What is the background to this research?
A valuable measure of the country’s potentially available water resources is through rainfall measurements. These measurements help with the quantitative definition of drought and possible climatic change, besides providing information for the design of structures such as spillways, bridges, culverts, channels, storm-water drains etc. On-line measurements of rainfall can help reduce the potential damage of floods; the calculations of runoff from flood-producing storms can be projected into the future to forecast flows in sensitive areas in flood-plains near rivers and through water-bodies stored behind dams.
Traditionally, rainfall measurement has been made using 120mm and 200mm diameter rain-gauges scattered randomly over the country. In South Africa the largest proportion of these were (and still are) read by volunteer observers at 8:00am each day and the daily totals collected by the South African Weather Bureau (SAWB). There was a time (1950) when 4500 or more such gauges were “live” on any particular day, but the number of daily read raingauges has dropped to 1750 in 1999 with less than 600 of these reporting daily for the public good; the numbers are still diminishing.
There is thus a need for remote sensing of rainfall using radar and satellite imagery. These technically advanced, relatively recent innovations provide detailed information about rain-rates over large areas in sequences of images which are typically at 5 minute intervals from radar and 30 minute intervals from satellite platforms. They thus have the potential to provide detailed information (in space and time) of rain rates which can be used in many applications.
Why is the String of Beads model good for rainfall modelling? What makes it a better model than others?
Space-time rainfall is a very complicated physical process and might seem to need a very complicated model to describe it. The advantage of the model described here is that it is relatively straightforward in that it is defined by a very few parameters and achieves remarkable realism in the synthetic sequences it generates. The result is that it is remarkably fast in execution, which is a necessary characteristic with all the detail it presents.
It was the first model of its sort to present spatial detail and realistic temporal development in one package. There are others which are based on point rainfall models which can capture the average behaviour over an area with time and those which can model rainfall events with similar detail, but none that did both before the String of Beads model was devised. It is also adaptable to a variety of rainfall regimes – it has been fitted to South African, Swiss, German and English data with success and has become well known internationally as a serious competitor in the rainfield modelling environment.
What did the research programme originally set out to achieve?
The contractual objectives were:
- To provide a means of describing rainfields in both time and space by means of fractals.
- To facilitate the modelling of rainfields with economical mathematical descriptors.
- To use the project as a means of enhancing collaboration between institutions in the KwaZulu Natal region.
The first of these objectives was modified from using fractals to Gaussian Random Fields, with better effect, and has been achieved successfully with the alternative approach. The second objective has been achieved if one interprets the words “with economical” as “parsimoniously with a small number of”, as was originally intended. The third objective was achieved in a different sense than was originally intended, because although little collaboration initially developed in KwaZulu Natal proper, other than through the participation of the University of Zululand in the steering committee, the collaboration between researchers in Civil Engineering at Natal University and in the Bethlehem Precipitation Research Programme has become strong.
Why model rainfall measured by radar in space-time?
Increasing demands are being made on our meagre water resources and because there is greater population pressure causing invasion of high-risk flood-prone areas, there is an increased need for more detailed information about the spatial and temporal distribution of rainfall to assist with flood warning and mitigation.
In general the purpose of modelling is two-fold: to summarise complex physical processes in a descriptive or mathematical way with as few parameters as possible (parsimony) and to gain as deep an understanding of the phenomena as is possible, relevant and cost effective under the circumstances (insight).
The reason for modelling the rainfall process is to provide as much parsimony and insight as is reasonable, so that possible future scenarios can be artificially generated to conduct “what-if” investigations. Because of the relatively recent advent of remote sensing of rainfall in South Africa, the radar-records are quite short (3 to 5 years) and do not provide, in themselves, an objective basis for some aspects of decision-making. However there is a relatively large archive of data collected from daily read rain gauges which are too sparse, in many instances to provide the spatial detail required in some analyses/studies but give the information needed to quantify the variability of the rainfall processes over a long time-base. It is therefore prudent and desirable to marry the long-term (but poor spatial) information in the archive of daily rain-gauge records with the finely detailed (but short) records obtained from radar.
What happened to the Fractals?
Fractal models have been proposed as appropriate candidates for modelling rainfields because, by their construction, they preserve certain statistical properties, amongst them self-similarity, scaling etc. The mathematics is difficult and the models used (in particular the multiplicative models) have had the tendency to produce “blocky” rainfields which are distinguishable from typical radar measured rainfields and not easily manipulated to give a movement of the rainfield. By contrast, the approach adopted here was to use the KISS principle, employing standard statistical analysis methods (yet exploiting what had been discovered about scaling fields in the search for appropriate fractal models) to craft a parsimonious, robust, adaptable, feasible model which is statistically correct and visually realistic when compared to typical radar images of areal rainfall. The outcome is a model which is relatively easy to understand, fit and apply, without becoming entangled in the thicket of arcane terminology which tends to make the subject of fractals unapproachable to all but the intrepid.
Why the “String-of Beads” model?
Rainfall events (storms, showers, light rain etc) are patchy in space and patchy in time but are (in a wide sense) “wet”. The time between rainfall events (when it is not raining on a given area) is evidently “dry”. Modelling the space-time process of rainfall over an area for a few days, months or years thus consists of defining the process of alternating wet and dry periods and modelling the time-series structure. This is a one-dimensional process in time. Once the rain starts falling on the area during a wet period, the spatial as well as the temporal behaviour of the rainfall phenomenon is of paramount interest:
- how does the rain cover the area?
- how quickly do storm cells grow and decay?
- what is the speed and direction of the storm?
are some of the questions that demand answers.
How is the model fitted to the data?
The one-dimensional “string” describing the wet/dry process is defined by the sequence of wet and dry days on an area as specified by the rain gauge records. This process can be thought of as the alternation of three types of “climate” (or weather) occurring on the days of the observed record. The types are dry, scattered rain (usually caused by convective storms) and general rain (usually caused by stratiform, large systems). The distinction is a matter of convenience because there are days when the weather is a mixture of convective and stratiform clouds producing rain. This process is modelled by a three-state Markov chain which was thoroughly validated as a good model in WRC contract No. 550 conducted by Pegram and Seed (Report No. 550/1/98).
The three-dimensional space-time “bead” of a given rain event is defined by a sequence of images called Constant Altitude Plan Position Indicators (CAPPIs) which are derived from weather radar measurements. These images are sampled at approximately 5-minute intervals. Where more than 3% of the area covered by the CAPPI records rainfall above 1mm/hr, the image is classified as wet, otherwise the image is part of the “dry” time. Each CAPPI image is analysed to extract three parameters characterising its 9000 or more data points. This parsimonious use of parameterisation was one of the objectives of the original contract. These statistics (s, ß and µ) are stored for use in the simulation/generation process.
How is a sequence of storms generated?
In generation, the process of analysis is reversed. First a plausible sequence of climate-type is generated. When (for example) a scattered rainday is encountered, a length of storm is randomly generated (usually between 2 and 4 hours in the late afternoon) and a reasonable set of values for the parameters s, ß and µ chosen. The rainfall event is then produced by power-law filtering (in the complex domain) some Fourier transformed white noise, which is then reverse transformed, scaled and exponentiated to produce a sequence of CAPPI images. These are manipulated to provide realistic velocities of the storm-cells across the design area.
Does the model work well?
The answer is that it does with surprising facility and veracity. The sequence of images of CAPPIs generated for a wet period, not only have the right statistics and clustering behaviour, they are visually indistinguishable from the real images they mimic. This may seem like a quaint comment, but the production of images which look like the “real thing” is very important for the end user.
How could the model be used in practice?
In order for the model to realise its potential usefulness in application, a link must be formed between the parameterisation of the model and the statistics of raingauges recording one day totals in a region of interest. The proportion of raingauges recording rain on a particular day within a circle of diameter 130km (which should contain about 50 raingauges) is a good indicator of the Wetted Area Ratio, which is likely to have a close link with the parameters of the String of Beads model. The follow-on project (K5/1010) will investigate these links and allow the model to be widely applied.