Multitemporal analysis of urban reflectance

Multitemporal analysis of urban reflectance

Christopher Small*

Lamont Doherty Earth Observatory, Columbia University, Palisades, NY 10964, USA

Received 12 June 2001; received in revised form 4 February 2002; accepted 4 February 2002

Abstract

Spatial and temporal changes in urban reflectance have a strong influence on energy flux through the urban environment. Optical

sensors on operational satellites provide self-consistent time series of urban reflectance variations, but quantitative analyses are

complicated by spectral heterogeneity at sensor instantaneous field of view (IFOV) scales and by temporal changes in illumination and

atmospheric conditions. These complications can be minimized by combining a multitemporal radiometric rectification with a physically

based reflectance analysis. Spectral Mixture Analysis (SMA) provides a physically based approach to quantifying spatial and temporal

changes in spectrally heterogeneous urban reflectance. Multitemporal analysis of Landsat Thematic Mapper (TM) imagery of the New

York metropolitan area suggests that urban reflectance can be described with a three-component linear mixture model spanned by high

albedo, low albedo, and vegetation endmembers. The topology of the spectral mixing space indicates that mixing fractions are well

constrained for the vegetation endmember and that nonlinear mixing occurs primarily between the high and low albedo endmembers.

Selection of pseudoinvariant (PIV) image endmembers allows radiometric rectification of multitemporal imagery to a common set of

endmembers, thereby minimizing variations in radiance that are unrelated to changes in surface reflectance. Inversion of the three-

component linear mixture model for the New York metro area produces robust, consistent fraction estimates for different combinations

of rectifications and inversion constraints. Temporal variation of the presumed invariant endmember sites provides a measure of

uncertainty for the endmember fraction estimates. The resulting vegetation fraction estimates agree with high-resolution reference

measurements to within 10% for a 1996 midsummer validation and PIV endmember fraction estimates vary by less than 7% over the

course of the 1996 growing season. In contrast, intraurban spatial variations in vegetation fraction span several tens of percent,

suggesting that the measured changes significantly exceed the uncertainty of the estimates. These results suggest that Landsat TM

imagery may be used to monitor seasonal to interannual variations in urban reflectance and vegetation abundance. D 2002 Published by

Elsevier Science Inc.

1. Introduction

Global urbanization is one of the primary forms of

environmental change directly impacting the human popu-

lation. Although cities occupy a small percentage of the

Earth’s land area, the physical conditions of the urban

environment exert a direct influence on almost half of the

world’s population (United Nations, 1999). In order to

understand the physical dynamics of the urban environment,

it will be necessary to quantify changes in certain key

environmental parameters. Many of the important envir-

onmental parameters in urban areas are best measured in

situ, but some parameters are more amenable to measure-

ment by remote sensing. Optical remote sensing measures

upwelling radiance, a parameter directly related to the

albedo and surface reflectance of the urban mosaic. Albedo

is a critical environmental parameter because it modulates

energy fluxes and can be influenced by choices of building

materials and landcovers.

Temporal variations in the albedo of the urban mosaic

exert a strong influence on the energy flux through urban

environments. A procedure to estimate shortwave urban

albedo from Landsat MSS imagery was devised by Brest

and Goward (1987) and used by Brest (1987) to quantify

seasonal variability in urban albedo for input to climate

models. One of the primary determinants of the albedo and

surface temperatures in urban and suburban environments

is the spatial and temporal distribution of vegetation

(Goward, Cruickshanks, & Hope, 1985). Fig. 1 shows an

example of the seasonal variation in Near Infrared (NIR)

reflectance resulting from vegetation phenology in the

New York metro area. By modulating reflection and

0034-4257/02/$ – see front matter D 2002 Published by Elsevier Science Inc.

PII: S0034 -4257 (02 )00019 -6

* Tel.: +1-845-365-8354; fax: +1-845-365-8179.

E-mail address: [email protected] (C. Small).

www.elsevier.com/locate/rse

Remote Sensing of Environment 81 (2002) 427–442

absorption of solar radiation and evapotranspiration of

water (Carlson & Boland, 1978), urban vegetation has a

direct impact on energy consumption and living conditions

in cities worldwide. The presence and abundance of

vegetation in urban areas may also influence air quality

and human health (e.g. Abdollahi & Ning, 2000; Nowak,

1994; Wagrowski & Hites, 1997). Conversely, urban

vegetation experiences both short- and long-term pheno-

logical changes and may itself be sensitive to subtle

changes in environmental conditions.

The objective of this study is to systematically quantify

spatial and temporal changes in urban reflectance using

broadband optical radiance measurements. A specific sensor

(Landsat Thematic Mapper (TM)) and location (New York

City) are used here, but the methodology discussed should

be applicable to other locations and suitable for other

moderate resolution sensors. The ultimate objective is the

quantitative multitemporal characterization of the visible

and infrared reflectance of the urban mosaic. Characteriza-

tion of urban reflectance as mixtures of spectral endmem-

bers has been investigated in a previous study (Small,

2001a). This study will attempt to extend the urban spectral

mixture model to a multitemporal analysis of the changes in

urban reflectance. The analysis will focus on the tasks of (1)

minimizing the effects of variations in atmospheric turbidity

and solar illumination on the apparent reflectance and (2)

quantifying the uncertainty introduced by these effects.

The effects of spatial scaling and the application of the

results to urban environmental analysis will be discussed in

separate studies.

2. Urban reflectance

Urban areas are generally recognized in remotely sensed

imagery by their geometric and textural characteristics.

Spectral characteristics of urban landcover are less dia-

gnostic than those of the rural periphery and unpopulated

areas such as deserts and forests. This is primarily a

consequence of the characteristic scale and spectral hetero-

geneity of surface reflectances in the urban mosaic (Small,

2001b). Urban bidirectional reflectance factor models have

been developed for low-resolution (500 m) imagery (Meis-

ter, Rothkirch, Spitzer, & Bienlein, 2001), but the diversity

of surface reflectances in the urban mosaic is an impediment

to generalization of urban reflectance at higher spatial

resolutions. In a study of the Washington, DC metro area,

Ormsby (1992) distinguished nine urban landcover classes

with significant spectral separability in Landsat imagery.

While these classes may be distinguishable when they occur

in homogeneous regions larger than the spatial resolution of

the sensor, this is rarely the case for moderate resolution

sensors like Landsat. Urban areas are therefore generally

characterized by spectral heterogeneity at scales approach-

ing Landsat pixel resolution.

Analysis of urban reflectance is fundamentally different

from many other applications of optical remote sensing.

Traditional hard classification methods, such as Maximum

Likelihood classification, have generally not proven effect-

ive in the urban environment because they are predicated on

the assumption that landcover classes are spectrally distinct

at pixel scales and therefore occupy separate regions of the

spectral feature space. The characteristic spatial scale of

urban landcover is comparable to the Ground Instantaneous

Field of View (GIFOV) of the most widely used operational

multispectral sensors (e.g. Landsat TM, Multispectral Scan-

ner, SPOT) (Small, 2001b; Welch, 1982). This results in a

large percentage of ‘‘mixed pixels’’ for which the observed

radiance is a mixture of distinct radiances from features

with different reflectances within the GIFOV. Mixed pixels

violate the cardinal assumption of hard classification meth-

ods, but, in some cases, they may be characterized by

simple combinations of target materials contributing to the

observed radiance.

The Spectral Mixture Analysis (SMA) methodology is

well suited to the quantitative characterization of urban

reflectance. SMA is based on the observation that, in some

situations, radiances from surfaces with different ‘‘endmem-

ber’’ reflectances mix linearly within the IFOV (Johnson,

Smith, Taylor-George, & Adams, 1983; Nash & Conel,

1974; Singer, 1981; Singer & McCord, 1979). This obser-

Fig. 1. Seasonal variation of upwelling NIR radiance for New York City

and its surrounding areas. Landsat TM band 4 DNs have been

radiometrically rectified to be consistent with an image acquired under

relatively clear atmospheric conditions on 4/15/96. The effect of

differences in solar irradiance and path radiance have been minimized,

but the atmospheric scattering and absorption of the 4/15/96 image have

not been compensated for and adjacency effects have not been corrected.

The overall change in NIR radiance seen here is primarily a result of the

seasonal phenology of deciduous vegetation within the urban mosaic.

Image area is 43� 46 km.

C. Small / Remote Sensing of Environment 81 (2002) 427–442428

vation has made possible the development of a systematic

methodology for SMA (Adams, Smith, & Johnson, 1986;

Gillespie et al., 1990; Smith, Ustin, Adams, & Gillespie,

1990) that has proven successful for a variety of quantitative

applications with multispectral imagery (e.g. Adams et al.,

1995; Elmore, Mustard, Manning, & Lobell, 2000; Pech,

Davies, Lamacraft, & Graetz, 1986; Roberts, Smith, Adams,

1993; Roberts, Batista, Pereira, Waller, & Nelson, 1998;

Smith et al., 1990). If a limited number of distinct spectral

endmembers are known, it is possible to define a ‘‘mixing

space’’ within which mixed pixels can be described by

linear mixtures of the endmembers. Given sufficient spectral

resolution, a system of linear mixing equations may be

defined and the best fitting combination of endmember

fractions can be estimated for an observed reflectance

spectrum. The strength of the SMA approach lies in the

fact that it explicitly takes into account the physical pro-

cesses responsible for the observed radiances and therefore

accommodates the existence of mixed pixels. Application of

linear mixing models for estimation of urban vegetation

fractions in New York City has yielded promising results—

showing agreement to within 10% with vegetation fractions

measured from high-resolution (2 m) imagery (Small,

2001a). The study described here focuses on the question

of whether the method is feasible for quantitative multi-

temporal analyses of urban reflectance.

Remote sensing of spatial and temporal changes in urban

reflectance poses two distinct challenges. The fundamental

challenge is to accurately determine the relative contribu-

tions of different endmembers to individual radiance meas-

urements. To this end, the success of SMA depends on an

accurate characterization of the mixing space and the

determination of distinct and separable endmembers. For a

single image, this allows endmember fractions to be esti-

mated for each pixel and results in a set of images showing

spatial distributions of endmember abundance at the time

the image was acquired. Extending the analysis into the time

dimension poses the additional challenge of compensating

for temporal variations in observed radiance that are not

related to changes in surface reflectance. The most pro-

nounced of these variations are related to differences in

atmospheric turbidity and solar illumination. In order to

make quantitative comparisons of temporal changes in

endmember fractions, it is necessary to compensate for, or

at least minimize, the effect of the changes in radiance that

are unrelated to variations in surface reflectance.

Multitemporal analyses of optical imagery must distin-

guish temporal changes in actual surface reflectance from

other processes responsible for changes in the observed

radiance. A methodology for multitemporal SMA has been

developed by Adams et al. (1995) and Roberts et al. (1998)

for application to landcover change in the Brazilian Ama-

zon. The study described here derives from these and earlier

seminal works but focuses specifically on the urban envir-

onment. One of the primary challenges in these applications

of SMA to landcover changes in the Amazon was related to

the characterization of the mixing space and determination

of spectral endmembers. In a sense, the application of SMA

to urban reflectance is a simpler problem because the

configuration of the broadband mixing space is less com-

plex. On the other hand, the characteristic spatial scale of the

urban mosaic introduces complications related to spatial

resolution that are less of an issue studies of deforestation.

The study described here focuses on the radiometric rec-

tification of multitemporal imagery and the combined effect

of rectification error and model definition on the accuracy of

temporal change estimates of endmember fractions. By

combining existing methodologies for radiometric rectifica-

tion with the SMA methodology discussed above, it is

possible to account for a number of sources of uncertainty

and to produce error bounds on the resulting fraction

estimates. This error analysis is essential for interpretation

of the spatial and temporal variations in endmember frac-

tions resulting from the SMA.

The urban mosaic is fundamentally different from the

landcover types typically considered in SMA. The char-

acteristic spatial scale of the urban mosaic is comparable to

the GIFOV of the Landsat TM sensor so uncertainties in

image coregistration will introduce additional sources of

error that must be taken into account before direct pixel-to-

pixel image comparisons can be made for different times.

This study will focus on temporally self-consistent estimates

of endmember fractions and the sources of error in those

estimates. The additional complications related to spatial

scale and image coregistration will be addressed in a

separate study. The overall strategy used for multitemporal

SMA of urban reflectance in this study consists of (1)

dimensional analysis of the spectral mixing space, (2)

radiometric rectification of multitemporal radiances to a

common apparent reflectance, (3) estimation of endmember

fractions, and (4) error analysis and validation.

3. Spectral dimensionality

The tractability of the SMA is determined by the spatial

scaling and spectral dimensionality of the mixing space. If

mixing among endmembers is linear, a complete description

is given by a system of linear mixing equations describing the

contribution of each endmember fraction in each wavelength

band (e.g. Adams et al., 1986; Johnson, Smith, & Adams,

1985; Johnson et al., 1983; Smith et al., 1990). Inversion of

the mixing equations for endmember fractions requires that

the number of endmembers be less than the number of spectral

bands (Adams et al., 1986; Settle & Drake, 1993). Dimen-

sional analysis of hyperspectral imagery suggests that the

Earth’s surface is characterized by hundreds of spectrally

distinct features (Green & Boardman, 2000) at a variety of

spatial scales. The spectral sampling performed by opera-

tional broadband sensors, like Landsat TM, results in a

projection of this high dimensional mixing space onto a lower

dimensional mixing space (Jackson, 1983; Boardman, 1993).

C. Small / Remote Sensing of Environment 81 (2002) 427–442 429

If the low dimensional projection has fewer endmembers than

the number of sensor bands then the resulting system of

mixing equations will be overdetermined (more equations

than unknowns) and amenable to the methods of linear

inverse theory (Boardman, 1989). In the case of Landsat

TM, the number of projected endmembers must be fewer

than the six reflected bands measured by the sensor.

The spectral and spatial resolution of the sensor and the

characteristic scale of the surface reflectance determine the

way the high dimensional mixing space that exists in reality

is projected onto the lower dimensional mixing space

detected by the sensor. If the differences between similar

spectral endmembers are not resolved by the broadband

sensor, they will appear to have the same reflectance in the

broadband imagery and will occupy the same portion of the

spectral mixing space. The spectral dimensionality of the

urban mosaic at scales of tens of meters is generally higher

than most natural environments at similar scales (Green &

Boardman, 2000). Multiresolution analyses of imaging

spectrometer data from a variety of urban settings indicate

that this higher dimensionality is consistently projected onto

a broadband mixing space in which a large majority of the

scene variance can be described by linear mixing between

three spectral endmembers (Small, 2001b). These are ana-

logous to, but different from, the image endmembers dis-

cussed by Adams et al. (1995) and Roberts et al. (1998). In

this analysis, the image endmembers are not calibrated to

specific laboratory or field spectra as in the previous studies

because the urban image endmembers are themselves com-

binations of a variety of spectrally similar materials that are

projected onto a composite endmember by the broadband

sensor. The rationale for estimating endmember fractions of

image endmembers rather than reference endmembers is

discussed in more detail below.

Comparative dimensional analyses of Landsat TM

imagery of New York and a number of other urban areas

worldwide indicates that urban reflectance can often be

accurately described as linear mixing between high albedo,

low albedo and vegetation endmembers. Principal compon-

ent analyses consistently show that the majority of the scene

variance for urban imagery is contained within the first two

principal components while the third principal component

provides some indication of the nonlinearity of the mixing

space (Small, 2001b). The configuration of the mixing space

Fig. 2. Spectral dimensionality estimates and mixing space topology for Landsat TM imagery of central Manhattan. Eigenvalue distributions show the variance

associated with each principal component given by a MNF principal component transformation. The eigenvalue distributions imply that the majority of the

spatially correlated information content for each of the eight images considered here is contained in the first two principal components of each six band image.

The configuration of the mixing space is indicated by the subplanar triangular distribution (side view) of transformed reflectances in the feature space defined

by the three primary MNF components. Spectral endmembers reside at the apexes of the distribution and the mixed pixels are contained within the interior of

the mixing space. Convexity of the distribution (top and end views) indicates varying degrees of nonlinear mixing among the endmembers.


within these three dimensions has a consistent planar

triangular structure analogous to the well-known ‘‘Tasseled

Cap’’ discovered by Kauth and Thomas (1976). The urban

mixing space is bounded by high albedo, low albedo, and

vegetative endmembers corresponding to the brightness,

wetness, and greenness dimensions of the Tasseled Cap.

Because urban areas are dominated by built, impervious

surface, the plane of soils discussed by Kauth and Thomas

takes the form of a more restricted cylindrical ‘‘gray axis’’

spanning the range between the high and low albedo

endmembers. The spectral dimensionality and mixing space

configuration of the New York metro area is shown in

Figs. 2 and 3.

In this study, the dimensionality of the imagery is

estimated using a Minimum Noise Fraction (MNF) trans-

formation. The MNF transformation is effectively a cascade

of principal component transformations (Green, Berman,

Switzer, & Craig, 1988) designed to accommodate the fact

that the noise components of some multispectral bands may

have larger amplitude than the signal components of other

bands (Lee, Woodyatt, & Berman, 1990). The MNF trans-

formation used here is the complement to the Maximum

Noise Fraction transformation described by Green et al.

(1988) but orders the resulting eigenimages by decreasing

signal/noise ratio (S/N) rather than increasing S/N as

described in the original formulation. The distribution of

eigenvalues greater than 1 gives an indication of the

dimensionality of the image while the mixing space topo-

logy is indicated by the scattergrams of the low order MNF

components. Consistency among the eigenvalue distribu-

tions and mixing spaces determines the validity of the

multitemporal mixing model.

The mixing space defined by the low order MNF

components of the New York images maintains a planar

triangular shape but also changes somewhat throughout the

year as a result of changes in surface reflectance, illumina-

tion, and atmospheric conditions. The projection of the two

primary dimensions of the mixing space (referred to here as

the side view) consistently show the triangular shape

described above (center column, Fig. 3). The projections

referred to as the end view and top view incorporate the

third dimension of the mixing space. Together, these three

projections of the three-dimensional cloud of pixels show

that the vast majority of the pixels lie within a subplanar

triangular mixing space defined by the three apexes seen in

the two primary dimensions. Some nonlinearity is apparent

from the cylindrical shape of the ‘‘gray axis’’ between the

high and low albedo endmembers, but the significance of

this dispersion of the distribution is exaggerated by the

logarithmic shading used in Figs. 2 and 3. The bimodal

distribution distinguishing the very low reflectance water

bodies from the dark endmember of the primary mixing

space is clearly visible at each date. The low albedo apex of

the mixing space is well defined by the sharp corner of the

water mode. The length of the upper high albedo limb of the

mixing space varies appreciably as a result of changes in

cloud cover. Using the apex of this limb to define the high

albedo endmember would produce inconsistent results since

it is controlled by atmospheric state rather than surface

reflectance. The high albedo endmember will be defined

by a stable, bright surface feature described below so the

cloud reflectances will lie outside the defined mixing space.

The vegetation endmember is well defined by the sharp

lower apex of the distribution. The seasonal shift of the

centroid of the mixing space for New York (Fig. 3) shows

the progressive green-up and senescence of the urban

vegetation (Fig. 1).

4. Radiometric rectification

In order to derive quantitative estimates of how surface

reflectances change with time, it is necessary to minimize

the effect of other processes that influence the measured

radiance. Aside from the actual changes in surface reflec-

tance, the three primary sources of this temporal variability

are changes in illumination, changes in the atmosphere, and

changes in the sensor itself. It is relatively straightforward to

predict and compensate for changes in illumination intens-

ity, but quantifying the combined effects of sensor drift and

atmospheric scattering and absorption on uncalibrated radi-

ance measurements is nontrivial. A variety of methods have

been developed to compensate for atmospheric effects.

When supplementary field measurements of atmospheric

conditions and composition is available, it is possible to

construct radiative transfer models of the atmospheric trans-

mission and convert measured radiances to scaled surface

reflectances (e.g. Gao, Heidebrecht, & Goetz, 1993; Tanre,

Deroo, Duhaut, Herman, & Mocrette, 1990). In most cases,

however, archival imagery are not calibrated, and adequate

field measurements of atmospheric turbidity are not avail-

able for the dates and times the archival imagery was

acquired. In this case, it is necessary to use relative rather

than absolute reflectance estimates.

The key to separating changes in surface reflectance from

the other factors influencing the at-sensor radiance is to

identify pseudoinvariant (PIV) features for which the reflec-

tance does not change significantly. If this can be done, then

it is possible to transform the measured radiances to a

common reference such that the PIV features all have the

same apparent reflectance. If the PIV features correspond to

the spectral endmembers discussed above then rectifying

them should result in rectification of the mixed reflectances

as well. This radiometric rectification of PIV endmembers

allows for the combined effects of changes in sensor

calibration, scene illumination and atmospheric scattering

and absorption to be minimized simultaneously. There is

evidence that, to first order, measurements of at-sensor

radiance vary linearly with ground reflectance within spec-

tral wavebands in the visible through Short Wave Infrared

(Conel, 1990; Moran et al., 2001) and that the slope and

intercept provide an indication of the atmospheric trans-


mittance and path radiance, respectively. This linear rela-

tionship can be extended to multidate images (Caselles &

Garcia, 1989; Hall, Strebel, Nickeson, & Goetz, 1991). If

PIV features can be identified, then a linear transformation

can be defined, which simultaneously minimizes differences

in solar irradiance and path radiance among radiance meas-


urements acquired at different times (Hall et al., 1991;

Moran et al., 2001; Schott, Salvaggio, & Volchok, 1988;

Smith & Milton, 1999). The identification of PIV features,

originally proposed by Schott et al. (1988), and the linear

radiometric transformation, proposed by Hall et al. (1991),

have been used successfully in conjunction with the ref-

erence endmember identification proposed by Adams,

Smith, and Gillespie (1993) to implement relative reflec-

tance retrieval in multitemporal spectral mixture analyses of

landcover changes in the Brazilian Amazon (Adams et al.,

1995; Roberts et al., 1998).

The rectification strategy used here is based on selection of

PIV endmember sites as discussed in the studies mentioned

above. It differs from earlier studies in that the PIV sites are

chosen to correspond to the endmembers used in the sub-

sequent inversions. In comparison to natural environments,

the relative abundance of maintained homogenous imper-

vious surface in the built environment greatly simplifies the

task of radiometric rectification. In this study, two different

rectification strategies are tested using the PIV sites chosen

for the three endmembers. Linear radiometric transforma-

tions for each band are defined using (1) average reflectances

from all three PIVendmembers and (2) using only reflectan-

ces from the high albedo and low albedo endmembers. The

vegetation endmember is spectrally bounded at all wave-

lengths by the high and low albedo endmembers so it can

serve as a check on the internal consistency of the rectifica-

tion that uses only the high and low albedo endmembers.

Alternatively, including the vegetation endmember may

improve the radiometric transformation by adding an addi-

tional constraint if the high albedo endmember is not truly

invariant. Raw Digital Numbers are transformed to exoatmo-

spheric reflectance prior to rectification to remove predictable

differences in solar irradiance (Markham & Barker, 1987)

and aspect between acquisition dates. The rectification pro-

cedure involves (1) selection of PIV endmember sites, (2)

estimation of the radiometric transformation parameters, and

(3) implementation of the transformation.

The success of the radiometric rectification depends

strongly on the selection of PIVendmember sites. The degree

to which the sites are truly spectrally invariant determines the

amount of error propagated through the analysis by the

radiometric rectification. Spectrally invariant sites in this

study were chosen on the basis of size, as well as consistency

of spectral brightness and temporal invariance across all

reflected TM bands. In order to maintain spectral consist-

ency, it is critical to avoid boundary pixels in the feature

selection. This can be facilitated by using multitemporal

mean and variance maps derived for each band as shown in

Fig. 4. The Low Albedo PIV site chosen for the New York

metro area was the reservoir in Central Park. The reservoir is

a noncirculating maintained body of water several meters

deep and � 600 m in diameter with negligible sediment

influx or biological productivity. It is considerably larger

than the GIFOV of the Landsat TM sensor and should be

spectrally flat and relatively invariant compared to natural

water bodies. The high albedo PIV site chosen for the New

York metro area was the roof of the U.S. Postal Service

Metro Bulk Mail Facility in Union City, NJ. This is a large

(230� 450 m) L-shaped warehouse that consistently appears

as one of the brightest features in the region on all the

reflected TM bands. The PIV vegetation endmember site

chosen for the New York metro area was the Sheep Meadow

in Central Park. The Sheep Meadow is a nearly circular lawn,

250 m in diameter and is the largest, most uniform, and

consistently maintained vegetative feature in the New York

Fig. 3. Seasonal variation of the spectral mixing space for New York City and its surrounding areas in 1996. The mixing space is represented by the projections

of the three-dimensional distribution of the three primary dimensions of the MNF transformed reflectances. Changes in the shape of the mixing space result

from changes in surface reflectance, as well as atmospheric and illumination effects. Progressive changes in vegetated area cause the centroid of the pixel

distribution to shift from the ‘‘gray axis,’’ between the high and low albedo endmembers, toward the vegetation endmember during green-up, and back toward

the gray axis during senescence. Nonlinear mixing along the gray axis is implied by the convexity of the pixel distribution between the high and low albedo

endmembers (seen in the end view). The tapering of the distribution approaching the vegetation endmember implies increasingly linear mixing and suggests

that the relative contribution of the vegetation endmember is well constrained.

Fig. 4. Selection of PIV endmember sites. Mean reflectance for the 1996

images shows the time average of albedos at visible blue (TM1) and NIR

(TM4) wavelengths. The variance of the seven 1996 images for these bands

indicates the temporal consistency of the reflectance. PIV endmember sites,

labeled as Bright (high albedo, 24 pixels), Green (vegetation, 16 pixels),

and Dark (low albedo, 42 pixels), are chosen on the basis of average

reflectance, low temporal variance, and their locations at the apexes of the

mixing space (Figs. 2 and 3). Higher variance around the border of the high

albedo site (B) is a result of high contrast and image coregistration error.

The large high variance regions result from clouds. The endmember

reflectance profiles shown in Fig. 5 are averages of several of the low

variance pixels near the centers of the sites.


metro area. Landsat-derived exoatmospheric reflectances for

all three endmember sites consistently occupy the apexes of

the spectral mixing space shown in Fig. 3. The endmember

reflectance profiles shown in Fig. 5 are averages of several of

the low variance pixels near the centers of the sites.

The combined effects of sensor drift, illumination differ-

ences, and atmospheric turbidity are apparent in the distri-

bution of apparent endmember reflectances shown in Fig.

5a. The curvature of the high albedo endmembers shows

varying degrees of wavelength dependent atmospheric

absorption while the curvature of the low albedo endmem-

bers show the effects of atmospheric scattering (path radi-

ance) at short wavelengths. This is especially pronounced

on the high and low albedo endmembers for the 8/20/96

image (thicker profiles) as a result of the exceptionally

turbid atmosphere at the time the image was acquired. No

attempt is made to remove these effects, but the rectification

procedure serves to minimize the temporal variability by

adjusting brightness levels to be consistent with those

observed in the 4/15/96 scene. Solar illumination data and

maximum unrectified NIR reflectance of the high albedo

endmember (BNIR) are given in Table 1.

The assumption of invariant reflectance must take into

account the Bidirectional Reflectance Distribution Function

(BRDF) of the features and the effect of changing solar

azimuth and zenith angle. This is especially important for

the high albedo endmember because the feature reflectance

will not generally be Lambertian and may actually be nearly

specular in some cases. This does not seem to be a problem

for the high albedo endmember site used in this study as the

amplitude of the vegetation endmember spectrum increases

in correspondence to the amplitude of the high albedo

endmember spectrum. Solar illumination does have some

influence on the amplitude variations of the endmembers,

but it is not the sole influence as the maximum amplitude of

the endmember spectra does not increase monotonically

with solar zenith angle (Fig. 5) The high albedo endmember

for the December image does, however, have considerably

lower amplitude that is likely related to poor scene illu-

mination. The very low solar zenith angle (21�) is close to

the 15� limit below which Landsat data are not collected.

The implications for this scene are discussed below.

The results of the two rectifications suggest that including

the vegetation endmember does not change the transforma-

tion significantly. The convergence of the PIV vegetation

endmember indicates the effectiveness of the radiometric

rectification using only the high and low albedo endmem-

bers. In both cases, the outlying vegetation endmember

corresponds to the December image with the exceptionally

low solar zenith. In this image, the low amplitude of the high

albedo endmember seems to have resulted in an overrectifi-

cation of the vegetation endmember.

Fig. 5. Effect of radiometric rectification on PIV endmembers. The upper

plot (a) shows the distribution of unrectified endmember spectra for seven

dates in 1996 and one date in 1988 (thin curves). Variability is related

primarily to atmospheric differences. Aside from the 12/27/96 scene (solar

zenith angle f= 21�), the amplitude of the high albedo endmember does not

vary consistently with solar zenith angle (labeled by decreasing amplitude).

The center plot (b) shows the results of radiometric rectification using a

linear transformation based on all three PIV endmembers and the lower plot

(c) shows the results of a rectification using only the high and low albedo

endmembers. Profiles denoted with filled symbols correspond to the 4/15

image to which the other images are rectified. The smooth thin curve shows

a field reflectance measurement for grass for comparison with the

vegetation endmember. The thicker curves correspond to the 8/5/96 scene

for which atmospheric effects were most pronounced. The outlying

vegetation profile results from overcorrection of the poorly illuminated

December image.

Table 1

Solar illumination data

Date Julian day Zenith angle Azimuth Maximum BNIR

4/15/96 106 49 129 0.50

5/1/96 122 53 125 0.53

6/2/96 154 59 116 0.53

7/20/96 202 57 116 0.54

8/5/96 218 54 121 0.37

10/24/96 298 33 150 0.47

12/27/96 362 21 152 0.38

6/28/88 180 63 118 0.68


The anomalous character of the endmember profiles from

the 1988 image (thin curves) may result, in part, from

degradation of the Landsat 5 TM sensor over the intervening

8 years. Both the vegetation and high albedo endmembers

from the 1988 image have noticeably larger amplitudes than

the 1996 endmembers. The increased amplitudes of the

1988 endmembers are not consistent with the magnitude

of the difference in solar zenith angle although the differ-

ence in illumination may contribute somewhat to the overall

difference in amplitude. While the amplitude difference may

be due, in part, to changes in the reflectance of the

warehouse roof used for the high albedo endmember, the

higher amplitude of the 1988 vegetation endmember sug-

gests that sensor degradation may also be partially respons-

ible. This could be tested by incorporating additional

imagery for the intervening years.

The temporal variation in the radiometrically rectified

reflectance of the PIV vegetation endmember gives an

indication of the effectiveness of the rectification, as

well as the expected error in the resulting vegetation

fraction estimates. The variation of the red edge ampli-

tude in the vegetation endmember for the 1996 images

spans a range of 0.05, significantly less than the 0.15

range in the unrectified vegetation endmembers. The

remaining variation in the rectified vegetation reflectance

suggests that either the vegetation endmember site is not

really invariant or that the effect of atmospheric turbidity

is not truly linear—or both. This is discussed in more

detail below.

5. Estimation of endmember fractions

Inversion of the urban three-component linear mixing

model for each pixel yields fraction estimates for each

endmember, as well as Root Mean Square (RMS) misfit

to the observed data. The linear three-component mixing

model is given in continuous form by (Eq. (1)):

RðlÞ ¼ fBEBðlÞ þ fGEGðlÞ þ fDEDðlÞ ð1Þ

where R(l) is the observed apparent reflectance profile, a

continuous function of wavelength l. The E(l)’s are the

spectra corresponding to the high albedo (Bright), vegeta-

tion (Green), and low albedo (Dark) endmembers. The

corresponding endmember fraction estimates that we seek

are fB, fG, and fD. The discrete implementation of the model

applicable to Landsat TM exoatmospheric reflectances is

given by (Eq. (2)):

fBe11 þ fGe12 þ fDe13 ¼ r1 ð2ÞfBe21 þ fGe22 þ fDe23 ¼ r2

fBe31 þ fGe32 þ fDe33 ¼ r3




where ri is the observed reflectance vector corresponding to

discrete estimates of apparent reflectance within the six

Landsat TM bands. The eij’s are the endmember reflectance

vectors corresponding to the high albedo (B), vegetation

(G), and low albedo (D) endmembers. This system has more

equations than unknowns and can be solved for an

‘‘optimal’’ set of endmember estimates chosen to minimize

misfit to the observed reflectance vector.

The overdetermined linear mixing model, incorporating

measurement error, can be written in matrix notation as

(Eq. (3)):

r ¼ Ef þ E ð3Þwhere E is an error vector, which must be minimized to find

the fraction vector f, which gives the best fit to the observed

reflectance vector r. There are a number of ways to solve

this type of problem (e.g. Menke, 1989; Pech et al., 1986;

Settle & Drake, 1993; Smith, Johnson, & Adams, 1985;

Smith et al., 1990). The procedure used to invert the urban

three-component linear mixing model for endmember

reflectances is described in detail in Small (2001a). A unit

sum constrained least squares inversion of the three-

component model was performed on all seven of the

1996 scenes and the 1988 scene using the common rectified

endmembers described above. The same procedure was

applied using scene-specific endmembers for comparison.

The endmember fraction and RMS misfit images are shown

in Fig. 6.

The RMS misfit distributions give one indication of the

mathematical validity of the three-component mixing

model. The overall distribution of RMS misfits is gen-

erally low (< 0.04) relative to the amplitude of the

reflectance vectors being modeled (� 0.4). The RMS

misfit images resemble the high albedo endmember frac-

tion images, suggesting that most of the misfit corre-

sponds to high albedo features. This is not surprising

since most of the nonlinearity evident in Fig. 3 is

associated with the high albedo corner of the mixing

space and greater amplitudes of the high albedo features

will have a greater influence on the RMS statistic. As in

the previous study (Small, 2001a), the maximum RMS

misfit diminishes with increasing vegetation fraction. The

consistent relationship of RMS misfit to vegetation frac-

tion in this study further verifies that the inversion is well

posed with respect to vegetation estimation.

Previous analyses verify the stability of the inversion of

the three-component model in the presence of endmember

variability. Stability analyses indicate that the constrained

linear least squares inversion produces stable, consistent

results for the urban reflectance model, even in the presence

of sensor artifacts and unmodeled endmembers such as

clouds and nonphotosynthetic vegetation (Small, 2001a).

The vegetation endmember is sufficiently distinct that the

results of the inversion are not sensitive to differences in the

vegetation endmember resulting from different endmember

selection strategies. In addition to endmember selection, it is


Fig. 6. Endmember fraction and RMS misfit images for Landsat TM imagery of New York City. Fraction estimates were produced by a unit sum constrained

least squares inversion of the three-component linear mixture model using the two endmember rectification. The linear gray shading of each endmember image

corresponds to the endmember fraction ranging from 0 (black) to 1 (white) for the low albedo (D) and vegetation (G) endmembers and to 0.5 (white) for the

high albedo (B) endmember. Shading of the RMS misfit images (M) ranges between 0 and 0.04. Note the temporal tradeoff between the vegetation and other

two endmember fractions. Larger misfits correspond to high albedo and unvegetated soil pixels while vegetated areas have consistently low misfits. Greater

RMS misfit for the 12/29 image suggests that the inversion is not well posed for the poorly illuminated winter scene.


also important to verify that small differences in the imple-

mentation of the inversion (i.e. radiometric rectification and

inversion constraints) do not result in large differences in the

resulting fraction estimates.

6. Error analysis and validation

Comparing solutions resulting from different combina-

tions of rectification and inversion constraints gives some

indication of how the implementation of the inversion influ-

ences the fraction estimates. The implementations considered

here result from the four combinations of the two radiometric

rectifications with the constrained and unconstrained inver-

sion discussed above. The 8/5/96 image is used for the

comparison because the extreme atmospheric turbidity pro-

vides a stringent test of the effectiveness of the rectification.

A consistent unitary linear relationship among vegetation

fraction estimates for different implementations of the inver-

sion demonstrates the stability of the solution. High correla-

tions (> .98) among vegetation endmember fractions for

different inversions quantify the linearity in the solutions

and imply low sensitivity of the resulting fraction to differ-

ences in the inversion constraints or endmember vectors.

Unity slopes (± 0.01) and zero intercepts (± 0.02) further

verify the equivalence of the correlated solutions. The non-

unitary slope (0.65) of the correlation with the unrectified

scene indicates that the rectification significantly influences

the vegetation fraction estimates (Fig. 7a).

Solution sensitivity can also be quantified as the range

(maximum–minimum) of fraction estimates resulting from

different implementations of the inversion for a given meas-

urement (pixel). Fraction estimate ranges plotted as a function

of average vegetation fraction for the 8/5/96 image are shown

in Fig. 7. The greatest sensitivity occurs at low vegetation

fractions (< 0.2) with ranges of less than 0.07 occurring over

the range of vegetation fractions (cumulative quartile con-

tours in Fig. 7b). The inversion method and vegetation

fraction estimates are therefore robust even under the

extreme atmospheric turbidity present in the 8/5/96 image.

Intercomparison of endmember fraction estimates of the

PIV sites provides complementary information on the

expected error in the estimates. Estimates of a particular

endmember fraction, in this case, vegetation, are subject to

two types of error. Errors of commission result when

reflectance amplitude unrelated to that endmember contrib-

utes to the estimate resulting in a tendency to overestimate

the fraction. Errors of omission result when reflectance

amplitude related to the endmember is attributed to the

other endmember fractions resulting in a tendency to under-

estimate the fraction of interest. Some indication of the

magnitude of these errors is given by the fraction estimates

of the PIV endmember sites shown in Fig. 8. The distribu-

tions of the fraction estimates in the PIV sites can be

Fig. 7. Vegetation fraction estimate sensitivity. (A) Density shaded scattergram shows linear correspondence of vegetation fraction estimates from the

unrectified 8/5/96 image to the average of the four inversions of the rectified images. The unrectified estimates are consistently � 35% lower with the expected

linear bias. (B) Distribution of vegetation fraction ranges of four different inversions as a function of mean vegetation fraction in the 8/5/96 image. The range

(maximum–minimum) of fraction estimates resulting from constrained and unconstrained inversions of both rectification methods gives an indication of the

sensitivity of the solution. The distribution indicates that, even under extremely turbid atmospheric conditions, the variability of solutions resulting from

different inversions and rectifications is consistently low. Contours show cumulative quartiles, indicating that median ranges are less than 0.05 and that 75% of

the points have ranges less than 0.07 over the range of vegetation fractions.


interpreted as a conservative indication of the probability of

these two types of error. If the inversion produced perfectly

accurate results, then the vegetation endmember site would

consistently have vegetation fractions of 1 and the high and

Fig. 8. Apparent temporal variability of endmember fractions for PIV sites. Distributions of endmember fractions are indicated by the mean (thick line) and

distribution of the individual estimates (circles) for each site at each date. The upper plot shows the distribution of vegetation fraction estimates for each of the

PIV sites. The lower plot shows the distribution of endmember fractions for the PIV vegetation site. The high albedo endmember distribution is shown by

dashed lines. The variability of the individual pixel estimates at each time is a result of the natural variability within the PIV site.


low albedo endmember sites would have vegetation frac-

tions of 0. Similarly, the high and low albedo fraction

estimates for the vegetation endmember site should be 0.

Fig. 8 shows that the distribution of vegetation fractions of

the PIV vegetation site is consistently within 0.06 of 1. The

distribution of vegetation fractions for the other two PIV

sites indicates that the expected value of the error is very

close to zero and that the probability of error is greatest at

the times when vegetation fractions are low, particularly in

the December image. The erroneous attribution of high and

low albedo fractions to the vegetation endmember site seem

to offset each other in the December image without influ-

encing the vegetation fraction estimate. These error esti-

mates are conservative because the inversion results should

be most accurate for the endmember sites. As such, they

represent lower bounds on the overall error of the estimate.

The mean PIV vegetation endmember fraction for the 1988

image was 0.87. This does not, however, provide a useful

error estimate for the 1988 image because the condition and

maintenance of the Sheep Meadow in Central Park is known

to have improved considerably between 1988 and 1996.

The ultimate measure of vegetation fraction estimate

accuracy is comparison with independent reference meas-

urements. Vegetation fractions estimated from the Landsat

TM data were validated by comparison with vegetation

fractions calculated from aerial photographs as described

by Small (2001a). Vegetation distribution was mapped at 2-m

resolution using aerial photographs acquired 9 days prior

to the 7/20/96 Landsat overpass. Fig. 9 compares distribu-

tions measured from the aerial photographs with the Landsat

derived vegetation fraction estimates for 34 validation sites

in central Manhattan. The results show that the Landsat

estimates generally agree to within 10% of the measured

fraction for both the scene-specific endmembers described

in the earlier study and the common rectified endmembers

discussed in this study. The vegetation fraction estimates

based on the common rectified endmembers are consistently

lower than the estimates based on the scene-specific end-

members, but the difference between the two estimates is far

less than the difference of either from the measured vegeta-

tion fraction. Some part of the disagreement between the

estimated and the measured fractions is expected to result

from the differences in illumination at the time the Landsat

image and aerial photographs were acquired. The important

result here is that using common rectified endmembers did

not significantly diminish the agreement between the Land-

sat estimate and the measured vegetation fraction. This is

consistent with the stability of the estimates to small

changes in the endmembers.

The results of the error analysis highlight several import-

ant points. The radiometric rectification has a significant

effect on the apparent reflectance of most of the scenes—

even those without obvious atmospheric turbidity. The two

endmember rectification did not result in perfect agreement

of the vegetation endmember spectra. If the vegetation

endmember is truly invariant, then the effect of atmospheric

turbidity on measured radiance is not truly linear. Alter-

natively, the converse may be true. If the atmospheric effects

are truly linear, then the PIV endmember is not truly

invariant. This is a fundamental uncertainty inherent in the

PIVendmember rectification method that cannot be resolved

retrospectively. Fraction estimates for the PIV vegetation

endmember varied within ± 0.06 of expected values provid-

ing a lower bound on vegetation fraction error. RMS misfit

Fig. 9. Comparison of estimated and measured vegetation fractions for Manhattan validation sites. Measurements are calculated from aerial photographs at 2-m

resolution as described by Small (2001a). Mean values generally agree to within ± 0.1 (gray diagonal) while the standard deviations (bars) of the distributions

of measured and estimated vegetation fraction within the validation sites are often greater than 0.1. The sizes of the validation sites are indicated by the shading

of the symbols ranging from < 10 (dark) to � 1000 (light) TM pixels. Disagreement between measured and estimated fractions results from errors in both the

measurement and estimation.


is most strongly correlated with the high albedo endmember,

presumably because of the nonlinearity indicated in the

mixing space. Rectified and scene-specific endmembers

produce results of comparable agreement with the reference

measurements— for low turbidity atmosphere. The discrep-

ancy may result more from the validation methodology than

actual disagreement in the estimates. In any case, radio-

metric rectification does not diminish the accuracy of the

validated result.

7. Temporal variation of urban reflectance

Before endmember fraction estimates from different

dates can be compared on a pixel-to-pixel basis, it is

necessary to consider the effects of image coregistration

and characteristic scale on the resulting fraction estimates.

Much of the vegetation distribution in the New York study

area has a characteristic scale comparable to the GIFOV of

the Landsat TM sensor (Small, 2001a). This means that

small differences ( < 30 m) in the location of the GIFOV can

result in significant changes in the amount of vegetation

contributing to the measured radiance for a particular pixel.

Since Landsat 5 images cannot be coregistered to subpixel

accuracy, positional uncertainties of less than 30 m can

result in significant differences in the vegetation fraction of

some pixels even if the vegetation distribution on the ground

has not changed between dates. For this reason, it is

necessary to compare distributions of estimates (Fig. 9)

rather than individual estimates. An example of registration

error can be seen in the difference image in Fig. 10. The

large differences in vegetation fraction seen at high contrast

boundaries (like the edge of Central Park) are a result of

image registration error and differences in the vegetated area

within the corresponding GIFOVs when the images were

acquired. These effects are most noticeable where the

vegetation fraction changes abruptly (like park boundaries),

but they occur wherever the vegetation fraction is not

homogeneous at the spatial scale of the GIFOV. At this

scale, the TM sensor’s spatial response characteristics

(Modulation Transfer Function) also have a strong influence

on the vegetation fraction estimates. Adjacency effects

further complicate the interpretation of individual fraction

estimates when the spatial distribution of fractions changes

significantly at spatial scales similar to the GIFOV of the

sensor. The combination of these confounding factors pre-

cludes a direct interpretation of individual pixel estimates of

endmember fractions.

Temporal changes in the overall distribution of endmem-

ber fractions provide quantitative estimates of seasonal and

interannual changes in the net albedo of the urban envir-

onment. Fig. 6 shows the effect of seasonal changes in

vegetation cover on the distribution of endmember fractions.

Endmember fraction distributions indicate that the low

albedo endmember dominates the New York City urban

environment—even when vegetation cover is at its seasonal

maximum. Correspondingly, the seasonal increase in vegeta-

tion cover has more effect on the distribution of the low

albedo endmember than the high albedo endmember. This

has a direct impact on the solar and thermal energy flux

through the urban environment. Vegetation reflects infrared

radiation, while it absorbs visible radiation and transpires

water. Low albedo surfaces absorb both visible and infrared

radiation and reradiate it at thermal infrared wavelengths.

These are two of the primary mechanisms by which the urban

heat island is maintained (Oke, 1982). The spatial and

temporal distribution of vegetation in the urban environment

therefore exerts a strong influence on the energetics of the

urban heat island. The spatial distribution of vegetation

influences the spatial extent and strength of the heat island

and presumably impacts mesoscale (100–10000 m) circula-

tion within the urban canopy layer. The temporal distribution

of vegetation cover should also contribute to the rate at which

Fig. 10. Full resolution comparison of spatial and temporal variation in

vegetation cover in 1996 and 1988. Grey shading shows vegetation fraction

ranging between 0 (black) and 0.75 (white). The 1996–1988 difference

image shows the change in vegetation fraction ranging from � 0.5 (black)

to 0.5 (white). The small difference in average vegetation fraction is likely a

result of images being acquired at different phases of each phenological

cycle, as well as temporal changes in the PIV vegetation endmember. The

six numbered sites indicate more pronounced changes resulting from

revegetation between 1988 and 1996. Image area is 15� 32 km.


seasonal changes in solar radiation heat the urban surface.

Satellite-derived estimates of spatial and temporal changes in

urban reflectance may ultimately provide boundary condi-

tions for studies of mesoscale and regional climate dynamics.

Temporal changes in the distribution of vegetation frac-

tion may reveal seasonal and interannual differences in the

rate of green-up and senescence of urban vegetation, as well

as long-term changes in the distribution of urban vegetation.

Fig. 10 shows several examples of urban revegetation

between 1988 and 1996. A more extensive data set, includ-

ing midsummer imagery from different years, would be

required for a systematic study of long-term changes in

urban vegetation. It is difficult to separate seasonal differ-

ences in phenological phase from year to year changes in

vegetation cover with only two dates. Imagery for several

years could show progressive changes in vegetation cover,

but full seasonal coverage would be required to constrain

the phase of the seasonal cycle. This type of analysis would

have application to studies of urban ecosystems to quantify

spatial and temporal dynamics of vegetation phenology. It is

well known that the distribution of urban vegetation mod-

ulates the development of the urban heat island, but it is not

known how the urban heat island effect impacts the pheno-

logy of urban vegetation. The results of this study suggest

that multitemporal SMA could provide a viable tool for

quantifying the spatiotemporal changes in urban vegetation

distribution. The principal conclusion of this study is that

the uncertainty in the vegetation fraction estimates is less

than the spatial and temporal changes observed in the New

York area. This suggests that the analysis described here

could provide meaningful quantitative estimates of actual

changes in vegetation cover and urban albedo.

8. Conclusions

The results of this study verify the temporal consistency

of the three-component linear spectral mixing model for the

New York metro area. The eigenvalue distributions indicate

that the majority of spatially coherent information can be

described with two principal components. The spectral

mixing space consistently shows a subplanar triangular form

with three distinct endmembers corresponding to high

albedo, low albedo, and vegetation endmembers. Seasonal

phenology produces the expected shifts of the centroid of

the mixing space toward and away from the vegetation

endmember following the phenological cycle. Nonlinear

mixing is present primarily along the gray axis between

the high and low albedo endmembers.

Pseudoinvariant endmember sites provide a basis for

linear radiometric rectification of apparent reflectances.

Appropriate sites are chosen on the basis of their locations

at the apexes of the mixing space and their temporal

statistics. Temporal mean and variance maps provide a

simple way to select PIV pixels with the lowest temporal

variance and to avoid pixels corrupted by registration error.

Radiometric rectification can eliminate more than 66% of

the temporal variability in PIV vegetation reflectance ampli-

tude. Some of the remaining variability may result from

actual changes in the reflectance of the vegetation target.

Multitemporal inversion of the three-component linear

mixing model produces consistent, stable estimates of

vegetation fraction that agree to within 10% of independent

measurements derived from aerial photography. This is

consistent with the temporal variability in the vegetation

fraction estimates for the PIV vegetation endmember site.

Sensitivity of the vegetation fraction estimates to the rec-

tification method and inversion constraints is less than 7%

for most estimates and diminishes with increasing vegeta-

tion fraction.

The multitemporal analysis approach described here

could be improved by refining the radiometric rectification

method. This analysis suggests that there may be significant

nonlinearity in the wavelength dependence of the combined

illumination and atmospheric effects. Spectral field cal-

ibration of the PIV sites would provide further constraints.

Collection of field spectra would also allow the modified

empirical line method of Moran et al. (2001) to be used. The

topology of the mixing space also indicates considerable

nonlinear mixing at low vegetation fractions. This is pre-

sumably a consequence of multiple scattering and shadow-

ing in sparsely vegetated areas. Nonetheless, both spatial

and temporal changes in vegetation fraction are consid-

erably greater than the uncertainty in the estimates, suggest-

ing that the methodology is suitable for monitoring

spatiotemporal dynamics of urban vegetation.

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Multitemporal analysis of urban reflectance

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