Shadow Fraction Research Articles

Regional mapping of gross light-use efficiency using MODIS spectral indices

Direct estimation of photosynthetic light-use efficiency (LUE) from space would be of significant benefit to LUE-based models which use inputs from remote sensing to estimate terrestrial productivity. The Photochemical Reflectance Index (PRI) has shown promise in tracking LUE at the leaf- to small canopy levels, but its use at regional to global scales still remains a challenge. In this study, we used different formulations of PRI calculated from the MODIS ocean band centered at 531 nm and a set of alternative reference bands at 488, 551, and 678 nm to explore the relationship between PRI and LUE where LUE was measured at eight eddy covariance flux towers located in the boreal forest of Saskatchewan, Canada. The magnitude and variability of LUE was significantly lower at the times when useful MODIS ocean band images were available (i.e. around midday under clear-sky conditions) relative to the rest of the growing season. PRI 678 (reference band at 678 nm) showed the strongest relationship ( r 2 = 0.70) with LUE 90a (i.e. 90-minute mean LUE calculated using Absorbed Photosynthetically Active Radiation, APAR), but only when all sites were combined. Overall, the relationships between the MODIS PRIs and LUE 90a were always stronger for observations closer to the backscatter direction and there were no significant differences in the strength of the correlations whether LUE was calculated based on incident PAR or on APAR. Predictions of ecosystem photosynthesis at the time of the MODIS overpasses were significantly improved by multiplying either PAR or APAR by MODIS PRI ( r 2 improved from 0.09 to 0.44 and 0.54 depending on the PRI formulation). We used our PRI-LUE model to create a regional LUE 90a map for the three cover types covering 47,500 km 2 around the flux sites. The MODIS PRI-derived LUE 90a map appeared to capture more realistic spatial heterogeneity of LUE across the landscape compared to a daily LUE map derived using the look-up table in the MODIS GPP (MOD17) algorithm. While our LUE map is only a snapshot of minimum regional LUE 90a values, with appropriate gap-filling methods it could be used to improve regional-scale monitoring of GPP. Moreover, the strong relationship between midday and daily LUE on clear days ( r 2 = 0.93) indicates that instantaneous MODIS observations of LUE 90a could be used to estimate daily LUE. Finally, pixel shadow fraction from the 5-Scale geometric-optical model was closely related to both MODIS PRI and tower-derived LUE suggesting that differences in stand leaf area and in diffuse illumination among flux sites play a role in the relationship we observed between LUE and MODIS PRI.

A shadow fraction method for mapping biomass of northern boreal black spruce forests using QuickBird imagery

We have developed and tested a method for mapping above-ground forest biomass of black spruce ( Picea mariana (Mill.) B.S.P.) stands in northern boreal forests of eastern Canada. The method uses QuickBird images and applies image processing algorithms to extract tree shadow fraction (SF) as a predictive variable for estimating biomass. Three QuickBird images acquired over three test sites and 108 ground sample plots (GSP) were used to develop and test the method. SF was calculated from the fraction of tree shadow area over the area of a reference square overlaid on the images. Linear regressions between biomass of GSP and SF from the images for each test site resulted in R 2 in the range from 0.85 to 0.87 (except one case at 0.41), RMSE of 11 to 18 t/ha and bias of 2 to 5 t/ha. Statistical tests demonstrated that local regressions for the three test sites were not statistically significantly different. Consequently, a global regression was calculated with all GSP and produced R 2, RMSE, and bias of 0.84, 14.2 t/ha and 4.2 t/ha, respectively. While generalization of these results to extended areas of the boreal forest would require further assessment, the SF method provided an efficient means for mapping biomass of black spruce stands for three test areas that are characteristic of the northern boreal forest of eastern Canada (boreal and taiga shield ecozones).

Mapping tree and shrub leaf area indices in an ombrotrophic peatland through multiple endmember spectral unmixing

Leaf area index (LAI) is an important parameter used by most process-oriented ecosystem models. LAI of forest ecosystems has routinely been mapped using spectral vegetation indices (SVI) derived from remote sensing imagery. The application of SVI-based approaches to map LAI in peatlands presents a challenge, mainly due to peatlands characteristic multi-layer canopy comprising shrubs and open, discontinuous tree canopies underlain by a continuous ground cover of different moss species, which reduces the greenness contrast between the canopy and the background. Our goal is to develop a methodology to map tree and shrub LAI in peatlands and similar ecosystems based on multiple endmember spectral mixture analysis (MESMA). This new mapping method is validated using LAI field measurements from a precipitation-fed (ombrotrophic) peatland near Ottawa, Ontario, Canada. We demonstrate first that three commonly applied SVI are not suitable for tree and shrub LAI mapping in ombrotrophic peatlands. Secondly, we demonstrate for a three-endmember model the limitations of traditional linear spectral mixture analysis (SMA) due to the unique and widely varying spectral characteristics of Sphagnum mosses, which are significantly different from vascular plants. Next, by using a geometric-optical radiative transfer model, we determine the nature of the equation describing the empirical relationship between shadow fraction and tree LAI using nonlinear ordinary least square (OLS) regression. We then apply this equation to describe the empirical relationships between shadow and shrub fractions obtained from mixture decomposition with SMA and MESMA, respectively, and tree and shrub LAI, respectively. Less accurate fractions obtained from SMA result in weaker relationships between shadow fraction and tree LAI ( R 2 = 0.61) and shrub fraction and shrub LAI ( R 2 = 0.49) compared to the same relationships based on fractions obtained from MESMA with R 2 = 0.75 and R 2 = 0.68, respectively. Cross-validation of tree LAI ( R 2 = 0.74; RMSE = 0.48) and shrub LAI ( R 2 = 0.68; RMSE = 0.42) maps using fractions from MESMA shows the suitability of this approach for mapping tree and shrub LAI in ombrotrophic peatlands. The ability to account for a spectrally varying, unique Sphagnum moss ground cover during mixture decomposition and a two layer canopy is particularly important.

Spectral mixture analysis of agricultural crops: Endmember validation and biophysical estimation in potato plots

The spectral reflectance of agricultural crops is affected significantly by sub-pixel scale spectral contributions of background soils and shadows as viewed by a remote sensing instrument. This has meant the potential of remote sensing imagery has not been fully realized for extracting biophysical information and assessing ecological stress using methods such as vegetation indices (VIs). In this paper, we address this problem explicitly using spectral mixture analysis (SMA) to quantify the area abundance of plants, soils and shadows at sub-pixel scales with the aim of improving extraction of plant biophysical and structural information from remote sensing data. Different measurement strategies were tested in the field for acquiring reference endmember spectra of crop vegetation, soil and shadows using a field spectroradiometer for a set of potato plots in western Canada. Endmember measurements included sunlit and shadowed spectra of in situ crop targets, optically thick stacks and data from excised leaves, as well as cultivated, rough and compacted soils. All possible combinations of crop, soil and shadow endmember spectra were analysed using SMA to derive sets of sub-pixel scale component fractions from radiometer spectra acquired from a boom truck over replicate plot samples with a sensor field of view of 1.05 m. Digital video image frames captured simultaneously with the radiometer data were used to determine ground proportions of crop, soil and shadow for independent validation of the SMA fractions. Endmember fractions derived from excised leaves, cultivated soil and shadowed vegetation spectra showed the best agreement with ground truth data, with differences of only ±3.3%. These sub-pixel scale fractions were used in regression analyses to predict leaf area index, biomass and plant width with an average r2 value of 0.85 from SMA shadow fraction, which was a substantial improvement over the best VI results from NDVI, NGVI and SR (average r2 = 0.53). Perspectives on SMA at different stages in the growing season and for different crop types are provided with a recommendation that further SMA research is warranted for local to regional scale agricultural crop monitoring programmes.

Multiple shadow fractions in spectral mixture analysis of a cotton canopy

Shadows are being used more frequently to estimate plant canopy biophysical characteristics. Typically, a zero value is assumed or a threshold value is derived from histogram analysis of imagery to determine the shadow endmember (EM). Here, two distinct shadow EMs were measured in situ for use in spectral mixture analysis of a cotton canopy on five dates in 2003. The four EMs used in the analysis were: sunlit green leaf, sunlit dry soil, self-shadowed leaf, shadowed dry soil. This 4-EM model was compared to a 3-EM model where a zero-value shade EM was used for unmixing with the two sunlit EMs. Multiple endmember spectral mixture analysis (MESMA) was used to allow EM composition to vary across each scene. The analysis and EMs were applied to fine-scale hyperspectral image data collected in the wavelength range, 440 to 810 nm. Ground data collected included percent cover, height, SPAD (a measure of leaf greenness), and chlorophyll a concentration. The normalized difference vegetation index (NDVI) was also compared to the unmixing results. Regression analysis showed that NDVI was equal to the 4-EM model for estimation of percent cover ( r 2 = 0.95, RMSE = 6.6) although the NDVI y-intercept was closer to zero. The 4-EM model was best for estimating height ( r 2 = 0.79, RMSE = 0.07 m) and chlorophyll a concentration ( r 2 = 0.46, RMSE = 7.0 μg/cm 2). The 3-EM model and NDVI performed poorly when estimating chlorophyll a concentration. Inclusion of two distinct shadow EMs in the model improved relationships to crop biophysical parameters and was better than assuming one, zero-value shade EM. Since MESMA operates at the pixel level and allows variable EM assignment to each pixel, mapping the spatial variability of shadows and other variables of interest is possible, providing a powerful input to canopy and ecosystem models as well as precision farming.

Minimization of the effect of the collecting grid in a solar cell based silicon

Abstract The solar cells collecting grids present a serious problem and more particularly under solar concentration. Our contribution in this article is to seek the best compromise between shadow effect and series resistance effect. The cell considered here is of Si (silicon) type, n+p with circular geometry (radius a = 4.9 cm ), a silver metallization ( ρ M = 1.6 × 1 0 - 6 Ω cm ), and a contact resistivity of ρ C = 1 0 - 5 Ω cm . Our calculations are made under the condition of AM1.5 with 1 sun concentration. The various power losses caused by this grid are: • losses due to the grid shadow, • losses in grain boundaries due to the metal/semiconductor contact, • power dissipated in the resistance of layer between bars, and • losses in the grid metallization. We summarize the results of our model by: • For a sheet resistance ρS=10 Ω/□:τoop(%) (optimum shadow fraction)=τcop(%) (optimum conduction fraction)=5.086% leading to θoptimum=0.0652 rad. • For a sheet resistance ρS=20 Ω/□:τoop(%)=τcop(%)=5.338% leading to θoptimum=0.05883 rad. • For a sheet resistance ρS=50 Ω/□:τoop(%)=τcop(%)=5.949%, θoptimum=0.0500 rad.

Relação entre índice de área foliar e frações de componentes puros do modelo linear de mistura espectral, usando imagens ETM+/Landsat

O índice de área foliar (IAF) é uma das mais importantes variáveis biofísicas da vegetação, estando relacionado diretamente com a evapotranspiração, com a produtividade da vegetação e com a interceptação da chuva pelo dossel. O objetivo deste trabalho foi analisar a relação do IAF de diversos tipos de cobertura do solo com Frações de Componentes Puros (FCPs) do Modelo Linear de Mistura Espectral (MLME). A área de estudo foi a microbacia hidrográfica do Ribeirão dos Marins, localizada no município de Piracicaba - SP. O IAF foi medido, no campo, com o equipamento LAI-2000, em 32 áreas com diferentes coberturas vegetais. A imagem utilizada foi do sensor ETM+ a bordo do satélite Landsat-7. No MLME, foram considerados três componentes puros (vegetação, solo e sombra), selecionados com o auxílio dos componentes principais. Como resultado, tem-se que o IAF variou de 0,47 a 4,48, quando consideradas todas as áreas. As relações do IAF com a fração do componente puro vegetação F VEG e com a fração do componente puro solo (F SOL) foram significativas, embora fracas. Ao considerar apenas dados de IAF de cana-de-açúcar, houve aumento da variação explicada tanto para F VEG como para F SOL, sugerindo que a estratificação da vegetação pelo tipo pode melhorar a estimativa do IAF.

Canopy shadow in IKONOS satellite observations of tropical forests and savannas

The biological and structural complexity of tropical forests and savannas results in marked spatial variation in shadows inherent to remotely sensed measurements. While the biophysical and observational factors driving variations in apparent shadow are known, little quantitative information exists on the magnitude and variability of shadow in remotely sensed data acquired over tropical regions. Even less is known about shadow effects in multispectral observations from satellites (e.g., Landsat). The IKONOS satellite, with 1-m panchromatic and 4-m multispectral capabilities, provides an opportunity to observe tropical canopies and their shadows at spatial scales approaching the size of individual crowns and vegetation clusters. We used 44 IKONOS images from the Large-Scale Biosphere–Atmosphere Experiment in Amazonia (LBA) data archive to quantify the spatial variation of canopy shadow fraction across a broad range of forests in the Brazilian Amazon and savannas in the Brazilian Cerrado. Forests had substantial apparent shadow fractions as viewed from the satellite vantage point. The global mean (±S.D.) shadow fraction was 0.25±0.12, and within-scene (e.g., forest stand) variability was similar to interscene (e.g., regional) variation. The distribution of shadow fractions for forest stands was skewed, with 30% of pixels having fractional shadow values above 0.30. Shadow fractions in savannas increased from 0.0±0.01 to 0.12±0.04 to 0.16±0.05 for areas with woody vegetation at low (<25% cover), medium (25–75%), and high (>75%) density, respectively. Landsat-like observations using both red (0.63–0.70 μm) and near-infrared (NIR) (0.76–0.85 μm) wavelength regions were highly sensitive to sub-pixel shadow fractions in tropical forests, accounting for ∼30–50% of the variance in red and NIR responses. A 10% increase in shadow fraction resulted in a 3% and 10% decrease in red and NIR channel response, respectively. The normalized difference vegetation index (NDVI) of tropical forests was weakly sensitive to changes in shadow fraction. For low-, medium-, and high-density savannas, a 10% increase in shadow fraction resulted in a 5–7% decrease in red-channel response. Shadows accounted for ∼15–50% of the overall variance in red-wavelength responses in the savanna image archive. Weak to no relationship occurred between shadow fraction and either NIR reflectance or the NDVI of savannas. Quantitative information on shadowing is needed to validate or constrain radiative transfer, spectral mixture, and land-surface models used to estimate material and energy exchanges between the tropical biosphere and atmosphere.

Shadow brightness and shadow fraction relations with effective leaf area index: importance of canopy closure and view angle in mixedwood boreal forest

This research examines the impacts of varying canopy closure and view angle on relations of high-resolution digital camera shadow fraction and shadow brightness with mixedwood boreal forest effective leaf area index (LAIe). Results from linear regression analyses revealed weak and insignificant positive relations between shadow fraction and LAIe. Considerable scatter was observed in the relationship and was attributed to differences in canopy closure among plots, as only forest with greater than 80% closure showed a positive relationship between shadow fraction and LAIe. These results emphasized that gap size and frequency were important factors in determining the shadow fraction. Relations between shadow brightness and LAIe were significantly improved over those with the shadow fraction because of decreased data point scatter when closure was less than 80%. Shadow brightness was not adversely affected by canopy closure, while remaining sensitive to species composition. The association of shadow fraction with LAIe was understood to exist through links to the projected shadow area of tree crowns; however, this relation became unstable in more open canopies. With shadow brightness, surrogate information on LAIe was implicitly linked to differences in the transmission of light through deciduous and coniferous tree crowns. An evaluation of view angle geometry effects suggested that bidirectional reflectance impacts on the shadow fraction ‐ LAIe relations were strongest in the forward-scattering direction, but had less effect on regression analysis in the backscattering direction and at nadir.

Three-dimensional measurement of void fraction in a bubble plume using statistic stereoscopic image processing

A method for measuring the instantaneous 3-D bubble distribution in bubbly flows is proposed by making use of stereoscopic image processing. The method applies to the case of higher rather than dilute void fractions, because it is based on the following two statistic-based procedures. The first process is to obtain the 2-D distribution of local void fraction from visualized images, where a correlation between the bubble shadow fraction and the projection void fraction is derived by introducing the new idea of overlapping probability of the bubble-images under various conditions. The second process is to reconstruct the 3-D distribution of the local void fraction. In this step, a logical method is proposed in order to obtain the bubble-existence probability function which simultaneously satisfies the two distributions of the two projection void fractions. The present method is applied for instantaneous measurement of the 3-D structure of a bubble plume. The measured results show its particular structure, such as the hollow distribution of the bubbles, and the local spiral structure, which cannot be captured by point-measurement experiments.

Compact Airborne Spectrographic Imager (CASI) used for mapping biophysical parameters of boreal forests

During the Boreal Ecosystem‐Atmosphere Study (BOREAS), which took place in Saskatchewan and Manitoba in 1994, the Compact Airborne Spectrographic Imager (CASI) acquired images of boreal forests. In this paper we present results of radiometric and geometric analysis of the CASI data for developing algorithms for retrieving leaf area index (LAI) and crown closure of the boreal forest. The images of over 30 sites, composed of black spruce, jack pine, and aspen stands, were acquired on different days and locations with various solar illumination and view geometries. The geometrical‐optical model, named “4‐Scale” [Chen and Leblanc, 1997, 1316–1337], was used to correct the images to a common solar zenith angle (35°) and a common view angle (nadir). The 4‐Scale model is also used for radiometric analysis based on spectral signatures of leaves and the background (moss, grass, and soil) acquired using various field and laboratory techniques. The red reflectance of all three cover types decreased with increasing LAI as expected. Similar but weaker decreasing trends were found in the near‐infrared (NIR) band for conifer stands in contrast to previous findings for cropland and grassland. No significant NIR response to LAI was found for aspen stands. It is shown from 4‐Scale that as LAI increases, the crown and ground shadow fractions of conifer forests increase, while the sunlit background fraction decreases and the sunlit crown fraction increases. The large change in the shadow fractions is the major factor controlling the behavior of red and NIR signals. Since boreal forests have abundant green moss and understory as the background, there is only a small difference in optical properties between the overstory and the background. The increases in the shadow fractions with LAI help strengthen the response of optical measurements to changes in LAI, providing a key mechanism for remote information retrieval. The implications of these findings on formulating/selecting vegetation indices and inversion models are discussed in this paper. Relationships of crown closure with CASI measurements are also included in the analysis.

Spectral Mixture Analysis and Geometric-Optical Reflectance Modeling of Boreal Forest Biophysical Structure

Abstract Biophysical structural information such as biomass, LAI, and NPP are important inputs to regional scale models of ecosystem processes and photosynthetic activity within boreal forests. However, traditional methods such as NDVI for deriving these variables from remotely sensed data have been inconsistent and unsatisfactory due to factors such as the confounding influence of background reflectance and canopy geometry on the overall pixel signal. To address this problem, we present new results which use spectral mixture analysis to determine areal fractions of sunlit canopy, sunlit background, and shadow at subpixel scales for predicting these biophysical variables. Geometric-optical reflectance models are used to estimate sunlit canopy component reflectance for input to the analysis together with field measures of background and shadow reflectance. In this article, we compare cylinder, cone, and spheroid models of canopy geometry and evaluate the importance of solar zenith angle variations in reflectance estimates for mixture fractions. These are computed from helicopter MMR radiometer data for 31 stands of black spruce along a gradient of stand densities near the southern fringe of the North American boreal forest. Component fractions are evaluated against ground data derived from dense-grid point analyses of coincident high resolution color photography and also for predicting biophysical variables. In general, the Li–Strahler spheroid model was better than the cone and cylinder models and the importance of correcting for solar zenith angle (SZA) was illustrated, with significant improvements noted for higher SZA as a result of corrections for canopy mutual shadowing. The best overall results were obtained from the shadow fraction using a spheroid model of canopy geometry at SZA 45°. Linear regression analyses showed biomass could be estimated with r2 values of 0.83 and a standard error (S.E.) of 1.7 kg/m2; LAI: r2=0.82, S.E.=0.46; and NPP: r2=0.86, S.E.=0.05 kg/m2/yr. These results were significantly higher than with NDVI for estimating biomass (r2=0.44), LAI (r2=0.60), and NPP (r2=0.56). Current and future areas of research are outlined towards improving our understanding of carbon cycling in large forested ecosystems as a function of variability in the physical climate system and environmental change.

Analyzing the effect of structural variability and canopy gaps on forest BRDF using a geometric-optical model

A geometric-optical reflectance model is presented which estimates the bidirectional reflectance distribution function (BRDF) of forest canopies by modeling four shadowing pattern components (i.e., illuminated crown, illuminated ground, shadowed crown, and shadowed ground. The model represents the forest canopy as a group of discrete tree crowns and uses a deterministic ray tracing procedure to generate a two-dinwnsional scene of shadow pattern components. It provides the type of shadow fraction information used to calculate the reflectance of forests when it is treated as an area-weighted sum of the shadow component reflectances. The model is designed to test the effect of canopy structure parameters on the bidirectional reflectance of forests and can easily be adapted to accommodate different crown shapes, crown sizes, stem densities, and distributions. The model was used to isolate and quantify the effects of crown shape, canopy cover, stein distribution pattern, and canopy gaps on the shadowing pattern and the canopy reflectance, using a set of statistically generated forest scenes and data gathered for a natural tropical lowland forest scene. The illuminated crown component and the illuminated ground component have the greatest impact on the reflectance values in the red and near-infrared region respectively. The red BRDF, mainly influenced by the illuminated ground component, proves to be sensitive to canopy cover, tree pattern distribution, and canopy gaps. The near-infrared BRDF, affected by the illuminated crown component, is sensitive to canopy cover, crown-center height distribution, and crown shape. The sensitivity of the BRDF to canopy gaps and crown clumping was investigated in more detail. Reflectance-measurements close to the hotspot are found to be insensitive to clumping while retaining sensitivity to crown cover. Forward scattering measurements are fcntnd to be more sensitive to gap size and frequency. The spatial effects of plantations, logging, or road features on the BRDF is further analyzed.

Unmixing the directional reflectances of AVHRR sub-pixel landcovers

Recent progress in canopy bidirectional reflectance distribution function (BRDF) model inversions has allowed accurate estimates of vegetation biophysical characteristics from remotely sensed multi-angle optical data. Since most current BRDF inversion methods utilize one-dimensional (1D) models, surface homogeneity within an image pixel is implied. The Advanced Very High Resolution Radiometer (AVHRR) is one of the few spaceborne sensors capable of acquiring radiometric data over the range of view angles required for BRDF inversions. However, its relatively coarse spatial resolution often results in measurements of mixed landcovers, and thus the data may not be ideal for BRDF inversions. The authors present a three-step spectral unmixing method for retrieving AVHRR sub-pixel directional reflectances in regions of high spatial heterogeneity. The reflectances of individual vegetation types are deconvolved using co-located Landsat TM and AVHRR data. The three major steps in the model include: (1) unmixing of vegetation endmember concentrations in TM imagery; (2) correction of dissimilar shadow fractions between TM and AVHRR data; and (3) unmixing of AVHRR sub-pixel reflectances of vegetation types for any Sun-sensor geometry. The authors tested the method using simulated TM and AVHRR data. A savannah landscape simulation, comprised of a canopy radiative transfer model and a crown geometric-optical model was used to create images containing mixed pixels of tree, grass, and shade endmembers. TM and AVHRR spectral response functions, viewing geometrics, and off-nadir pixel shape calculations were incorporated into the simulations. Following the successful testing of the unmixing method on error-free simulations, random noise representing atmospheric perturbations and co-registration inaccuracies was added to the data. The method is stable when errors resulting from either the first unmixing step or image co-registration inaccuracies are introduced. Potential errors in the AVHRR data may result in inaccurately retrieved reflectances if the image scene contains a spatially homogeneous mix of landcovers. A method for detecting and mitigating this problem is presented.

Remote sensing of biophysical variables in boreal forest stands of Picea mariana

Using two hybrid radiative transfer models to represent conifer canopies and stands, algorithms to infer several important structural parameters of stands of black spruce (Picea mariana), the most common boreal forest dominant, were developed and evaluated. Spectral mixture analysis and multi-spectral reflectance data for 31 black spruce stands of varying density and structure were used to infer the values for the areal proportions of sunlit canopy, sunlit background and shadow fraction, which we call radiometric elements, and the areal proportions of these radiometric elements were strongly related to leaf area index, biomass density, and annual above ground net primary productivity. The best overall correspondence between the radiometric elements and biophysical variables was found from the shadow fraction obtained with the cone-based canopy reflectance model corrected for variations in solar zenith angle.

Remote Sensing of Forest Biophysical Structure Using Mixture Decomposition and Geometric Reflectance Models

Using geometric shadow and linear mixture models we develop and evaluate an algorithm to infer several important structural parameters of stands of black spruce (Picea mariana), the most common boreal forest dominant. We show, first, that stand reflectances for this species can be represented as linear combinations of the reflectances of more elemental radiometric components: sunlit crowns, sunlit background, and shadow. Secondly, using a geometric model, we calculate how the fractions of these radiometric elements covary with each other. Then, using hand‐held measurements of the reflectances of the sunlit background, sphagnum moss (Sphagnum spp.), and assuming shadow reflectance to be that of deep, clear lakes, we infer the reflectance of sunlit crowns from the geometric shadow model and low‐altitude reflectance measurements acquired by a helicopter‐mounted radiometer. Next, we assume that the reflectance for all black spruce stands is simply a linear combination of shadow, sunlit crown, and sunlit background reflectance, weighted in proportion to the relative areal fractions of these pixel elements. We then solve a set of linear equations for the areal fractions of these elements using as input helicopter observations of total stand reflectance. Using this algorithm, we infer the values for the areal proportions of sunlit canopy, sunlit background, and shadow for 31 black spruce stands of varying biomass density, net primary productivity, etc. We show empirically and theoretically that the areal proportions of these radiometric elements are related to a number of stand biophysical characteristics. Specifically, the shadow fraction is increasing with increasing biomass density, average diameter at breast height, leaf area index (LAI), and aboveground net primary productivity (NPP), while sunlit background fraction is decreasing. We show that the end member fractions can be used to estimate biomass with a standard error of ≈ 2 kg/m2, LAI with a standard error of 0.58, dbh with a standard error of ≈ 2 cm, and aboveground NPP with a standard error of 0.07 kg·m‐2·yr‐1. We also show that the fraction of sunlit canopy is only weakly correlated with the biophysical variables and are thus able to show why a popular vegetation index, Normalized Difference Vegetation Index (NDVI), does not provide a useful measure of these biophysical characteristics. We do show, however, that NDVI should be related to the fraction of photosynthetically active radiation incident upon and absorbed by the canopy. This work has convinced us that a paradigm shift in the remote sensing of biophysical characteristics is in order–a shift away from direct inference of biophysical characteristics from vegetation indices and toward a two‐step process, in which (1) stand‐level reflectance is approximated in terms of linear combinations of reflectance‐invariant, spectrally distinct components (spectral end members) and mixture decomposition used to infer the areal fractions of these components, e.g., shadow, sunlit crown, and sunlit background, followed by (2) the use of radiative transfer models to compute biophysical characteristic values as a function of the end member fractions.

Theory and applications for optimization of every part of a photovoltaic system

A general method is presented for quantitatively optimizing the design of every part and fabrication step of an entire photovoltaic system, based on the criterion of minimum cost/Watt for the system output power. It is shown that no element or process step can be optimized properly by considering only its own cost and performance. Moreover, a fractional performance loss at any fabrication step within the cell or array produces the same fractional increase in the cost/Watt of the entire array, but not of the full system. One general equation is found to be capable of optimizing all parts of a system, although the cell and array steps are basically different from the power-handling elements. Applications of this analysis are given to show (a) when should Si wafers be cut to increase their packing fraction; and (b) what are the optimum dimensions for solar cell metallizations. The optimum shadow fraction of the fine grid is shown to be independent of metal cost and resistivity as well as cell size. The optimum thicknesses of both the fine grid and the bus bar are substantially greater than the values in general use, and the total array cost has a major effect on these values. By analogy, this analysis is adaptable to other solar energy systems.