Continuous Mapping of the Standardized Precipitation Index

Continuous mapping is the best mapping method that allows extracting data for each grid point or a grid cell and calculating a spatial average without discontinuity zones and artificial borders. The challenge in continuous mapping is how to get a reliable estimation of a variable between the measuring stations. When dealing with a small country like Israel (~25,000 km^2) that encompasses three different climatic zones associated with different rainfall regimes and sharp precipitation gradients, the challenge in mapping climatic variables is even greater.

The present work describes the continuous mapping of the Standardized Precipitation Index (SPI), a drought index based on the precipitation measurements only. The SPI is a normalized index therefore allowing a comparison of the severity of a meteorological drought in different climatic regions. This makes the SPI an optimal drought index for Israel.

Calculating the SPI entails three steps:

STEP 1 - composing the time series of the precipitation totals for an identical calendar month or sequence of months over a long period.

STEP 2 - fitting a probability distribution (parent distribution) to this time series and calculating the cumulative distribution probability (CDF) value corresponding to each precipitation total value.

STEP 3 - calculating the SPI as a number of standard deviations of the standard normal distribution N(0,1), i.e. z-score, corresponding to each CDF value.

In order to scheme continuous SPI maps in a 1 km^2 resolution, we have decided to utilize any possibility available throughout STEP 1 and STEP 2, as described next.

STEP 1. Continuous mapping of precipitation totals.

From a climatological point of view, the longer the time series the more stable the fitted parent distribution family along with its parameter estimates. We used a 65-y long period, to construct the monthly precipitation totals' time series for each pixel. There were two possibilities: (1) interpolating the monthly totals only of the rain gauges with the full 65-y records or (2) meticulous mapping of the monthly precipitation totals year by year and then treating each pixel as a measuring station with its own time series extracted from the precipitation maps. The second possibility is much more suitable for Israel, considering that there are only about 100 long-term rain gauges with only a few in the arid areas, in comparison to about 400 to 500 rain gauges operated in each specific year or month during this 65-y period.

In order to overcome a problem of sharp rainfall gradients and rain shadow effects, we decided to interpolate (with the IDW method) not the actual precipitation totals for each month, but their ratios to the long-term monthly averages (1981-2010). We then multiplied the interpolated ratio map with the mean monthly rainfall layer to get the actual monthly map. This method significantly reduces the mapping errors in comparison to the simpler method of direct interpolation of the precipitation totals themselves.

Once producing the monthly precipitation maps for 65 years (1952/1953 to 2016/2017 rainfall seasons), ~18,000 precipitation time series were extracted (for each pixel in the map with 1 km^2 resolution) for each SPI time scale. The extreme arid region (mean annual precipitation < 100 mm) was excluded from the SPI mapping because the term "drought" is irrelevant for this dry region.

STEP 2. Searching for the best parent distribution.

Upon composing the 65-y rainfall time series for calculating the SPI-3 to SPI-7 for the rainy period extending between October and April, a search for the best parent distribution was initiated. In order to avoid spatial discontinuity, we searched for a common distribution family suitable for the entire study area, among all of the distribution families applying MATLAB.

The null hypothesis for each distribution family was tested with the Anderson-Darling goodness-of-fit test. This test was chosen because it assigns more weight to the tails of a distribution, which is the point in the SPI application. For each SPI time scale, each pixel got its p-value. At first, the distribution families with p-values below 0.05 by even one pixel were filtered out. Then, the comparison was carried out among the p-value maps derived for the rest of the distribution families. It turned out that the GEV family had the highest p-values over the entire study area, steadily throughout all SPI time scales. The commonly recommended and widely used two-parameter Gamma distribution family has received p-values lower than GEV, especially for the short SPI time scales. An option for using the three-parameter Gamma (non-zero shift parameter) was rejected (by checking the first four moments) for all pixels, which is understandable for the semi-dry areas where there always is a potential for a zero Gamma shift parameter.

STEP 3. Calculating, and mapping the SPI on a monthly basis

The parameter estimates for the GEV distribution family for all SPI-covered pixels and for all relevant Eastern Mediterranean SPI time scales were saved in the Israel Meteorological Service's (IMS) database. These estimates were then implemented during the 2017/2018 rainfall season and are to be useful for the few next ones, assuming that an addition of 1 to 5 annual data would not affect the validity of the parameter estimates (periodical re-estimation of the distribution parameters is to be considered). As for now, on the 1st day of each rainy season month, starting from Jan 1^st for SPI-3 (OND), upon collecting and mapping the accumulated rainfall data for the preceding month, the SPI is being calculated and published as a continuous map in the IMS web site http://ims.gov.il/IMS/CLIMATE/SPI/2018/.

References:

1. McKee, T.B., N.J. Doesken and J. Kleist, 1993: THE RELATIONSHIP OF DROUGHT FREQUENCY AND DURATION TO TIME SCALE. In: Proceedings of the 8-th Conference on Applied Climatology, Anaheim, CA, Jan 17–22, 1993. Boston, American Meteorological Society, 179–184.
2. WMO, 2012: STANDARDIZED PRECIPITATION INDEX, User Guide. No. 1090