Radiative transfer and cloud structure

Last month our paper on small-scale cloud structure and radiative transfer using a state-of-the-art 3-dimensional Monte Carlo radiative transfer model was published. It was written together with two radiative transfer specialists: Sebastian Gimeno García and Thomas Trautmann. The paper introduces the new version of this model called MoCaRT, but the interesting part for this blog on variability are the results on the influence of small-scale variability on radiative transfer. Previously, I have written about cloud structure, whether it is fractal and the processes involved in creating such complicated and beautiful structures. This post will explain, why this structure is important for radiative transfer and thus for remote sensing (for example for weather satellites) and the radiative balance of the earth (determining the surface temperature). I will try to do so also for people not familiar with radiative transfer.

As an aside, the word radiation in this context should not be confused with radioactive radiation. (It is rumored that the Earth Radiation satellite Mission had to be renamed to the EarthCARE to be funded, as the word radiation sounds negative due to its association with radioactivity.)

Radiative transfer

In theory, radiative transfer is well understood. The radiative transfer equation is long know and describes how electromagnetic radiation (intensity) propagates through a medium and is scatter and emitted by it. Climatologically important are solar radiation from the sun and infrared (heat) radiation from the earth's surface and the atmosphere. For remote sensing of the atmosphere also radio waves are important.

In practice, radiative transfer through the atmosphere is difficult to compute. This starts with the fact that the equation is valid for one frequency of the electromagnetic wave only, while the optical properties of the atmosphere can depend strongly on the frequency. To compute the radiative balance of the earth, a large number of frequencies in the solar and infra red regime thus need to be computed (such models are called line-by-line models). More efficient are computations in broader frequency bands, but then approximations need to be made.

The situation is still more complicated because only for some simple configurations an analytic solution of the radiative transfer equation is known. In other cases, the equation needs to be solved numerically. This is often the case for atmospheric problems, especially in the presence of clouds or aerosols. Fortunately, we have so-called Monte Carlo (MC) radiative transfer models, which simulate the random movement of photons (radiation packages) through a medium without the need to make any approximations. Such MC model can in principle be arbitrarily accurate, one "just" needs sufficient photons. Simulating a sufficient number of photons makes these Monte Carlo models computationally very expensive. Furthermore, important for this study is that for an accurate solution, the atmosphere also needs to described with sufficient spatial resolution.

Cloud structure

The reason that we need information about the cloud structure at a very high spatial resolution is because radiative transfer is highly nonlinear and non-local. Let's illustrate this with the simple example to the right, a 1-dimensional atmospheric column with the sun overhead and ignoring absorption. This column contains one scattering cloud layer with 50 % cloud cover. Imagine the cloudy half reflects 90% of the radiation and lets 10% through. In the cloud free half, no radiation is reflected and 100% is let through. Thus in total, 45% is reflected and 55% reaches the ground.

Now let's double the amount of cloud water in this layer. We could double the amount of water in the block cloud. This is similar to putting a second cloud below the initial one. Of the 10% transmitted through the initial cloud, approximately another 10% will get through the lower cloud. Just one percent reaches the ground in the cloudy part of the column, but in the cloud free part still 50% reaches the ground. The total reflected (49%) and transmitted radiation (51%) thus does not change much compared to the first case.

The third case is doubling the amount of water by increasing the cloud cover to 100%. Now 90 of the radiation is reflected in total and 10% reaches the ground. The same average amount of cloud water can thus have very different effects on radiative transfer. The distribution of the cloud water amount, which not only describes the mean, but also the variability, is clearly important for radiative transfer.

Nonlinearity

In this simple example, the reflected radiation is linear in the cloud cover; if you double the cloud cover, also the reflected radiation doubles. However, the reflected radiation (for 100% cloud cover) is in a first approximation proportional to the logarithm of the cloud optical depth. Because of this nonlinearity, we do not only need to know the mean optical depth of the atmospheric column, but the entire optical depth distribution.

The contrast in the cloud water content in case of a broken cloud field is very strong and consequently the effect of this variability on the reflectance. You will, however, get a similar, but smaller, effect if part of the cloud has a low cloud water amount and another part has a high amount. In other words, also for an overcast closed cloud layer, the distribution of the cloud water can be important.

Non-local

Next to the distribution, also the size of the clouds matters. This is because inside the clouds, the photons are scattered in all directions. The photons that are scattered side ways and manage to leave the cloud, will most likely not be scattered back again and thus have a higher chance of reaching the ground. In small clouds, whose sides are relatively large, more photons will leave the cloud through the sides as for large cloudy patches. This effect is strongest if the size of the patches is similar to the depth of the cloud field.

Again, you will see a similar phenomenon in case of overcast clouds with variability in the amount of cloud water. In this case the photons will preferentially scatter towards the optically thin regions. The spatial scale of the variability is now described by the spatial auto-correlation function, by how fast the correlation decreases as a function of distance.
The effect is strongest if the correlation length, the distance at which the correlation becomes smaller than a threshold (about 0.37), is in the same order as the depth of the cloud. If the correlation length is much smaller (“noisy” cloud), the cloud appears quite homogeneous to the photons. In this case there are no optically thin regions in which the photons will gather as they will easily scatter back to or transfer into optically thicker regions. If the correlation length are much larger than the cloud depth, moving all the way from the optically thick to the optically thin parts becomes a long travel and thus less likely.

Concluding, for radiative transfer we need 3-dimensional clouds at a higher resolution and we need to know the distribution (variability) of the cloud water and its spatial structure.

Downscaling

Now "back" to the article. In the article we have compared young fair-weather cumulus cloud fields, the kind you see developing at a nice summer day, at two spatial resolutions. These cloud were generated by a high resolution dynamical cloud model, that is able to resolve the large turbulence eddies. The model has a horizontal resolution of 100m. The reflectivity of one of these clouds can be see at the right.

From these clouds, we computed a coarser set of clouds with 400m resolution by simply averaging over 4x4 pixels. The coarser version of the cloud presented above can be seen to the right. It clearly reflects more solar radiation.

To be able to generate realistic high-resolution cloud field we already developed a downscaling (or disaggregation) method in a previous study (Venema et al., 2010). This statistical method, generates a cloud field that has the same cloud amount at the coarse scale (in this case 400m) and adds small-scale variability to reduce biases in the radiative transfer. The downscaled version of our example cloud can be seen to the right. Because we started with high-resolution clouds, we can study how well the downscaling algorithm performed. In a real application, you would start with coarse resolution cloud fields, from a coarser resolution cloud resolving or weather prediction model or derived from a satellite measurement. That is the situation you would need such a downscaling algorithm.

In Venema et al. (2010), the downscaling algorithm was validated with respect to the radiative fluxes (upward and downward radiation in all directions) needed to compute the radiative balance of the earth, in this new study (Gimeno García et al., 2012) we now also validated the algorithm with respect to radiances (radiation traveling into one specific direction), which are important for (satellite) remote sensing. In both cases, the biases are strongly reduced and no longer statistically significant.

Variability

Such downscaling algorithms are not only important for radiative transfer, but for all fields showing clear small-scale variability and for applications showing nonlinear behavior. Similar methods are, for example, also frequently used for the downscaling of precipitation (Maraun et al., 2010). It makes a difference whether the same daily precipitation amount in a region comes down in a single downpour in a small sub-region or as drizzle over the entire region for the risk of flooding, the amount of water that is absorbed by the ground and the spread of fungi on plants.

In general, if the process you are interested in is strongly nonlinear, you will likely have to take the variability into account.

Related post

On cloud structure

Clouds are not spheres or cubes in the sky, but are better described as fractals. However, part of the structure is also non-fractal.

References

S. Gimeno García, T. Trautmann, and V. Venema. Reduction of radiation biases by incorporating the missing cloud variability by means of downscaling techniques: a study using the 3-D MoCaRT model. Atmos. Meas. Tech., 5, doi: doi: 10.5194/amt-5-2261-2012 , pp. 2261-2276, 2012.

D. Maraun, F. Wetterhall, A.M. Ireson, R.E. Chandler, E.J. Kendon, M. Widmann, S. Brienen, H.W. Rust, T. Sauter, M. Themeßl, V.K.C. Venema, K.P. Chun, C.M. Goodess, R.G. Jones, C. Onof, M. Vrac, and I. Thiele-Eich. Precipitation downscaling under climate change. Recent developments to bridge the gap between dynamical models and the end user. Reviews in Geophysics, 48, RG3003, doi: 10.1029/2009RG000314, 2010.

Venema, V.K.C., S. Gimeno García, and C. Simmer. A new algorithm for the downscaling of cloud fields. Quart. J. Royal. Meteorol. Soc., 136, no. 646, pp. 91-106, doi: 10.1002/qj.535, 2010.