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Earth's reflectivity is among the key parameters of climate research. National Aeronautics and Space Administration (NASA)'s Earth Polychromatic Imaging Camera (EPIC) onboard National Oceanic and Atmospheric Administration (NOAA)'s Deep Space Climate Observatory (DSCOVR) spacecraft provides spectral reflectance of the entire sunlit Earth in the near backscattering direction every 65 to 110 min. Unlike EPIC, sensors onboard the Earth Orbiting Satellites (EOS) sample reflectance over swaths at a specific local solar time (LST) or over a fixed area. Such intrinsic sampling limits result in an apparent Earth's reflectivity. We generated spectral reflectance over sampling areas using EPIC data. The difference between the EPIC and EOS estimates is an uncertainty in Earth's reflectivity. We developed an Earth Reflector Type Index (ERTI) to discriminate between major Earth atmosphere components: clouds, cloud-free ocean, bare and vegetated land. Temporal variations in Earth's reflectivity are mostly determined by clouds. The sampling area of EOS sensors may not be sufficient to represent cloud variability, resulting in biased estimates. Taking EPIC reflectivity as a reference, low-earth-orbiting-measurements at the sensor-specific LST tend to overestimate EPIC values by 0.8% to 8%. Biases in geostationary orbiting approximations due to a limited sampling area are between −0.7% and 12%. Analyses of ERTI-based Earth component reflectivity indicate that the disagreement between EPIC and EOS estimates depends on the sampling area, observation time and vary between −10% and 23%.

NASA’s Earth Polychromatic Imaging Camera (EPIC) onboard NOAA’s Deep Space Climate Observatory (DSCOVR) satellite observes the entire sunlit Earth every 65 to 110 min from the Sun–Earth Lagrangian L1 point. This paper presents initial EPIC shortwave spectral observations of the sunlit Earth reflectance and analyses of its diurnal and seasonal variations. The results show that the reflectance depends mostly on (1) the ratio between land and ocean areas exposed to the Sun and (2) cloud spatial and temporal distributions over the sunlit side of Earth. In particular, the paper shows that (a) diurnal variations of the Earth’s reflectance are determined mostly by periodic changes in the land–ocean fraction of its the sunlit side; (b) the daily reflectance displays clear seasonal variations that are significant even without including the contributions from snow and ice in the polar regions (which can enhance daily mean reflectances by up to 2 to 6% in winter and up to 1 to 4% in summer); (c) the seasonal variations of the sunlit Earth reflectance are mostly determined by the latitudinal distribution of oceanic clouds.

Vegetation growing season and maximum photosynthetic state determine spatiotemporal variability of seasonal total gross primary productivity of vegetation. Recent warming induced impacts accelerate shifts on growing season and physiological status over Northern vegetated land. Thus, understanding and quantifying these changes are very important. Here, we first investigate how vegetation growing season and maximum photosynthesis state are evolved and how such components contribute on inter-annual variation of seasonal total gross primary productivity. Furthermore, seasonally different response of northern vegetation to changing temperature and water availability is also investigated. We utilized both long-term remotely sensed data to extract larger scale growing season metrics (growing season start, end and duration) and productivity (i.e., growing season summed vegetation index, GSSVI) for answering these questions. We find that regionally diverged growing season shift and maximum photosynthetic state contribute differently characterized productivity inter-annual variability and trend. Also seasonally different response of vegetation gives different view of spatially varying interaction between vegetation and climate. These results highlight spatially and temporally varying vegetation dynamics and are reflective of biome-specific responses of northern vegetation to changing climate.

We use the spectrally invariant method to study the variability of cloud optical thickness τau and droplet effective radius r(sub eff) in transition zones (between the cloudy and clear sky columns) observed from Solar Spectral Flux Radiometer (SSFR) and Shortwave Array Spectroradiometer-Zenith (SASZe) during the Marine ARM GPCI Investigation of Clouds (MAGIC) field campaign. The measurements from the SSFR and the SASZe are different, however inter-instrument differences of self-normalized measurements (divided by their own spectra at a fixed time) are small. The spectrally invariant method approximates the spectra in the cloud transition zone as a linear combination of definitely clear and cloudy spectra, where the coefficients, slope and intercept, characterize the spectrally invariant properties of the transition zone. Simulation results from the SBDART (Santa Barbara DISORT Atmospheric Radiative Transfer) model demonstrate that (1) the slope of the visible band is positively correlated with the cloud optical thickness τ while the intercept of the near-infrared band has high negative correlation with the cloud drop effective radius r(sub eff)even without the exact knowledge of τau; (2) the above relations hold for all Solar Zenith Angle (SZA) and for cloud-contaminated skies. In observations using redundant measurements from SSFR and SASZe, we find that during cloudy-to-clear transitions, (a) the slopes of the visible band decrease, and (b) the intercepts of the near-infrared band remain almost constant near cloud edges. The findings in simulations and observations suggest that, while the optical thickness decreases during the cloudy-to-clear transition, the cloud drop effective radius does not change when cloud edges are approached. These results support the hypothesis that inhomogeneous mixing dominates near cloud edges in the studied cases.

Amazon forests experienced two severe droughts at the beginning of the 21st century: one in 2005 and the other in 2010. How Amazon forests responded to these droughts is critical for the future of the Earth's climate system. It is only possible to assess Amazon forests' response to the droughts in large areal extent through satellite remote sensing. Here, we used the Multi-Angle Implementation of Atmospheric Correction (MAIAC) Moderate Resolution Imaging Spectroradiometer (MODIS) vegetation index (VI) data to assess Amazon forests' response to droughts, and compared the results with those from the standard (Collection 5 and Collection 6) MODIS VI data. Overall, the MAIAC data reveal more realistic Amazon forests inter-annual greenness dynamics than the standard MODIS data. Our results from the MAIAC data suggest that: (1) the droughts decreased the greenness (i.e., photosynthetic activity) of Amazon forests; (2) the Amazon wet season precipitation reduction induced by El Niño events could also lead to reduced photosynthetic activity of Amazon forests; and (3) in the subsequent year after the water stresses, the greenness of Amazon forests recovered from the preceding decreases. However, as previous research shows droughts cause Amazon forests to reduce investment in tissue maintenance and defense, it is not clear whether the photosynthesis of Amazon forests will continue to recover after future water stresses, because of the accumulated damages caused by the droughts.

A strong positive correlation between vegetation canopy bidirectional reflectance factor (BRF) in the near infrared (NIR) spectral region and foliar mass-based nitrogen concentration (%N) has been reported in some temperate and boreal forests. This relationship, if true, would indicate an additional role for nitrogen in the climate system via its influence on surface albedo and may offer a simple approach for monitoring foliar nitrogen using satellite data. We report, however, that the previously reported correlation is an artifact - it is a consequence of variations in canopy structure, rather than of %N. The data underlying this relationship were collected at sites with varying proportions of foliar nitrogen-poor needleleaf and nitrogen-rich broadleaf species, whose canopy structure differs considerably. When the BRF data are corrected for canopy-structure effects, the residual reflectance variations are negatively related to %N at all wavelengths in the interval 423-855 nm. This suggests that the observed positive correlation between BRF and %N conveys no information about %N. We find that to infer leaf biochemical constituents, e.g., N content, from remotely sensed data, BRF spectra in the interval 710-790 nm provide critical information for correction of structural influences. Our analysis also suggests that surface characteristics of leaves impact remote sensing of its internal constituents. This further decreases the ability to remotely sense canopy foliar nitrogen. Finally, the analysis presented here is generic to the problem of remote sensing of leaf-tissue constituents and is therefore not a specific critique of articles espousing remote sensing of foliar %N.

Townsend et al. (1) agree that we explained that the apparent relationship (2) between foliar nitrogen (%N) and near-infrared (NIR) canopy reflectance was largely attributable to structure (which is in turn caused by variation in fraction of broadleaf canopy). Our conclusion that the observed correlation with %N was spurious (i.e., lacking a causal basis) is, thus, clearly justified: we demonstrated that structure explained the great majority of observed correlation, where the structural influence was derived precisely via reconciling the observed correlation with radiative-transfer theory. What this also suggests is that such correlations, although observed, do not uniquely provide information on canopy biochemical constituents.

The single scattering albedo omega(sub O lambda) in atmospheric radiative transfer is the ratio of the scattering coefficient to the extinction coefficient. For cloud water droplets both the scattering and absorption coefficients, thus the single scattering albedo, are functions of wavelength lambda and droplet size r. This note shows that for water droplets at weakly absorbing wavelengths, the ratio omega(sub O lambda)(r)/omega(sub O lambda)(r (sub O)) of two single scattering albedo spectra is a linear function of omega(sub O lambda)(r). The slope and intercept of the linear function are wavelength independent and sum to unity. This relationship allows for a representation of any single scattering albedo spectrum omega(sub O lambda)(r) via one known spectrum omega(sub O lambda)(r (sub O)). We provide a simple physical explanation of the discovered relationship. Similar linear relationships were found for the single scattering albedo spectra of non-spherical ice crystals.

The concept of canopy spectral invariants expresses the observation that simple algebraic combinations of leaf and canopy spectral reflectance become wavelength independent and determine two canopy structure specific variables the recollision and escape probabilities. These variables specify an accurate relationship between the spectral response of a vegetation canopy to incident solar radiation at the leaf and the canopy scale. They are sensitive to important structural features of the canopy such as forest cover, tree density, leaf area index, crown geometry, forest type and stand age. This paper presents the mathematical basis of the concept which is linked to eigenvalues and eigenvectors of the three-dimensional radiative transfer equation.

This note shows that for water droplets at weakly absorbing wavelengths, the ratio omega(sub O lambda(r))/omega(sub O lambda(r(sub O)) of two single scattering albedo spectra, omega(sub O lambda(r) and omega(sub O lambda(r (sub O)), is a linear function of omega(sub O lambda(r). The slope and intercept of the linear function are wavelength independent and sum to unity. This relationship allows for a representation of any single scattering albedo omega(sub O lambda(r) via one known spectrum omega(sub O lambda(r(sub O)). The note provides a simple physical explanation of the discovered relationship. In addition to water droplets, similar linear relationships were found for the single scattering albedo of non-spherical ice crystals.

Elevation data acquired from radar interferometry at C-band from SRTM are used in data fusion techniques to estimate regional scale forest height and aboveground live biomass (AGLB) over the state of Maine. Two fusion techniques have been developed to perform post-processing and parameter estimations from four data sets: 1 arc sec National Elevation Data (NED), SRTM derived elevation (30 m), Landsat Enhanced Thematic Mapper (ETM) bands (30 m), derived vegetation index (VI) and NLCD2001 land cover map. The first fusion algorithm corrects for missing or erroneous NED data using an iterative interpolation approach and produces distribution of scattering phase centers from SRTM-NED in three dominant forest types of evergreen conifers, deciduous, and mixed stands. The second fusion technique integrates the USDA Forest Service, Forest Inventory and Analysis (FIA) ground-based plot data to develop an algorithm to transform the scattering phase centers into mean forest height and aboveground biomass. Height estimates over evergreen (R2 = 0.86, P < 0.001; RMSE = 1.1 m) and mixed forests (R2 = 0.93, P < 0.001, RMSE = 0.8 m) produced the best results. Estimates over deciduous forests were less accurate because of the winter acquisition of SRTM data and loss of scattering phase center from tree ]surface interaction. We used two methods to estimate AGLB; algorithms based on direct estimation from the scattering phase center produced higher precision (R2 = 0.79, RMSE = 25 Mg/ha) than those estimated from forest height (R2 = 0.25, RMSE = 66 Mg/ha). We discuss sources of uncertainty and implications of the results in the context of mapping regional and continental scale forest biomass distribution.

The Multi-angle Imaging SpectroRadiometer (MISR) instrument has been collecting global Earth data from NASA's Terra satellite since February 2000. With its nine along-track view angles, four visible/near-infrared spectral bands, intrinsic spatial resolution of 275 m, and stable radiometric and geometric calibration, no instrument that combines MISR's attributes has previously flown in space. The more than 10-year (and counting) MISR data record provides unprecedented opportunities for characterizing long-term trends in aerosol, cloud, and surface properties, and includes 3-D textural information conventionally thought to be accessible only to active sensors.

Advanced climate models require a more realistic description of canopy radiation with reasonable computational efficiency. This paper develops the mathematics of scattering from a spherical object conceptualized to be a spherical bush to provide a building block that helps to address this need of climate models. It is composed of a homogeneous distribution of individual smaller objects that scatter isotropically. In the limit of small optical depth, incident radiation will scatter isotropically as the sum of that scattered by all the individual scatterers, but at large optical depth the radiation leaving the spherical bush in a given direction is reduced by mutual shadowing of the smaller objects. In the single scattering limit, the scattering phase function and so the albedo are obtained by simple but accurate analytic expressions derived from analytic integration and numerical evaluation. Except in the limit of thin canopies, the scattering and hence albedos are qualitatively and quantitatively different than those derived from 1-D modeling.

The evaluation of a new global monthly leaf area index (LAI) data set for the period July 1981 to December 2006 derived from AVHRR Normalized Difference Vegetation Index (NDVI) data is described. The physically based algorithm is detailed in the first of the two part series. Here, the implementation, production and evaluation of the data set are described. The data set is evaluated both by direct comparisons to ground data and indirectly through inter-comparisons with similar data sets. This indirect validation showed satisfactory agreement with existing LAI products, importantly MODIS, at a range of spatial scales, and significant correlations with key climate variables in areas where temperature and precipitation limit plant growth. The data set successfully reproduced well-documented spatio-temporal trends and inter-annual variations in vegetation activity in the northern latitudes and semi-arid tropics. Comparison with plot scale field measurements over homogeneous vegetation patches indicated a 7% underestimation when all major vegetation types are taken into account. The error in mean values obtained from distributions of AVHRR LAI and high-resolution field LAI maps for different biomes is within 0.5 LAI for six out of the ten selected sites. These validation exercises though limited by the amount of field data, and thus less than comprehensive, indicated satisfactory agreement between the LAI product and field measurements. Overall, the intercomparison with short-term LAI data sets, evaluation of long term trends with known variations in climate variables, and validation with field measurements together build confidence in the utility of this new 26 year LAI record for long term vegetation monitoring and modeling studies.

The generation of multi-decade long Earth System Data Records (ESDRs) of Leaf Area Index (LAI) and Fraction of Photosynthetically Active Radiation absorbed by vegetation (FPAR) from remote sensing measurements of multiple sensors is key to monitoring long-term changes in vegetation due to natural and anthropogenic influences. Challenges in developing such ESDRs include problems in remote sensing science (modeling of variability in global vegetation, scaling, atmospheric correction) and sensor hardware (differences in spatial resolution, spectral bands, calibration, and information content). In this paper, we develop a physically based approach for deriving LAI and FPAR products from the Advanced Very High Resolution Radiometer (AVHRR) data that are of comparable quality to the Moderate resolution Imaging Spectroradiometer (MODIS) LAI and FPAR products, thus realizing the objective of producing a long (multi-decadal) time series of these products. The approach is based on the radiative transfer theory of canopy spectral invariants which facilitates parameterization of the canopy spectral bidirectional reflectance factor (BRF). The methodology permits decoupling of the structural and radiometric components and obeys the energy conservation law. The approach is applicable to any optical sensor, however, it requires selection of sensor-specific values of configurable parameters, namely, the single scattering albedo and data uncertainty. According to the theory of spectral invariants, the single scattering albedo is a function of the spatial scale, and thus, accounts for the variation in BRF with sensor spatial resolution. Likewise, the single scattering albedo accounts for the variation in spectral BRF with sensor bandwidths. The second adjustable parameter is data uncertainty, which accounts for varying information content of the remote sensing measurements, i.e., Normalized Difference Vegetation Index (NDVI, low information content), vs. spectral BRF (higher information content). Implementation of this approach indicates good consistency in LAI values retrieved from NDVI (AVHRRmode) and spectral BRF (MODIS-mode). Specific details of the implementation and evaluation of the derived products are detailed in the second part of this two-paper series.

Radiation reflected from vegetation canopies exhibits high spatial variation. Satellite-borne sensors measure the mean intensities emanating from heterogeneous vegetated pixels. The theory of radiative transfer in stochastic media provides the most logical linkage between satellite observations and the three-dimensional canopy structure through a closed system of simple equations which contains the mean intensity and higher statistical moments directly as its unknowns. Although this theory has been a highly active research field in recent years, its potential for satellite remote sensing of vegetated surfaces has not been fully realized because of the lack of models of a canopy pair-correlation function that the stochastic radiative transfer equations require. The pair correlation function is defined as the probability of finding simultaneously phytoelements at two points. This paper presents analytical and Monte Carlo generated pair correlation functions. Theoretical and numerical analyses show that the spatial correlation between phytoelements is primarily responsible for the effects of the three-dimensional canopy structure on canopy reflective and absorptive properties. The pair correlation function, therefore, is the most natural and physically meaningful measure of the canopy structure over a wide range of scales. The stochastic radiative transfer equations naturally admit this measure and thus provide a powerful means to investigate the three-dimensional canopy structure from space. Canopy reflectances predicted by the stochastic equations are assessed by comparisons with the PARABOLA measurements from coniferous and broadleaf forest stands in the BOREAS Southern Study Areas. The pair correlation functions are derived from data on tree structural parameters collected during field campaigns conducted at these sites. The simulated canopy reflectances compare well with the PARABOLA data.

The analysis of two years of Collection 3 and five years of Collection 4 Terra Moderate Resolution Imaging Spectroradiometer (MODIS) Leaf Area Index (LAI) and Fraction of Photosynthetically Active Radiation (FPAR) data sets is presented in this article with the goal of understanding product quality with respect to version (Collection 3 versus 4), algorithm (main versus backup), snow (snow-free versus snow on the ground), and cloud (cloud-free versus cloudy) conditions. Retrievals from the main radiative transfer algorithm increased from 55% in Collection 3 to 67% in Collection 4 due to algorithm refinements and improved inputs. Anomalously high LAI/FPAR values observed in Collection 3 product in some vegetation types were corrected in Collection 4. The problem of reflectance saturation and too few main algorithm retrievals in broadleaf forests persisted in Collection 4. The spurious seasonality in needleleaf LAI/FPAR fields was traced to fewer reliable input data and retrievals during the boreal winter period. About 97% of the snow covered pixels were processed by the backup Normalized Difference Vegetation Index-based algorithm. Similarly, a majority of retrievals under cloudy conditions were obtained from the backup algorithm. For these reasons, the users are advised to consult the quality flags accompanying the LAI and FPAR product.

The validation of moderate-resolution satellite leaf area index (LAI) products such as those operationally generated from the Moderate Resolution Imaging Spectroradiometer (MODIS) sensor data requires reference LAI maps developed from field LAI measurements and fine-resolution satellite data. Errors in field measurements and satellite data determine the accuracy of the reference LAI maps. This paper describes a method by which reference maps of known accuracy can be generated with knowledge of errors in fine-resolution satellite data. The method is demonstrated with data from an international field campaign in a boreal coniferous forest in northern Sweden, and Enhanced Thematic Mapper Plus images. The reference LAI map thus generated is used to assess modifications to the MODIS LAI/fPAR algorithm recently implemented to derive the next generation of the MODIS LAI/fPAR product for this important biome type.

An extensive verification of cloud property retrievals has been conducted for two algorithms using zenith radiances measured by the Atmospheric Radiation Measurement (ARM) Program ground-based passive two-channel (673 and 870 nm) Narrow Field-Of-View Radiometer. The underlying principle of these algorithms is that clouds have nearly identical optical properties at these wavelengths, but corresponding spectral surface reflectances (for vegetated surfaces) differ significantly. The first algorithm, the RED vs. NIR, works for a fully three-dimensional cloud situation. It retrieves not only cloud optical depth, but also an effective radiative cloud fraction. Importantly, due to one-second time resolution of radiance measurements, we are able, for the first time, to capture detailed changes in cloud structure at the natural time scale of cloud evolution. The cloud optical depths tau retrieved by this algorithm are comparable to those inferred from both downward fluxes in overcast situations and microwave brightness temperatures for broken clouds. Moreover, it can retrieve tau for thin patchy clouds, where flux and microwave observations fail to detect them. The second algorithm, referred to as COUPLED, couples zenith radiances with simultaneous fluxes to infer 2. In general, the COUPLED and RED vs. NIR algorithms retrieve consistent values of tau. However, the COUPLED algorithm is more sensitive to the accuracies of measured radiance, flux, and surface reflectance than the RED vs. NIR algorithm. This is especially true for thick overcast clouds where it may substantially overestimate z.

The first significant processing of Terra MODIS data, called Collection 3, covered the period from November 2000 to December 2002. The Collection 3 leaf area index (LAI) and fraction vegetation absorbed photosynthetically active radiation (FPAR) products for broadleaf crops exhibited three anomalies (a) high LAI values during the peak growing season, (b) differences in LAI seasonality between the radiative transfer-based main algorithm and the vegetation index based back-up algorithm, and (c) too few retrievals from the main algorithm during the summer period when the crops are at full flush. The cause of these anomalies is a mismatch between reflectances modeled by the algorithm and MODIS measurements. Therefore, the Look-Up-Tables accompanying the algorithm were revised and implemented in Collection 4 processing. The main algorithm with the revised Look-Up-Tables generated retrievals for over 80% of the pixels with valid data. Retrievals from the back-up algorithm, although few, should be used with caution as they are generated from surface reflectances with high uncertainties.

By analyzing aircraft measurements of individual drop sizes in clouds, it has been shown in a companion paper that the probability of finding a drop of radius r at a linear scale l decreases as l(sup D(r)), where 0 less than or equals D(r) less than or equals 1. This paper shows striking examples of the spatial distribution of large cloud drops using models that simulate the observed power laws. In contrast to currently used models that assume homogeneity and a Poisson distribution of cloud drops, these models illustrate strong drop clustering, especially with larger drops. The degree of clustering is determined by the observed exponents D(r). The strong clustering of large drops arises naturally from the observed power-law statistics. This clustering has vital consequences for rain physics, including how fast rain can form. For radiative transfer theory, clustering of large drops enhances their impact on the cloud optical path. The clustering phenomenon also helps explain why remotely sensed cloud drop size is generally larger than that measured in situ.

Because of their radiative interactions, the vegetation canopy and the atmosphere are coupled together; each serves as a boundary condition to the radative transfer equations in the adjacent medium. To better understand radiative processes in these media we need an accurate description of their interactions. This presentation outlines a technique needed to describe interactions between vegetation and clouds and exploits it to retrieve cloud optical depth from ground-based radiance measurements. measurements of zenith radiance in the RED and NIR spectral regions. In addition to an algebraic combinations of spectral radiances such as Normalized Difference Cloud Index (NDCI) that can result in poor retrievals due to its insensitivity to cloud fraction, both RED and NIR radiances as points on the "RED vs. NIR" plane are proposed to be used for retrieval. The proposed method is applied to a multi-channel sunphotometer (as a part of AERONET) measurements at the ARM site in Oklahoma. In addition to cloud optical depth, the new method also infers a "radiatively effective" cloud fraction.

The analysis of aircraft measurements of individual drop sizes in clouds suggests that for sufficiently small volumes the mean number of cloud drops with a given radius is proportional to volume powered by a drop-size dependent exponent. For abundant small drops present, the exponent is 1 as assumed in conventional approach. However, for rarer large drops, the exponents fall below unity. We show striking examples of the spatial distribution of large cloud drops using models that simulate the observed power laws. In contrast to currently used models that assume homogeneity and therefore a Poisson distribution of cloud drops, these models show strong drop clustering, the more so the larger the drops. The degree of clustering is determined by the observed exponents. The strong clustering of large drops arises naturally from the observed power-law statistics. This clustering has vital consequences for rain physics explaining how rain can form so fast and also helps explain why remotely sensed cloud drop size is generally biased.

Knowledge of the directional and hemispherical reflectance properties of natural surfaces such as soils and vegetation canopies is essential for classification studies and canopy model inversion.

Neutral particle interaction for photons traveling in media consisting of finite-dimensional scattering centers is investigated. A leaf canopy, a typical example of such media, is idealized as a binary medium consisting of gaps (voids) and regions with phytoelements (turbid phytomedium). The leaf canopy is represented by a combination of all possible open oriented spheres. The extinction coefficient at any phase-space location in a leaf canopy is the product of the extinction coefficient in the turbid phytomedium and the probability of absence gaps at that location. An expression for the differential scattering coefficient is derived using the same approach. The effect of canopy parameters and direction of photon travel on the extinction coefficient is illustrated by numerical examples.

The physics of neutral particle interaction for photons traveling in media consisting of finite-dimensional scattering centers that cross-shade mutually is investigated. A leaf canopy is a typical example of such media. The leaf canopy is idealized as a binary medium consisting of randomly distributed gaps (voids) and regions with phytoelements (turbid phytomedium). In this approach, the leaf canopy is represented by a combination of all possible open oriented spheres. The mathematical approach for characterizing the structure of the host medium is considered. The extinction coefficient at any phase-space location in a leaf canopy is the product of the extinction coefficient in the turbid phytomedium and the probability of absence gaps at that location. Using a similar approach, an expression for the differential scattering coefficient is derived.