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The main goal of this work is a feasibility study for the Bayesian Processor of Output (BPO) method applied to tropical convective precipitation regimes over Central and West Africa. The study uses outputs from the Weather Research and Forecasting (WRF) model to develop and test the BPO technique. The model ran from June 01 to September 30 of 2010 and 2011. The BPO method is applied in each grid point and then in each climatic zone. Prior (climatic) distribution function is estimated from the Tropical Rainfall Measuring Mission (TRMM) data for the period 2002-2011. Many distribution functions have been tested for the fitting. Weibull distribution is found to be a suitable fitting function as shown by goodness of fit (gof) test in both cases. The rain pattern increases with the value of the probability p. BPO method noticeably improves the distribution of precipitation as shown by the spatial correlation coefficients. It better detects certain observed maxima compared to the raw WRF outputs. Posterior distribution (forecasting) functions allow for a simulated rainfall amount, to deduce the probability that observed rainfall falls above a given threshold. The probability of observing rainfall above a given threshold increases with simulated rainfall amounts.

Economies of sub-Saharan Africa largely depend on agriculture. The agriculture is essentially rain-fed. Precipitation is the most important and most widely studied weather variable ( [

For these reasons, there is a great deal of research activities to improve quantitative precipitation forecast (QPF) and weather centers continuously evaluate their operational high-resolution limited-area models to trace error sources. QPF is particularly challenging over Equatorial Africa, especially capturing small convective cells that constitute most of the rain events ( [

Furthermore, QPFs obtained from a single numerical weather prediction (NWP) model are deterministic, and thus do not convey any information about the uncertainty about the prediction, which is a shortcoming in weather-related decision-making [

In this paper, Bayesian Processor of Output for probabilistic quantitative precipitation forecasts is used. The Bayesian Processor of Output (BPO) is a theoretically-based technique for probabilistic forecasting of weather variates. It processes output from a numerical weather prediction (NWP) model and optimally fuses it with climatic data in order to quantify uncertainty about a predictand. The theoretical structures of the BPO are derived from the laws of probability theory.

As is well known, Bayes theorem provides the optimal theoretical framework for fusing information from different sources and for obtaining the probability distribution of a predictand, conditional on a realization of predictors, or conditional on an ensemble of realizations [

The objective of this work is a feasibility study for the Bayesian Processor of Output (BPO) method applied to tropical convective precipitation regimes over Central and West Africa. The paper is organized as follows: In Section 2, the model and experimental design are described, followed by the data used in this study. In Section 3, BPO techniques are briefly presented followed by the verification of BPO forecasts in Section 4. In Section 5, results for probabilistic forecasts of daily precipitation accumulation over the Central and West Africa is presented. Section 6 is devoted to the conclusion.

We performed simulations using version 3.3 of the Advanced Research Weather Research and Forecasting (ARW-WRF) model [

Hong et al. [

Microphysics | WRF single-moment 6-class microphysics (WSM6) |
---|---|

Radiation | Rapid Radiative Transfer Model (RRTM) longwave radiation scheme |

Surface layer scheme | Monin-Obukhov surface layer scheme |

Surface physics | Noah Land Surface model (LSM) |

PBL scheme | YSU PBL scheme |

Cumulus scheme | Tiedtke cumulus parameterization scheme |

moisture convergence, while that used in this version is based on the convective available potential energy (CAPE) modified by [

The model is run from June 01 to September 30 of 2010 and 2011. The initial and boundary conditions are provided by the National Center for Environmental Prediction (NCEP) Global Forecasting System (GFS) three hourly products. We use the 0000 UTC cycle and run the WRF model for 48 hours starting at 0000 UTC. The model is set at a horizontal grid resolution of 25 km × 25 km and has 41 vertical levels. Data analysed are total precipitation amount for the 24-hourperiod starting at 0600 UTC, thus having 6 hours of spinup (from 00 UTC to 0600 UTC).

The study area extends from 15˚W to 30˚E and 10˚S to 30˚N (

Six distinct main climatic regions (

Region 2 covers arid (Sahara Desert) and semiarid (Sahel) zones over Mauritania, Mali, Niger, Chad and parts of Sudan, Cameroon and Nigeria. In the northern part of this region the climate is uniformly dry, with most areas receiving less than 130 mm/year of rain, some getting none at all for some years.

The southern part serves as a transition zone between the arid Sahara and the wetter savanna region further south. Annual rainfall averages between 100 and 200 mm received from June to September (

For the purpose of verification we used Tropical Rainfall Measuring Mission (TRMM) data as ground truth. TRMM data show that the JJAS seasons 2010 and 2011 were wet and dry respectively (

Although the 0.25˚ grid spacing of TRMM data is close to WRF’s 25 km, they were regridded in order to achieve coincidence of both grids points.

The 1DD GPCP data set is a 1˚ × 1˚ longitude/latitude precipitation product

from Global precipitation Climatology Project. The GPCP algorithm combines precipitation estimates from several sources, including infrared (IR) and passive microwave (PM) rain estimates, and rain gauge observations [

Following the ideas of Bayes, if we have a set of forecasts and past observations, we can use this prior information to improve future forecasts. Based on past couples of forecasts and observations, we can construct a model to link each forecast amount to the probability of observed amount. Example, determine the probability of observing a rainfall amount greater than 10 mm, knowing that the forecast amount is 1 mm. The concept is illustrated in

The following algorithm deﬁnes the input elements, outlines the estimation procedure, and details the calculation of the posterior parameters (the parameters of the forecasting equations).

Step 0: Given are two samples, the climatic sample of the predict and W, and the joint sample of the predictor vector and the predict and (X, W), respectively:

where

Step 1: Using the climatic sample, the prior (climatic) distribution function G of predict and W is estimated, such that

Let g denote the corresponding prior (climatic) density function of W.

Step 2: Using the marginal sample

(The bar over signifies that this is only an initial distribution function of, which need not cohere to the specified prior distribution function of W and the yet-to-be-constructed family of likelihood functions. This detail is accounted for in the derivation of the meta-Gaussian BPO, and thus need not be considered in application.)

Step 3: The normal quantile transform (NQT) of the predictand and of every predictor is defined:

where Q is the standard normal distribution function, and

Step 4: Using the transformed joint sample, we estimate the following moments. For the transformed predictand

For every transformed predictor

The estimates of variances and covariances should be the maximum likelihood estimates (i.e., they should be calculated using N as the divisor).

Step 5: We form two I-dimensional column vectors

Step 6: The values of the posterior parameters are calculated as follows:

where

Given a prior distribution function G of predict and W and given a realization

For any number p such that

Given also a prior density function g of predict and W, the meta-Gaussian posterior density function of predict and W is specified by the equation

In the current work, one predictor is used. When there is only one predictor (I = 1), its subscript is omitted. Thus

In the following, processing will be done by grid point and climatic zones.

The prior distribution function G of precipitation amount W is conditional on precipitation occurrence:

The single predictor X is the estimate of the 24-h total precipitation. The marginal distribution function

Once the five elements

From the definition, the number p is the probability that the value of the precipitation is less than or equal to w p. In this section, the number p is simply interpreted as the probability that the rain is equal to w p. Only values of p for which the spatial distribution of precipitation is close to the observations will be presented.

BPO method introduces a noise, that is, it introduced rains in some parts of

the field, compared to observations.

For p = 0.4 (

The observed intensity is 25 mm instead of 15 mm as forecasted by the BPO method for p = 0.4. For p = 0.45, other maxima are found over West Cameroon and northern Burkina Faso. It is generally found that when the probability p increases, the areas that have the maxima are preserved with the difference that intensity also increases. For p = 0.6, some observed maxima are located by the BPO method. These include the maximum observed on the north of the Central African Republic and the south-eastern Nigeria.

Given the foregoing, it is found that BPO method introduced background noise. It provides low rainfall almost throughout the study area especially when the value of the probability p increases. This led us to subtract the average daily climatology (8.8 mm) over the entire region to get rid of this noise.

In the following, the analysis will be conducted in each of the five regions (Region 2 to Region 6) that cover the study area (see

posterior distributions functions based on three different realizations: 1 mm, 10 mm and 25 mm of predictor X. For simulated value of 1 mm of precipitation, the probability of observing an amount less than or equal to 20 mm of rainfall at any point in the Region 2 is 0.75. This means that there is 75% chance to observe at any point of this region an amount of rain less or equal to 20 mm. We deduce that the probability of observing rainfall amount greater than 20 mm is 0.25, that is there is only 25% chance to observe rain greater than 20 mm in intensities.

For simulated value of 10 mm of rainfall, the probability of observing rainfall less than or equal to 20 mm is 0.65. That is 65% chance to observe rainfall ≤20 mm when the model simulates 10 mm of precipitation at a point. We deduce from the above that the probability of observing rainfall greater than 20 mm is 0.35.

Thus, there is 35% chance of observing rainfall intensities greater than 20 mm at a point when the model simulates 10 mm of rainfall. For a simulated value of 25 mm, the probability of observing rainfall ≤ 20 mm is 0.58, that is there is 58% of chance of observing rainfall ≤ to 20 mm when the WRF model simulates 25 mm of rainfall at a point. The probability to observe rainfall intensity greater than 20 mm is 0.42; that is 42% of chance.

Based on the above analysis, it appears that the probability of observing rainfall above a given threshold increases with simulated rainfall amounts. This result is consistent with that of Tanessong et al. [

Thus the probability of observing rainfall greater than 20 mm is 0.25; yielding 25% chance. For simulated rainfall amount of 25 mm, the probability of observing rainfall ≤ 20 mm is 0.7; 70% chance. The chances of observing the rainfall greater than 20 mm are 30%.

Unlike Region 2, we note that the chances of observing rainfall greater than a given threshold increase weakly when the simulated rainfall amount increase in

Region 3. This could be due to complex topography of Region 3. This region includes the Niger valley, the west highlands of Cameroon, the Adamawa Plateau of Cameroon and Mount Cameroon. The climate of this region is very diverse and complex.

posterior distributions functions. For simulated rainfall amount of 1 mm, the probability of observing less than or equal to 20 mm rainfall is 0.8; that is 80% chance.

The probability of observing rainfall amount greater than 20 mm is 0.2; 20% chance.

For simulated rainfall amount of 10 mm, the probability of observing rainfall ≤ 20 mm is 0.6; that is 60% chance. The probability to observe rainfall amount greater than 20 mm is 0.4. The probability of observing rainfall amounts ≤ 20 mm knowing that the simulated rainfall amount is 25 mm is 0.5 and the probability to observe rainfall greater than 20 mm is 0.5.

For simulated rainfall amount of 1 mm, the probability of observing rainfall ≤ 20 mm is 0.8 and that to observe rainfall greater than 20 mm is 0.2 (see

The Bayesian Processor of Output method was used to produce Probabilistic Quantitative Precipitation Forecast over Central and West Africa. It processes output from a NWP model and optimally fuses it with climatic data in order to quantify uncertainty about a predictand. Outputs from the Weather Research and Forecasting (WRF) model were used to develop and test the BPO technique. The model ran from June 01 to September 30 of 2010 and 2011. The BPO method was applied in each grid point and then in each climatic zones. Prior (climatic) distribution function was estimated from the Tropical Rainfall Measuring Mission (TRMM) data for the period 2002-2011. Many distribution functions have been tested for the fitting. Weibull distribution was found to be a

suitable fitting function as shown by goodness of fit (gof) test in both cases. BPO method noticeably improves the distribution of precipitation as shown by the spatial correlation coefficients, reliability diagrams and relative operating cha-

racteristic curves. It better detects certain observed maxima compared to the raw WRF outputs. Posterior distribution (forecasting) functions allow for a simulated rainfall amount, to deduce the probability that observed rainfall falls above a given threshold. The probability of observing rainfall above a given threshold increases with simulated rainfall amounts. The forecasting functions determined in the present paper can be used by forecasters as guidance for issuing probabilistic forecasts from a single deterministic forecast. In addition, this forecasting tool might assist forecasters throughout the season in a wide variety of weather events.

WRF simulations were done on a workstation provided by Dr Serge Janicot of LOCEAN (Paris), in the framework of the PICREVAT project, funded by the French government. WRF was provided by the University Corporation for Atmospheric Research website (for more information see http://www2.mmm.ucar.edu/wrf/users/download/get_source.html). GPCP data were obtained from the NOAA website http://www.esrl.noaa.gov. TRMM data were provided online by NASA at http://mirador.gsfc.nasa.gov.

Tanessong, R.S., Vondou, D.A., Igri, P.M. and Kamga, F.M. (2017) Bayesian Processor of Output for Probabilistic Quantitative Precipitation Forecast over Central and West Africa. Atmospheric and Climate Sciences, 7, 263- 286. http://dx.doi.org/10.4236/acs.2017.73019