Relating Insolation to Cloud Cover

Research question

Some thoughts about how to answer the research question

In other research project descriptions, we have made use of a clear-sky model for insolation. This provides a way to make various kinds of comparisons between observed insolation under actual sky conditions compared to what the insolation would be if there were no clouds at all. One interesting research problem is to derive a relationship between measured insolation and "cloudiness." Then, given some measure of cloud cover, one could predict the insolation in some statistical sense. Conversely, given the insolation, one could know something about the cloud cover.
      It is sometimes the case that looking at older research papers through more modern eyes can provide an interesting starting point for research, and that might be the case for this project. A 1977 paper by R. K. Reed gave an empirical relationship between the ratio of observed insolation to clear-sky insolation over open ocean as a function of cloud cover and solar elevation angle:

S_Elev = 1353s•(0.61 + 0.20s)
S/S_Elev = 1 – 0.62C + 0.0019Elev_noon

where s is the sine of the solar elevation at a particular time where C is the cloud cover in tenths and Elevnoon is the solar elevation angle at solar noon.
      Comparisons with other equations relating insolation to cloud cover are also presented in Reed's paper:

Kimball (1928)	S/SElev = 1 – 0.71C
Berliand (1960), cited in Kondratyev (1969)	S/SElev = 1 – aC + 0.38C2 a varies with latitude
Laevastu (1960)	S/SElev = 1 – 0.60C3
Tabata (1964)	S/SElev = 1 – 0.716C + 0.00252Elev

      The interesting thing about this old paper is that it was written before sophisticated computing capabilities were available to everyone. How can we make use of these capabilities? One obvious advantage is that we can very quickly calculate the solar elevation angle anytime, anywhere. More importantly, we may be able to use image processing software to obtain a value that is related to cloud cover.
      The lefthand image below is a photo of a partly cloudy sky, saved as a bitmap (.bmp) image. The righthand image is that same image resaved as a two-color (black or white) bitmap file, using the popular freeware image processing program IrfanView. We no longer have to estimate cloud cover in tenths, based on human observations. Instead, it remains only to count the black and white pixels in this image and calculate the ratio of white pixels to the total number of pixels.

Original color image.	Image reduced to two-color black and white file.

      IrfanView will be consistent when it decides what should be counted as a white or black pixel, using criteria built into the software, but how it does that and the implications of the decisions it makes are not clear. For the example shown above, with cumulus clouds against a clear blue sky, the conversion seems entirely reasonable. For other sky conditions, including lightly or heavily overcast skies, or cirrus clouds, IrfanView's performance can be more puzzling! The lefthand image below is an overcast sky, taken just after 4:00 pm local time, and the righthand image is IrfanView's conversion to a two-color black/white file. Note that the "darker" part of the sky is now classified as white and the "lighter" part as black! This is a puzzling outcome that deserves further investigation based on a series of sky images! But in any case, the black/white separation is probably meaningless for the righthand image because the sky is overcast.

Original color image.	Image reduced to two-color black and white file.

      Writing software to count black and white pixels in a .bmp image requires knowledge of the structure of .bmp files. Fortunately, the structure of these files is relatively simple. If you search online for "bitmap file format" you will find many references. Also, there is a reference below to a web page I wrote some years ago. Although image files aren't "text" files, they can nevertheless be treated like a long string of bytes to be read as "characters" and interpreted one at a time. The beginning of a .bmp file contains information about the image and these "header" records are followed by image data. Two-color (black/white) .bmp files are small compared to color files. A full-color .bmp image requires three bytes (24 bits) for each pixel, but each pixel in a black/white file requires only 1/8 of a byte — 1 bit — per image pixel. If a bit is a 1, then the pixel is black. If it is 0, then the pixel is white.
      Here's another possible approach to analyzing sky images using ImageJ image processing software (a free download from the National Institutes of Health). Starting with the original color images, use the Image-->Color-->Split tool to separate the image into three separate images for the red, green, and blue channels. (All colors are defined by the "intensity" of the red, green, and blue values in each pixel.) Clear blue sky is nearly black in the red image, but lightly to moderately overcast skies have higher values in all three channels. For a heavily overcast dark sky, the average value in all three channels will go down. Once you have separated the colors, you can apply ImageJ's Analyze-->Histogram tool to the red channel as a way to separate clear sky and clouds. Although it might not be immediately obvious what to do with these histograms (you can save the values in a text file), it is certainly clear that overcast skies look completely different from partly cloudy skies with cumulus!

Two sky images processed with ImageJ's Split and Histogram tools.

      It is also interesting to note that there may be information contained in the brigtness of a clear sky. This brightness should be a function of atmospheric conditions and sun angle. Under the same sun conditions, a very clean sky (deep blue) will have a lower average pixel value than a hazier sky. Here is ImageJ processing of two sky images (not taken under the same sun conditons). Remember that the histogram is a "grayscale" brightness – the average of the RGB values – for the pixels inside the rectanglular area selected. (You can select circular or irregularly shaped areas, too.)

Histograms for clear sky under two different conditions, inside selected area.