# Performance of Using Cascade Forward Back Propagation Neural Networks for Estimating Rain Parameters with Rain Drop Size Distribution

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

^{−1}) (Z-R relationship). The observed precipitation’s variability is due to the raindrop size distribution whose instability is reliable for some factors like speed, collision or agglomeration of rain drops [4,5]

**Figure 1.**Radar reflectivity factor Z vs. rain rate R as deduced from the 1-min rain RDSD (Rain Drop Size Distribution) observed with the JWD (Joss-Waldvogel Disdrometer) in Dakar (Senegal), during 1997–2002, and fitted curve.

## 2. Theoretical Background on Artificial Neural Networks

#### 2.1. Basic Principles

#### 2.2. Model of a Neuron in a Neural Network

**n**, with b as bias, of this integrator, is transformed by a transfer function

**f**in order to produce an output

**a**. The matrix representation of the neuron is shown on Figure 2b, with S supposed to be equal to the unity, to signify that there is only one neuron [16].

**R**entries of the neuron correspond to the vector P = [P

_{1}P

_{2}… P

_{R}]

^{T}while W = [W

_{11}W

_{12}… W

_{1R}]

^{T}represents the vector weights of the neuron. The output

**n**of the integrator is then described by Equation (1):

**f**becomes zero or positive, the activation level reaches or is superior to the bias b, if not, it is negative [16,17].

^{T}P − b)

#### 2.3. Construction of a Neural Network and its Learning Process

^{k}, where k designates the index of the layer. Thus, the vectors

**b**

^{k},

**n**

^{k}and

**a**

^{k}are associated to the layer

**k**. To specify the neural network structure, the number of layers and the number of neurons in each layer must be chosen. The learning step is a dynamic and iterative process which consists in modifying the parameters of the network after receiving the inputs from its environment. The learning type is determined by the way the change of the parameters occurs [14]. In almost all neural architectures encountered, the learning results in the synaptic modification of the weights connecting one neuron to the other. If w

_{ij}(t) designates the weight connecting the neuron i to its entry j and at time t, a change ∆w

_{ij}(t) of the weights can simply be expressed by the Equation (3):

_{ij}(t) = w

_{ij}(t + 1) − w

_{ij}(t)

_{ij}(t + 1) represents the new entry value of the weight w

_{ij}.

#### 2.4. The Feed Forward Back Propagation Method Used by a Multilayer Neural Network

^{k}= f

^{k}[W

^{k}a

^{k −1}− b

^{k}], k = 1, …, M

_{q}, d

_{q})}, q = 1, …, Q, where P

_{q}designates an entry-vector and d

_{q}a desired output-vectors, we can forward propagate at each instant t, an entry-vector P(t) through the neural network in order to get an output-vector a(t).

**Figure 3.**An example of synoptic representation of a 3 layer-neural network in a feed forward back propagation form.

^{T}(t) · e (t)]

## 3. Localities and Data Used for the Study

Town (Country) | Collection Period of the RDSD | Total Number of the RDSD (min) |
---|---|---|

Abidjan (Ivory Coast) | June; September to December: 1986 | 23,126 |

Boyele (Congo) | March to June: 1989 | 18,354 |

Debuncha (Cameroon) | May to June 2004 | 39,113 |

Dakar (Senegal) | July to September: 1997, 1998 et 2000 | 15,145 |

Niamey (Niger) | July to September: 1989 | 4468 |

**Table 2.**Values of the exponent factor p and of the multiplicative factor a

_{p}in relation to the parameter studied ( ).

Parameter | Symbol P | Exponent Factor p | Unit | Multiplicative Factor a_{p} |
---|---|---|---|---|

Rain rate | R | 3.67 | mm∙h^{−1} | 7.1 × 10^{−3} |

Liquid water content | W | 3 | g∙m^{−3} | (π/6) × 10^{−3} |

Radar reflectivity factor | Z | 6 | mm^{6}∙m^{−3} | 1 |

^{−1}∙m

^{−3}), D (mm) is the diameter of the equivalent sphere, and a

_{P}a coefficient which depends on the type of parameter considered and on the chosen units, so as shown in t Table 2. The stated disdrometer observation normally has large errors for small drop size and large drop size [27]. The coefficient p in Equation (13) is issued from the definition of the parameters; Z and R are respectively defined as the moment of order 6 and 3.67 of the diameter D [28].

## 4. Methodology

- Initialize the weights with small random values;
- For each combination (p
_{q}, d_{q}) in the learning sample:- Propagate the entries p
_{q}forward through the neural network layers:a^{0}= p_{q}; a^{k}= f^{k}(W^{k}a^{k −1}− b^{k}), k = 1, …, M - Back propagate the sensitivities through the neural network layers:δ
^{M}= −2F^{′M}(n^{M}) (d_{q}− a^{M}); δ^{k}= F^{′k}(n^{k}) (W^{k +1})^{T}δ^{k+1}, k = M − 1, …, 1 - Modify the weights and biases:∆W
^{k}= −η δ^{k}(a^{k −1})^{T}, k = 1, …, M,

∆b^{k}= η δ^{k}, k = 1, …, M

- If the stopping criteria are reached, then stop; if not reached, they permute the presentation order of the combination built from the learning database, and begin again at Step 2.

#### 4.1. Description of the Databases and their Use in the Neural Models

_{1}, x

_{2}, …, x

_{25}]

^{T}. The base Y is a matrix of 23,126 × 3 dimensions whose three rows represent three different vectors, Y

_{1}, Y

_{2}and Y

_{3}. These three last vectors are known as the water content (M(g∙m

^{−3})), the rain rate (R(mm∙h

^{−1})) and the radar reflectivity (Z(mm

^{6}∙m

^{−3})), each retrievable individually, from the X database. With each sample j of the first database is then associated a real number y

_{ij}(i = 1,2,3; j = 1,2,…,23126). In this work, we have to retrieve these parameters. The different neural networks have as entries the parameters x

_{k}(k = 1,2,…,25) and as outputs the parameter y

_{i}(i = 1,2,3). The neural model is a cascade forward back propagation (CFBP) one, with a hidden layer of 20 neurons (Figure 7a,b). We develop for each locality a model which will be applied to other localities before building another model, which will integrate all five localities. This last model will be applied on all five localities.

**Figure 7.**ANN models: (

**a**) description of the model; (

**b**) the faster CFBP model, which was used for our simulations.

- -
**R**entry’s number (25 in our case)- -
**S1:**Number of neurons in the first layer (hidden layer): 20 in our case- -
**S2:**Number of neurons in the output layer: 1 in our case- -
**b1:**Bias’ vector of the first layer with the dimension S^{1}× 1 (20 × 1 dimension in our case)- -
**b2:**Bias’ vector of the output layer with the dimension S^{2}× 1 (1 × 1 in our case)- -
**W1:**Matrix of weights according to the neurons of the first layer with the dimension S^{1}× R (20 × 25 dimension in our case);- -
**W2:**Matrix of weights according to the neurons of output layer with the dimension S^{2}× S^{1}(1 × 20 dimension in our case)

#### 4.2. Construction and Use of the Different Neural Networks

_{i}. The initial 5000 entries and the primary 5000 desired outputs, except Niamey with only 4468 sample data, are used for the training of the neural networks, and the 5000 following sample data (from the 5001st to 10,000th) have served as test data. After the training, we have obtained acceptable performance curves and have stored the LANN, that we designate by the indices, i, to confer them the role of retrieving the parameter y

_{i}. Each LANN i of a given locality is then invited to predict the parameters y

_{i}over the other localities. In this case, we have used as LANN inputs the entries obtained in the locality subjected to the predictive process. We estimate for the samples from the 5001st–10,000th the outputs that we compare with the available experiment present in this locality.

_{i}(M, R, Z) a polyvalent model (PANN): we extract from the databases of each locality the first 1200 sample data and constitute a training set composed of 6000 training data combinations. Each PANN i (i indicating its role of retrieval of parameter Y

_{i}) is then used to predict the parameters Y

_{i}over all localities. As follows, we use the designations from Table 3 and the notations M, R and Z, respectively, for the variables Y1, Y2 and Y3.

Locality | Neural Model Developed by Data from the Locality |
---|---|

Abidjan (Ivory Coast) | A |

Boyele (Congo Brazzaville) | B |

Debuncha (Cameroon) | C |

Dakar (Senegal) | D |

Niamey (Niger) | N |

All localities | PANN |

## 5. Results and Discussions

#### 5.1. Capability of a Neural Network to Estimate the Parameters in Various Other Localities

#### 5.2. Estimation of the Values of M, R and Z with the PANN

**Figure 8.**Neural prediction in Abidjan (Ivory Coast) of the liquid water content M (

**a**), the rain rate R (

**b**) and the radar reflectivity Z (

**c**) by a LANN constructed with data issued only from Debuncha (Cameroon).

#### 5.3. Evaluation of the Root Mean Square Errors (RMSE)

_{i}: Error (observed value−Neural Network estimate); and N: Number of observed values.

#### 5.3.1. RMSE between Measured Parameters in the Localities and Estimates Delivered by the LANN A, B, C, D and N

LANN | Abidjan | Boyele | Debuncha | |||||||

Y1 (M) | Y2 (R) | Y3 (Z) | Y1 (M) | Y2 (R) | Y3 (Z) | Y1 (M) | Y2 (R) | Y3 (Z) | ||

A | 0.01 | 0.01 | 0.58 | 4.87 | 0.35 | 870.16 | 17.60 | 1.25 | 3144.06 | |

B | 0.04 | 0.14 | 1.27 | 0.01 | 0.01 | 0.24 | 0.02 | 0.03 | 0.58 | |

C | 0.02 | 0.06 | 1.05 | 0.01 | 0.01 | 0.49 | 0.01 | 0.01 | 0.33 | |

D | 0.83 | 0.15 | 1.91 | 0.17 | 0.01 | 0.36 | 0.48 | 0.06 | 0.66 | |

N | 0.07 | 0.54 | 3.77 | 0.01 | 0.01 | 0.46 | 0.03 | 0.13 | 0.83 | |

LANN | Dakar | Niamey | ||||||||

Y1 (M) | Y2 (R) | Y3 (Z) | Y1 (M) | Y2 (R) | Y3 (Z) | |||||

A | 6.11 | 0.44 | 1091.45 | 3.57 | 0.26 | 638.14 | ||||

B | 0.01 | 0.01 | 0.35 | 0.01 | 0.01 | 0.25 | ||||

C | 0.01 | 0.01 | 0.62 | 0.01 | 0.01 | 0.45 | ||||

D | 0.01 | 0.01 | 0.15 | 0.13 | 0.01 | 0.22 | ||||

N | 0.02 | 0.02 | 0.38 | 0.01 | 0.01 | 0.13 |

**Table 5.**Choice of the two best LANN for retrieval of the parameters in the different localities (A, B, C, D and N).

Abidjan | Boyele | Debuncha | Dakar | Niamey | ||||||
---|---|---|---|---|---|---|---|---|---|---|

1st | 2nd | 1st | 2nd | 1st | 2nd | 1st | 2nd | 1st | 2nd | |

LANN | LANN | LANN | LANN | LANN | LANN | LANN | LANN | LANN | LANN | |

M | A | C | B | C | C | B | D | C | N | C |

R | A | C | B | D | C | B | D | C | N | B |

Z | A | C | B | D | C | B | D | B | N | D |

#### 5.3.2. RMSE between Values Measured by the Disdrometer and Estimates Delivered by the PANN

Abidjan | Boyele | Debuncha | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

M | R | Z | M | R | Z | M | R | Z | |||||

0.02 | 0.01 | 0.66 | 0.01 | 0.01 | 0.24 | 0.02 | 0.01 | 0.52 | |||||

Dakar | Niamey | ||||||||||||

M | R | Z | M | R | Z | ||||||||

0.01 | 0.01 | 0.25 | 0.01 | 0.01 | 0.15 |

LANN | Abidjan | Boyele | Debuncha | |||||||||||

Y1 (M) | Y2 (R) | Y3 (Z) | Y1 (M) | Y2 (R) | Y3 (Z) | Y1 (M) | Y2 (R) | Y3 (Z) | ||||||

A | 0.010 | 0.010 | 0.130 | 0.940 | 0.052 | 67.49 | 1.730 | 0.125 | 118.5 | |||||

B | 0.010 | 0.010 | 0.075 | 0.010 | 0.010 | 0.043 | 0.010 | 0.010 | 0.063 | |||||

C | 0.010 | 0.010 | 0.130 | 0.010 | 0.010 | 0.098 | 0.010 | 0.010 | 0.075 | |||||

D | 0.010 | 0.010 | 0.100 | 0.015 | 0.010 | 0.089 | 0.010 | 0.061 | 0.360 | |||||

N | 0.020 | 0.010 | 0.092 | 0.010 | 0.010 | 0.066 | 0.012 | 0.058 | 0.220 | |||||

LANN | Dakar | Niamey | ||||||||||||

Y1 (M) | Y2 (R) | Y3 (Z) | Y1 (M) | Y2 (R) | Y3 (Z) | |||||||||

A | 1.100 | 0.074 | 52.45 | 1.450 | 0.089 | 124.0 | ||||||||

B | 0.010 | 0.010 | 0.051 | 0.010 | 0.010 | 0.058 | ||||||||

C | 0.010 | 0.010 | 0.120 | 0.010 | 0.010 | 0.120 | ||||||||

D | 0.010 | 0.010 | 0.060 | 0.010 | 0.010 | 0.070 | ||||||||

N | 0.010 | 0.010 | 0.110 | 0.010 | 0.010 | 0.05 |

#### 5.3.3. Computation of the RMSE after Distinction of Stratiform and Convective Rains

**Stratiform Rainfall**

**Table 8.**RMSE between values measured by the disdrometer and those estimated by the PANN: Case of the stratiform rains.

Abidjan | Boyele | Debuncha | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

M | R | Z | M | R | Z | M | R | Z | |||||

0.010 | 0.010 | 0.065 | 0.010 | 0.010 | 0.060 | 0.010 | 0.010 | 0.016 | |||||

Dakar | Niamey | ||||||||||||

M | R | Z | M | R | Z | ||||||||

0.010 | 0.010 | 0.066 | 0.010 | 0.010 | 0.072 |

LANN | Abidjan | Boyele | Debuncha | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Y1 (M) | Y2 (R) | Y3 (Z) | Y1 (M) | Y2 (R) | Y3 (Z) | Y1 (M) | Y2 (R) | Y3 (Z) | ||||||

A | 0.011 | 0.010 | 0.140 | 13.91 | 1.030 | 2916.0 | 42.35 | 3.160 | 9009.7 | |||||

B | 0.076 | 0.310 | 3.040 | 0.020 | 0.015 | 0.7731 | 0.045 | 0.053 | 1.6500 | |||||

C | 0.030 | 0.130 | 2.510 | 0.015 | 0.020 | 1.6100 | 0.020 | 0.010 | 0.9500 | |||||

D | 1.670 | 0.316 | 4.610 | 0.480 | 0.010 | 1.150 | 0.480 | 0.010 | 1.6000 | |||||

N | 0.134 | 1.150 | 9.110 | 0.025 | 0.027 | 1.500 | 0.065 | 0.280 | 2.3000 | |||||

LANN | Dakar | Niamey | ||||||||||||

Y1 (M) | Y2 (R) | Y3 (Z) | Y1 (M) | Y2 (R) | Y3 (Z) | |||||||||

A | 14.80 | 1.120 | 3410 | 10.17 | 0.760 | 2051.6 | ||||||||

B | 0.025 | 0.026 | 1.070 | 0.025 | 0.010 | 0.7700 | ||||||||

C | 0.020 | 0.017 | 1.890 | 0.020 | 0.010 | 1.4000 | ||||||||

D | 0.020 | 0.015 | 0.400 | 0.390 | 0.010 | 0.6700 | ||||||||

N | 0.040 | 0.187 | 1.130 | 0.014 | 0.012 | 0.3720 |

**Convective Rainfall**

**Table 10.**RMSE between values measured by the disdrometer and those estimated by the PANN: case of the convective rainfall.

Abidjan | Boyele | Debuncha | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

M | R | Z | M | R | Z | M | R | Z | ||||

0.022 | 0.020 | 1.600 | 0.020 | 0.010 | 0.770 | 0.030 | 0.010 | 1.420 | ||||

Dakar | Niamey | |||||||||||

M | R | Z | M | R | Z | |||||||

0.020 | 0.010 | 0.750 | 0.020 | 0.010 | 0.420 |

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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## Share and Cite

**MDPI and ACS Style**

Tengeleng, S.; Armand, N.
Performance of Using Cascade Forward Back Propagation Neural Networks for Estimating Rain Parameters with Rain Drop Size Distribution. *Atmosphere* **2014**, *5*, 454-472.
https://doi.org/10.3390/atmos5020454

**AMA Style**

Tengeleng S, Armand N.
Performance of Using Cascade Forward Back Propagation Neural Networks for Estimating Rain Parameters with Rain Drop Size Distribution. *Atmosphere*. 2014; 5(2):454-472.
https://doi.org/10.3390/atmos5020454

**Chicago/Turabian Style**

Tengeleng, Siddi, and Nzeukou Armand.
2014. "Performance of Using Cascade Forward Back Propagation Neural Networks for Estimating Rain Parameters with Rain Drop Size Distribution" *Atmosphere* 5, no. 2: 454-472.
https://doi.org/10.3390/atmos5020454