**Space-Time Modelling of Rainfall using the String of Beads Model: Integration of Radar and Raingauge Data
1010/1/02**

January 2002

**Executive Summary**

**What was the rationale for the research?**

The String of Beads model is a stochastic rainfall model based on the combined observations of a large network of daily raingauges and an S-band weather radar situated near Behtlehem, South Africa. The model draws on the ideas and observations of many other rainfall modelers (mentioned at appropriate places in the report) to form an efficient means of simulating rainfall over a wide range of spatial and temporal scales.

**What were the aims of the research and were they achieved?**

The aims in the original proposal were:

- Extend the model to where only rain-gauge data are available
- Validate the model against continuous raingauge data.
- Extend the model's capacity to allow for topographic features and large meteorological systems
- Incorporate information from satellites
- Make the model available to the hydrological community as a simulation tool in a user-friendly package

Three of the five aims listed above were achieved. One of the exceptions was the intention to incorporate satellite data; this was transferred to another WRC contract, K5/1153, the Spatial Interpolation and Mapping of Rainfall (SIMAR) where progress is satisfyingly rapid. The other exception is the use of continuous raingauge data. As will be seen in the sequel, difficulty was encountered with the more recent radar data and its correspondence with the historical raingauge data (the model's primary calibration source) so the continuous data were not relevant at this stage.

The most successfully achieved aim was the sharing of the strengths and simplicity of the String of Beads model (SBM) with the Hydrological, in particular, the international community. The list of technology transfer opportunities realised at the end of this Executive Summary attests to that. As a result of these initiatives, in July 2001, the rainfields on two foreign catchments (one in Australia the other in Switzerland) are being modelled with the SBM. In principle, the model can be adapted to varying topography and these ideas are under further development.

One of the unexpected successes of the model is that it can be used for short-term rainfall forecasting -more about this later in the summary -so although there is more to do in improving the calibration, there are already practical spinoffs which will bear fruit from this research contract.

**What is this new version of the String of Beads Model? **

Weather radar measures rainfall rates over a large area in fine spatial detail and very frequently. This translates into a circular area up to 200km across with 1 km resolution every 5 minutes. In this study, the diameter of the circle is reduced to l28km for precision and manageability, even so, each image representing the estimated rain rate over the area is described by up to 12868 numbers. A year's data can quickly fill a computer disk unless it is managed cleverly. What to do with all the information that these data represent? The answer is to summarize it using a model.

The String of Beads model was introduced in WRC contract K5/752 when the "Bead" (the model of the rainfall process while it is raining) was developed. In this study the model has been refined and extended to improve the "Bead" process and model the arrival of the beads of space-time rain on the "String" of time.

The new version of the model has the particularly useful property that it can be calibrated on historical daily raingauge data, even in the absence of a radar in the area, although this would be an added advantage. The daily raingauges (if there is a reasonably good network covering more than 40 years) characterize the inter-annual and intra-annual variability of the rainfall. This means that the model can be used in places where radar data is short or even absent.

The model is designed to perform two important modelling tasks. The first is simulation of long sequences (of many years duration) to conduct "what if' studies of rainfall over the target area. The primary variables of interest are the mean rainfall rate over the area and the percentage of the area which is wetted by the rainfall (which are called the Image Mean Flux, or IMF, and Wetted Area Ratio, or WAR, in this report). The output from such simulations matches the refined gridded data of digital elevation models now being used in catchment rainfall/runoff studies. The second task to which the model can be adapted is to give short-term forecasts (nowcasts in the literature) of future rainfall for use in real-time flood forecasting in either rural or urban environments. Using the String of Beads model to forecast rainfall on a catchment up to one or two hours ahead has the benefit that managers (catchment, flood or disaster) have extra information to enable them to make anticipatory decisions -they can be proactive rather than reactive when a flood threatens.

**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 sorts of Simulations are envisaged?**

The crucial characteristics that a space-time model of rainfall must capture are

- the storm arrivals,
- their duration and intensity while it is raining and also
- move realistically in the right directions and at the right velocities.

Each of these characteristics must be quantified and their mutual dependence and variability accounted for. The way the new model works is to examine the "String" of time and identify the alternating wet and dry periods on it. These lengths vary randomly (and we have discovered, independently of each other) but can be I characterized by the lognormal probability distribution function whose two parameters vary periodically over the year. This is the basic process.

Once it is raining, the "String" supports a "Bead" of space-time rainfall. The "Beads" have the lengths of the wet periods on the "String" and their structure in space and time defines the behaviour of the rainfall event. These descriptors are its velocity and direction and whether it describes, on the one hand, patchy thunder-storm rainfall or, on the other hand, widespread rain of less variability but longer duration on average. The new model achieves all this and passes the stringent statistical and visual tests to check whether the result is good.

**Can the "String of Beads" model be used for Forecasting?**

Yes. The "String of Beads" model in its new form is particularly suitable for forecasting. This is because of its efficient structure based on time series ideas. In the previous formulation, the set of images comprising the "Bead" were all generated simultaneously using Fourier transforms in a box of space-time. Although mathematically elegant, this was not practical because it was difficult to generate wet periods of the correct length and the method did not lend itself to forecasting applications.

The new model, by contrast, is based on time series ideas, in that images are constructed in series using linear combinations of the images in the recent past with added noise, both at image scale and at pixel scale. This is convenient for simulation because the wet period can be defined to be any length to suit the "String" process and the computation method is more efficient and quicker than previously. This is because the dependence structure is relatively short -only the previous five images are needed to construct the new one.

Forecasting is a natural extension of this formulation of the model. At any time that the data are available (which is possible with a radar dedicated to the catchment concerned) the previous five images are used to construct the possible future an image at a time until the forecast horizon has been achieved. It is found that forecasts are good to start with, deteriorating gradually so that they are still relatively useful an hour ahead, but useless after two hours. For urban applications in particular the methodology has particular promise for "buying time" to help managers to anticipate disaster.

**What of Capacity Building?**

This research was conducted in the Civil Engineering Programme of the University of Natal in Durban, in whose final year class of 46 in 2001, a mixed group of students can be found. The classification of the class is 5 Black (1 woman), 17 Indian (2 women) and 24 White (2 women). Two of the final year dissertations in 2001 involve aspects of rainfall measured and modelled by radar and all these students are taught about rainfall patterns and relationships based on the results of this research -this was not so 4 years ago. There are other spinoffs which feed other WRC research contracts, in particular SIMAR (K5/1153) and Flood Nowcasting in the Umgeni (K5/1217), recently commenced.