Utility of Assimilating Surface Radiometric Temperature Observations for Evaporative Fraction and Heat Transfer Coefficient Retrieval

Recent advances in land data assimilation have yielded variational smoother techniques designed to solve the surface energy balance based on remote observations of surface radiometric temperature. These approaches have a number of potential advantages over existing diagnostic models, including the ability to make energy flux predictions between observation times and reduced requirements for ancillary parameter estimation. Here, the performance of a recently developed variational smoother approach is examined in detail over a range of vegetative and hydrological conditions in the southern U.S.A. during the middle part of the growing season. Smoother results are compared with flux tower observations and energy balance predictions obtained from the two-source energy balance model (TSM). The variational approach demonstrates promise for flux retrievals at dry and lightly vegetated sites. However, results suggest that the simultaneous retrieval of both evaporative fraction and turbulent transfer coefficients by the variational approach will be difficult for wet and/or heavily vegetated land surfaces. Additional land surface information (e.g. leaf area index (L_AI) or the rough specification of evaporative fraction bounds) will be required to ensure robust predictions under such conditions. The single-source nature of the variational approach also hampers the physical interpretation of turbulent transfer coefficient retrievals. Intercomparisons between energy flux predictions from the variational approach and the purely diagnostic TSM demonstrate that the relative accuracy of each approach is contingent on surface conditions and the accuracy with which L_AI values required by the TSM can be estimated.     

The Effect of Source Distribution on Bulk Scalar Transfer between a Rough Land Surface and the Atmosphere

We investigate the effect of source distribution on the bulk transfer of passive scalars between rough, vegetated land surfaces and the atmosphere, using data from a wind-tunnel experiment in which passive heat was emitted from both the underlying surface and canopy elements of a three-dimensional regular bluff-body array. The experimental results are compared with a simple one-dimensional, two-source model for scalar transfer. We find that: (1) the observed scalar transfer resistance across the boundary layer at the underlying surface is simply related to flat-plate theory by a constant of 0.62, despite the complexity of the turbulent flow within the wind-tunnel canopy; (2) one-dimensional gradient-transfer theory, even with extensions to account for the non-local nature of turbulent transfer within the canopy, does not describe the observed details of scalar concentration gradients in the highly three-dimensional canopy flow, but does provide a reasonable framework for bulk scalar transfer between the composite ground-canopy surface and the flow above the canopy; (3) the kB ⁻¹ parameter (which accounts for bulk excess resistance to scalar transfer over momentum transfer) is highly sensitive to scalar source partition between ground and canopy.     

Wave-dependence of friction velocity,roughness length,and drag coefficient over coastal and open water surfaces by using three databases

The parameterization of friction velocity, roughness length, and the drag coefficient over coastal zones and open water surfaces enables us to better understand the physical processes of air-water interaction. In context of measurements from the Humidity Exchange over the Sea Main Experiment (HEXMAX), we recently proposed wave-parameter dependent approaches to sea surface friction velocity and the aerodynamic roughness by using the dimensional analysis method. To extend the application of these approaches to a range of natural surface conditions, the present study is to assess this approach by using both coastal shallow (RASEX) and open water surface measurements (Lake Ontario and Grand Banks ERS-1 SAR) where wind speeds were greater than 6.44 m s^-1. Friction velocities, the surface aerodynamic roughness, and the neutral drag coefficient estimated by these approaches under moderate wind conditions were compared with the measurements mentioned above. Results showed that the coefficients in these approaches for coastal shallow water surface differ from those for open water surfaces, and that the aerodynamic roughness length in terms of wave age or significant wave height should be treated differently for coastal shallow and open water surfaces.     

Using a One-and-a-Half Order Closure Model of the Atmospheric Boundary Layer for Surface Flux Footprint Estimation

A knowledge of the distribution of the contribution of upwind sources to measurements of vertical scalar flux densities is important for the correct interpretation of eddy covariance data. Several approaches have been developed to estimate this so-called footprint function. Here a new approach based on the ensemble-averaged Navier—Stokes equations is presented. Comparisons of numerical results using this approach with results from other studies under a range of environmental conditions show that the model predictions are robust. Moreover, the approach outlined here has the advantage of a potential wide applicability, due to an ability to take into account the heterogeneous nature of underlying surfaces. For example, the model showed that any variations in surface drag, such as must occur in real life heterogeneous canopies, can exert a marked influence of the shape and extent of flux footprints. Indeed, it seems likely that under such circumstances, estimates of surface fluxes will be weighted towards areas of highest foliage density (and therefore quite likely higher photosynthetic rates) close to the measurement sensor. Three-dimensional footprints during the day and night were also determined for a mixed coniferous forest in european Russia. A marked asymmetry of the footprint in the crosswind direction was observed, this being especially pronounced for non-uniform plant distributions involving vegetation types with different morphological and physiological properties. The model also found that, other things being equal, the footprint peak for forest soil respiration is typically over twice the distance from the above canopy measurement sensor compared to that for canopy photosynthesis. This result has important consequences for the interpretation of annual ecosystem carbon balances by the eddy covariance method.     

Radiation Exchange Between Stratus Clouds and Polar Marine Surfaces

The radiative energy exchange between arctic sea-ice and stratiform clouds is studied by means of aircraft measurements and a two-stream radiation transfer model. The data have been obtained by flights of two identically instrumented aircraft during the Radiation and Eddy Flux Experiments REFLEX I in autumn 1991 and REFLEX II in winter 1993 over the arctic marginal ice zone of Fram Strait. The instrumental equipment comprised Eppley pyranometers and pyrgeometers, which measure the solar and terrestrial upwelling and downwelling hemispheric radiation flux densities, and a line-scan-camera on one aircraft to monitor the surface structure of the sea-ice. An empirical parametrization of the albedo of partly ice-covered ocean surfaces is obtained from the data, which describes the albedo increasing linearly with the concentration of the snow-covered sea-ice and with the cosine of the sun zenith angle at sun elevations below 10°. Cloud optical parameters, such as single scattering albedo, asymmetry factor and shortwave and longwave height-dependent extinction coefficient are determined by adjusting modeled radiation flux densities to observations. We found significant influence of the multiple reflection of shortwave radiation between the ice surface and the cloud base on the radiation regime. Consistent with the data, a radiation transfer model shows that stratus clouds of 400 m thickness with common cloud parameters may double the global radiation at the surface of sea-ice compared to open water values. The total cloud-surface-albedo under these circumstances is 30% larger over sea-ice than over water. Parametrizations of the global and reflected radiation above and below stratus clouds are proposed on the basis of the measurements and modeling. The upwelling and downwelling longwave emission of stratus clouds with thicknesses of more than 500 m can be satisfactorily estimated by Stefan's law with an emissivity of nearly 1 and when the maximum air temperature within the cloud is used.     

The extension of a density current model of katabatic winds to include the effects of blowing snow and sublimation

A density current model was extended for use in katabatic flow over the steep slopes of Antarctica through the inclusion of the Coriolis effect and weight flux terms corresponding to blowing snow and cooling caused by sublimation. The model was calibrated and tested against data obtained during two flights in Adelie Land, Antarctica, along a trajectory starting about 170 km inland and extending to Dumont d'Urville. The predicted trend in water vapor flux agrees with measurements of this flux, lending support to empirical formulae for both snow flux and sublimation rate. Model predictions of velocity were in good agreement with measured quantities when reasonable estimates of radiation divergence and surface heat exchange were provided as input to the model. The potential temperature gradient above the katabatic layer was found to play a major role in flow stability for high velocity and deep katabatic flows. Velocity predictions were in better agreement with the data when a locally determined value was used for the coefficient in the empirical snow flux expression.     

Bulk parameterization for a vegetated surface and its application to a simulation of nocturnal drainage flow

Two simple models are presented for describing the surface energy budget above vegetated surfaces. One is the traditional single-source model that includes only one energy budget equation for the entire canopy-soil system, and the other is the double-source model that includes separate energy budget equations for the vegetation canopy and the underlying soil surface. In both models, the bulk transfer coefficients needed to solve the energy budget equations are parameterized as functions of leaf area index, leaf transfer coefficients, and soil surface roughnesses to obtain the best fit to values calculated by a standard multilayer-canopy model. The validity of these models was tested by comparing their performance with that of the multilayer-canopy model for simulation of the surface energy balance and nocturnal drainage flow above vegetation. Results show that the double-source model gives reliable estimations for all cases ranging from sparse to dense vegetation covers; the single-source model is only applicable to dense, fully-covered vegetation. It is also shown that sparse vegetation weakens nocturnal drainage flow, since it isolates the cool underlying soil surface from the atmosphere above the canopy. This phenomenon cannot be described by a traditional single-source model incorporated commonly in many atmospheric models; however, the double-source model adequately describes this process.     

Experimental study of the transfer velocity for urban surfaces with a water evaporation method

A major problem in urban climate modelling is determining how the heat fluxes from various canyon surfaces are affected by canyon flow. To address this problem, we developed a water evaporation method involving filter paper to study the distribution of the convective transfer velocity in urban street canyons. In this method, filter paper is pasted onto a building model and the evaporation rate from the paper is measured with an electric balance. The method was tested on 2D (two-dimensional) street canyon models and 3D model arrangements. Moreover, in this technique, it is easy to restrict the flux within an arbitrary surface in question. That is, the evaporation distribution on a surface can be studied by using several small pieces of filter paper. In the 2D case, the wall transfer velocity was strongly dependent on the canyon aspect ratio for perpendicular wind directions and it varied widely with height within both windward and leeward wall surfaces. For 3D cubic arrays, the relation to canyon aspect ratio was largely different from that of the 2D canyon. And, as a case study, the variation of wind direction was investigated for a city-like setting. The area-averaged transfer velocity was insensitive to wind direction but its local deviation was significant. Finally, we measured the transfer velocity for a clustered block array surrounded by relatively wide streets. The effect of spatial heterogeneity on the transfer velocity was significant. Moreover, for a fixed total building volume, the transfer velocity was considerably larger when the building height varied than when it was uniform. Therefore, the water evaporation method with filter paper is expected to be useful for studying the transfer velocity and ventilation rates in urban areas with various canyon shapes.     

Estimation Of Sensible Heat Flux From The Surface Temperature Wave And One-Time-Of-Day Air Temperature Observation, Part Ii – Heterogeneous Surfaces

By means of the algorithm presented in Part I of this study, the temporal course H(t) and the daily mean H of the sensible heat flux H can be determined from measurements of the thermodynamic surface temperature (as a function of time) and from a one-time-of-day air temperature observation. Inaddition to these temperatures, one needs estimates of daily mean wind speed,of the roughness lengths of momentum and heat transfer, and of the displacementheight. In Part I, the algorithm was derived for areas with homogeneous surfaces,i.e., with uniform surface temperature, and the method was verified with measurements taken during several field campaigns. The root mean square error for the temperature difference between surface and air, in the comparison between measurement and model, amounted to one or two kelvin, and the error of H was 10 to 25 per cent. The method can be used to determine the sensible heat flux from measurements of surface temperatures by satellites, but can also be applied to ground based measurements.In Part II, the procedure is generalized for areas that consist of various surface types (sub-regions) with different surface temperatures, and can be usedwhen only a few (at least one) air temperature measurements per day are available over only one of the different sub-regions. This generalization should allow improvements to the estimates for H(t) by means of temperature measurements from, e.g., NOAA/AVHRR or LANDSAT/TM, taking into account the heterogeneity of the area contained in one METEOSAT pixel. Criteria are given as to whether effective (areal mean) surface temperatures and roughness lengths may be used for the computation of H or if the above mentioned generalized procedure has to be applied. The new algorithm is verified by measurements sampled during the field campaigns EFEDA 91 (Spain) and HIBE 89 (Hildesheimer Börde in Germany), and by using synthetic data (due to the lack of measured data) for one further combined surfacetype [soil and water (lakes)].     

Bulk transfer velocity to and from natural and artificial surfaces

A wide set of published mass and heat transfer data is reviewed in terms of the stochastic renewal theory. A simple model is then proposed describing the transfer in terms of the fluid bulk properties. Two expressions are presented and they enable us to evaluate the bulk transfer velocity as a function of the turbulence of the flow for smooth and fully-rough surfaces in large ranges of Schmidt (or Prandtl) and Reynolds numbers. The transfer over transitional surfaces can also be evaluated by using a simple criterion for the choice of the right expression. The proposed relationships agree satisfactorily with data from laboratory and atmospheric measurements, for both solid and liquid surfaces.     

Effect of surface layer thickness on simultaneous transport of heat and water in a bare soil and its implications for land surface schemes

Abstract

A physically‐based multi‐layer numerical model is developed to determine the coupled transport of heat and water in the soil and in the soil‐atmosphere boundary layer. Using inputs of standard weather data and initial soil conditions the model is capable of predicting the surface energy balance components as well as water content and temperature profiles in the soil. It is used to predict these variables for a bare silt loam soil under two tillage treatments, viz. culti‐packed and left loose after disc‐harrowing, and the predicted results are compared with measurements. Very good agreement between the model predictions and measured evaporation and heat fluxes and soil water and temperatures for a ten‐day period shows that the model is capable of simulating the coupled transport of soil heat and soil water and their transfer across the soil surface‐atmosphere interface adequately.

Model predictions were compared with those of CLASS (Canadian Land Surface Scheme). It is shown that CLASS, version 2.6, provides good estimates of evaporation and hence the latent heat flux density, Q_E, under wetter soil conditions, but overestimates Q_E at moderately wet soil conditions and underestimates it under dry soil conditions. Under dry to moderately wet soil conditions the calculation of evaporation from bare soil is very sensitive to the thickness of the top layer particularly as the thickness approaches 10 cm.     

三维城市地表反射率计算模式

地表反射率是控制地表能量平衡的一个重要参数。城市建筑物的分布具有较大的不均一性,这不仅给地基观测城市地表反射率带来了很大困难,数值模拟城市地表反射率也是非常困难的。作者开发了一个三维城市地表反射率模式city_photo,该方法结合了蒙特卡洛方法和几何光学方法的优点,具有较高的精度和较快的计算速度。通过引入城市地图的概念,该模式能够计算具有不同结构的城市的地表反射率。2002至2004年晴空MODIS（MODerate Resolution Imaging Spectroradiometer,中分辨率成像光谱辐射计）1~7通道可见光和近红外通道地表反射率资料被用来检验模式的有效性,位于北京朝阳区的中国科学院大气物理研究所的AERONET站点观测得到气溶胶光学特性和水汽资料,6S（Second Simulation of Satellite Signal in the Solar Spectrum）大气辐射传输模式被用来对其进行大气订正。模式计算的北京城市地表反射率个例与MODIS 7个通道地表反射率观测结果具有很高的相关性,相关系数在0.80~0.93之间,说明模式能够较好地模拟城市地表反射率随太阳和观测角度的变化情况。最后讨论了城市结构对城市地表反射率的影响。     

Multi-year Simulations and Experimental Seasonal Predictions for Rainy Seasons in China by Using a Nested Regional Climate Model (RegCM_NCC). Part Ⅰ: Sensitivity Study

A modified version of the NCAR/RegCM2 has been developed at the National Climate Center (NCC), China Meteorological Administration, through a series of sensitivity experiments and multi-year simulations and hindcasts, with a special emphasis on the adequate choice of physical parameterization schemes suitable for the East Asian monsoon climate. This regional climate model is nested with the NCC/IAP (Institute of Atmospheric Physics) T63 coupled GCM to make an experimental seasonal prediction for China and East Asia. The four-year (2001 to 2004) prediction results are encouraging. This paper is the first part of a two-part paper, and it mainly describes the sensitivity study of the physical process parameterization represented in the model. The systematic errors produced by the different physical parameterization schemes such as the land surface processes, convective precipitation, cloud-radiation transfer process, boundary layer process and large-scale terrain features have been identified based on multi-year and extreme flooding event simulations. A number of comparative experiments has shown that the mass flux scheme (MFS) and Betts-Miller scheme (BM) for convective precipitation, the LPMI (land surface process model I) and LPMII (land surface process model Ⅱ) for the land surface process, the CCM3 radiation transfer scheme for cloud-radiation transfer processes, the TKE (turbulent kinetic energy) scheme for the boundary layer processes and the topography treatment schemes for the Tibetan Plateau are suitable for simulations and prediction of the East Asia monsoon climate in rainy seasons. Based on the above sensitivity study, a modified version of the RegCM2 (RegCM_NCC) has been set up for climate simulations and seasonal predictions.     

Multi-year Simulations and Experimental Seasonal Predictions for Rainy Seasons in China by Using a Nested Regional Climate Model （RegCM_NCC）. Part Ⅰ： Sensitivity Study

A modified version of the NCAR/RegCM2 has been developed at the National Climate Center （NCC）, China Meteorological Administration, through a series of sensitivity experiments and multi-year simulations and hindcasts, with a special emphasis on the adequate choice of physical parameterization schemes suitable for the East Asian monsoon climate. This regional climate model is nested with the NCC/IAP （Institute of Atmospheric Physics） T63 coupled GCM to make an experimental seasonal prediction for China and East Asia. The four-year （2001 to 2004） prediction results are encouraging. This paper is the first part of a two-part paper, and it mainly describes the sensitivity study of the physical process paraxneterization represented in the model. The systematic errors produced by the different physical parameterization schemes such as the land surface processes, convective precipitation, cloud-radiation transfer process, boundary layer process and large-scale terrain features have been identified based on multi-year and extreme flooding event simulations. A number of comparative experiments has shown that the mass flux scheme （MFS） and Betts-Miller scheme （BM） for convective precipitation, the LPMI （land surface process model I） and LPMII （land surface process model Ⅱ） for the land surface process, the CCM3 radiation transfer scheme for cloud-radiation transfer processes, the TKE （turbulent kinetic energy） scheme for the boundary layer processes and the topography treatment schemes for the Tibetan Plateau are suitable for simulations and prediction of the East Asia monsoon climate in rainy seasons. Based on the above sensitivity study, a modified version of the RegCM2 （RegCM_NCC） has been set up for climate simulations and seasonal predictions.     

Observations on the use of analytical and numerical models for the description of transfer to porous surface vegetation such as lichen

Convective deposition of submicron-size aerosol to porous surface vegetation was studied by electrochemical simulation, under Reynolds and Schmidt similarity, to a rectangular array of closely-packed lichen and artificial wire roughness layers. Results, showing an approximate tenfold increase in deposition velocity over that of a flat plate placed at the same position, were compared with predictions made on the basis of various rough-surface transfer models, including those based on statistical eddy renewal, as well as with numerical solutions of the diffusion equation in statistically-renewed surface cavities. Most analytical models could be made to fit the observed data, at least for a limited range of flow velocities, but poorly known and poorly defined parameters limit their usefulness for predictive purposes; and their validity across a large variation in molecular diffusivity (or Schmidt number Sc) is generally not assured. Numerical models also depend on poorly substantiated physical assumptions but the effect of such assumptions on transfer can be calculated for a wider range of conditions than those permitting an analytical solution. This allows more direct feedback between model assumptions and calculated or observed transfer. Numerically calculated values for deposition velocity in air for Sc from 0.7 to 7000 and flow velocities from 0.2 to 5 m s^-1 are presented for different model assumptions, with values ranging from < 0.01 to > 1 cms^-1.     

An Analytical one-Dimensional Second-Order Closure Model of Turbulence Statistics and the Lagrangian Time Scale Within and Above Plant Canopies of Arbitrary Structure

An analytical one-dimensional second-order closure model is developed to describe the within canopy velocity variances, turbulent intensities, dissipation rates, Lagrangian time scale and Lagrangian far field diffusivities for vegetation canopies of arbitrary structure and density. The model incorporates and extends the model of momentum transfer developed by Massman (1997) and the model of within canopy velocity variances developed by Weil (unpublished) from the second-order closure model of Wilson and Shaw (1977). Model predictions of within and above canopy velocity variances, turbulent intensities, dissipation rates and the Lagrangian time scale are in reasonable agreement with previously measured or estimated values for these parameters. The present model suggests that the Lagrangian time scale and the far field diffusivity could be strongly dependent upon foliage structure and density through the foliage effects on the velocity variances. A simple formulation for the Lagrangian time scale at canopy height is derived from model results. Taken as a whole, the present model may provide a relatively simple way to incorporate turbulence parameters into models of soil/canopy/atmosphere mass transfer.     

A theory for the scalar roughness and the scalar transfer coefficients over snow and sea ice

Although the bulk aerodynamic transfer coefficients for sensible (C _H ) and latent (C _E ) heat over snow and sea ice surfaces are necessary for accurately modeling the surface energy budget, they have been measured rarely. This paper, therefore, presents a theoretical model that predicts neutral-stability values of C _H and C _E as functions of the wind speed and a surface roughness parameter. The crux of the model is establishing the interfacial sublayer profiles of the scalars, temperature and water vapor, over aerodynamically smooth and rough surfaces on the basis of a surface-renewal model in which turbulent eddies continually scour the surface, transferring scalar contaminants across the interface by molecular diffusion. Matching these interfacial sublayer profiles with the semi-logarithmic inertial sublayer profiles yields the roughness lengths for temperature and water vapor. When coupled with a model for the drag coefficient over snow and sea ice based on actual measurements, these roughness lengths lead to the transfer coefficients. C _E is always a few percent larger than CH. Both decrease monotonically with increasing wind speed for speeds above 1 m s^–1, and both increase at all wind speeds as the surface gets rougher. Both, nevertheless, are almost always between 1.0 × 10^–3 and 1.5 × 10^–3.     

A Method for Monitoring the Heat Flux from an Urban District with a Single Infrared Remote Sensor

The proposed methodology relies on the modelling capabilities of the thermo-radiative model Solene to simulate the heat and radiation energy exchanges between an actual urban district and the atmosphere. It is based on the comparison of the simulated upward infrared and sensible heat flux diurnal cycles that may be measured by elevated sensors above the three-dimensional scene, as a function of sensor position: the heat flux is a function of an equivalent surface temperature given by the infrared sensor and an equivalent heat transfer coefficient deduced from Solene simulations with the actual geometry. The method is tested against measurements obtained in the city centre of Toulouse, France during an experimental campaign in 2004–2005. To improve the computation of the heat exchanges between air and building surfaces a new algorithm is first implemented, based on an empirical model of the wind distribution within street canyons. This improvement is assessed by a direct comparison of the simulated brightness surface temperatures of the Toulouse city centre to measurements obtained with an airborne infrared sensor. The optimization of the infrared remote sensor position is finally analyzed as a function of its height above the mean roof level: it allows evaluation of the heat flux from an urban district when the three different classes of surfaces (roofs, walls, grounds) have similar contributions to the infrared flux towards the sensor, and to the heat flux into the atmosphere.     

The Kwinana Coastal Fumigation Study: II – Growth of the Thermal Internal Boundary Layer

Aircraft measurements of potential temperature and turbulent kinetic energy are used to examine the growth of the thermal internal boundary layer (TIBL) in sea-breeze flows on four selected days of a coastal fumigation study performed in 1995 at Kwinana in Western Australia. The aircraft data, together with radiosonde measurements taken on the same days, show a multi-layered low-level onshore flow in the vertical with a superadiabatic layer extending to about 50 m above the water surface on all four days. On the first three days the layer above the superadiabatic layer was neutral, typically 200 m deep, capped by a stably stratified region, whereas on the remaining day it was fully stable. The occurrence of the neutral layer on most experimental days contrasts with the more usual situation involving an entirely stable onshore flow. A composite approach based on both temperature and turbulence data is used to provide a pragmatic but self-consistent definition of the TIBL height. The data for the first three days indicate that the TIBL grows rapidly into the neutrally stratified region to the top of the region within about 2 km from the coast, with a very slow subsequent growth into the stable stratification aloft. On the other hand, the TIBL grows only to about 200 m within a distance of 7 km from the coast on the fourth day due to a strong stable stratification.An existing numerical TIBL model based on the slab approach, capable of describing the TIBL growth in both neutral and stable environments, and a recent analytical model, more efficient for operational use, are used to simulate the aircraft TIBL observations. The predictions by both models agree reasonably well with the data.     

A simplified scheme of the generalized layered radiative transfer model

In this paper, firstly, a simplified version （SGRTM） of the generalized layered radiative transfer model （GRTM） within the canopy, developed by us, is presented. It reduces the information requirement of inputted sky diffuse radiation, as well as of canopy morphology, and in turn saves computer resources. Results from the SGRTM agree perfectly with those of the GRTM. Secondly, by applying the linear superposition principle of the optics and by using the basic solutions of the GRTM for radiative transfer within the canopy under the condition of assumed zero soil reflectance, two sets of explicit analytical solutions of radiative transfer within the canopy with any soil reflectance magnitude are derived： one for incident diffuse, and the other for direct beam radiation. The explicit analytical solutions need two sets of basic solutions of canopy reflectance and transmittance under zero soil reflectance, run by the model for both diffuse and direct beam radiation. One set of basic solutions is the canopy reflectance αf （written as α1 for direct beam radiation） and transmittance βf （written as β1 for direction beam radiation） with zero soil reflectance for the downward radiation from above the canopy （i.e. sky）, and the other set is the canopy reflectance （αb） and transmittance βb for the upward radiation from below the canopy （i.e., ground）. Under the condition of the same plant architecture in the vertical layers, and the same leaf adaxial and abaxial optical properties in the canopies for the uniform diffuse radiation, the explicit solutions need only one set of basic solutions, because under this condition the two basic solutions are equal, i.e., αf = αb and βf = βb. Using the explicit analytical solutions, the fractions of any kind of incident solar radiation reflected from （defined as surface albedo, or canopy reflectance）, transmitted through （defined as canopy transmittance）, and absorbed by （defined as canopy absorptance） the canopy and other properties per