The Forecast Interpretation Tool a Monte Carlo technique for blending climatic distributions with probabilistic forecasts

Transcription

1 INTERNATIONAL JOURNAL OF CLIMATOLOGY Int. J. Climatol. (2009) Published online in Wiley InterScience ( The Forecast Interpretation Tool a Monte Carlo technique for blending climatic distributions with probabilistic forecasts Gregory J. Husak, a * Joel Michaelsen, a Phaedon Kyriakidis, a James P. Verdin, b Chris Funk, a,b and Gideon Galu c a Department of Geography, University of California, Santa Barbara, CA, USA b U.S. Geological Survey Earth Resources Observation and Science (EROS) Center, Sioux Falls, SD, USA c Famine Early Warning System Network, Nairobi, Kenya ABSTRACT: Probabilistic forecasts are produced from a variety of outlets to help predict rainfall, and other meteorological events, for periods of 1 month or more. Such forecasts are expressed as probabilities of a rainfall event, e.g. being in the upper, middle, or lower third of the relevant distribution of rainfall in the region. The impact of these forecasts on the expectation for the event is not always clear or easily conveyed. This article proposes a technique based on Monte Carlo simulation for adjusting existing climatologic statistical parameters to match forecast information, resulting in new parameters defining the probability of events for the forecast interval. The resulting parameters are shown to approximate the forecasts with reasonable accuracy. To show the value of the technique as an application for seasonal rainfall, it is used with consensus forecast developed for the Greater Horn of Africa for the 2009 March-April-May season. An alternative, analytical approach is also proposed, and discussed in comparison to the first simulation-based technique. Copyright 2009 Royal Meteorological Society KEY WORDS rainfall; forecast; probability; Monte Carlo simulation Received 13 July 2009; Revised 16 November 2009; Accepted 21 November Introduction The continent of Africa has a large population of people walking a precarious line balancing water needs and precipitation totals. Shortages or excessive amounts of rainfall can result in impacts that are difficult to measure precisely, but can be reflected in the number of lives lost, amount of food produced, economic indicators, or changes in the landscape (Hulme et al., 1992; Dube and Pickup, 2001; Gbetibouo and Hassan, 2005; Funk et al., 2008). Understanding the likelihood that precipitation will be within a particular range during a future period can help prepare people for the potential outcomes in any number of ways, such as rationing water, distributing seed with a greater chance of succeeding given the predicted conditions, or mobilizing relief efforts to list a few. Techniques for probabilistic forecasting have been developed and are implemented for different regions of Africa, and globally, on a regular basis. While forecasting is a critical component of providing people with an early warning of food insecurity, oftentimes a lack of understanding of the implications of a given forecast limits the ability of governments or non-governmental * Correspondence to: Gregory J. Husak, Department of Geography, University of California, Santa Barbara, CA 93106, USA. husak@geog.ucsb.edu organizations (NGOs) to assist the people of a region. Providing policy-makers with better tools for interpretation of probabilistic forecasts can lead to better-informed decisions about various aspects of management of grain reserves, crop seed distribution, reservoir levels, livestock operations, economic indicators, and a variety of other rain-dependent activities. This article presents a methodology for updating climatic distribution parameters accounting for probabilistic forecasts, with a specific eye towards applications in Africa. The result is a set of probability distribution parameters that can be queried by decision makers to find the likelihood of a particular amount of rainfall being exceeded, rainfall being within a specified range, or a particular scenario such as the one-in-ten event. The proposed methodology serves to contextualize a probabilistic forecast and express its implications in terms that may be easier to interpret. An example of the technique, as applied to a rainfall forecast for the Greater Horn of Africa (GHA), is presented. With this example are the types of analysis which are available using the proposed technique. 2. Background Probabilistic forecasts are currently produced by many agencies for various parts of the globe at varying spatial Copyright 2009 Royal Meteorological Society

2 G. J. HUSAK ET AL. resolutions from sub-national vector maps (Buizer et al., 2000) to gridded global numerically modelled forecasts (Barnston et al., 2003; Goddard et al., 2003). These forecasts relate the probability typically expressed as a percent of rainfall being below-normal, near-normal, or above-normal for a specific interval (Buizer et al., 2000). In fact, these categories represent the low, middle and upper terciles (thirds) of the climatologic probability distribution for a defined forecast period, and the sum of their forecast probabilities is always one (Buizer et al., 2000; Goddard et al., 2003). The significance of a forecast is typically defined based on the magnitude and tercile bin assignment of the largest forecast value. This redistribution of the probabilities provides a conceptual understanding of expectations to allow decision makers to anticipate potential rainfall outcomes. A linear shift in the probabilities within each tercile based on the forecast weights would result in discontinuities at the tercile boundaries (Wilks, 2000). The goal of this research is to define a smooth probability distribution, which maintains the forecast probabilities without the discontinuities. Barnston et al. (2000) explore the information provided by probabilistic forecasts about expected rainfall given a known climatology. In that work, an example of the combination of historically derived seasonal rainfall and a probabilistic forecast is presented. The methodology for achieving the resulting probability distribution is described as the estimation of the shift in the central location of the distribution, combined with a possible change (decrease) in the dispersion of the distribution. (Barnston et al., 2000) Whether the resulting distribution was quantitatively defined or estimated is not further specified. In addition, Wilks (2000) provides a set of equations that allow the user to update the moments of the distribution, which can then be used to derive gamma distribution parameters. The research presented here proposes a new technique to fuse a probabilistic forecast with historical rainfall characterized by a gamma probability distribution using a Monte Carlo resampling approach. The outcome of the new technique is also tested to investigate how well it reflects the forecast probabilities. The continent of Africa has numerous rainfall regimes making the description of rainfall patterns across its large area difficult (Thompson, 1975). Previous efforts to fit distributions to rainfall records have shown the conditional gamma distribution to be a good fit (Walden et al., 1992; Juras, 1994). The gamma distribution uses three parameters to define a rainfall distribution function: shape (α), scale (β), and the probability of no rainfall (q) during the interval. In this distribution, the shape parameter represents the skewness of the distribution, while the scale is a measure of the spread of values relative to the mean. Distribution parameters are estimated using maximum likelihood estimation(thom, 1958; Öztürk, 1981; Wilks, 1990, 1995). Applying the gamma distribution to African monthly rainfall (Funk et al., 2003) shows the gamma to perform well for over 97% of the continent (Husak, 2005; Husak et al., 2007). 3. Methods The technique presented here to quantitatively perform the adjustment described in previous work (Barnston et al., 2000; Wilks, 2000) relies on a Monte Carlo resampling of the climatologically derived probability distribution in proportion to the forecast probabilities. This technique, termed the Forecast Interpretation Tool (FIT), estimates new distribution parameters defining the probability distribution for the forecast interval. This new distribution can be used in assessing the likelihood of specific events during the forecast interval at specific stations or for rainfall fields so long as there is a reasonable climatologic distribution. The proposed technique requires a reasonable estimate of the climatologic probability distribution of the variable for a select interval and a probabilistic forecast for the corresponding interval. Because seasonal forecasts may not be valid for every shorter interval within the season, nor for the neighbouring periods, the time periods of the climatic distribution and the forecast should be identical. However, the technique can be applied to historical data and forecasts of any duration so long as the intervals match. In contrast, probabilistic forecasts are typically regions with unique forecast distributions, where the forecast may be applied to spatial subsets of the region. Buizer et al. (2000) note that there is spatial variance within a forecast polygon; however, it can be assumed that the best estimate of the forecast distribution at any point within a region is the forecast for that region. In this way, forecasts at points can be extracted from areal forecasts. A single Monte Carlo simulation begins with the selection of samples from the historical distribution. More specifically, probabilities (v) within each tercile are selected from a uniform distribution [U(0,1/3), U(1/3,2/3), and U(2/3,1), respectively]. Using the climatologic distribution parameters, these probabilities (v) are converted to corresponding quantities, which serve as the sample value used to estimate the new distribution parameters. A schematic of the FIT is shown in Figure 1. The number of samples drawn in each simulation is a function of available computational resources, but the distribution of the samples between the terciles must be proportional to the forecast probabilities. To reduce the uncertainty that the resulting parameters represent the new probability distribution, multiple simulations should be performed with the median parameter values selected from the simulated values. Increasing the number of samples drawn in each simulation as well as the number of simulations will improve the robustness of the resulting parameter estimate. Sample FIT output probability distribution functions for parameters derived from two forecasts are shown in Figure 2. These distributions were derived by drawing

3 FORECAST INTERPRETATION TOOL QUANTIFYING PROBABILISTIC FORECASTS USING MONTE CARLO RESAMPLING Figure 1. Sample cumulative distribution function for a gamma distribution with (α = 10.0, β = 10.0) tercile boundaries shown. The x-axis is of arbitrary rainfall units for a specified time interval (i.e. mm/month, inches/year, etc.) An example of a randomly drawn probability being converted to a quantity is shown in the lower left of the plot. Because the gamma parameters apply only to nonzero events, in a location with a non-zero probability of no rainfall the resampling technique must be altered. There are many ways to handle this case to create a meaningful probability distribution; this article suggests drawing probabilities from the uniform distribution in proportion to the forecast as previously described. Those samples less than q are discarded (as they represent a no-rainfall event) and the remaining samples are transformed by subtracting q and multiplying the difference by (1 q ). The resulting value (v ) can then be converted to a quantity as before and distribution parameters can be estimated from the resulting sample of quantities to reflect the new rainfall distribution after accounting for the change in the probability of no rainfall. The median parameter estimates from all trials represent the post-fit probability distribution function for rainfall outcomes. There are a number of ways to assess how well the post-fit distribution reflects the forecast probabilities in accordance with the historical terciles. This study will look at the tercile boundaries for a select set of distribution parameters and compare these with the cumulative probabilities at those tercile boundaries after the FIT to measure how responsive that new distribution is to the forecast. 4. Results and discussion Figure 2. Probability distribution curves for climatologic conditions (α = 10.0, β = 10.0) and two unique forecasts. The dry forecast corresponds to a 45/35/20 (dry/mid/wet) forecast, whereas the wet forecast is a 20/35/45 forecast samples per simulation, for simulations. For a single station these large values are not computationally prohibitive for a single station, but if performing this for a grid of rainfall a smaller number should be used. The probability of no rainfall must also be altered to reflect the forecast probabilities. This is shown in Equation (1), where ˆq is the probability of no rain estimated from historical data and q is the probability of no rain after the forecast is applied, with BNF, NNF, and ANF representing the below-normal forecast probability, the near-normal forecast probability, and the above-normal forecast probability, respectively. q = ˆq 1/3, ˆq BNF ( ( 3 ) ) 1/3 < ˆq 2/3,BNF + ˆq 1/3 NNF ( 3 ( ) ) 2/3 < ˆq,BNF + NNF + ˆq 2/3 ANF 3 (1) Before discussing any results in how the post-fit distribution reflects the input forecast, we first look at how different forecasts affect the distribution parameters. As described earlier, the output of the FIT is a new set of probability distribution parameters that represent seasonal forecast probabilities. An analysis of the post-fit shape and scale values for different input parameters and forecasts shows that typically the shape parameter increases while the scale parameter decreases. This phenomenon can be explained by considering the sampling of the original distribution function. Because a portion of the original distribution is being sampled more heavily than the rest, the output distribution is more likely to have a higher density in the dominant forecast tercile, reducing the variance in the output distribution and therefore a reduction in the scale parameter. By plotting the shape and scale parameters for a sample input distribution and a variety of forecasts, it is possible to visualize the change in the parameters. The example in Figure 3 shows a climatologic distribution with α = 10.0 and β = 10.0, and the sample output parameters for eight unique forecasts. The forecasts are listed by BNF/NNF/ANF probabilities. The output parameters generally show an increase in shape parameter and a decrease in scale parameter. The two exceptions to this are the forecasts that have a middle-tercile forecast, which is less than the climatology. Our approach for error analysis is to measure the FIT-determined probability of being within the old

4 G. J. HUSAK ET AL. Figure 3. Plot of gamma distribution parameters for a sample climatic distribution and eight forecasts defined by their dry/mid/wet likelihood. tercile ranges, a test which should result in the forecast probabilities. In this analysis we investigated a range of shape values, but kept the input scale value at 1.0, as the post-fit scale parameter will just be the climatologic scale times this new value. Our analysis shows probabilities in each tercile are relaxing away from the forecast and towards climatology, with the more extreme forecast terciles experiencing larger errors. That being said, the maximum errors were typically on the order of 2 3%. To put this in forecast terms, if a forecast called for 40/35/25 dry/mid/wet conditions, the results of the FIT would yield a distribution that had tercile values of 38/34/28 for the interval. This tendency towards climatology was relatively stable within a single forecast for all tested shape values (integers from 1 to 50). In fact, over the range of tested parameters, the error in post-fit tercile probabilities had a standard deviation on the order of one half of 1%. This stability for tested shape values reassures the user that the post-fit distribution parameters reasonably representthe forecast, regardlessof climatologic distribution, which gives confidence in implementing the algorithm for a variety of locations. Table I shows the average error in the cumulative distribution at the climatologic tercile boundaries, as it is the cumulative probability at these quantities that is defined by the probability forecast. For the shape values from one through 50, the cumulative probability of the post-fit distribution at the climatologic tercile boundaries was calculated and compared to the forecast probability. The sum of the absolute value of the error from the first and second boundary were added and then divided by two to arrive at the error value for each unique forecast and shape value. This average error approximates the overall difference in the post-fit distribution from the forecast probability in a single value. The table shows select values and then the average for a variety of forecasts. Analysis of the table shows that the errors for a particular forecast tend to be consistent regardless of the input shape. The shape will not impact the error in probability at the tercile boundaries, however, it will input the magnitude of the resulting error in terms of rainfall quantity. A few points of interest about Table I are that the errors found for opposite forecasts were similar in magnitude, and also those for forecasts with a dominant nearnormal tercile were especially poorly modelled. The first point is the result of a continuous representation of a discontinuous distribution. A solution to these errors would be the analytical solution that will be discussed later, but the FIT smoothes out these discontinuities. The further the forecast is from the climatology, the more the FIT-estimated probabilities differ from the forecast probabilities. Poor fitting of forecasts with a dominant middle tercile show the inability of the gamma distribution to capture this type of pattern. The few, but relatively extreme samples in the tails result in a distribution that overexaggerates the likelihood of being in the tails to compensate for the spread of these samples. Table I. Errors in cumulative probability for different shape values and forecasts. The bottom row represents the average error for integer shape values ranging from 1 to 50. Shape Forecast values (dry/mid/wet) 25/30/45 25/35/40 20/35/45 40/35/25 45/35/20 45/30/25 25/45/30 30/45/ Average

5 FORECAST INTERPRETATION TOOL QUANTIFYING PROBABILISTIC FORECASTS USING MONTE CARLO RESAMPLING When framing the magnitude of the errors shown in Table I, it is appropriate to consider the confidence in the climatologic distribution parameters. Distribution parameters may be based on any number of rainfall observations or estimates although in practice it is usually 30 or more observations and the uncertainty in those parameters may lead to cumulative distribution functions that are less certain than the magnitude of the errors presented in this section. This is not to suggest that the errors in the FIT should be ignored, just that their impact may be within the uncertainty of the climatologic distribution. An analytical solution for arriving at forecast-updated probabilities performs a linear transformation of the climatologic probabilities within each tercile in proportion to the forecast probability for that tercile. The result is a distribution that perfectly retains the forecast probabilities, but has discontinuities in the probability distribution function at the climatologic tercile boundaries (Wilks, 2000). It is somewhat a matter of taste whether to retain the forecast probabilities or have a continuous probability distribution function. We believe that the use of an updated probability distribution function defined by the FIT distribution parameters allows for a distribution that is more readily usable for analysis and scenario building, and therefore of better service to decision makers. In addition, as there is no physical reason for the discontinuity in probabilities at the tercile boundaries, it seems preferable to attempt to smooth the discontinuities as is done in the FIT resampling approach. The FIT allows for a variety of analyses to explore how forecasts impact the likelihood of any number of rain-related issues such as successful crops, the potential for flooding, or the availability of drinking water. Caution should be used when looking at events in the extremes of the distribution as these may be poorly represented by the FIT output distribution. The flexibility of the FIT allows it to be applied to station data or gridded rainfall estimates. Also, the FIT may allow consensus-based forecasts to be adjusted before their release to match the expectations of experts regarding rainfall in certain regions. For instance, experts may realize that the median rainfall should increase by approximately 30 mm at a location, and then suitably adjust the forecast to reflect this intuitive understanding of rainfall expectations. The consensus seasonal rainfall forecast is based on considerations of the prevailing atmospheric-oceanic conditions and an ensemble of seasonal climate forecasts generated by leading global climate centres; European Centre for Medium Range Forecasts (ECMWF), UK Meteorological Office (UKMO), International Research Institute (IRI), and the regional IGAD Climate and Applications Centre (ICPAC). This consensus forecast is produced as part of the GHACOF process, and is based on discussions by many interested parties. The definition of the boundaries between the forecast regions is a process of that discussion. We are using this consensus forecast as an example of the type of forecast which can be used by the FIT technique. Figure 4 shows the 2009 MAM rainfall forecast map with three delineated regions: Regions I and III; with increased likelihood of belownormal (45%) to normal (35%) probabilities and decreased chances of above-normal (20%) probabilities (shaded in yellow), Region II; with increased likelihood for above-normal (40%) to below-normal (35%) and relatively less likelihood for below-normal rainfall of 25% probability (green shaded areas), and Region IV; mainly climatology was forecasted in this region (shaded in grey), which does not receive any significant rainfall during the MAM period. These probabilistic forecasts serve as advisories and are very difficult for end-users to interpret in terms of quantifiable climatic hazard and beneficial impacts (location and severity) in the region. Hence, they are of limited use as decision support tools. Because of the need of end-users to translate these forecasts into meaningful impacts, the FEWSNET Agroclimatology Toolkit (FACT) was developed utilizing the FIT algorithm. Since 2003, the GHACOF has used the FACT to translate these probabilistic rainfall forecasts into potential rainfall amounts and anomalies based on the 5. Case study The 23rd Climate Outlook Forum for the Greater Horn of Africa (GHACOF23) issued a consensus seasonal rainfall forecast for the period of March May, 2009 (MAM). March to May constitutes an important rainfall season over the equatorial parts of GHA and has significant implications on the socio-economic activities of the region, especially in revising food security strategies, contingency, and response planning. GHA is one of the highly food insecure regions of the world, with a current estimated population of more than 10 million people requiring food and non-food aid. Figure 4. Probabilistic forecast from GHACOF23 for MAM 2009.

6 G. J. HUSAK ET AL. Figure 5. GHACOF output maps showing seasonal rainfall median anomalies (left) and the ratio of the forecast probability of receiving more than 400 mm over the historical probability, describing the relative change in the chance for favourable maize conditions for the 2009 MAM rainfall season. FACT s built-in Collaborative Historical African Rainfall Model (CHARM) dataset (Funk et al., 2003) and the associated gamma probability distribution function. The FACT allows the users to generate probability maps, indicating locations that are likely or unlikely to attain critical rainfall amounts (thresholds) conducive to crop production, such as maize. These maps help inform decisions for various sectors that rely on seasonal rainfall performance. In addition, FACT-generated maps lend their usefulness in determining the geographical areas that are at risk of significant rainfall deficit or surplus, with increased likelihood of drought or flood occurrence, depending on the area s specific climatic vulnerability. This information is helpful for a variety of applications such as water resources availability, diseases and pest risk resurgences, escalating resource-based conflicts and their impacts on food production and accessibility in the region. It is against this background that GHACOF constituted a regular Food Security Outlook Forum (FSOF) in August 2004, with simple guidelines to assist in interpreting the rainfall forecasts, especially for the food security user community composed of the early warning, response and contingency planners at sub-national to regional level. These basic guidelines were intended to assist the user community to develop food security outlook scenarios based on well-informed assumptions with a lead time of 3 6 months. In a qualitative context, the food security outlooks have served well as food security decision support tools for contingency and response planning with a lead time of 3 6 months. The Famine Early Warning Systems Network (FEWSNET) has modified the FSOF guidelines into their own guidelines, including assessment of the current food security situation, the probabilistic forecast, anticipated Copyright 2009 Royal Meteorological Society outcomes, and key monitoring indicators. The targeted user of the FEWSNET product is mainly decision/policy makers at national, regional, and international levels and also donor agencies. This information only reaches farmers through media and partner organizations that have direct mandate to disseminate information at the lowest levels. Applying this process to the 23rd GHACOF forecast, we can construct maps for the anomaly resulting from the median rainfall performance as well as the change in probability of favourable maize growing conditions using the FACT, as shown in Figure 5. These maps were created using 600 samples drawn for 100 trials at each grid cell in the climatology. Because we use the gamma distribution to define the rainfall, the median rainfall amount is commonly used because it is the amount that will be exceeded half the time. The median anomaly is the difference between the 50th percentile rainfall amount derived from the FACT-generated distribution for this season and the climatic median. The change in probability map shows the ratio of the probability of receiving 400 mm (an amount associated with good maize yield) for the 2009 MAM season divided by the historical probability of receiving that amount. These products, among others, form the basis for the identification of the location and severity of the expected climatic hazard or benefit. The median anomaly rainfall performance map indicates that there is increased likelihood for meaningful rainfall deficit, or meteorological drought risk, in the eastern sector of the GHA region. The drought risk hazard is expected to adversely impact the marginal agricultural and vulnerable pastoral communities living in southeastern and northern Kenya and the neighbouring eastern Ethiopia, and south and central Somalia. These areas Int. J. Climatol. (2009)

7 FORECAST INTERPRETATION TOOL QUANTIFYING PROBABILISTIC FORECASTS USING MONTE CARLO RESAMPLING have, in general, remained highly food insecure as a result of four to six successive seasons of rainfall failures since Further, there is evidence that these areas of concern are less likely (reduced probabilities) to receive rainfall amounts of at least 400 mm, meaning an increase in the likelihood of unfavourable local maize conditions. To put into local context, these MAM rainfall events account for over 50% of their normal annual rainfall in the pastoral areas on the eastern sector of the GHA and if they fail again, the food security situation will certainly worsen, increasing the food insecure population in the areas of concern. Despite this bad projection for the eastern portion of the GHA, the key agricultural areas of western Kenya, central Uganda, and parts of the Belg crop growing areas of Ethiopia are likely to have favourable crop conditions and subsequently good prospects for agricultural production. It is also a common practice at the GHACOF to generate maps indicating the expected timeliness of the season and crop conditions for a variety of crops based on the use of crop models with climatic inputs derived from identified analogue years. The resulting maps provide important information for the agricultural communities in terms of land preparation and crop options for better crop production prospects. 6. Summary This article presents a technique for quantitatively modifying the climatologic rainfall distributions with information from probabilistic forecasts to create new distribution functions defining the likelihood of rainfall for the forecast interval. Overall, the error analysis gives promising results for all tested forecasts and climatologic parameters. The fact that errors are reasonably stable for all tested shape values shows that the distribution parameters resulting from the FIT algorithm generally represent the probabilities described by the forecast, with a slight systematic error in the result towards climatology. Testing the FIT output with this level of depth may be incongruent with the precision involved in creating the forecast probabilities themselves; the errors, however, are sufficiently small to give confidence in the FIT output. An example of the FIT, being put into practice in the GHA, shows the potential value of turning the probabilistic forecasts into practical products, which can be used by the food security community. References Barnston A, He Y, Unger D A forecast product that maximizes utility for state-of-the-art seasonal climate prediction. 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