Sampling uncertainties in surface radiation budget calculations in RADAGAST


  • 14 October 2008


[1] In the Radiative Atmospheric Divergence Using ARM Mobile Facility GERB and AMMA Stations (RADAGAST) project we calculate the divergence of radiative flux across the atmosphere by comparing fluxes measured at each end of an atmospheric column above Niamey, in the African Sahel region. The combination of broadband flux measurements from geostationary orbit and the deployment for over 12 months of a comprehensive suite of active and passive instrumentation at the surface eliminates a number of sampling issues that could otherwise affect divergence calculations of this sort. However, one sampling issue that challenges the project is the fact that the surface flux data are essentially measurements made at a point, while the top-of-atmosphere values are taken over a solid angle that corresponds to an area at the surface of some 2500 km2. Variability of cloud cover and aerosol loading in the atmosphere mean that the downwelling fluxes, even when averaged over a day, will not be an exact match to the area-averaged value over that larger area, although we might expect that it is an unbiased estimate thereof. The heterogeneity of the surface, for example, fixed variations in albedo, further means that there is a likely systematic difference in the corresponding upwelling fluxes. In this paper we characterize and quantify this spatial sampling problem. We bound the root-mean-square error in the downwelling fluxes by exploiting a second set of surface flux measurements from a site that was run in parallel with the main deployment. The differences in the two sets of fluxes lead us to an upper bound to the sampling uncertainty, and their correlation leads to another which is probably optimistic as it requires certain other conditions to be met. For the upwelling fluxes we use data products from a number of satellite instruments to characterize the relevant heterogeneities and so estimate the systematic effects that arise from the flux measurements having to be taken at a single point. The sampling uncertainties vary with the season, being higher during the monsoon period. We find that the sampling errors for the daily average flux are small for the shortwave irradiance, generally less than 5 W m−2, under relatively clear skies, but these increase to about 10 W m−2 during the monsoon. For the upwelling fluxes, again taking daily averages, systematic errors are of order 10 W m−2 as a result of albedo variability. The uncertainty on the longwave component of the surface radiation budget is smaller than that on the shortwave component, in all conditions, but a bias of 4 W m−2 is calculated to exist in the surface leaving longwave flux.

1. Introduction

[2] Of the solar radiation that arrives at the top of the atmosphere, some 30% is reflected directly back to space, approximately half is absorbed at the surface, and the remainder is absorbed by the atmosphere. These are global averages, and there is some uncertainty over the totals. Kiehl and Trenberth [1997], in perhaps the most widely quoted figure, calculated the amount absorbed by the atmosphere as 67 W m−2. However, other estimates range as high as 93 W m−2 [Wild et al., 1995]. Unfortunately, we cannot directly observe the divergence of radiative flux across the atmosphere, and it has to be calculated as a residual: we must estimate flux budgets at the top and bottom of an atmospheric column and take their difference. The flux budget at the top of the atmosphere can only be obtained using measurements taken from space, and as most Radiation Budget instruments have until recently been located on low earth orbit satellites, it has been difficult to achieve adequate sampling of the diurnal cycles of the fluxes. The satellite-based measurements also define the horizontal scale of the column we must deal with. This is typically tens of kilometers. It is not feasible to measure the surface radiation budget accurately at more than a few places within such a large footprint, and an uncertainty therefore exists in the final divergence because of the consequent spatial sampling problem. Heterogeneity in the atmosphere, principally subfootprint variations in cloud cover, means that the downwelling surface fluxes retrieved from a single surface site will not be the same as the area average corresponding to the satellite footprint, although we might hope that if we average for long enough, the mean atmospheric conditions seen at a point will be close to those averaged over the required area. Over the land surface a second difficulty arises from the variability of the radiative properties of the surface over relatively short scales. This means that reflected shortwave flux, and emitted longwave flux, are likely to be systematically different at the surface from the large-area averages of those quantities. Thus, if we use the point fluxes in our calculations, without any attempt to blend these with other data in some way to estimate the appropriate area averages, we can expect to suffer from a degree of sampling error in our estimate of the surface radiation budget, which will immediately pass over to our estimate of the flux divergence.

[3] In this paper we characterize these uncertainties in the surface radiation budget (SRB) for the particular case of the Radiative Atmospheric Divergence Using ARM Mobile Facility GERB and AMMA Stations (RADAGAST) study, although to understand them completely we would need to know far more about the large-area SRB than we can measure. Satellite-derived data sets of surface properties are used to characterize the surface heterogeneities that are the main cause of the uncertainty in the upwelling surface fluxes, and we infer details of the spatial variability of downwelling fluxes from studying the similarities and differences in two sets of flux measurements separated by an appropriate distance.

[4] In this rest of this paper, unless otherwise stated, all fluxes are daily averages of the broadband (longwave and/or shortwave) surface fluxes.


[5] The RADAGAST project [Miller and Slingo, 2007] was set up with the intention of enabling more accurate calculations of atmospheric divergence (and heating rates, etc.). The project ran during the last months of 2005, and for the whole of 2006, and was closely associated with the regional AMMA (African Monsoon Multidisciplinary Analyses) project, an extensive collaboration between European, African and American scientists to study aspects of the West African Monsoon [Redelsperger et al., 2006]. The idea of RADAGAST was to take advantage of two recent developments that would, together, enable us to tie down better the bounding radiation budgets that define the atmospheric divergence without being compromised by poor diurnal sampling of the fluxes. One of these developments was the deployment of a radiation budget instrument in geostationary orbit, so that the diurnal flux cycle could be captured. This, the Geostationary Earth Radiation Budget instrument [Harries et al., 2005], is a broadband radiation sensor that acquires shortwave (directional) flux and total flux; by subtraction the longwave flux component is obtained. These are acquired every 5 min for the whole of that part of the globe visible to the instrument, and so can capture the diurnal variability of the top-of-atmosphere fluxes. The instrument is aboard satellites of the Meteosat Second Generation series, in geostationary orbit close to the zero meridian, so the instrument obtains fluxes over the Southern Atlantic ocean, the whole of Africa, Europe and the Middle East; it has a footprint of some 50 km at the subsatellite point. The platform carries, in addition, a multispectral imager called SEVIRI [Schmetz et al., 2002] which measures narrowband radiance in 11 channels and has a footprint of 3 km at the subsatellite point. The SEVIRI data are used for accurate geolocation of the GERB data, for scene identification (needed for radiance to flux conversion) and are also used to generate a high-resolution flux product at 9 km resolution [DeWitt et al., 2008].

[6] A number of fixed stations exist around the world that measure, more or less continuously, surface radiative fluxes, both upward and downward, longwave and shortwave. The World Climate Research Program oversees the operation of the Baseline Surface Radiation Network (BSRN) which deploys 30–40 flux stations around the world, but mostly in Europe and North America. An important contribution to this capability is provided by the U.S. Atmospheric Radiation Measurement (ARM) program, which in addition to taking the standard broadband flux measurements also maintains at each of its sites a variety of additional instruments for the study of clouds and aerosols. Unfortunately, none of the fixed ARM stations can be seen by GERB, and the second development that makes RADAGAST possible is the recent creation by ARM of a mobile facility, comprising an extensive suite of active and passive instruments in addition to the broadband flux instrumentation. This, the ARM Mobile Facility (AMF) was deployed at the main airport at Niamey, republic of Niger (lat:13.482 N, long: 2.184E) from late 2005, and measurements continued into early 2007. The main reasons for the selection of Niamey are the wide range of atmospheric conditions experienced there in any single year, and the excellent logistic support [Miller and Slingo, 2007].

[7] In addition to the main deployment, a second set of surface flux instruments was located at Banizoumbou (lat: 13.541N, long: 2.665E), located some 53 km to the east of Niamey. In the rest of this document, the label NIA will be used to denote measurements at the main site, and BAN to label those from the auxiliary site. The relative locations of the two sites are shown in Figure 1. The background of Figure 1a is a MODIS temperature map of the area, taken on the last day of 2006, and Figure 1b shows the distribution of albedo over the same area during the second half of March 2006. The two large squares delineate the GERB pixels that contain each of the sites; the small square containing the cross marking Niamey airport is the corresponding resolution-enhanced 9 km cell.

Figure 1.

(top) Locations of the two AMF deployments and outlines of two adjacent GERB pixels. Background provided by MODIS land surface temperatures, 31 December 2006. (bottom) Albedo variations over the same area, late March 2006.

1.2. Spatial Sampling

[8] Bringing these two elements, GERB and the AMF, together in the RADAGAST project makes possible the first (temporally) well-sampled broadband flux measurements, simultaneous at the top and bottom of the atmosphere, with sufficient frequency to capture the diurnal cycle in radiative flux, so eliminating many of the uncertainties previously associated with flux divergence calculations. The principal sampling effect that remains for us to consider is that of the mismatch in spatial scale between the measurements made at the surface, and those made at the top of the atmosphere. The latter correspond to a surface footprint of some 50 km × 50 km, whereas the surface instruments can see only a small part of the upwelling surface flux field that contributes to the fluxes seen by GERB. The downwelling surface flux measurements are also limited to a small sample of the large-area flux, and instantaneous values are affected by the immediate distribution of clouds and aerosol. It is expected that with advection, (and over a suitable period of time) the time-averaged downwelling flux at the sensor is a good surrogate for the time-averaged flux averaged spatially over the GERB footprint. However, it is difficult to quantify this assertion, and little work seems to have been done on the uncertainty to be expected in the point-area comparison. We begin to address some of these issues by exploiting the fact that there were two sets of AMF flux instruments. The two sites are separated by a distance similar to that between the centers of two adjacent GERB pixels, so the correspondence in the two sets of irradiance measurements permits us to come to some judgment on how well either represents the required spatial average of the same quantity.

1.3. Data Processing

[9] Upwelling and downwelling surface fluxes, both longwave and shortwave, are generated every minute at the AMF stations. For each of these four fluxes, the mean was calculated for each day at each site, and these daily values are the principal data sets used here. The average was taken from midnight to midnight for the longwave instruments, and from sunrise to sunset for the shortwave. For most days, there is a complete set of data at both sites. For estimating some contributions to sampling error we needed to calculate the correlations of the longwave and shortwave irradiances at the two sites. Because these are naturally high when a year's worth of daily data are taken, owing to geometrical/astronomical factors for shortwave fluxes, and seasonal variations in atmospheric water vapor for the longwave fluxes, the correlations were calculated using departures from a smooth curve running through the time series of daily means. The method used, explained in detail in Appendix A, finds a curve that balances a simple (least squares) measure of fidelity to the original data with a requirement for local smoothness. The same algorithm was used for interpolating missing values in daily time series of surface temperature (explained further in section 4).

[10] The processing chain for the GERB instrument makes use of the higher-resolution SEVIRI imager, and this also allows the generation of two main products: one of these corrects for the point spread function of the instrument, so that the resulting flux corresponds to flux arising equally form all points within a defined area, while the other retains the spatial response function of the GERB instrument. A third, resolution-enhanced product known as SHI, is created by distributing the observed flux spatially according to the observed spatial radiance distributions seen by the SEVIRI; this product has a spatial resolution of 9 km, but is not used in this study.

[11] In the next section the flux time series at the two sites are compared. In section 3, we describe and develop the tools that are used to characterize the sampling error for the downwelling fluxes. These include an estimate of the error variance in the irradiances that depends on the correlation coefficient between the two sites, and an upper estimate based on simpler statistics that limits the effects of nonstationarity in the irradiances. In section 4 the various contributions to the sampling error, both downwelling and upwelling are considered separately. For the upwelling fluxes these are based on spatial data sets mapping the albedo and emissivity of the area, and of the space-time variation in surface temperatures. In section 5 we present estimates of the total uncertainty, month by month, for the major part of 2006.

2. Mean Statistics of the Surface Fluxes

2.1. Shortwave Fluxes

[12] The average daily shortwave irradiances could be calculated for most days in 2006. However, the instruments at Banizoumbou were closed down on 8 December 2006, and in addition there were a number of days when no, or only limited, shortwave (SW) data were recorded at that site. These days are ignored in the comparisons presented here. On some other days a few data points were missed, but not enough to affect significantly the mean daily fluxes observed. Linear interpolation was used to fill in those short gaps before calculation of the daily average. The daily fluxes at the main site at Niamey are analyzed in depth by Slingo et al. [2008].

[13] There were 331 days in 2006 for which both sites had useful measurements for shortwave fluxes, and the basic statistics of these fluxes are shown in Table 1. The daily shortwave irradiances, and their differences are plotted in Figure 2. The general features of the plot are driven by illumination geometry, responsible for the increase in irradiance in the early part of the year, and cloud cover, which is responsible for the recurring drops during the dry season, and the very rapid fluctuations during the monsoon months of July, August and September.

Figure 2.

(top) Daily SW downwelling fluxes in 2006 and (bottom) their differences.

Table 1. Statistics for Daily SW Surface Fluxes (331 Days)
 Downwelling (W m−2)Upwelling (W m−2)

[14] We see that in the early part of the year there are occasional dips in the irradiance, which happen on cloudy days, but that the two irradiance curves are well matched: the daily averages at 53 km separation follow each other closely, which leads us to expect that the daily average at a point is indeed a good estimate of the daily average over a 50 km square area. The sampling error appears intuitively to be quite small here (and will be quantified later). The summer, monsoon season sees greater variability in the shortwave flux, but the two sites are still well matched as far as the daily average goes (Figure 3). However, on several days differences of over 30 W m−2 are seen, and occasionally these differences are larger than 50 W m−2 (Figure 3, bottom).

Figure 3.

(top) Daily SW downwelling fluxes in August and September 2006 and (bottom) their differences.

[15] Smoothed versions of the curves of shortwave irradiance are shown in Figure 4, and the difference of the two smoothed curves in Figure 5. The detailed shape of these curves varies with a smoothing parameter, and heavier smoothing of the original curves reduces the oscillations in the difference curve, while keeping the mean difference constant. It is probably not a good idea to read too much into the shape of the graph, although it is clear that the difference is not constant throughout the year, and that the biggest variations are associated with the later part, and end of, the monsoon period.

Figure 4.

Daily SW irradiances, smoothed.

Figure 5.

SW irradiance difference, BAN–NIA (smoothed).

[16] The daily SW irradiances for the two sites are shown as a scatterplot in Figure 6. The two lines either side of the 1–1 line are 10 W m−2 different from that line; points lying outside are those that differ by more than 10 W m−2, or in other words lie more than 7 W m−2 from the nearest point on the 1–1 line. There are more of these (69) below the set of lines than above it (37), consistent with the greater mean shortwave irradiances seen at site BAN, and the slightly higher level of cloud cover seen at NIA.

Figure 6.

Scatterplot of daily SW irradiances. The heavy line is the 1-1 line, and the lines parallel to it are displaced by 10 W m−2.

2.2. Longwave Fluxes

[17] There were 323 days in 2006 with complete or nearly complete longwave (LW) flux measurements at both sites. A small number of days had large gaps in the 60-s data from which the averages were calculated, and those days were not used in these comparisons. The statistics for the downwelling LW fluxes for those 323 days are shown in Table 2.

Table 2. Statistics for Daily LW Surface Fluxes (323 Days)
 Downwelling (W m−2)Upwelling (W m−2)Net (W m−2)

[18] A scatterplot of the daily mean downwelling LW flux at the two sites (Figure 7) shows a systematic difference, with the fluxes at the airport (NIA) being some 7 W m−2 higher, on average, than those at the auxiliary site (BAN). This difference is almost independent of time of year, yet the daily means of the near-surface temperatures recorded by the met instruments at the two sites are within a degree of each other for most of the year. A study prepared for the DOE [Long et al., 2008] found no plausible explanation for this difference in terms of atmospheric conditions, but noted anomalous temperature readings for the instrument case and dome temperatures, which are used in the calibration of the flux. The likelihood is that the downward LW fluxes at BAN are systematically too low, and that the real difference in the means is much smaller than the 7 W m−2 observed.

Figure 7.

Comparison of daily LW downward fluxes. Correlation = 0.995.

[19] The relationship between the upward and downward fluxes at either site shows a great deal more scatter, as we would expect, but it is noticeable that the scatterplots for the two sites are quite similar (Figure 8), suggesting that the same things are going on at both sites. A comparison of the upward fluxes, which we might expect to be nuanced given the different thermal environments at the two sites, instead shows high correlation (Figure 9). Again, there is an offset of about 7 W m−2 between the two. So, despite the apparently complex relationship between downwelling and emitted fluxes manifest in Figure 8, the uncorrected net fluxes are found to be very highly correlated (Figure 10).

Figure 8.

Upward and downward daily LW fluxes, 2006.

Figure 9.

Comparison of daily upwelling LW fluxes. Correlation = 0.96.

Figure 10.

Comparison of daily net LW fluxes. Correlation = 0.97.

[20] The longwave component of the surface radiation budget is typically between 50 and 150 W m−2 for the daily means. There is a noticeable seasonal variation to this, tied to variations in atmospheric humidity and cloud cover, as we can see in Figure 11, which shows the daily net LW fluxes for the two sites. The maximum in the net flux occurs at the end of the dry season, when the surface temperature is high and the moisture content of the atmosphere at its lowest. The minimum occurs during the monsoon, when increased cloud cover and humidity greatly increase the downwelling LW flux. The shape of the curve closely follows the variations in the range of surface temperature (daily maximum minus daily minimum) as measured by an Infrared Thermometer (IRT) at Niamey, and shown in Figure 12. In fact the correlation is strong enough that a simple linear function of the temperature range can predict the net LW flux with an RMS error of less than 14 W m−2..

Figure 11.

Daily net LW fluxes.

Figure 12.

IRT surface temperature range (daily maximum minus daily minimum) at NIA (K).

[21] The mean difference in the net LW flux between the two sites is just 0.3 W m−2, and the root mean square difference is 6.8 W m−2, less than the mean difference in either upward or downward fluxes. The 7.5 W m−2 difference between the sites is consistent for both upward and downward fluxes: it is on average the same for daytime and nighttime fluxes, and shows no significant variation with season. This is strong evidence of a modest calibration problem at the secondary site for at least the downwelling fluxes.

2.3. Cloud Cover

[22] Clouds can be detected reliably during the day from the SEVIRI instrument, as described in Slingo et al. [2008]. We took the cloud flags for the 5 by 5 block of SEVIRI pixels centered on each AMF site, to give an estimate of fractional cloud cover every 15 min. A daily weighted average was taken: the weights were zero for any time period between 1800 UT and 0600 UT, one for 1200 UT, and varying linearly between 0600 and 1200 UT, and between 1200 and 1800 UT. Effectively, a triangle shaped filter is run over the cloud estimates that cover daylight. This gives a cloud index which is more closely related to solar irradiance than a simple average would be, but also eliminates certain biases that are seen in the cloud product when the sun is low in the sky. The annual average of this cloud index is 0.301 at BAN, and 0.314 at NIA, some 4% higher, so at least some of the higher mean irradiance at the BAN site may be explicable in terms of the lower cloud cover there.

2.4. Monthly Averages

[23] Monthly statistics for downward fluxes at both sites, and for the difference between them, are summarized in Table 3. In Table 3, Nd denotes the number of days' fluxes that were available for the statistics, SD denotes the standard deviation, and the units are W m−2. We see a tendency for the longwave fluxes to be most variable when the shortwave variability is relatively low, and vice versa. The SW variability is largely determined by variations in the extent and timing of cloud cover (although some will be due to aerosols), and is therefore greatest in the monsoon season at the time of the largest diurnal changes in cloud cover. Strangely, the greater variability in longwave irradiance in the winter months is probably also due to cloud cover. In dry conditions, small amounts of cloud tend to have a larger effect on the longwave flux then than on the shortwave flux. In humid and cloudy conditions the emissivity of the clear sky is similar to that of the clouds, so varying amounts of cloud then have less effect on the downwelling longwave flux.

Table 3. Statistics of the Monthly Downwelling Fluxesa
  • a

    Flux units are W m−2.


3. Sampling Uncertainty

[24] Our overarching concern is to characterize the difference between the surface radiation budget at the AMF, which is used in divergence calculations, and that which applies across the larger area which contains the measurement site: in our case a given square 50 km on a side. The downwelling fluxes are modulated by cloud cover, and this is a factor that is essentially independent of the position of the instrumentation site. The changes in irradiance at a point can be quite fast (within seconds in the case of a cloud passing across the sun) but the randomness of inhomogeneities in the atmosphere, and their advection to and from the site, gives us confidence that the time-integrated irradiance at the AMF is a good estimate of the time integrated irradiance over a nearby area, provided that area is small enough and the period of integration long enough. Therefore, we will assume that variations in irradiance across the area are indeed random (no one part of it is more likely to be cloudy than any other part, say) so that if the irradiance is regarded as a random variable the point measurement and its corresponding area average have the same expected value. While there may be no bias in the downwelling terms, there will generally still be a difference between the point and areal values over any given period of time, and the best we can hope to do here is characterize the expected mean square difference between the two values. We cannot estimate this precisely, but the existence of the second set of measurements at Banizoumbou helps us to place bounds on that variance. This forms only a part of the sampling uncertainty, being concerned only with the irradiances (although a large part, as we will see). Other factors controlling the surface radiation budget (SRB) vary much more slowly in time. These arise from surface heterogeneity and are shortwave albedo, longwave bulk emissivity and the surface temperature. These can vary quite markedly on spatial scales much smaller than the scale of the GERB footprint. The second set of surface flux measurements is of very limited help here, and we must exploit other sources of information to estimate the corresponding sampling uncertainties.

3.1. Modeling the Sampling Error

[25] Let us consider first the shortwave contribution to the SRB. This is

equation image

where a is the albedo (this is essentially a definition of that quantity). At any given point the surface will have distinctive spectral variation to its reflectance properties, and the surface is not usually Lambertian, so the albedo generally varies with the directional distribution of the irradiance, the spectral distribution of the irradiance, and under the clearest skies will vary with time of day, and the daily average will therefore vary with time of year. There is also some variation with soil moisture content (wet soil is darker than dry soil) so the albedo may change more quickly in the monsoon season than during the dry season. Despite these considerations, the daily albedo (the average of the total reflected SW flux to the total SW irradiance) changes little from day to day during the dry season at either of the AMF sites and so can be considered a stable parameter which varies mainly with position. However, that is not the case throughout the year. Figure 13 plots the daily albedo at NIA (i.e., the ratio of the all-day reflected flux to the all-day shortwave irradiance), and shows a smoothed version through it. Month-by-month statistics for the mean value, and mean square deviation from the smoothed curve are shown in Table 4. Also shown is an indicative measure of the mean daily precipitation (in mm: this is from an experimental instrument and should be regarded as no more than a comparative value). The above-average albedo spike at the beginning of March occurred during a notable dust storm [Slingo et al., 2006]. During this period, the direct solar illumination dropped to near zero, and almost all the shortwave irradiance was diffuse; Slingo et al. [2006] stress the higher uncertainty of the shortwave fluxes at this time. It would be surprising if, during this severe dust storm, there were not some deposition of dust on the upward pointing pyranometers, which would have the effect of raising the apparent albedo. The sharp dip at the beginning of June happens in the middle of a 4 day rain event, and the variability in albedo during the remainder of the monsoon is probably driven by variations in soil moisture. Table 5 shows the same statistics for the BAN data, together with a measure of rainfall.

Figure 13.

Variation of the daily albedo at NIA, raw and smoothed time series.

Table 4. NIA Albedo, Monthly Mean and Standard Deviation, and Rainfall
μ (W m−2)0.2520.2550.2670.2580.2330.2250.2090.1890.1980.2260.2470.256
σ (W m−2)0.0030.0040.0080.0030.0060.0170.0220.0270.0230.0050.0020.004
ppt (mm/d)0.1230000.3341.44613.6511.003.7800.3310.0210
Table 5. BAN Albedo, Monthly Mean and Standard Deviation, and Rainfall
μ (W m−2)0.3130.3090.3180.3180.3040.2910.2650.2380.2250.2410.280
σ (W m−2)0.0030.0040.0110.0060.0070.0110.0160.0130.0090.0040.004
ppt (mm/d)0.0010.0010.0020.0010.3813.34219.6517.917.1420.0260.010

[26] The variability in albedo from month to month at either site is quite well correlated (r > 0.8) with the mean monthly rainfall. The small variations during the dry season are possibly derived from instrumental uncertainties, but the albedo varies somewhat with the proportion of diffuse irradiance, and so some variability can be expected from the variability in that proportion. The small variation seen in the dry seasons is consistent with the small differences in the white-sky (100% diffuse) albedo and black-sky (0% diffuse) albedo products derived from the Moderate Resolution Imaging Spectroradiometer (MODIS).

[27] The longwave upward flux consists of reflected downward flux, as for the shortwave, and the radiative flux emitted by the ground. Simple parameterizations of the instantaneous values of these terms are given respectively by

equation image

where Ts is the surface temperature and ɛ the emissivity (so that 1-ɛ is the longwave albedo). These expressions are correct for a gray Lambertian surface. The instantaneous net flux is

equation image

or, using the expressions given above,

equation image

and the net surface fluxes used in our divergences studies are these values, integrated over a 24 h period. To simplify the notation, the variable g will be used for the daily average of σT4; g is the upwelling blackbody flux for the given temperature time series. We will use S and L for the shortwave and longwave time-averaged irradiances, a subscript N to denote value obtained at the main AMF site, a subscript B to represent any flux measured at the auxiliary site, and a subscript A to denote an average of the variable over the appropriate GERB footprint. Then our observation of the daily mean flux, and what we really should be using to combine with the TOA fluxes are

equation image


equation image

respectively. We denote by U the arithmetic difference of these two, which represents the uncertainty associated with the limited sampling, which we may also call the sampling error. We split U up into the sum of 5 terms, U1 to U5:

equation image

Each of these 5 terms is the product of a difference between some quantity and its area average, times some parameter value at the AMF, or times an area averaged flux. Specifically: U1 is the contribution to the uncertainty of variability in shortwave irradiance across the area of interest, and U2 that from the difference in the surface albedo at the site and the average albedo across the GERB footprint. U3 results from spatial variations in downwelling longwave flux, U4 from spatial variations in surface temperature, and U5 the result of systematic difference in emissivity. It is done this way in the hope that these five terms will have some statistical independence, and that we can understand the greater part of the sampling uncertainty by understanding the corresponding problems for these five terms. In practice, there is some correlation between these terms which generally acts to reduce the error slightly.

3.2. Irradiance Uncertainties

[28] It is in estimating the mean square values of U1 and U3 that the value of the second AMF suite of radiation instruments proves most valuable. The difference in SW irradiance between the two can put an upper bound on U1 squared, and the correlation between the two sites leads to another estimate which is smaller, and perhaps more optimistic, but which imposes slightly more restrictive assumptions in the irradiance field. The same applies to U3.

3.2.1. A Possible Lower Bound to the Irradiance Uncertainty

[29] For one estimate of the sampling uncertainty we can use a result from Settle [2004], where an expression was derived for precisely this quantity, under certain constraints. These amount to treating the irradiance as a random variable with a mean and variance independent of position, and an autocorrelation structure that is a function only of the distance between two points, divided by a scale length. The expression, in terms of integrals of the autocorrelation function, was derived for the case when the comparison point is located at the center of a square area. The required expression, for cases where the correlation length is large, is

equation image

where V is the variance of F at a point, γ(equation image) is the spatial autocorrelation function (assumed isotropic) of F at distance r, D is the length of the side of the area A, and R (≫D) is the autocorrelation length, defined such that

equation image

We will see that the correlations (ρ, say) are indeed very high between the two sites, so we have, expanding the correlation function out to first order:

equation image

where L is the distance between the two measurements. Therefore

equation image

so that

equation image

V and ρ are measured, and D and L are known. In practice the distance L between the sites is 53 km, D = 50 km for the GERB pixel.

[30] The numerical constant, 0.24, appearing in equation (10) holds for point measurements made at the center of a bounding square. The AMF sites are not exactly located at the centers of the GERB pixels containing them, and the numerical constant will be slightly greater than this. If the distance, d say from the AMF to the true center of the corresponding GERB footprint is modest, then the necessary adjustment is that the constant increases from 0.24 to 0.24 + 1.76(d/D)2.

3.2.2. An Upper Bound to the Irradiance Uncertainty

[31] There must be some doubt that the assumptions inherent in the just-derived estimate of sampling uncertainty are fully realized. The stationarity implicit in the derivation is evidently not well honored through the year, as there are minor differences in the mean irradiance at the two sites (Figure 4); the assumption of an isotropic autocorrelation function, that depends on a single scale parameter, is unproven, and a small correction is needed to account for the off-center locations of the flux stations. The estimate we have derived above is likely therefore to be on the optimistic side. However, the following argument gives us an upper bound to the sampling error, which gives a larger estimate than the one we have just derived, but not greatly so. We can relax the requirement of global stationarity, and need assume nothing about the autocorrelation structure. Our main assumption is that the measurement at either site be an unbiased estimate of the corresponding area average.

[32] That upper bound to the variance is half the mean square difference between the values at the two sites. To see this, let the time-averaged fluxes at the two sites, Niamey and Banizoumbou, be denoted by FN and FB respectively. Consider areas based on each of these: let the spatial average of the corresponding time-averaged fluxes over those areas be AN and AB. We are interested in the mean square difference between F and the corresponding A, and we can relate it to the observed flux difference thus:

equation image

We now treat the differences in a statistical manner, replacing each term with its expectation. Statistically, E(F) = A, so the fourth and fifth terms on the right hand side can be set to zero. The last term is also essentially zero, as it makes no sense for point values at the two sites to be consistently both higher or lower than their respective area averages. So, applying expectations, we have

equation image

where E denotes expectation value. Taking UNUB we have the following formula for the mean squared uncertainty:

equation image

say. Replacing equation image {(FNFB)2} with observed values gives us the required upper limit.

[33] The usefulness of this limit depends on the magnitude of the mean square difference between the two area-averaged fluxes (i.e., (ANAB)2). The smaller the areas, the more likely those area averages are to be different, the greater their mean square difference, and the more pessimistic and less valuable the derived upper limit. The estimate is most accurate when the areas are large, and the expected difference between two area averages much smaller than the sampling error on either. However, if the areas are too large, and overlap significantly, then we are no longer justified in ignoring the term in equation (11) containing (FNAN)(FBAB) which effectively approaches the flux covariance between the two sites. The size of the GERB pixel is comparable to the distance between the two sites, so for this specific case there would be no overlap, the upper limit is valid, and some difference can be expected between AN and AB, although this is difficult to quantify without a model for the spatial variability in irradiance.

[34] For genuinely random fields with the appropriate autocorrelation structures, the upper bound to the sampling error variance given by equation (11) is about four times that implied by equation (10) (so that the associated flux uncertainties differ by a factor of about 2). We will call these the pessimistic and optimistic estimates, respectively, of the relevant contribution to the sampling uncertainty. We will later use a value of 3/4 of the upper bound, midway between the optimistic and pessimistic estimates, as our preferred estimate of the true uncertainty. Despite being based on time series at just two locations, it is unlikely to be far from the real sampling uncertainty.

[35] These arguments can only be applied to the downwelling fluxes, as they require the AMF value to be an unbiased estimate of the area average to simplify the arithmetic. This simply is not true for some of the remaining terms, which depend on surface variability that may have much shorter correlation length scales, and for which time averaging is of no benefit.

4. Quantifying the Sampling Uncertainty in RADAGAST: Bias and Mean Square Uncertainty

[36] In the previous section we decomposed the sampling error, the difference between the observed net surface flux observed at the AMF, and the average of the same quantity over the GERB footprint, into five separate terms. Of these terms those, U1 and U3, that are proportional to the differences in irradiance have, regarded as random variables, expectation zero, as (we have assumed that) the irradiance measured at the AMF is an unbiased estimate of that over the GERB square. The remaining terms, each dependent on the difference between a point value of some surface variable and its area average, represent possible sources of bias. Ignoring U4 for now, we can combine the remaining bias terms thus:

equation image

SA is generally between 200 and 250 W m−2, and the term gALA about 100 W m−2, so provided the emissivity and albedo at the AMF site are not more than about 0.02 different from the corresponding area average, these contributions to the bias should be less than 5 W m−2. However, as we shall see, while any bias in the emissivity term is likely to be small, it seems that the difference between the albedo of the immediate area of the AMF, and that of the GERB pixel as a whole, could be about 0.04, with the GERB value being the higher. This term alone could thus introduce a systematic error of 8–10 W m−2 into our calculations of the SRB.

[37] The remaining term, U4, effectively the difference between g at the AMF and averaged over the GERB footprint, is more difficult to assess. As will later be shown, it appears from satellite data that the variability in g across the GERB area may be no more than a few W m−2 under clear skies. The combined effect is a bias that is probably limited to less than 10 W m−2 for the daily average of the net surface flux. We will return to this in the relevant section below.

[38] The bias may be small, but the day-to-day variability need not be small if the variance of the unbiased terms, U1 and U3 should be large. The mean square uncertainty is

equation image

assuming that there is no significant correlation between the individual components. In fact, there is some correlation, which generally tends to lower the uncertainty For example, during the dry season increasing cloud cover tends to increase longwave irradiance and decrease shortwave irradiance. The effects are generally small, however.

4.1. Variability of Shortwave Irradiance (U1)

[39] The error that arises when the shortwave irradiance measured at the instrument point is not the same as the average irradiance over the required area is given by

equation image

In cloudy conditions shortwave irradiance can vary rapidly at a point, and from point to nearby point, according to whether the sun is obscured or not. With no reason to believe that our measurement site is any more or less cloudy, on average, that anywhere else in the GERB footprint, the AMF value, SN, is taken to be an unbiased estimate of the required value, SA, while the factor (1 − aN) will be regarded as essentially constant over a period of weeks. The mean square difference of SNSA we bound using the methods described earlier. The contribution U1 is then about three quarters of the uncertainty on SW↓ once the albedo contribution (1 − a ∼ 0.75) has been factored into the calculation. Over the whole of 2006, the mean square difference between the daily average of SW↓ at the two sites is 222 (W m−2)2. Halving this, and taking the square root gives an uncertainty, from this source alone, of 10.5 W m−2 on the daily average, which is considerably larger than the presumed bias in the measurement. The corresponding error U1 is then about 7 W m−2 when we include the albedo. As our preferred estimate of irradiance uncertainty is three quarters of the upper bound (roughly halfway between the upper and lower bounds), our overall estimate of uncertainty is about 5 W m−2 in the shortwave. But this is an average over the whole year of course; this upper bound will be higher when there is plenty of cloud about, and lower in cloud-free conditions. Taking just the months of January and February, for example, we find the mean square difference in SW↓ to be 22.6 (W m−2)2, giving an upper limit to the root mean square error for U1 of just 2.5 W m−2 and a preferred estimate of 2 W m−2..

[40] Estimates of the error, based on the monthly mean albedo and monthly statistics of the flux differences between the sites, are shown in Table 9 in section 5.

4.2. Spatial Variability in Albedo (U2)

[41] The second source of error,

equation image

according to the simple view (equation (5)), is the fact that the arbitrary siting of the radiometers means the albedo folded into the measurements of reflected flux is almost certainly different from the mean albedo of the larger area, which is the quantity relevant for estimating divergence. Because the effect is systematic, it is not possible to use the two data sets to estimate the scale of the error, and in fact no ARM measurements are useful for such a purpose. We must use some other source of data to examine the spatial variability. The most sensible option seems to be the estimates of surface albedo generated from the MODIS instruments on the Terra and Aqua satellites [Schaaf et al., 2002], given their high spatial resolution and the significant amount of development, validation and testing that has been devoted to that product. In the first instance we will not aim for a direct comparison of the surface albedos formed from the point fluxes with those in the satellite product. Instead, we try to assess how the satellite albedo inferred for the immediate vicinity of the AMF compares to that inferred for the GERB pixel, found by averaging over the many MODIS pixels that fit into a GERB pixel.

[42] For the period of interest, the available products correspond to various spectral and broadband albedos evaluated over 16 day periods, at 500 m spatial resolution (version 5). They are generated every 8 days, so there is some oversampling of the daily MODIS data. There are ten thousand such estimates within each GERB footprint, and the observed variability of those albedo estimates can be used to characterize the heterogeneity of the reflected fluxes at sub-GERB scale. The MODIS albedo product in fact consists of much more than a single number, as it gives spectral albedo over a number of wavelengths regions, allows calculation of albedo at different times of day (different values of solar zenith angle) and also distinguishes between a black-sky albedo (that which would apply in the complete absence of diffuse irradiance) and a white-sky value, which assumes that the irradiance is completely and uniformly diffuse. In practice, the numerical difference between these values is small, and a “blue-sky” albedo, which lies between them, will not vary much from either. For the sake of estimating heterogeneity, and the typicality of the AMF locations, we have taken the black-sky, broadband albedo evaluated at local solar noon.

[43] Table 6 shows the mean MODIS-derived albedo for GERB pixels containing the two sites, the variability of the MODIS albedo within the GERB pixel (the standard deviation of the retrieved albedos of up to 10,000 MODIS pixels) and the albedo of the MODIS pixel that contains the instrument site. In Figure 14 the difference between the latter value and the area average is plotted, and the standard deviations drawn on. The BAN site appears to be a faithful representative of the GERB pixel which contains it, perhaps more than we have a right to expect, and the difference in the albedo values, BAN pixel and GERB pixel, are usually only a fraction of the scatter observed in the GERB pixel. The 500 m × 500 m area around the main site, however, is clearly not so typical of other similar subdivisions of the corresponding GERB pixel, or the area as a whole. Had we chosen such a point at random, the albedo error would have a standard deviation of about 0.025 (and a mean of zero, of course), whereas the actual site is different by 1 1/2 to 2 times this. If the few square meters seen by the pyranometer measuring upward flux should be typical of that 500 m square area, then there is a bias of some 8–12 W m−2 contributing to the sampling uncertainty of the NIA shortwave radiation budget.

Figure 14.

Albedo “errors” from MODIS 500 m albedo retrievals. Symbols represent the differences (AMF value minus area mean); curves show ±1 standard deviation of the MODIS values within a GERB pixel.

Table 6. MODIS Albedos: AMF Pixel and GERB Area Averages for 16-Day Periods During 2006a
  • a

    On two occasion (blank entries) MODIS was unable to retrieve a value for the pixel containing the AMF.

9 Jan0.2450.0280.2050.0400.2660.0110.2620.004
25 Jan0.2520.0280.2100.0420.2800.0120.2780.002
10 Feb0.2630.0280.2220.0410.2920.0120.294−0.002
26 Feb0.2740.0270.2320.0420.2990.0120.2930.006
14 Mar0.2810.0250.2410.0400.3080.0120.313−0.005
30 Mar0.2820.0260.2440.0380.3090.0140.318−0.009
15 Apr0.2740.0280.2310.0430.3010.0150.309−0.008
1 May0.2670.0270.2280.0390.2980.0190.2980
17 May0.2570.0260.2030.0540.2830.0210.2810.002
2 Jun0.2700.0210.2170.0530.2920.0160.296−0.004
18 Jun0.2770.0280.2390.0380.3130.0170.316−0.003
4 Jul0.2570.0240.2060.0490.2890.022 
20 Jul0.2460.023 0.2740.0220.300−0.026
5 Aug0.2350.0230.1930.0420.2550.0180.271−0.016
21 Aug0.2410.0220.1930.0480.2600.0170.277−0.017
6 Sep0.2290.0240.1860.0430.2510.0150.263−0.012
22 Sep0.2340.0280.1980.0360.2600.0160.2570.003
8 Oct0.2400.0270.2050.0350.2650.0120.267−0.002
24 Oct0.2490.0310.2080.0410.2730.0130.275−0.002
9 Nov0.2610.0310.2160.0450.2870.0130.295−0.008
25 Nov0.2640.0340.2170.0470.2900.0130.2870.003
11 Dec0.2680.0330.2230.0450.2950.0130.2930.002

[44] All this assumes that the spatial variations in the MODIS albedo product are a reliable guide to the true variability on the ground. We can see that they are, at least for 1 day in January 2006, by comparison with albedo measurements from a low-flying aircraft. We compare the MODIS albedo calculated for the 16 day period between 16 January and 31 January 2006, with albedos obtained from a flight on 19 January 2006 of the Facility for Airborne Atmospheric Measurements (FAAM), run jointly by the UK's Natural Environment Research Council and the UK Met Office during the DABEX campaign [Haywood et al., 2008]. The flight took place between the vicinity of the NIA site (1040 UT) and the BAN site (1050 UT), at a height of 500 ft. Upwelling and downwelling broadband fluxes were recorded every second. Given the speed of the aircraft, these measurements overlap heavily and there might be up to four FAAM albedo measurements for each MODIS pixel on the transect. The albedos at the start and end of the flight agree quite well with the corresponding AMF albedos, and are rather higher than the MODIS values. This applied along the length of the transect, but consistently so: if the MODIS black-sky albedos for this period are multiplied by 1.1352, they fall nicely into line with the FAAM values (Figure 15).

Figure 15.

Albedo comparison, from aircraft (FAAM) and MODIS V5. NIA is at the beginning of the flight (left of plot), and BAN is at the end. The comparison is for 19 January 2006.

[45] The good match of the fine detail gives us a lot of confidence that the MODIS albedo product can be used to identify how typical the locality of the AMF is for the much larger GERB area.

[46] We have concentrated here on characterizing the spatial variability of the albedo, assumed more-or-less constant over the 8 day period of the MODIS retrievals. As we saw in section 3.1, this appears to be a good assumption during the dry seasons, although less so during the monsoon (at which time, in addition, the MODIS retrieval struggles to find enough cloud-free data to generate its albedos).

4.3. Variability in Longwave Irradiance (U3)

[47] The longwave irradiance term,

equation image

can be treated in a manner similar to that used for U1. We shall assume that there is a flat calibration offset of 7.5 W m−2 to be applied to the flux difference; when this is done, the average over the year of the squared difference in LW↓ is (3.3 W m−2)2, giving an upper limit of 3.3 ÷ √2 ∼ 2.3 W m−2. From Table 7 we see that the variance of this flux is generally much smaller than that for the shortwave, while the correlation is higher. These factors mean that our two bounds for longwave irradiance are very small: the error based on the correlation is never more than 2 W m−2, which is probably within the accuracy of the pyrgeometer. The largest monthly error value that we derive is the upper limit applied to the April fluxes; this is just 3.45 W m−2.

Table 7. Upper and Lower Limits of LW Daily Irradiance Sampling Error, by Montha
  • a

    Unit is W m−2. Upper limits are based on flux differences, and lower limits are derived from correlations (ρ).


4.4. Variability in Emissivity (U5)

[48] This term is given by

equation image

The mean net flux at NIA is 103 W m−2 (with a scatter of 28 W m−2) so every 0.01 uncertainty in emissivity translates into a systematic error of 1 W m−2 or so in the LW uncertainty. The root mean square variation in emissivity can never be less than half the range in its values, and these sensibly lie between about 0.9 for bare rock and 0.98 for water. This sets an upper limit of about 0.04 to the 1σ error in emissivity (the real value will probably be lower still) and so an upper limit of about 5 W m−2 to this systematic component of the longwave sampling uncertainty. The true value (which contributes to the bias in our estimate of the SRB) is likely to be smaller than this.

[49] Satellite-derived estimates of emissivity are very rarely produced, although reasonable knowledge of spectral emissivity is required to estimate land surface temperature from satellite infrared measurements. To estimate the variability of emissivity in our study we took data from the global IR land surface emissivity database [Seemann et al., 2008]. This has a 5 km resolution, and is available as monthly global data. The statistics of these values for January and September 2006, for the GERB pixel containing the NIA station, are given in Table 8.

Table 8. Variability of Emissivity in the NIA GERB Pixel

[50] The variability of the emissivity, a scatter value of 0.003 or so, means that there is probably no significant contribution to the sampling error from this source. It could be that the emissivity varies more appreciably at scales below that of the 5 km data set, but that is beyond the scope of our investigation.

4.5. Variability in Blackbody Ground Flux (U5)

[51] For this we can reasonably assume the emissivity is equal to the area value of 0.935 from the land surface emissivity database (previous section) and just work with the variability of the blackbody flux, g:

equation image

This component of the SRB uncertainty is the most difficult to characterize with the observations from the AMF sites. In the first instance, we strictly speaking have no observations of the surface (skin) temperature at all, although at the main site an IRT, which covers the 8–13 μm atmospheric window, is used to calculate an approximation to the required value (see also N. A. Bharmal et al., Simulation of surface and top of atmosphere thermal fluxes and radiances from the RADAGAST experiment, submitted to Journal of Geophysical Research, 2008). Under clear skies the downwelling flux in this window is relatively small, and so the reflected contribution to the measured flux is the product of two small amounts and can be neglected. The temperature value determined still has some assumptions about the local spectral emissivity folded into it. These numbers are available only at the main site, so we can construct just one time series of g. We could use MODIS temperature retrievals, which give spatial images of temperature fields twice a day, with an 8 day product also defined, to try to characterize the spatial variability of g, but this must entail some assumptions about the actual shape of g as a function of time of day. The MODIS temperatures are more suited toward mitigating the effects of spatial heterogeneity for radiative transfer calculations at the appropriate time of day (Bharmal et al., submitted manuscript, 2008). These high-resolution maps of surface temperatures can only be derived in the absence of cloud cover, and assumptions still need to be made about the surface spectral emissivity. With these caveats in mind, we shall nevertheless try to estimate the variability of g from satellite data.

[52] Retrievals of land surface temperature can be made from a number of satellites using so-called “split window” algorithms [e.g., Price, 1984; Becker and Li, 1990; Sobrino and Romaguera, 2004], but for most the number of temperature maps thus obtained is woefully short of the number needed to make a decent estimate of the daily average of temperature's fourth power. SEVIRI, however, takes 96 sets of thermal images each day, once every 15 min, and the LandSAF (Land Surface Analysis Satellite Applications Facility, generates estimates of land surface temperature at each of these times, in a product that has 5 km ground resolution [Trigo et al., 2008]. Of course, cloud cover affects one's ability to recover surface temperature, and there are many gaps in this temperature product. Sometimes these are spatially very extensive, as when there is large-scale cloud cover or an image acquisition failed; at other times the dropouts are patchy. The cloud-clearing algorithm used at one stage in the product appears to be rather conservative, as there are extremely few days where a complete run of 96 temperatures are obtained over most of the area.

4.5.1. Clear Day of 16 December 2006

[53] One such clear day was 16 December, when 85 of the 90 pixels had full temperature sets of 96 values (the remaining 5, which are always missing in these data sets at this time, have probably been masked out as being “water” rather than “land”). The time series of g for temperatures retrieved for the 5 km SEVIRI sample containing the AMF was compared to the values of g derived at 15 min intervals using the IRT temperatures. The comparison of temperatures is shown in Figure 16 (left), and the match up of the corresponding blackbody flux is shown in Figure 16 (right). The value of g for this day is 464.6, calculated from the IRT temperatures. That for the 5 km cell containing the AMF is 460.9, a difference of 3.7 W m−2.

Figure 16.

(left) Surface temperature and (right) blackbody flux from the infrared thermometer at the AMF (solid line) and from SEVIRI retrieval (dots).

[54] The hour-to-hour differences seen may well be real of course, as the SEVIRI retrieval is generated for a 5 km by 5 km square area, while the IRT sees just a few square meters of soil. The SEVIRI temperature includes contributions from vegetation, but also from airport buildings and runways. For each of the 96 images the mean and standard deviation of σT4 were calculated, and are plotted in Figure 17 (the mean data are the solid black curve, the green curves correspond to the mean curve ±1 standard deviation and the values for the pixel containing NIA are shown as dots). The cell containing the NIA site lines up quite well with the mean curve of all the cells; the average daily flux (g) for the NIA cell is 460.9 W m−2 and that for the mean curve is 459 W m−2. When we calculate g for all of the 85 cells in the 16 December set, we find the standard deviation is just 3.2 W m−2 (the mean is still 459, of course). This suggests that the NIA cell is typical of the GERB footprint as a whole; the difference in g between the 5 km NIA cell and the GERB mean is just 0.6 of a standard deviation of the SEVIRI values.

Figure 17.

Mean and mean ±1 standard deviation curves of SEVIRI-derived σT4 and values calculated from the IRT.

[55] Whether the AMF data provide a good representation of what is happening in the NIA cell is a more subtle question, but the reasonable correspondence in g, despite the difference in the range of temperatures seen (Figure 16), suggests it is, at least for the daily averages we are considering here. We can get down to slightly better resolution by using the SEVIRI data themselves, and a very simple split-window estimate of the land surface temperatures:

equation image

where T10.8 and T12.0 are the brightness temperatures for the split-window SEVIRI channels, equation imageS is the estimate of LST, and a is a parameter (typically about 4 for split window algorithms). For any given value of a this allows us to generate a simulated series of surface temperatures, and hence of g, at the higher spatial resolution. Varying a from 2 to 6 makes little difference to the spatial variance of g, which is typically between 3 and 4 W m−2 for the set of SEVIRI pixels defining the GERB NIA pixel. The value of g evaluated for the 3 km pixel containing the AMF is within 0.2 W m−2 for the BAN pixel, but about 1 standard deviation different for the NIA pixel, that pixel being “hotter” by 3–4 W m−2 than the GERB area it represents. That is in the sense we would expect, as the albedo there is lower than the average, meaning more shortwave heating of the surface takes place. We will therefore assume that the area around NIA has values of g that are on average about 1 standard deviation greater than the mean of the corresponding GERB pixel.

4.5.2. Estimating the Variability of g on Less Clear Days

[56] Evidence from the clear day of 16 December 2006 suggests that the area around Niamey is fairly typical of the larger GERB area; blackbody fluxes of a 5 km area containing the AMF lie within one half to two thirds of one standard deviation of such subarea fluxes, and a 3 km area around is similarly typical of the 50 km × 50 km area around it (1 standard deviation difference). The match between the diurnal temperature plot at the AMF and the SEVIRI estimates of temperature show discrepancies, with the SEVIRI retrieval being hotter at midday and cooler at night; nevertheless, the blackbody fluxes represented by the two time series are less than 2 W m−2 apart. At a few other times in the year, where all 96 temperature estimates were available over most of the area, the variability in the daily average of σT4 was again relatively small. However, these occasions were in the dry season, and no useful complete sequences were available during the monsoon period. Extending the evaluation from the dry season is problematic, given that cloud cover may severely reduce the number of satellite LST estimates we have to work with.

[57] To extend the number of temperature sets significantly the following approach was taken. For any given day, for each of the 90 5 km by 5 km SEVIRI cells for which LST was generated, the number of error-free, cloud-free estimates was counted. If this was greater than a certain threshold, then any missing values would be estimated by interpolation, and the time average then taken. The number of true values employed is used to define a simple confidence parameter for the estimate: it is zero when the number of true values is at the threshold for acceptance, is one if all 96 temperature values exist, and otherwise varies linearly. For example, if the threshold is 60, an estimate generated from a set of 78 (halfway between 60 and 96) retrieved temperatures has an assigned confidence value of 0.5 for that pixel. This process gives up to 90 estimates of the daily average of σT4 across the GERB pixel: the standard deviation of these numbers is taken as the sampling uncertainty, and the average confidence parameter is calculated as an uncertainty parameter on that sampling uncertainty.

[58] Figure 18 shows the standard deviation in the daily average of σT4 around Niamey when the threshold for a calculation is that 60 of the 96 possible LST values should exist. Blue diamonds in this plot represent sampling error calculations for which the confidence parameter is higher than 80%. The red circles are for those that have confidence values less than 0.5, and the green squares represent the in-between values. During periods of lowest cloud cover, at either end of the year, this uncertainty is seen to be no more than about 3–4 W m−2. There appears to be a slight dependence on season of the spatial variability of g; the values corresponding to the blue symbols in the middle of the year being a little higher than at the beginning and end of the year, but we find that any calculations that suggest the scatter could be more than about 7 W m−2 are those in which we have least confidence. When we have most confidence in our calculation of sampling error, that error is smaller. This may reflect upon the interpolation method chosen, or it may indeed be the case that in cloud-free conditions, which is when we expect the confidence parameter to be high, there is little variation in the time average of σT4.

Figure 18.

Variability of σT4 around NIA, variation in time. The confidence parameter is related to the number of SEVIRI LST values (maximum 96) contributing to the calculation for pixels.

[59] However, there does seem to be a tendency for the variability of g to be slightly higher outside the dry season, and there are physical reasons to think that it should be so: surface temperature during the day depends on surface insolation, and the more spatially variable the latter is, the more spatially variable the g factor will be. If we examine the fluctuations (residuals about a smooth curve) in the daily values at NIA of g (from the IRT) and of the downwelling shortwave flux, we see a correlation coefficient of ∼0.54 for a statistical relationship of the form

equation image

for which the best fit suggests α ∼ 0.2. If this is typical of the coupling strength between the irradiance and the blackbody heat flux across the GERB pixel, then we might infer that on nonclear days there is an additional, random uncertainty on g which is about one fifth of the corresponding random uncertainty on SW↓, and so is limited to very small values in the dry season, and to no more than 3 W m−2 at the height of the monsoon. The fact that the relationship is a positive one means that the variations in g compensate variations in SW↓ as far as their contribution to the SRB is concerned. To establish this link more quantitatively we would need to model the surface energy budget across the GERB pixel, not simply the net surface radiation budget.

[60] Working from the SEVIRI data therefore suggests that the spatial variability in g is about 4 W m−2 in clear sky conditions, which is rather less than we might expect given variations in temperature that are evident at any one time. The SEVIRI pixel containing the BAN site appears to be completely typical for the surrounding area for the g function, just as it is for the albedo. NIA runs a little hotter, and a bias of about 4 W m−2 can perhaps be expected there.

5. Surface Radiation Budget at Niamey, by Month, and the Corresponding Sampling Error

[61] The uncertainties estimated for each month are tabulated in Table 9. The numbers represent the uncertainty for a day at random in the given month, and correspond to the AMF value, minus the corresponding area average, and units are W m−2. The numbers are presented in the form a ± b, where either a or b may be missing; a corresponds to systematic differences, while b represents the root mean square difference of a quantity whose systematic error is taken to be negligible. To recall: U1 gives that part of the total uncertainty associated with variations in shortwave irradiance across the area of interest; it is calculated by multiplying together the observed monthly average coalbedo and three quarters of the upper bound for the root-mean-square uncertainty of irradiance. U2 gives that associated with the difference between the albedo at the AMF and the required area average of albedo; for this we have assumed fixed albedo errors of −0.04 at NIA and +0.005 at BAN, and multiplied these by the observed mean SW irradiance. U1 + U2 gives the sampling uncertainty that is attached to the shortwave component of the surface flux.

Table 9. Contributions to the Sampling Errora
  • a

    Apart from albedos, all entries have units of W m−2 and estimate the sampling error for daily fluxes. Numbers preceded by ± give the standard deviation of the random component; other numbers are biases.

Jan0.2520.313±2.06±1.898.45−2.14±1.70±1.70−404.45 ± 2.26−2.14 ± 2.20
Feb0.2550.309±2.35±2.1810.04−2.56±1.86±1.86−406.04 ± 2.21−2.56 ± 2.07
Mar0.2660.317±3.44±3.2010.58−2.71±2.30±2.30−406.58 ± 3.20−2.71 ± 3.00
Apr0.2590.318±3.85±3.5411.45−2.92±2.85±2.85−407.45 ± 3.40−2.92 ± 3.15
May0.2340.304±7.80±7.5210.39−2.62±2.66±2.66−406.39 ± 7.89−2.62 ± 7.20
Jun0.2260.293±5.60±5.1210.07−2.60±2.32±2.32−406.07 ± 6.03−2.60 ± 5.57
Jul0.2220.269±9.12±8.579.34−2.38±1.64±1.64−405.34 ± 9.34−2.38 ± 8.79
Aug0.1960.240±12.00±11.358.87−2.24±1.63±1.63−404.87 ± 12.2−2.24 ± 11.6
Sep0.2000.225±6.89±6.679.06−2.30±2.65±2.65−405.06 ± 7.00−2.30 ± 6.79
Oct0.2260.240±4.77±4.6810.08−2.47±2.15±2.15−406.08 ± 4.66−2.47 ± 4.54
Nov/Dec0.2480.282±3.25±3.109.67−2.40±1.29±1.29−405.67 ± 3.10−2.40 ± 2.96

[62] U3 is from variations in longwave irradiance across the area of interest, and is calculated as three quarters of the upper bound multiplied by 0.935, the assumed area emissivity; for BAN, an assumed calibration correction of 7.5 W m−2 has been applied. U4 gives the uncertainty which arises from variations in surface temperature, based on SEVIRI land surface temperature products. This is the clear sky value, which is consistently around 4 W m−2. Any random element associated with this through varying cloud cover will be smaller that the uncertainty in shortwave irradiance, probably about 20% of that value and of the opposite sign; it has been ignored in Table 9. U5 is the uncertainty that which is associated with variations in emissivity; the spatial data sets of Seemann et al. [2008] lead us to conclude that this term is less than 1 W m−2, and so is not shown in Table 9. The sum U3 + U4 + U5 is the sampling error for the longwave flux. Finally, U gives the combined uncertainty from these contributions. Correlations that exist between the five terms are usually negative, an increase in one term being partly compensated for by a decrease in another. The calculated random component of U is therefore probably a conservative estimate of the true value.

[63] A plot of the daily values of the SRB at Niamey, downwelling LW + SW flux minus the upwelling LW + SW flux, is shown in Figure 19. In Table 10 the monthly mean SRB, and the scatter within the month, are shown together with the random component of the uncertainty (taken from Table 9). During the dry season months the SRB values are about 50 W m−2, and there is a steady increase from January to June as the solar irradiance increases. With the arrival of the monsoon, the values are generally held to about 100–110 W m−2, with occasional excursions to over 150 W m−2. The day-to-day variability is very high during the summer, having a standard deviation of ∼35–40 W m−2 for days in July, August and September. This variability is much greater than the uncertainty on the values.

Figure 19.

Net surface radiation budget at NIA, daily values and smoothed time series.

Table 10. Estimated Monthly Mean Surface Radiation Budget (Net Downwelling Minus Upwelling) for the GERB Pixel Containing NIA, the Day-to-Day Variation Each Month (1 Standard Deviation), and U, the Random Error on the Daily Value of the SRB
σ (SRB)

[64] The values correspond to the expected error for any given day. We might expect the random elements to be smaller if we were to average over a period of several days. No attempt has been made here to characterize the errors on anything other than daily data, but an idea of the benefit to be gained may be obtained by examining the autocorrelation of the irradiance data at short lags. Thus, for instance, for the months of January to April, 1 day's residual (daily average minus smoothed values) has a correlation with the next day's of about 0.5 (SW) and 0.7 (LW). This suggests that averaging over 3 days will reduce the random component of the LW uncertainty not by a factor of √3 (1.7), but by a little less than 1.15, and by about 1.28 for SW. In the months of July, August and September the correlation is actually negative (−0.3) for SW fluxes (the weather systems appear to have shorter time constants then) and averaging over several days will evidently reduce the random errors, which are dominant at this time, considerably. However, a complete study over different integration periods is beyond the scope of this initial attempt to characterize the sampling uncertainty.

6. Summary and Conclusions

[65] One of the principal aims of the RADAGAST project is to calculate the flux divergence across an atmospheric column in the Sahelian region of Africa. This is achieved by taking the difference between the net radiative flux at the top of the atmosphere, calculated from geostationary satellite data, and that at the surface, calculated using the instrumentation of the AMF, a new mobile facility of the Atmospheric Radiation Measurement program. The top-of-atmosphere fluxes correspond to a surface footprint of about 50 km square, but the surface radiation budget is obtained effectively at a single point. The uncertainties in using these point fluxes to represent the required fluxes over the larger area have fixed and variable components. The fixed errors result from heterogeneities in surface albedo, emissivity and temperature, and the random component from variations in the atmosphere, principally cloud cover. The fixed sampling uncertainties have been estimated from a number of surface data products, and it has been possible to estimate the random components by using a second set of surface flux measurements located at some distance form the main AMF deployment. Consideration of the differences between the downwelling fluxes, and their correlation, allows us to put upper and lower bounds to the uncertainty (root mean square difference between the point and large-area average) of the irradiance contributions. The upper bound is generally about twice that of the lower bound, and we have used the mean of these two as a representative estimate.

[66] Table 9 shows that the biggest contributions to the sampling error are from the variability of shortwave irradiance during the monsoon period, and the systematic albedo difference during the dry seasons. The greatest random uncertainty in the SW irradiance is realized in August, reaching 11–12 W m−2, and drops to less than 5 W m−2 during the dry season. The systematic component is between 4.5 and 7.5 W m−2, the energy absorbed at the AMF being higher by this amount than across the area as a whole. Longwave irradiance is much more uniformly distributed across incoming directions than shortwave irradiance, and the instantaneous pyrgeometer flux is already a smooth average. The shortwave irradiance variations are generally dominated by whether or not there is a direct beam, i.e., whether the sun is instantaneously obscured by cloud at the measurement site. Thus the uncertainties associated with LW irradiance are almost always smaller than those for the shortwave, although our analysis of this has been somewhat hampered by an apparent calibration problem. Surface temperature variations across our main GERB pixel can be several degrees near midday, as the MODIS temperature map (Figure 1) shows. However, thanks to a unique land surface temperature product generated from SEVIRI, we have seen that there is evidently some compensation in that areas hotter at midday may be colder at midnight, and so the average emitted flux, integrated over the day, shows less relative variation than may be implied by a temperature map generated at a single time of day. The location bias in emitted longwave flux appears to be no more than about 4 W m−2, and offsets some of the shortwave bias arising from the albedo.

7. Discussion

[67] The difference between the SRB observed at Niamey and the average across the corresponding GERB footprint has, when we deal with daily fluxes, a fixed element of the order of 6 W m−2, and a variable element whose root-mean-square value can be over 10 W m−2. Some simple strategies could reduce these uncertainties. The sampling uncertainty in the SW daily irradiance is greatest in August, when it is as high as 11–12 W m−2. One way to reduce this uncertainty is to calculate the flux budget not on a daily basis, but over an extended period, as the SW irradiance on successive days is anticorrelated (ρ ∼ −0.3), so that the effect of averaging over n days is to reduce the uncertainty by more than a factor of √n (how much more depends on the autocorrelation at lags of more than 1 day). The simplest way to characterize this better is probably to work with irradiances averaged over the appropriate time period, and repeating the comparison of the differences and correlations at the two sites to bound the sampling variance. Another way to reduce the uncertainty during the monsoon would be avoid making SRB estimates on days where the AMF irradiance is likely to be very different from the average over the GERB footprint. For example, in August there are 10 days on which the observed flux difference between the two sites was at least 29 W m−2, and all these numbers have been included in the statistics. On such days the large difference between the two sites presents us with a priori evidence that the daily average SW irradiance varies significantly over a distance comparable to the ARG footprint. Similarly, days with heavy rainfall are also included, as the only criterion for inclusion in the statistical calculation was that we had an all-day flux measurement at each of the two sites. We might be circumspect about making calculations of the daily divergence when we have such indications of extreme variability. Nevertheless, even under these more difficult conditions, the uncertainty in the SRB for a single day is smaller than the scatter in the daily values, so that the daily SRB, once corrected for albedo bias, is accurate enough to be used with top-of-atmosphere fluxes for flux divergence calculations.

[68] The systematic albedo error could be vitiated to some extent if we were to combine the downwelling shortwave flux with a suitable mean area albedo. At the moment, the MODIS product appears to represent very well the spatial variability in the albedo, but there are sufficient differences between it, and the observed albedo at the AMF, and their covariation through the year, that require further attention. A possibility for studies at Niamey would be simply to increase the observed albedo by 0.04, which would correct for most of the systematic error. Further development of this study should concentrate on characterizing the errors over time scales other than daily, and on establishing a correction for the albedo effect. The high-frequency fluctuations in albedo that are evident during the monsoon season at the surface stations are not picked up in the MODIS product, and albedo products from SEVIRI, which can capture more rapid changes, should also be considered.

[69] Measurement of a physical variable only has scientific value when we know the uncertainty associated with that measurement. Even if the precision and accuracy of such a measurement should be high, we need to exercise caution if it is being used as an implicit surrogate for some other quantity. The flux measurements at AMF sites are of the highest quality, and give very accurate values, but have limited spatial support. Treating them, as we do in RADAGAST, on an equal basis to satellite data that cover much wider areas leaves us with a tricky sampling issue, only part of which can be tackled by characterizing, and compensating for, the heterogeneities of the land surface within that area that affect our estimate of upwelling fluxes. In our particular case a second set of ground flux measurements was found to be invaluable in helping to understand and bound the likely estimation error for the downwelling fluxes. It is unlikely that the presence of a third or fourth station would help much more, beyond enabling us to test some of the assumptions about stationarity that were used in deriving the lower bound of our sampling error variance of the irradiance fields. Most of the benefit comes from just the first additional station, and the relatively small additional cost of a secondary site during such an ambitious project seems a small price to pay for the ability to place an error bar on our flux calculations. The use of satellite data to characterize spatial variability is not new, but we think the use of paired sets of flux measurements to bound sampling errors is a new, if simple, technique that should find application in many similar studies. Thanks are due to the ARM team for their farsightedness in including this additional resource in the RADAGAST project.

Appendix A:: Data Smoothing

[70] Data were smoothed in this work for two reasons. One was to estimate the correlation in surface irradiance at the two sites. Simply taking the raw correlations over a longish time period, that is, comparing the daily difference between observed and long-term mean values, gives misleadingly high correlation values for shortwave flux, as the length of day and the maximum elevation of the sun mean that irradiances at the sites are high together and low together for that reason alone; this tells us nothing of how we might expect the irradiances to vary spatially over a more limited time period. Smoothing those time series, and dealing with the differences between the smoothed value and the observed values, gives more useful results as it eliminates the astronomical effect. A running average could be done for this purpose, provided missing days' values were first interpolated; special treatment might be needed for the beginning and end of the time series. The other time we smooth data is when calculating the spatial variation in the daily average of the blackbody ground leaving flux, σTs4, where Ts is the instantaneous surface temperature. For any given day, and for any given SEVIRI sample contributing to the GERB footprint, there may be from 0 to 96 estimates of surface temperature at 15 min intervals. To generate some numbers to work with, we needed an interpolation method that could fill in extended gaps as well as the occasional dropout points. The smoothing method we outline below was used for both purposes.

[71] We assume we have a set of data points {xi}, i = 1,2.N, and a set of weights, {wi} describing the confidence we have in the value (for the work described here, these were just unity if we have a value, and zero if the value is missing. However, the SEVIRI temperature values came with a confidence value, which we could have used). We want to find a smooth curve through these points, that is, we look for a set of numbers {si} that minimizes a weighted mean square difference between them and the data values, while retaining certain smoothness properties. Specifically, we want to find s values to minimize:

equation image

Subject to

equation image

for some k, not known a priori. The smaller k is, the closer the s values must be to a straight line through the data. The problem is a fairly standard one, solved with the aid of a Lagrange multiplier λ. We look for the unconditional minimum of

equation image

yielding (after differentiating with respect to each si in turn, and setting the results to zero):

equation image

where W is the diagonal matrix whose ith diagonal element is wi, s and x are vectors whose ith elements are si and xi respectively, and S is a symmetric 5-diagonal matrix. S has the form:

equation image

Note that the elements in each column sum to zero, therefore if j is a vector of 1s, we have

equation image

so that the operation “conserves mass”; the weighted sum of the smoothed values is the same as the weighted sum of the data values.The system of equations (A1) can be solved efficiently for s, as λS + W is a band diagonal matrix. The smoothed series were used to define the residuals (sx) used for evaluating correlations, and to interpolate missing values in the daily series provided by the SEVIRI land surface temperature product. In the examples shown in this paper, where each of the numbers xi tends to be of the order 200–400, acceptable results were obtained when λ was taken to be 10−4 (acceptable in that values of 10−3 or 10−5 gave virtually the same results).


[72] Several rounds of thanks are owed to the Atmospheric Radiation Measurement program: first, for providing the Mobile Facility and managing to keep it running almost continuously for over 12 months, often in difficult conditions, and further, for the rapid availability of the data and the deceptive simplicity of its distribution through a most excellent web interface. Discussions with Mark Miller and Chuck Long were of enormous benefit in understanding instrumental errors. Ben Johnson and Jim Hayward of the UK Met Office kindly made available to us the albedo data used for Figure 15, acquired during the DABEX campaign. Special thanks to Chrystal Schaaf and the MODIS team at the university of Boston for generating the version 5 albedo data for Niamey ahead of its scheduled upgrade. Finally, the careful work of anonymous reviewers has helped us to eliminate a number of obscure passages and to improve significantly the final submission.