## Abstract

The authors examine the temporal variability of measured 415- and 611.7-nm spectral solar irradiance in cloudy stratocumulus conditions. This is accomplished by normalizing measured data by the equivalent irradiance for cloudless conditions and the same solar zenith angle. Spectral and other analyses of the time series exhibit fractal behavior in agreement with the multifractal model of Schertzer and Lovejoy. A three-dimensional cloud model with dimensions of 6 km × 38 km is constructed that has these fractal properties, and a Monte Carlo radiative code is applied to obtain irradiances in 50-m grid elements at the surface. Model output is used to test the ability of satellites to calculate hourly irradiance with one, two, three, four, five, and six observations per hour. Root-mean-square errors are substantial—between 17% and 43% for one single satellite observation per hour. The smallest errors of around 5% are obtained with six scans per hour. These results argue that a higher frequency of satellite observations is needed to estimate hourly surface solar irradiances with acceptable accuracy.

## Introduction

Modeling downward surface solar radiation fluxes in cloudy environments is a considerable challenge at short time scales. Yet, there is a clear need for this information to understand boundary layer phenomena occurring at intervals from 10 min to several hours. Applications such as evapotranspiration estimates for irrigation scheduling, boundary layer growth models, and modeling synoptic weather systems all rely on well-defined radiation and surface energy budgets at scales matching those of the processes being studied. Clouds, being the dominant atmospheric depletion agent for solar radiation, introduce high spatial and temporal variability in the radiation fluxes at short time scales, therefore, making the modeling task much more difficult (Long and Ackerman 2000).

One well-established approach is to use satellite data to provide information on cloud characteristics that can yield surface radiation fluxes. Validation is performed by relating pixel radiance as seen by the satellite to surface solar radiation flux measurements in the same pixel (Pinker and Laszlo 1992; Pinker et al. 1995; Bishop et al. 1997). Nevertheless, problems remain at short time scales.

Figure 1 presents published rms errors for 21 studies in the last 20 yr, employing satellite data to estimate surface solar radiation fluxes. Averaging periods are plotted in the *x* axis and standard errors are in the *y* axis. Both physically based and statistically based models perform equally poorly when hourly estimates, based on one satellite pass per hour, are compared with measured hourly data (see also reviews by Sellers et al. 1990; Schmetz 1991; Nunez and Kalma 1996). Clearly, the regional average satellite pixel radiance does not encompass all of the temporal variability in the measured surface solar radiation fluxes using surface pyranometers. This is partly due to the scale mismatch between the regional view of the satellite and point measurements of surface radiometers (Li et al. 1993), but is also due to variability in cloud properties (Stephens 1988; Boers et al. 2000; Szczap et al. 2000). For various reasons, these short-term results produce an rms error of 20%–30% or 25–50 W m^{−2}, which is quite unacceptable for climate work where concern is for a typical radiative forcing of 4 W m^{−2} resulting from a doubling of atmospheric carbon dioxide (Houghton et al. 2001).

Figure 1 suggests that the structural properties of clouds are somehow adding to the overall error at short time scales. To examine this feature we use the considerable body of work that argues that cloud properties behave as fractals (Albers et al. 1999; Cahalan et al. 1994; Fienberg and Nunez 2002; Szczap et al. 2000; Romanova 1998). In this study we use a multifractal model to examine the effects of cloud spatial structure on surface solar radiation fluxes. From surface observations we present temporal radiometric measurements taken under stratocumulus cloud, and show that derived statistics of the surface radiation field are totally consistent with fractal behavior. Last, the fractal model is used to examine errors associated with different frequencies of temporal observations that are potentially obtained from satellite data.

This study will deal exclusively with the temporal averaging issue as it is applicable to satellite estimation of surface solar radiation fluxes. In other words, the assumption is made that the satellite offers a perfect method of estimating the radiation fluxes under its instantaneous field of view. Errors then occur because the satellite is incapable of observing the cloud field continuously. A separate but subordinate problem not addressed in this study is the accuracy of the retrieval process and the radiation parameterization.

## Cloud and radiation modeling

Much like atmospheric turbulence, fractal clouds are characterized by the fact that their general structures and level of detail are unaffected by changes of resolution, which is to say that they are scale invariant. This is not altogether surprising because cloud formation is dependent on convective processes in the lower atmosphere. As in other turbulent processes, a fractal cloud transfers energy from large scales to smaller ones in a cascade behavior, but with added complications. These may arise from such processes as inertial behavior in droplets, evaporation, condensation, sinking, and many others. Therefore, the radiation field in cloudy conditions may be viewed as turbulent, but with a loose link to classical turbulence behavior in the atmosphere.

Given the above considerations, The Fractionally Integrated Flux (FIF) multifractal model was used in this study. It was developed from turbulent velocity cascade processes (Schertzer and Lovejoy 1987, 1991, where the properties of the governing equations lead to a cascade process of energy transport from large to small scales. Model details are given in appendix A.

There are three numbers that totally define the fractal field in the FIF model. These are popularly addressed as *H*, *α*, and *C*_{1}. The first term *H* is linearly related to the slope of the power spectrum of the field (Lovejoy et al. 2001). Additionally, when *H* is positive the field is nonstationary. The typical behavior of *C*_{1} and *α* is shown in the simulated fields of Fig. 2. Figure 2a shows the effect of varying *α* while *C*_{1} and *H* are constant, while Fig. 2b shows the result of varying *C*_{1} while the other parameters are unchanged. In both cases the fields are vertically offset for ease of viewing, and the seed of the random number generator was kept constant so that the difference between the series is due only to the different parameters. This shows that *C*_{1} is the measure of intermittency or mean inhomogeneity—as *C*_{1} increases, the average departure from the mean increases, with each peak in Fig. 2b becoming larger. A totally homogenous field would have *C*_{1} = 0. By contrast, Fig. 2a demonstrates the nature of *α* in determining the distribution of departures of the mean—as *α* increases (with *C*_{1} constant) there are fewer peaks, but those that are there are higher.

The model was used to analyze liquid water in stratocumulus clouds collected from 98 aircraft flights in Tasmania, Australia. Details are given in appendix B, and the properties of the liquid water collected gave fractal numbers of *H* = 0.3, *α* = 1.47, and *C*_{1}= 0.108, indicating a prevalence of high peaks (*α* = 1.47), fluctuations with low departures from the mean (*C*_{1} = 0.108) and nonstationarity (*H* = 0.3). The fractal properties were used to generate model clouds with cells of 50-m side length and 0.5-km thickness, with a base of 1.0 km above the ground and horizontal dimensions of 6.4 km × 38.4 km. These dimensions were chosen to represent the typical passage of a regional cloud structure above a sensor in a period of 1 h. Monte Carlo radiative transfer calculations were performed on the model cloud, and surface solar radiation fluxes were estimated at a grid resolution of 50 m, with solar zenith angles of 30° and 50° [see also the paper by Kuchinke et al. (2004) for model validation]. These datasets formed the basis of the subsequent analysis. Appendix C provides further detail of the radiative modeling.

As an example of the radiation model output, Figs. 3a and 3b show the liquid water and resultant transmission, respectively, for a 50 × 50 subset of one realization, using a mean liquid water content of 0.1 g m^{−3}. The transmission Tr is defined as the ratio of the measured cloudy irradiance *K _{C}* to that for cloudless conditions

*K*

_{0,}measured under the same zenith angle:

Notice the high transmission value of over 0.5 in the background, corresponding to the low vertical liquid water path of Fig. 3a. Low transmission values of less than 0.2 equate to the high liquid water peaks seen in the foreground of Fig. 3a. Note that these peaks are consistent with fractal behavior and that *α* is equal to 1.47 (see also Fig. 2). It is interesting to note that unlike the pattern seen in liquid water, transmission peaks are damped. This is likely to be a result of multiple scattering within the cloud structure.

Optical depths for this model cloud gave values equal to 0.5 as indicative of a clear-sky threshold (see appendix C). Values greater than this implied an effective cloud cover. Note that we do not explore the relationship between the fractal model and stratocumulus cloud cover in this paper. We simply choose a mean optical depth that is representative of the cloud, then generate a certain mean liquid water path and distribute it according to the measured fractal parameters. As a result, some areas have very little cloud and can be thought of as effectively clear. It is also interesting to note that in other areas of the domain (not shown), marked by partly cloudy conditions, transmission values exceed 1.0 in response to downward photon reflection from cloud sides.

## Radiometric measurements of cloud temporal structure

The approach followed here is to observe Tr using radiometric measurements under stratocumulus clouds. Any temporal change in Tr will mostly result from the changes in cloud optical properties as the cloud moves over the fixed radiometer location. To estimate Tr we have used data from a Multi-Filter Rotating Shadowband Radiometer (MFRSR) (Yankee Environmental Systems), which is presently operating at our field site on the roof of the School of Geography and Environmental Studies at the University of Tasmania, in Hobart, Australia (42.90°S latitude, 147.33°E longitude). Two channels centered at 415.0 and 611.7 nm, respectively, are examined, with each being 10 nm in bandwidth. The 611.7-nm channel is close to the midpoint of the satellite visible channel (550–750 nm), which is used in the estimation of surface solar irradiance (Nunez and Kalma 1996). It is largely free from water vapor and ozone absorption bands (Valiente 1996), although there is some absorption by ozone. By contrast, the 415.0-nm channel is totally free of absorbers and, as a result, it is also used in this analysis for comparison purposes.

The first step involves the calculation of *K*_{0} for both channels. Three cloudless days in August and another three in November are examined. They have been checked for consistency in being able to produce a very similar cloudless irradiance value as a function of solar zenith angle. The following statistical relationships are obtained for the two channels when pooling these data together:

Thus, it is possible to calculate *K*_{0} for any period in the 4 months, provided that the solar zenith angle is known. Spectral transmissions (Tr) are obtained by taking measurement from the MFRSR every 3 min and dividing by *K*_{0,} using Eqs. (2a) or (2b). While this sampling rate is not fast, it is considered appropriate because fluxes change much more slowly than does radiance, even under clouds.

Ten separate Tr series have been selected in July and August 2002, and another nine in November 2002 with cloudy stratocumulus skies. These correspond to winter and late spring conditions, respectively, in Tasmania. Of these 19 series, 8 are overcast (8 octas) and the other 11 represented partly cloudy conditions (2–7 octas). Cloud cover data have been obtained every 15 min using a digital sky camera also installed at our field site. In selecting these series, care has been taken to ensure that cloud cover was consistent and did not vary consistently during the measurement sequence by more than 1 octa. Measurements were terminated when cloud cover exhibits a significant and consistent change. Out of the 19 series, 15 are over 5 h in length, equating with over 100 spectral data points per channel at a measurement interval of 3 min. The lowest number of elements in a run was 66, with data collected on 13 July, and representing a run of 198 min in duration.

Spectral transmissions as in Eq. (1) are estimated for all selected data. A fast Fourier transform is then applied to each transmission series, and power spectra were plotted against the natural logarithm of frequency. To eliminate some of the noise, the data output is averaged in frequency intervals that increase logarithmically (i.e., <0.04, 0.04–0.08, 0.08–0.16, 0.16–0.32, etc.). The final step involves normalizing all of the data series by taking differences from the mean, thus, allowing data from different time periods to be plotted together:

and the angle brackets denote a time average for the series.

Figure 4 presents the results for the two bands examined. First, the relationship is linear in all cases, which argues that the transmission statistics behave as fractals. That is, substitution of higher or lower frequencies in the regression equations of Fig. 4 will only change the logarithm of power in a linear fashion, indicating similarity in the processes operating at various scales. Figure 4a shows that the slope of the power spectrum for the 415 transmissions and partly cloudy conditions is −1.52 ± 0.07. For overcast conditions it is −1.45 ± 0.08, which is within the statistical error of the partly cloudy figure. Figure 4b shows the 611.7-nm power spectra for both the same partly cloudy and overcast conditions as in Fig. 4a. The slope of the partly cloudy and overcast condition, at −1.52 and −1.48, respectively, are also statistically similar to each other and to the 415 transmissions.

To test further the multifractal behavior of the measured cloud transmittance data (Tr), and to determine the parameters of the multifractal model, we use the Double Trace Moment (DTM) method (Lavallée et al. 1991; Tessier et al. 1993). Values for *α*, *C*_{1,} and *H* are given in Table 1 for the 415-nm transmittance, the 611.7-nm transmittance, and the liquid water obtained from aircraft measurements of stratocumulus clouds.

Results give similar fractal values for all three cloud types, with *α* occupying a range between 1.49 and 1.51, *C*_{1} varying between 0.110 and 0.135, and *H* varying between 0.27 and 0.37. These values of *α* from our previous discussion indicate a substantial number of smaller peaks departing from the mean, but with occasional high peaks (Fig. 2a). The *C*_{1} value is associated with a rather low departure from the mean (Fig. 2b), and the positive value of *H* indicates nonstationarity.

These fractal numbers also show agreement with the liquid water data taken from the aircraft. Marshak et al. (1995) used Monte Carlo radiative transfer calculations for a model fractal cloud to show that both transmittance and reflectance have the same power series slope and the same fractal scaling statistics as that of cloud optical depth. Similar results for model fractal clouds have been reported by Davis et al. (1991) and Naud et al. (1996). Nevertheless, to our knowledge no direct transmission measurements similar to ours have been reported. Although it is possible to argue that the radiation transmission field as defined in this work is related to the liquid water field, the relationship might be quite complex and nonlinear to the extent that close agreement in fractal numbers may be unjustified. On a qualitative basis, the two data fields indicate fractal behavior, a marked degree of nonstationarity, and substantial departure from the mean. Further analysis linking the two datasets is beyond the scope of this study.

Of interest in remote sensing is the nonstationarity of the radiation field, implying that short-term averaging is not meaningful. This point can be further illustrated by examining Tr for 2 days separate from the series shown in Fig. 4. The first day, 29 July 2002, recorded Tr under mostly overcast conditions as observed by the cloud camera (Fig. 5). The 411.0-nm channel records a highly variable transmission from just over 1.0 at the beginning of the measurement sequence to values under 0.5 for most of the morning and increasing again in the afternoon. Further insight may be gained by examining the cumulative mean calculated from the beginning of the sequence. For illustration purposes, the mean of the sequence has been normalized to 1.0. As may be noticed, the cumulative mean approaches 1.0 only near the end of the time series, after 1575 LST. Therefore, a representative mean is only obtained after averaging throughout an interval close to the entire sequence. This is one prominent feature of nonstationarity. Another typical pattern of nonstationarity is exhibited on 10 August 2002, under mostly 5-octa conditions. Both daily transmissions and the cumulative mean show a gradual increase throughout the day.

Theory also argues that fractal behavior is maintained at shorter time intervals. This is illustrated by selecting 1-h averages—a typical interval over which satellite-derived radiation fluxes may be desired. Cumulative 1-h means are investigated for the same 2 days as used in the previous section. For illustration purposes, hourly fractions are examined when the cumulative mean reaches to within 5% of the hourly mean. Table 2 shows the result. There is a wide variability in the hourly decimal fraction. For the partly cloudy day, 10 August 2002, it ranges from 0.35, obtained between 12 and 13 h, to 0.75, obtained between 9 and 10 h. The variability is even larger for the overcast day, 29 July 2002. This ranges from 0.05 for 11–12 h to 0.88 for 13–14 h. Therefore, nonstationary behavior is maintained even at hourly intervals.

## Implications for satellite modeling of surface solar radiation fluxes

In this section we examine typical sampling errors associated with satellite estimates of surface solar radiation fluxes on a model cloud using the cloud and radiation parameterizations described in section 2. The cloud field is assumed to be “frozen”—unchanging in form but moving over the ground at a rate of 10.55 m s^{−1}. Statistics are built from the Monte Carlo radiative model output for 68 multifractal cloud realizations: 34 with a solar zenith angle (SZA) of 30°, and 34 with an SZA of 50°. The Monte Carlo radiative transfer calculation produces a value for cloud transmission in each 50 m × 50 m grid square in the 6 km × 38.4 km system. The first step in the analysis involves 20 × 20 averaging of the radiation matrix to simulate a satellite resolution of 1 km. Rounding off by eliminating some cells at the edge, a 6 × 38 matrix is obtained, moving at a speed of 10.55 m s^{−1} above the earth’s surface (Fig. 6).

Averaging all of the elements in a column *i* would essentially represent the true hourly surface transmission located in a 1 km × 1 km grid in the same column. We shall denote this value as *T*_{0}* _{i}*, where 0 denotes an average transmission and

*i*refers to any column between 1 and 6. The objective is to compare

*T*

_{0}

*—the true hourly average transmission for column*

_{i}*i*—with an average transmission for the same column, based on a limited number of satellite-derived samples (see Fig. 6).

Six different sampling frequencies are selected for one, two, three, four, five, or six observations per hour. An hourly surface solar radiation flux estimate is obtained for each sampling frequency, and this value is then compared with the true value *T*_{0i}. All possible combinations are sampled for each observation frequency and an rms error is calculated between the satellite estimate and the true estimate. That is,

where RMS_{i} represents the rms for column *i*, TF* _{ij}* is the average transmission obtained at a frequency

*F*, column

*i,*and with the first sample occurring at row

*j*. For example, for

*F*equal to three observations per hour, column

*i*, and

*j*varying between 5 and 7, we would have (see also Fig. 6)

and the row spacing is equidistant and corresponds to the frequency of hourly sampling. When *j* reaches the column end, the spacing sequence wraps around and continues at the bottom of the column. Thus,

The final step involves normalizing the rms by division by the true estimate *T*_{0}* _{i}* and averaging over all six columns [Eq. (7)]:

Figures 7a and 7b present the results for the 68 cases—SZA 30° and 50°, with 34 series for each. The error range is highest at a sampling frequency of one per hour, with mean rms errors of 0.29 and 0.32 for the 30° and 50° zenith angles curves, respectively. The range of rms values is also highest at the 1-h sampling, with a standard deviation of 0.06 and 0.07 for the 30° and 50° curves, respectively. It is interesting to compare this result with model values reported in the literature for the 1-h prediction, as shown in Fig. 1. The two error ranges are in close agreement and further substantiate the fractal nature of stratocumulus clouds considered here. Errors drop substantially at higher frequencies, with a range of 27%–10% for 2 h^{−1}, 19%–6% for 3 h^{−1}, 11%–3% for 4 h^{−1}, and 6%–2% for 6 h^{−1}. At the highest frequency, the span of all six series is very small, only 4%, indicating that errors are consistently small at this frequency.

It may be argued that spatial averaging may improve the rms error when single hourly satellite scans are used. While spatial averaging is not specifically examined in this temporal paper, it is appropriate to mention that there are two considerations if spatial averages are done. First, there is the need to examine how well hourly fluxes are estimated from a single satellite scan as compared to the true spatial average value over the hour. A second point that is just as important deals with ground truthing the satellite-derived spatially averaged fluxes using pyranometers. In fact, errors arise not only from inadequate sampling of the temporal field, as in Fig. 7, but also from the assumption that one or more pyranometer measurement may properly represent an hourly surface irradiance in an area of several square kilometers.

To illustrate this point, six radiation series from the dataset in Fig. 6 have been selected for analysis—three series with a zenith angle of 30°, and three with 50°. As before, each 50 m × 50 m cell was assumed to properly represent the transmission. These cells were spatially averaged into 100 × 100 arrays, thus, representing 5 km × 5 km averaging. The center point of the array was varied along the long *j* axis to encompass all of its 760 (50 m × 50 m) elements, and averages were calculated for all 760 locations of the array (Fig. 6b). As in the previous example, data were “wrapped around” when the upwind boundary was exceeded, with the missing data being substituted by data from the downwind boundary. We calculate two relevant statistics called RMS_{1} and RMS_{2}:

where *T*_{j} is a mean transmission for the array with its center point located in the *j*th row, *T _{jk}* is the transmission for an individual cell belonging to

*T*

_{j}, and

*T*

_{m}is the true average of all 100 × 760 cells (5 km × 38 km). RMS

_{1}then measures the average within-array variability while RMS

_{2}measures the variability of individual

*T*

_{j}values. In other words,

*T*

_{j}in Eqs. (8) and (9) represents a single spatially averaged transmission over the hour, while

*T*

_{m}represents the true hourly average value that is required. Table 3 shows the results.

As may be observed, spatial averaging lowers the rms error of the array transmission (RMS_{2}) when compared with using a single 1-km pixel. However, within-pixel variability (RMS_{1}) increases, meaning that use of a single pyranometer in the 5 km × 5 km array will create added rms errors. Of course, RMS_{1} may be lowered using a comprehensive pyranometer network, but this is impractical. Therefore, spatially averaging the satellite signal lowers slightly the rms error, but not to a satisfactory level, and additional errors in validation of the spatial signal are introduced.

## Discussion and conclusions

This study has examined the variability of surface solar radiation fluxes at short time scales in cloudy stratocumulus conditions. Using stratocumulus clouds, we have examined measurements of spectral radiation transmission, aircraft-derived cloud liquid water, and output from a Monte Carlo radiation model of a fractal cloud. Multifractal parameters from the experimental measurements (Table 1) all show low mean inhomogeneity (low *C*_{1}), highly intermittent areas with values a long distance from the mean (high *α*), and nonstationarity (*H* > 0). It is also interesting to note that there is no significant difference between the statistics under broken cloud and those in overcast conditions, which seems to imply that it is possible to treat these as a single ensemble.

Results from the fractal model showed that rms errors are substantial, between 17% and 38% of the estimated irradiance for a single satellite observation per hour, and decrease to 5% for six observations per hour. No amount of improvement to satellite retrieval methods will yield better results for a single satellite observation per hour given the nonstationary nature of the fractal cloud field. It is at six satellite observations per hour, where the rms error approaches the instrument accuracy, that model improvements may be directed since basic information on cloud variability has been obtained.

The target objective of 5% error at the hourly time scale is clearly not met with our present operational satellites, but the outlook appears bright. Polar orbiting satellites, such as the National Oceanic and Atmospheric Administration (NOAA) series, are not useful because they scan the earth’s surface approximately once daily using the Advanced Very High Resolution Radiometer (AVHRR) (Rao et al. 1990). Geostationary satellites give the highest scanning frequencies twice hourly, but they may be programmed to cover smaller areas at higher frequencies (Komajda 1994; Tanahashi et al. 2000). Higher frequencies are planned for Geostationary Operational Environmental Satellite (GOES)-R (Schmidt et al. 2003), and presently the Geostationary Earth Radiation Budget (GERB) instrument aboard the new generation of Meteosat satellites provides data every 15 min (Lopez-Baeza et al. 2003).

Given the above limitations in satellite data, there is very little published corroboration of the results presented here. One exception is the study of Tanahashi et al. (2000), who used the *Geostationary Meteorological Satellite* (*GMS*)*-5* data in a dedicated mode to show that there is marked improvement in the estimation of hourly solar radiation when the frequency of satellite scans is increased from one to three per hour—rms errors decreased from 19.6% to 16.8% when using a 4 × 4 pixel average around the surface pyranometer.

The work presented in this study is preliminary in nature and much work remains to be done to assess the short-term variability of solar radiation in cloudy conditions. Transmission in other spectral bands and for other cloud types must be examined. While radiation flux calculations under stratocumulus clouds appear large, they may be small by comparison with the more active cumulus or cumulonimbus cloud systems in the atmosphere. Further complications occur in that stratocumulus clouds may undergo diurnal variability in fractal behavior (Cahalan et al. 1994), and it is possible that seasonal changes might be occurring as well. Until these issues of high-frequency radiation estimation are resolved, we will continue to have an important gap in our understanding of surface climates at short time scales.

## Acknowledgments

This project was supported by a grant from the Antarctic Science Advisory Committee. Ms Moya Kilpatrick typed the manuscript. The authors are grateful for comments made by three unknown reviewers.

## REFERENCES

### APPENDIX A

#### Multifractal Cloud Model

In this process, the (mean) energy flux is conserved with change of scale, but the velocity fluctuations decrease with decreasing resolution (Kolmogorov 1949). Because the cloud spatial structure is created through the transport of liquid water by the velocity field, this should form a similar cascade of liquid water content. As with the velocity cascade, there is a flux field related to liquid water content, which is canonically conserved with a change of resolution. The relationship between this flux *ϕ* and the liquid water concentration *ρ* is

where *l* is the length of the grid at the current scale and Δ*ρ* = *ρ*(*x+l*) − *ρ*(*x*) (Schertzer and Lovejoy 1987). Because the ensemble average of the flux 〈*ϕ*〉 is constant for all values of *l*, the scalar parameter *H* determines how much the fluctuations in liquid water content vary with changes of scale. By the relationship *β* = 1 + 2*H*, *H* is related to the slope *β* of the power spectrum of the field (Lovejoy et al. 2001). For a positive value of *H* the liquid water field *ρ* is nonstationary, while the field *ϕ* is always stationary (Davis et al. 1994). This implies that it is often easier to work with the flux field as opposed to directly using the density field.

The flux can be described by its probability (Pr) distribution, given by the relationship

where *c*(*γ*) is known as the codimension and *λ* is the scale relationship of the series, defined by *L*_{max}/*l*, with *L*_{max} being the total length of the data series and *l* the current resolution. The function *c*(*γ*) is a scale-invariant way to describe the probability distribution and can be used to find the probability of obtaining *ϕ _{λ}* greater than a threshold value. According to the FIF model, multifractal fields have underlying conserved fluxes

*ϕ,*with the following codimension functions and universal scaling exponents (Schertzer and Lovejoy 1991):

with 1/α + 1/α′ = 1. Therefore, the two parameters *C*_{1} and *α* in Eq. (A3) fully define the statistics of the conserved flux field in the FIF model. The liquid water field can then be found from the known flux using Eq. (A1). Hence, there are three scalar parameters—*H*, *C*_{1}, and *α*—that fully define the multifractal fields in the scheme.

To simulate fractal fields described by the FIF model, the numerical algorithms of Pecknold et al. (1994) were used [see also Naud et al. (1996) for a summary of the generation method]. This involved first generating a random series of numbers drawn from a Lévy distribution (Feller 1971), then applying a Fourier filter to ensure the correct scaling behavior described by Eq. (A2), and finally taking the exponential. The result of this process is the flux field *ϕ*. From Eq. (A1) it can be shown that a power-law filter of order −*H*, otherwise known as a fractional integration of order *H*, may then be applied to the flux *ϕ* to obtain the fractal liquid water content *ρ* (Wilson et al. 1991). This fractional integration lends the FIF multifractal model its name.

### APPENDIX B

#### Aircraft Liquid Water Measurements

Liquid water content was measured in cloud fields over northern Tasmania, Australia, with an aircraft-mounted King hot-wire probe (King et al. 1978). A total of 98 flights spanning 3 yr (1999, 2000, 2001) were conducted, encompassing altostratus, stratocumulus, and stratus cloud types (Fienberg and Nunez 2002). Individual horizontal measurement series lasted between 0.5 and 1.5 h, with data being collected at a frequency of 1 Hz. This collection frequency gave a resolution of approximately 100 m with aircraft speeds being used as a distance surrogate. There was no correction for cloud motion. It is likely that the sampling frequency of liquid water may alter slightly because of cloud motion. However, Davis et al. (1996) and Marshak et al. (1997) measured cloud liquid water using a variety of instruments operating at different frequencies. They showed that the slope of the power spectrum and other fractal statistics did not vary between instruments or change with sampling frequency. Additionally, the fractal statistics showed little difference when individual series, lasting up to 1.5 h, were broken up into 10-min intervals. Similar results are reported by Cahalan and Snider (1989).

### APPENDIX C

#### Monte Carlo Radiation Model

The FIF model was used to simulate clouds for use in Monte Carlo radiative transfer calculations. The cloud fields are taken to be 6.4 km × 38.4 km in horizontal dimensions and 0.5 km thick, with a base 1.0 km above the ground. These dimensions corresponded to a typical passage of a cloud layer through a ground sensor in a period of 1 h. The model grid consists of cubes with side lengths of 50 m. Horizontal variations in liquid water were generated using the FIF multifractal model, as outlined above. Liquid water content was kept constant in the vertical but allowed to vary horizontally because vertical variations are relatively small in comparison with the effects of horizontal structure in typical stratocumulus conditions (Marshak et al. 1998; Romanova 1998).

A Monte Carlo radiative transfer technique (Marchuk et al. 1981) was used to calculate the radiation field under stratocumulus clouds generated by the FIF model. Droplet radius distribution and scattering phase function was assumed to be constant throughout the cloud while the liquid water content varied. Therefore, the scattering phase function was also constant and the extinction coefficient was proportional to the liquid water content. A Henyey–Greenstein phase function was used with an asymmetry factor of 0.858, which was calculated using the parameterization of Hu and Stamnes (1993), with an effective droplet radius of 10 *μ*m and a wavelength of 0.6 *μ*m. The parameterization of single scattering albedo in the same work yielded a value of 0.999 97, or almost no absorption.

The simulations were each done using 10^{9} photons. Horizontal boundaries were periodic and the model was run at SZAs of 50° and 30°. The mean cloud optical depth was set at 8 over the entire cloud domain, which is equivalent to a mean liquid water content of 0.1 g m^{−3}. For all non-cloud-attenuating species, a midlatitude summer profile was used. Ground albedo was assumed to be 0.05. For each simulation the surface solar radiation flux was recorded at each 50-m pixel.

The final product involved a cloud transmission value Tr that approximates the ratio of the radiation flux density at the surface in cloudy conditions to an equivalent value for a cloudless sky. Note that the operation of the model for clear skies gave global transmissions of 0.8. Adding a cloud field optical depth of 0.5 gave global transmissions that were approximately 97% of the clear-sky value. This provided a clear-sky threshold and, hence, optical depths greater than this constituted a coincidental average cloud cover (over 68 realizations) of 0.81.

## Footnotes

*Corresponding author address*: Manuel Nunez, School of Geography and Environmental Studies University of Tasmania, GPO Box 252-78, Hobart 7005, Australia.nunez@utas.edu.au