An Approach to Correcting AggregateArea Flux Biases
Sea ice and climate models have relatively coarse grids, typically on the order of 100250 km. In the computation of radiation fluxes, surface and atmospheric properties are treated as being horizontally homogeneous and average grid cell properties are used. We found previously that using mean surface and cloud properties to compute surface radiative fluxes in a grid cell results in substantial errors. Here we propose a simple approach to correct the fluxes for errors that result from horizontal variability. The correction can reduce the average bias (error) to nearly zero and can be easily implemented in numerical models.
Data from the Advanced Very High Resolution Radiometer (AVHRR) are used in this study. Surface and cloud properties and surface radiative fluxes were retrieved for the SHEBA period and area, September through August 1997 in the western Arctic. Fourteen hypothetical GCM grid cells sizes were used: 15x15 km^{2}, 25x25 km^{2}, 35x35 km^{2}, 55x55 km^{2}, 85x85 km^{2}, 105x105 km^{2}, 155x155 km^{2}, 205x205 km^{2}, 255x255 km^{2} , 305x305 km^{2}, 355x355 km^{2}, 405x405 km^{2}, 455x455 km^{2}, and 505x505 km^{2}.
Grid cell radiative fluxes are computed in two ways. One is to calculate the fluxes with average surface and cloud properties over a grid cell, analogous to what is done in models. The other way is to calculate fluxes for every 5x5 km^{2} pixel within the hypothetical grid cell then average those fluxes over the cell. We refer the former method as the areaaverage method, and the latter as the pixelaverage method. In both cases the surface and cloud properties are from the satellite retrievals.
We expect that the fluxes calculated with these two methods will be different because (a) surface and cloud parameters exhibit spatial variability on scales less than that of a typical climate or ice model grid cell, (b) the magnitude of the variability differs for each parameter, and (c) the relationship between some parameters and surface radiation is nonlinear. To test that hypothesis, both methods were used to compute the flux bias in the downwelling shortwave (SWD) and longwave (LWD) radiative flux calculations, where the bias is defined as the areaaverage flux minus the pixelaverage flux. We found previously that the bias can be as large as 22% (Figure 1) for both of downwelling shortwave and longwave fluxes, indicating the correction should be taken into account when the parameterization is used to calculate model grid cell fluxes.
Examinations of all surface and cloud properties revealed that combinations of cloud amount, cloud optical depth, surface broadband albedo and solar zenith angle for SWD flux biases influence the magnitude of the biases to a much greater degree than other parameters (Figure 2). For the LWD flux biases the cloud amount, surface skin temperature and their combination contribute the most to the explanation of bias variance (Figure 3). For SWD flux the bias tends to change from negative to positive when the cloud optical depth becomes large. When the cloud amount is small and/or surface skin temperature is low, the LWD bias is negative. With the increases of the cloud amount and/or temperature the LWD flux bias changes become positive.
An analysis of the relationship between the biases and geophysical parameters are performed by using multivariate regression analysis (Liou, 1979) of the form




where SWDBias and LWDBias denote the downwelling shortwave (SWD) and downwelling longwave (LWD) flux biases, respectively; b_{ijkl }and b_{ij} are the regression coefficients for the SWD and LWD flux biases correction, C is cloud amount, t is cloud optical depth, m is cosine of the solar zenith angle, a is surface broadband albedo and T is surface skin temperature.
Based on these relationships, the five parameters used to estimate the SWD bias are cloud amount, cloud optical depth, the product of cloud amount and optical depth, the product of cloud amount, optical depth, the cosine of solar zenith angle, and surface broadband albedo. The multiple correlation coefficient for five parameters used to estimate SWD flux bias is 0.63. For the LWD flux biases three parameters are used: cloud amount, surface skin temperature, and the product of cloud amount and surface skin temperature. The multiple correlation coefficient for the LWD flux biases is 0.62.
The analysis and correlation imply that it may be possible to correct the fluxes for errors resulting from subgrid cell variability. The regression analysis was performed on the SHEBA AVHRR retrievals. Regression equations were developed to estimate the SWD and LWD flux biases under all weather conditions, but with only those terms in equations (1) and (2) that passed the statistical significance test (at a significance level of 0.05.) The regression equations are:
SWDBias = 5.6519 + 0.33793t + 18.246C –3.9539Ct + 3.3107Ctm + 2.1057Cta (3)
LWDBias = 77.403 + 73.115C + 0.28325T – 0.26119CT (4)
Figures 4 and 5 show the uncorrected and corrected bias frequency distributions. The bias correction equations are applicable to all 14 hypothetical model grid cell sizes, from 15x15 km^{2 } to 505x505 km^{2}. The results show that the overall mean biases for SWD and LWD fluxes are small, but the standard deviations of SWD and LWD flux biases are 14.40 and 6.25 W/m^{2}, respectively, before the correction. After the correction the overall standard deviations are reduced to 8.84 and 3.57 W/m^{2} for SWD and LWD flux biases. The largest biases generally occur in the spring and fall seasons when surface characteristics change most rapidly.
Fig. 1. The percentage bias in downwelling shortwave (SWD) and longwave(LWD) fluxes. The bias is defined as areaaverage flux minus pixelaverage flux, therefore the percentage bias is equal to bias divided by the pixelaverage flux. The solid line is SWD, and dotted line is LWD for area size of 505x505 km^{2}. Biases for other cell sizes are similar.
Fig. 2. The relationship between the parameters used in regression equation for SWD flux bias correction and the SWD flux percentage biases. Variable 1 is Ct (C is cloud amount, and t is cloud optical depth.), variable 2 is t, variable 3 is Ctm (m is cosine of the solar zenith angle) and variable 4 is Cta (a is surface broadband albedo).
Fig. 3. The relationship between the parameters used in the regression equation for the LWD flux bias correction and the LWD flux percentage biases. Variable 1 is C (C is cloud amount.), variable 2 is T ( T is surface skin temperature) , variable 3 is C·T and variable 4 is t ( t is cloud optical depth).
Fig. 4. Relative frequency of flux differences for the area average and pixelaverage flux computations before correction. Values shown are for the period September 1997 through August 1998. Downwelling shortwave and longwave radiation fluxes are denoted by SWD and LWD, respectively.
Fig. 5. Relative frequency of flux differences for the area average and pixel average flux computations after correction. Values shown are for the period September 1997 through August 1998. Downwelling shortwave and longwave radiation fluxes are denoted by SWD and LWD, respectively.