, sensor, and atmospheric conditions. The results of these studies not only demonstrate the importance of scene, sensor, and atmospheric properties on classification accuracies, but also illustrate the utility of realistic scene simulation modeling for studying the impact of such data acquisition parameters on the imagery produced by space-based remote sensing instruments.

Scaling the Relationship between NDVI and Biophysical Variables 235

Remote Sensing and Biophysical Modeling The development of techniques to estimate land surface parameters from remotely sensed data has been the focus of extensive research over the past two decades [see Asrar (1989) for a detailed review of these techniques]. In particular, the relationship between spectral reflectance data and plant biophysical variables such as LAI and FPAR has received much attention (Asrar et al., 1989; 1992; Sellers, 1985; 1987; Sellers et al., 1992a; Goward and Huemmrich, 1992; Bonan, 1993; Nemani et al., 1993). Currently, significant efforts are focusing on the use of spectral vegetation indices (SVIs) in general, and the normalized difference vegetation index (NDVI) in particular, for studying critical global change issues such as monitoring land cover change (Townsend and Justice, 1988; Townsend et al., 1991), estimating global net primary production (Fung et al., 1987; Box et al., 1989; Prince, 1991; Potter et al., 1993; Ruimy et al., 1994), and modeling land surface energy balance (Choudhury, 1991; Hall et al., 1992).

The basis for much of this work lies in the the assumption of a function, and therefore invertible, relationship between surface parameters of interest such as LAI and FPAR, and SVIs such as the NDVI. Much effort is currently focused on exploiting relationships between FPAR and NDVI to estimate net primary production from satellites (e.g., Ruimy et al., 1994; Potter et al., 1993). Increasingly, plant canopy radiative transfer models are being used to both explore these functional relationships and to invert remotely sensed data for canopy attributes (e.g., Goward and Huemmrich, 1992; Sellers et al., 1992a). These studies have used one- dimensional (i.e., horizontally homogeneous laterally infinite) canopy models, and have demonstrated significant potential for the use of SVIs for global change research.

Subpixel Heterogeneity and Scaling Recently, researchers have begun to examine relationships among spatial heterogeneity in land surfaces, scale invariance of surface radiance fields, and remotely sensed measures of land surface biophysical variables. Myneni et al. (1992), fi~r example, developed a three- dimensional (3-D) radiative transfer model to account for weaknesses in traditional one-dimensional (l-D) models that treat canopies as being horizontally homogeneous and laterally infinite. Using this 3-D model, Asrar et al. (1992) demonstrate significant errors in simulated radiances and SVIs based on 1-D formula- tions. Specifically, they observed a strong interaction between LAI and vegetation clump structure that sub- stantially complicates the relationship between NDVI and LAI. At the same time, they observed a linear relationship between absorbed PAR and NDVI that was independent of spatial heterogeneity, but sensitive to the soil or background reflectance.

In a related study, Hall et al. (1992) present a framework to examine scale invariance in remote sensing algorithms, radiance fields, and vegetation indices. They identify two key questions: 1) Is the variable of interest linear with respect to the SVI and 2) is the SVI itself linear with respect to radiance? In this framework, an algorithm, Aj, which infers the value of a variable p from radiance ~ emitted from an homogeneous patch of land surface is scale-invariant over a region R only if

pR = ~w,Aj(~u,) = Aj(~w,~u,) = Aj(~u~), (1)

where w~ is the proportion of area R occupied by the mean value for p over R.

Hall et al. (1992) also suggest that ratio based SVIs such as the NDVI are linear with respect to radiance only if the correlation between the numerator and the inverse of the denominator of the SVI is 0. For example, for NDVI to be linear with respect to radiance,

r = corr [(PN,, - ,Ovis), ( f f m a -I- P v , s) - 1] = 0 , (2)

where p~, and pv,s are the near-infrared and visible reflectances of the surface, respectively. Using this framework, it was shown that while NDVI appears to be linear with respect to FPAR, NDVI is not scale-invariant with respect to radiance and, by extension, estimation of FPAR from low resolution data may be problematic. Hall et al. (1992) also point out that the relationship between FPAR and NDVI is further complicated by sensor noise and atmospheric effects.

To examine scaling effects in data acquired from space-based platforms, it is also necessary to consider sensor and atmospheric effects. In analytical terms, the response of a sensor over a ground area R corresponding to the sensor instantaneous field of view (IFOV) is determined by the spatial distribution of upwelling radiance within R, ~u(x,y), the point spread function (PSF) of the atmosphere (PSFa), and the PSF of the sensor (PSFs). Assuming that the PSF~ is known and that the PSFa can be approximated, then (monochromatically) the upwelling radiance over area R measured by a space-based sensor is the convolution of the PSF,, the PSFa, and the spatial distribution of surface radiance ~u(x,y):

= PSr (x,y),VSFo(x,y),v/(x,y), (3)

where ~u~ is the value of ~ measured by the sensor over ground area R and * denotes the spatial convolution operator. Note that this relationship also depends on sensor and solar geometry.

Complexities introduced by factors such as spatial heterogeneity and anisotropic surface reflectance, atmospheric effects, and interactions between sensor spatial and spectral resolution among others (Duggin and Rob- inove, 1990) have precluded a precise analytical formulation defining the scaling function between the mean

236 Friedl et al.

Scene Simulation Model 2-D Autoregressive Process Simulator

LAI distribution GROUND SCENE ~ topography

reflectance model background reflectance leaf optical parameters solar~sensor geometry

radiative transfer ATMOSPHERE ~=_ path radiance

transmission

SENSOR ~=- spatial sensitivity calibration (noise)

OUTPUT: SIMULATED REMOTELY

SENSED IMAGE (TM,AVHRR,ETC)

1 Generate matrix of uncorrelated Gaussian random numbers

2. 2-D FFT to frequency space

3. Multiply by frequency response for desired autoregressive process

4. Inverse transform to 2-D image

5. Map Gaussian random numbers to desired frequency distribution

Figure 1. Schematic diagram showing the basic structure, inputs, and data flow within the scene simulation model.

surface upwelling radiance, and the top of atmosphere upwelling radiance. In this context, physically based scene simulation models represent a mechanism to study these effects by simulating satellite imagery based on high resolution ground scenes and examining the manner in which data scale from high to low resolutions as a function of ground scene, atmospheric, and sensor properties.

MODEL DESCRIPTION

The simulation model described below is designed as a physically based model to realistically simulate remotely sensed images from a high spatial resolution input scene that explicitly accounts for effects associated with land surface attributes, the atmosphere, and sensor electrical-optical engineering. The model follows the conceptual design described by Strahler et al. (1986) and imple- mented by Kerekes and Landgrebe (1989b), and is composed of three submodels: a ground scene model, an atmospheric model, and a sensor model (Fig. 1).

Ground Scene Model The ground scene model consists of a high resolution two-dimensional array of LAI values overlaid on a digital elevation model (DEM) of the terrain to be simulated, and is designed to simulate realistic spatial structure associated with natural landscapes. To this end, we

Figure 2. Flow diagram showing the data flow in 2-D autoregressive process simulator.

model spatial structure in vegetation using a two-dimensional (2-D) process model that simulates patch structure and spatial autocorrelation in vegetated surfaces. In this framework, each high resolution ground scene element is treated as being composed of uniform surface properties (and therefore has uniform surface reflectance), where the dimension of each scene cell is specified by the user. For example, the simulations performed for this work used ground scene cell dimensions roughly 0.11 times the size of the sensor IFOV (i.e., 92 ground cells per simulated sensor pixel). Surface directional reflectance at each point in the ground scene is calculated using a canopy reflectance model parameterized using surface biophysical parameters including LAI, canopy geometric and optical properties, and soil background reflectance. Further, we account for topographic effects on incident and upwelling surface radiation using a topographic correction scheme proposed by Dubayah (1992). In this framework, adjacency effects and interactions due to multiple scattering between cover types are treated as second-order effects and are not explicitly modeled.

Spatial Distribution of LAI The basic routine underlying the scene simulation pro- cedure is a two-dimensional autoregressive process simulator containing five steps (Fig. 2). This routine takes an input frequency distribution for LAI, and distributes


random values from thi,'; frequency distribution within the ground scene in a spatially coherent manner. To do this, a matrix of independent Gaussian random numbers is generated. Next, the matrix is transformed to frequency space using a 2-D fast Fourier transform and multiplied by a user-specified frequency response for the desired autoregressive process defined by an autoregressive parameter (ARP) ranging from 0 to 0.24. This parameter effectively represents the correlation between adjacent grid cell~ in the ground scene. This transformed matrix is then inverse-transformed to produce a 2-D image with a spatial structure approximating the spatial variation of natural landscapes. Finally, the Gaussian random numbers are mapped to the specified frequency distribution through a ,;imple matching of ranks. The output grid is therefore a 2-D field of values for LAI, exhibiting coherent spatial structure and a specified frequency distribution.

Surface Reflectance A key objective of our model is to account for the effects of differing satellite and solar geometries on imagery produced by space-borne sensors. Clearly, such effects are anisotropic and any realistic simulation must account for surface directional reflectance as characterized by the surface bidirectional reflectance distribution function (BRDF) (Nicodemus et al., 1977). To simulate surface directional reflectance, we employ a canopy reflectance model that calculates directional reflectance as a function of solar and viewing geometry, and surface biophysical properties. At the present time we are using SAIL (Verhoef, 1984), a reflectance model explicitly developed for turbid medium canopies such as grasslands, and which is therefore appropriate for simulating surface directional reflectances for the tallgrass prairie vegetation at FIFE site. Note that the SAIL model does not provide for the shadow hiding effects that occur when illumination and viewing positions coincide (i.e., the hot-spot effect).

Topography Topography imposes two main effects on the upwelling radiance from land surfaces viewed by satellites. First, local variation in slope angle and azimuth produce variation in incident beam and diffuse radiation (Dubayah, 1992). Second, the upwelling radiance viewed by remote sensing devices varies as a function of local slope and view zenith angle (Dozier, 1989). We simulate these effects in two stages. We first account for spatial variation in incidence radiation due to topography. To do this, we normalize the downwelling irradiance at each point on the grid by the irradiance for equivalent atmospheric conditions on a horizontal surface using a topographic correction factor (Kt). This normalization factor is calculated from a digital elevation model and knowl- edge of atmospheric properties (Dubayah, 1992):

K, F~ r ~ ± e-rluo[lt~ Va] = vd + J ' (4/ , L/to

where (monochromatically) F& is the downwelling irradiance incident on a sloping surface, F&u0 is the downwelling irradiance over a horizontal surface, Va is a sky-view factor accounting for the portion of the overlying hemi- sphere visible to the sloped surface, Ct is a terrain configuration factor that accounts for direct and diffuse irradiance from adjacent terrain, p is the reflectance of the surface, r is the optical depth of the atmosphere, T, is the directional hemispherical transmittance through the atmosphere (Dozier, 1989),/l, is the cosine of the local solar zenith angle, and/~0 is the cosine of the solar illumination angle on a horizontal surface.

In addition to correcting for topographic effects on downwelling radiance, upwelling radiance at each point is adjusted to account for the satellite view zenith angle using a cosine correction, where the cosine of the sensor view zenith ~o) is calculated as a function of the view zenith and azimuth, and the local slope angle and azimuth. In this way, topographic effects on upwelling radiance are included in the model by adjusting the "fiat-surface" directional reflectance for topographically induced variation in incident and upwelling radiation. The upwelling radiance directly above the surface viewed from a space-born sensor is then

~ ( ~r,Or) -~ e&l~oKt~,Lvp( ~oOo;~t~r, Ov), (5)

where Ft(gr,0r) is the upwelling radiance directly above the surface in direction 9r,0r, and p(~o,Oo;~Or, Or) is the directional reflectance of the surface for solar and view zeniths and azimuths ~o,0o,~,0~, respectively.

Atmospheric Model The objective of this component of the simulation model is to accurately simulate the atmospheric effect on upwelling surface radiance as viewed by a satellite at the top of the atmosphere. Myneni and Asrar (1994) present a systematic analysis of the combined effects of atmospheric and surface directional reflectance effects on a variety of SVIs using a coupled canopy-atmosphere radiative transfer scheme, and clearly demonstrate the importance of accounting for the additive effects of surface directional reflectance and atmospheric interactions in satellite derived SVIs. For a specified spectral band, surface reflectance, atmospheric model (e.g., mid- latitude, summertime, continental), visibility, and sensor/solar geometries we compute top-of-atmosphere (TOA) radiance using the 5S radiative transfer model (Tanr6 et al., 1990). 5S is specifically designed for satellite remote sensing applications and includes sensor spectral weighting functions in the calculation of simulated TOA radiances. This model has been widely used and its accuracy has been well documented elsewhere in the literature (Markham et al., 1992). Using 5S, we

238 Friedl et al.

Table 1. Sensor Model Parameterization Data for TM Bands 3 and 4 from Kerekes and Landgrebe (1989a) a

Input TM3 TM4

Full scale radiance (Wm -2 #m -l sr l) Min radiance (Wm -2/zm -1 sr -l) Sensor gain state Number of radiometric bits Sensor altitude (m) Across scene interval ~rads) Down scene interval (prads) Number of angles LSF response Step size of LSF response ~rads)

204.3 206.2 - 1 . 2 - 1 . 5

1.0 1.0

8 8 705000.0 705000.0

0.000043 0.000043 0.000043 0.000043 9 9 0.000004778 0.000004778

a Note that noise effects were not included in the simulation runs, and that the spatial point spread function (not included) was parameterized using a cosine approximation truncated at the half-amplitude response.

calculate TOA radiance as a function of atmospheric conditions, solar and sensor geometry, surface reflectance derived from SAIL, and topography.

Sensor Model

Goward et al. (1991) examined the variation in the precision of surface reflectances and the NDVI derived from AVHRR data caused by factors such as sensor radiometry, target brightness, and solar and viewing geometries, and showed that the relationship between surface biophysical state and the NDVI from AVHRR is highly dependent on these effects. To simulate the electrical-optical engineering and associated imaging process of satellite-based remote sensing devices, we have adapted a set of subroutines developed by Kerekes and Landgrebe (1989a). This submodel explicitly accounts for sensor spatial response, as well as for noise effects introduced by sensor electrical-optical engineering including sensor shot and thermal noise, and noise introduced by analogue to digital (A/D) conversion of continuous radiance measures to digital numbers (DNs). The inputs to this model are the TOA radiance image (generated by the ground scene and atmospheric models), and a suite of parameters describing the sensor spatial, radiometric, and calibration characteristics (Ta- ble 1). It is important to note that sensor spectral point spread functions are accounted for by 5S, and are therefore not included in the sensor model. Based on this sensor description, the model applies the sensor spatial response to the TOA radiance image, converts the spatially integrated radiances to digital numbers, and adds noise to the image data. Thus, the output from this model is a simulated digital image for the specified scene, atmosphere, solar/sensor geometry, sensor, and spectral band.

METHODS

Data We used data from FIFE to simulate TM data for sensor and solar geometries corresponding to 6 June 1987 at the FIFE site. To parameterize the ground scene model

and SAIL, we used LAI and canopy biophysical data (leaf optical properties, background reflectance) acquired on or near this date. Leaf angle distributions within SAIL were parameterized using a spherical model, and LAI frequency distributions were derived using kernel density techniques. We estimated distributions for the main grassland grazing and burning treatments present within the FIFE site: unburned, ungrazed grassland; burned, ungrazed grassland; and burned, grazed grassland (Fig. 3).

The scene simulation model described in the previ- ous section explicitly accounts for topographic effects in remotely sensed imagery and therefore requires an accurate DEM to incorporate these effects. A DEM was

Figure 3. Plots of LAI frequency distributions for burned, grazed grasslands (BG), burned, ungrazed grasslands (BUG), and unburned, ungrazed grasslands (UBUG) in early June 1987 at the FIFE site.

LAI Frequency Distributions O0 0 0

~O 0 0

¢0 c"

a .

. d o

2 a .

o

0 0

[] Burned, Grazed I 0 Bumed, Ungrazed I A Unburned, Ongrazed I

n 0

0 0

0

D Dn 00000000000000

.A ' ' ~ - O- "-t'~'5'AA ~ 0 0 , , ,o , o o °

/ ooO° \ "%,,

0.5 1.0 1.5 2.0 2.5

LAI


produced for FIFE by the U.S. Army Corps of Engineers by digitizing contour lines on portions of four U.S. Geological Survey 7.5.-min topographic quadrangles (Swede Creek, Wamego SW, Volland, White City NE) and interpolating to a 25-m grid. A 30-m grid was produced from this DEM to conform with the spatial resolution of Landsat TM using bilinear interpolation. For this work, however, we simulated a ground scene with a grid cell resolution (~-3.3 rn) significantly below the 30-m horizontal resolution of this DEM. To realistically simulate the variation in topography within the FIFE site at this scale, we applied a low pass filter to the site DEM to generate a high resolution elevation data set with topographic properties similar to those present at the Konza Prairie. To simulate atmospheric effects, we used a midlati- tude, summertime, continental model with a visibility of 45 km using 5S. Finally, we parameterized the sensor model for TM Bands 3 ;and 4 based on the engineering specification data for TM given in Kerekes and Landgrebe (1989a). In this context, because we were interested in isolating scaling effects, sensor noise effects were not included in our simulations. That is, with the exception of quantization errors due to A/D conversion, all other electrical-optical sources of noise were set to zero.

Study Site The FIFE experimental site is located in the Flint Hills of northeastern Kansas and consists of 25,600 ha of native bluestem prairie :mixed with lower lying riparian zones and cultivated lands. The northwest portion of the site includes the 3487-ha Konza Prairie Long Term Ecological Research site (KPLTER) located 8 km south of Manhattan, Kansas (39°9'N, 96040'W). The site is a dissected plateau, with level uplands and steep dendritic drainages in which total relief is 50-75 m. The vegetation is dominated by C4 grasses and shrubs. Unplowed native bluestem prairie covers much of the area, and the dominant grasses include big bluestem (Andropogon gerardii), little bluestern (Andropogon scoparious), and Indian grass (Sorghastrum nutans). Controlled treatments on the Konza prairie include grazed and ungrazed, and burned and unburned areas in various an- nual rotations. Some areas left unburned for many years are undergoing invasion by trees and shrubs, notably smooth sumac (Rhus glabra), American elm (Ulmus americana), buckbrush (Symphoricarpos orbiculatus), and red cedar (Juniperus virginiana). Oak forests (Ouercus spp.) occupy the lower ends of some drainages and many steep north facing slopes. Outside of the KPLTER site, uplands are largely used for cattle grazing, and cereal crops are cultivated in the broader stream valleys.

Analyses The objectives of the analyses presented below are to simulate Landsat TM imagery using scene properties

and solar and viewing geometries from FIFE, and to use these simulated data to examine the effect of subpixel spatial autocorrelation in LAI on the relationship between simulated TM NDVI data and LAI and FPAR. To do this, we performed a set of simulations where the scale of spatial variation in ground scenes was varied from high frequency spatial variation well below the spatial resolution of the simulated sensor IFOV, to low frequency spatial variation with autocorrelation length scales roughly equivalent to the spatial resolution of the sensor IFOV. We used identical atmospheric and sensor parameterizations for all simulation scenarios, varying only the spatial arrangement and frequency distribution of LAI in the ground scene for each simulation.

In total, nine simulated scenes were used to analyze relationships between NDVI and LAI and FPAR. To do this, we defined the high resolution ground scene as being composed of uniform cells with a spatial dimension of 3.33 m, or 81 (92) ground scene elements per 30-m TM pixel. Using this format, we simulated three sets of images for each frequency distribution with ARP values of 0.05, 0.15, and 0.24. These ARP values produced ground scenes with autocorrelation length scales of roughly 5 m, 15 m, and 30 m, respectively. Figure 4 shows sample images of these 2-D LAI scenes in which the differing spatial structure as a function of ARP is apparent.

Values for FPAR were calculated for each cell in the high resolution ground scenes using an expression based on Beer's law:

FPAR = FPAR~.[1 - exp( - K" LAI)], (6)

where FPAR® is the FPAR for an infinitely thick canopy (~ 0.94) and K is an extinction coefficient that depends on solar geometry and the leaf angle distribution of the canopy (Montieth and Unsworth, 1990). Based on this framework, 30-m spatial averages for both LAI and FPAR were calculated for each simulation scenario using the high resolution ground scene data for each variable. These values were then compared with simulated NDVI data and used to determine regression models.

To measure the uncertainty due to regularization and scaling effects, we used a statistic described by Baret and Guyot (1991) called the relative equivalent noise (REN). This statistic estimates the uncertainty associated with a variable (in this case LAI or FPAR) inverted from a SVI such as the NDVI. This estimate is based on the variance in the NDVI associated with a given LAI or FPAR value and the slope of the relationship between NDVI and the variable in question. For example, for LAI the REN is calculated as

R E N ~'LAI O'NDvI[d(NDVI)1-1 LAI ~-i-[ ~ - J ' (7)

where a is the standard deviation of the subscripted

240 Friedl et al.

ARP=0.05

i iiiii!! ii! ii iiii i

0 100 200 300 400 500

O

O O

O O 0 3

O O C'X,I

O O

O

ARP=0.24

i

0 100 200 300 400 500

Q

t.O

UJ O o

4. Simulated LAI (518 x 518) >~ Figure images using two different ARP val- _, d ues (0.05, 0.24) to parameterize the m~ ¢u d model. Both images have LAI values o9 derived from the burned, grazed fre- ":. quency distribution (Fig. 3). Also o plotted are semivariograms for each ~. simulation showing the effect of dif- o fering ARP values on the autocorrelation length scales within the images.

0 20 40

LAG(m)

tq. 0

tO LU 0

Z "¢ < ,:5 r r

m m

CO

0

0

0

60 0

t

d'

20 40 60

LAG(m)

variable. For this work, we calculated the slope of the relationship between NDVI and FPAR or LAI using empirical models estimated from the simulated data (see below).

RESULTS

Sensor Regularization Effects

Results from our simulations show several well-defined patterns arising from interactions between sensor spatial resolution and the frequency of spatial variation in ground scenes. At the simplest level, the amount of variance in LAI and FPAR explained by NDVI (R g) increases monotonically with autocorrelation length scale in the ground scenes (Table 2). For example, for the burned, grazed frequency distribution R 2 varies from 0.49 to 0.81 for ground scenes with high (ARP = 0.05) and low (ARP = 0.24) frequency spatial variation, respectively. Note that patterns in the relationship between NDVI and both LAI and FPAR are highly similar, and that a sharp increase in R 2 is observed as the autocorrelation length scale in the ground scene approaches the spatial resolution of the simulated TM sensor. The slope

coefficients in these models show similar patterns where the estimated slopes increase sharply for ARP--0.24 reflecting higher sensitivity in the NDVI to change in the variable in question.

Figure 5 plots the relationship between NDVI and both FPAR and LAI at both the land surface [bottom of atmosphere (BOA)] and at the top of atmosphere (TOA). In this plot, the BOA NDVI data are spatial means of high resolution ground scene NDVI values averaged to the 30-m resolution of the TM sensor, and are plotted against 30-m spatial averages of LAI and FPAR from the ground scene. The TOA values are simulated NDVI data based on modeled planetary reflectance values for TM Bands 3 and 4 including both atmospheric and sensor effects. As has been widely illustrated elsewhere, the effect of the atmosphere is to depress the TOA NDVI data relative to BOA values. More importantly, these plots illustrate that sensor regularization introduces significant scatter to the relationship between the NDVI and the variables in question.

Figure 5 also plots linear regression models (dashed lines) to predict LAI and FPAR from TOA NDVI along with 95% confidence intervals about those predictions.


Table 2. Summary Table Showing Regression Coefficients and R 2 for Each of the Simulation Runs Performed

Variable Frequency Dist. ARP Intercept Slope R 2

I_,AI Burned, grazed 0.05 - 0 . 1 3 2.46 0.49 0.15 - 0 . 1 2 2.41 0.61

0.24 - 0.36 3.09 0.81

Burned, unfrazed 0.05 - 0.47 3.44 0.45 0.15 - 0.52 3.52 0.52

0.24 - 0.82 4.08 0.75

Unburned, ungrazed 0.05 - 0.05 2.58 0.38 0.15 - 0 . 1 9 2.84 0.48 0.24 - 0.50 3.47 0.71

FPAR Burned, grazed 0.05 0.02 0.79 0.50

0.15 0.02 0.77 0.60

0.24 - 0.06 1.00 0.81

Burned, ungrazed 0.05 0.06 0.83 0.46

0.15 0.03 0.87 0.54 0.24 - 0.03 0.99 0.78

Unburned, ungrazed 0.05 0.10 0.70 0.40 0.15 0.05 0.79 0.49

0.24 - 0.02 0.95 0.72

Clearly, a significant proportion of predicted LAI values lie outside of the 95% confidence bounds. Note that the scales on the axes of these plots are not consistent. In this context, Figure 5 also demonstrates that the ranges and distributions of NDVI, LAI, and FPAR values at the spatial scale of the sensor IFOV are strongly controlled by the spatial autocorrelation in the ground scenes.

This effect is illustrated in Figure 6 which plots frequency distributions estimated from random samples (n = 200) of the 30-m LAI, FPAR, and NDVI data. These plots show that the effect of regularization is to reduce the range and variance of LAI, FPAR, and NDVI values relative to the actual range and distribution of these data within the ground scene. For scenes with high frequency spatial variation (ARP = 0.05), the majority of the spatial variance in the simulated ground scenes is at subpixel scales. Consequently, the spatially averaged LAI, FPAR, and simulated TM NDVI data capture only about 15% of the actual range of values present in the high resolution ground scene data. For scenes with low frequency spatial variation (ARP -- 0.24), ground scene areas are more uniform at the scale of the simulated TM pixels. Nonetheless, the ranges for LAI, FPAR, and simulated TM NDVI capture only about 60%, 72%, and 75%, respectively, of the ranges of data actually present in the ground scene. As a result, spatial averaging tends to make these frequency distributions more Gaussian in form by smoothing out secondary modes, and by truncating the tails of the ground scene frequency distributions.

Relative Equivalent Noise To calculate REN, we stratified random samples (n = 200) of simulated NDVI data into groups corre-

sponding to distinct LAI and FPAR increments for each of our simulation scenarios. We then used these subsets of NDVI to calculate REN for each LAI and FPAR increment. A representative sample of these results is plotted in Figure 7 for the burned, grazed LAI frequency distribution. Again, note that the axes on these plots are not consistent. Also, note that because of the frequency distribution truncation effect discussed above, the REN values for simulations with high frequency spatial variation in ground scenes (ARP = 0.05 and 0.15) cover a smaller range of LAI and FPAR values than those for the simulation with low frequency spatial variation (ARP = 0.24).

REN values for LAI are consistently larger than those for FPAR. For both variables, REN tends to be larger for smaller values of the variable in question. This result is due to the fact that the REN measures the relative uncertainty in remotely sensed estimates of LAI or FPAR. Because the variance in NDVI is fairly constant across all values of LAI and FPAR (at least for the simulations considered here), REN varies inversely with the magnitude of LAI or FPAR.

While there is some variation associated with specific values of LAI and FPAR, REN values tend to increase for larger values of ARP. For LAI, mean REN values were 0.12, 0.16, and 0.31 for ARP values of 0.05, 0.15, and 0.24, respectively. For FPAR, mean REN values were 0.09, 0.11, and 0.22 for ARP values of 0.05, 0.15, and 0.24, respectively. Intuitively, one might expect that REN would vary inversely with the R 2 between NDVI and the variable in question. However, because the variance in NDVI varies directly with ground scene ARP (due to regularization effects), REN also tends to vary directly with ARP. In this context, the magnitude of uncertainty in estimates of LAI and

2 4 2 Friedl et al.

I% ¢5

o.

to

~ °

t~ d

LAI: A R P = 0.05

÷ ......"" + TOA I ÷ ./..

+ + ." A . ,~. ++ ,.. A

~ "~

...." +÷÷÷÷ . . / ~ /

0.30 0.35 0.40 0.45 0.50

N D V 1

k~l: ARP = 0.15

+ +. Z..': ~

0.25 0.35 0.45 0.55

N D V I

LAI: A R P = 0.24

/ / , . .

0.2 0.4 0.6 0.8

N D V I

=..

==

FPAR: A R P = 0.05

÷ ,"' + TOAI ÷ + . + ... &a

÷÷+ ~ ~ ÷

÷÷÷ +

/ ' ¢ ,~ + / . ÷ - + .

0.30 0.35 0 .411 0.45 0.50

NDVI

FPAR: A R P = 0.15

÷ ÷ ,...~ . :.÷7~" ,¢ e

÷

0.25 0.35 0.45 0.55

N D V I

FPAR: A R P = 0.24

0.1 0.2 0.3 0.4 0.5 0.6 0 . 7

N D V I

Figure 5. Scatter-plots showing the relationship between NDVI and LAI and FPAR at three different scales of subpixel heterogeneity. The plots show the relationship between NDVI and the variable in question at the bottom of the atmosphere using data derived from the ground scene only, and the relationship between NDVI and the variable in question at the top of the atmosphere• Also plotted are the estimated regression relationships for the top of atmosphere data and their associated 95% confidence intervals.

FPAR inverted from the NDVI data (10-30%) are espe- cially significant given the relatively simple ground scene description and narrow range of LAI values used to generate the simulated TM data.

DISCUSSION AND CONCLUSIONS

The results from our simulations demonstrate a strong interaction between sensor resolution and the scale of variation in ground scenes. In particular, the effect of sensor regularization is shown to weaken the statistical relationship between NDVI and both FPAR and LAI,

LAI Probability Density N D V I Probability Density

,:5

r~ d

o o [ ~ . ~ ARP=0.05

o o [] ARP=O~.4

0.0 0.5 1.0 1.5 2•0 2.5

L A I

FPAR Probability Density

-

. _ o • • o • •

o = •

o • o - ;

Q. • • o

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0.0 0.2 0.4 0.6 0.8

F P A R

D o

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d

Hi-Res ARP=0.0E

[] ARP=0.2=

0.2 0.4 0.6 0.8

N D V I

Figure 6.. Empirically estimated probability density functions showing the effect of ARP values on the frequency distributions for LAI, NDVI, and FPAR at the spatial resolution of the sensor IFOV.

and to truncate and modify the frequency distribution of inverted LAI and FPAR values relative to the actual distribution of these variables on the ground. These types of patterns have been previously demonstrated in studies examining the relationship between three-dimensional scene structure and image variance (Woodcock and Strahler, 1987; Jupp et al., 1989). The results presented here further this research domain by exploring the relationship between remotely sensed data and specific land surface biophysical variables using a physically based model.

To this end, our results suggest that the remote sensing process introduces substantial scatter to the TOA relationship between NDVI and land surface biophysical variables relative to that observed for spatially averaged BOA data. This scatter can be explained by a combination of three mechanisms:

i. Nonlinearity in the relationship between NDVI and LAI or FPAR: The relationship between LAI and NDVI is distinctly nonlinear, and it is well established that this relationship is not scale invariant over space [Eq. (1)]. However, because of the relatively narrow range of LAIs used to generate the ground scenes, errors of this nature were relatively minor. Similarly, because the relationship between NDVI and FPAR is nearly linear, scaling error introduced


Burned Grazed Burued Grazed B u r n e d , G r a z e d - A R P 0 . 2 4 A R P = 0 . 0 5 A R P = 0 . 0 5

o , 0 , 0 , 0 , 0 0 °2 . . . . . 0 , 0 ~ ~ LAI FPAR

A R P = 0 . 1 5 A R P = 0 . 1 5 Il l

0.40.~.~.'.~ .eO.g1.01.1 I ~ LAI

A R P = 0 . 2 4 I d o A R P = 0 . 2 4 ...I

i l l 7

0.~,~,~.81.01 ,~1,41.6

<5 , Figure 7.. REN for both LAI and FPAR as a function ARP for the burned, grazed frequency distribution. (Note that these values are expressed as proportions; i.e., REN x 100 = % uncertainty.)

by spatial averaging of NDVI and FPAR introduced little uncertainty to this relationship. These effects are illustrated by examining the relatively low scatter in the BOA relationships between NDVI and both LAI and FPAR.

ii. The NDVI itself is not scale invariant: Spatial scaling of this variable therefore introduces bias to the relationship between the arithmetic mean of high resolution NDVI in the ground scene and TOA NDVI calculated from low resolution reflectance data [Eq. (2)]. To illustrate this effect, we have plotted spatially averaged (30-m) high resolution BOA NDVI versus NDVI calculated from spatially averaged BOA reflectances (Fig. 8). This result shows that NDVI calculated from low resolution reflectance data tends to slightly underestimate the true spatial mean, but that scaling in the NDVI is a relatively minor contributor to the total scatter observed in the TOA data in Figure 5.

iii. Sensor regularization produces scatter in the relationship between TOA and BOA reflectances: To illustrate this effect, we have plotted spatial averages of BOA reflectances in TM Bands 3 and 4 from the high resolution ground scene versus TOA reflectances from the simulated TM data (Fig. 9). This plot illustrates that sensor regularization is the dominant source of uncertainty in the simulated TOA relationship between NDVI and both LAI and FPAR.

0.2 0.3 0.4 0.5 0.6

Mean NDVI

Figure 8. Scatter plots showing the effect of nonlinear scaling in NDVI. The horizontal axis plots the arithmetic mean of high resolution NDVI values, and the vertical axis plots NDVI calculated from low resolution BOA reflectances for the burned, grazed frequency distribution.

A primary goal of much current research is to use remotely sensed data to monitor and model key biophysical parameters and processes such as plant photosynthesis, transpiration, and net pr imary production over large areas encompassing significant heterogeneity in land cover. The ultimate utility of these efforts, however, will depend in part on the sensitivity of such biophysical

Figure 9. Scatter plot showing pixe] scale reflectances for TM Bands 3 and 4 from the ground scene (i.e., true mean reflectance at BOA) versus TOA reflectances estimated from from the simulated TM data (TOA regularized). Also note that atmospheric effects were minimal in the TM3 data because a very clear atmosphere (visibility = 45 km) was used for these simulations.

<5

d

I-- o

d

o

T M 3 - B O A vs T M 3 - T O A

~. A t` ~ A t`

0 .06 0.08 0.10 0.12 0.14 0.16

BOA

T M 4 - B O A vs T M 4 - T O A

A A A

A A A

/ . , , . ~ t ; t t . " r / . #P A A A ~

0.22 0.26 0.30 0.34

BOA

2 4 4 Friedl et al.

models to uncertainty in remotely sensed inputs. In this context, it is important to note that despite the well-controlled and simplified simulation scenarios examined in this article, uncertainties in pixel scale LAI and FPAR values estimated from simulated NDVI data were substantial ( 2 10-30%). For real world applications where ground scenes are considerably more complex, scaling effects associated with subpixel heterogeneity in variables such as soil reflectance and land cover will introduce significantly higher uncertainties than those produced by the model simulations presented here.

FUTURE DIRECTIONS

With these issues in mind, a variety of model refine- ments and analyses are currently planned to improve the realism of the scene simulation model and to further our understanding of scaling processes in remotely sensed data. In particular, soil background effects have been widely shown to have a strong influence on SVIs (Huete, 1988; Goward and Huemmrich, 1992; Huete and Liu, 1994). An important goal in the near future is to include provision for variable soil background reflectances within the model. Furthermore, the optical properties of real world vegetation canopies are not uniform in space, and we intend to allow for both spatial variation in dead (brown) versus live (green) LAI within canopies, as well for spatial covariance between canopy attributes and variables such as topography and land cover. Also, while the current version of the model accounts for atmospheric transmission and path radiance, the atmospheric PSF is not explicitly included. Clearly, this factor is relevant to future studies of scaling processes in remotely sensed data, and inclusion of this effect will be a priority in future model development. Finally, we plan to couple this modeling framework with a land surface biophysical model, and to use this framework to perform an end-to-end assessment of the impact of scaling efforts in remotely sensed data on the outputs produced by models designed to use remotely sensed input data.

This work was supported through NASA Grant NAG5-917. We acknowledge the excellent support provided by Don Strebel and the entire FIFE information system staff. Fred Huemmrich provided source code and valuable advice regarding SAIL. Mike Bueno at UCSB helped in preliminary coding of some of the Perl scripts used in the model. We thank David Landgrebe for providing access to source code for the sensor model. Alan Strahler, Guido Salvucci, Crystal Schaaf, and anonymous re- viewers provided useful comments on earlier drafts of this article.

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scaling and uncertainty in the relationship between the

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