Forecasting the spatiotemporal variability of soil CO₂ emissions in sugarcane areas in southeastern Brazil using artificial neural networks

Abstract

Carbon dioxide (CO₂) is considered one of the main greenhouse effect gases and contributes significantly to global climate change. In Brazil, the agricultural areas offer an opportunity to mitigate this effect, especially with the sugarcane crop, since, depending on the management system, sugarcane stores large amounts of carbon, thereby removing it from the atmosphere. The CO₂ production in soil and its transport to the atmosphere are the results of biochemical processes such as the decomposition of organic matter and roots and the respiration of soil organisms, a phenomenon called soil CO₂ emissions (FCO₂). The objective of the study was to investigate the use of neural networks with backpropagation algorithm to predict the spatial patterns of soil CO₂ emission during short periods in sugarcane areas. FCO₂ values were collected in three commercial crop areas in the São Paulo state, southeastern Brazil, registered through the LI-8100 system during the years 2008 (Motuca), 2010 (Guariba city), and 2012 (Pradópolis), in the period after the mechanical harvesting (green cane). A neural network multilayer perceptron with a backpropagation algorithm was applied to estimate the FCO₂ in 2012, using data from 2008 and 2010 as training for the neural network. The neural network initially presented a mean absolute percentage error (MAPE) of 18.3852 and a coefficient of determination (R²) of 0.9188. Data obtained from the observed and estimated values of FCO₂ present moderate spatial dependence, and it is observed from the maps of the spatial pattern of the CO₂ flow that the results from the neural network show considerable similarity to the observed data. The model results identify the higher and lower characteristics in sample points of CO₂ emissions and produce an overestimation of the range of spatial dependence (0.45 m) and an underestimation of the interpolated values in the field (R² = 0.80; MAPE = 12.0591), when compared to the actual soil CO₂ emission values. Therefore, the results indicate that the artificial neural network provides reliable estimates for the evaluation of FCO₂ from data of the soil’s physical and chemical attributes and describes the spatial variability of FCO₂ in sugarcane fields, thereby contributing to the reduction of uncertainties associated with FCO₂ accountings in these areas.

Backpropagation algorithm

The initial weights are usually adopted as random numbers (Widrow and Lehr 1990). The backpropagation (BP) algorithm consists of adapting the weights such that the network quadratic error is minimized. The sum of the instantaneous quadratic error of each neuron of the last layer (network output) is given by (Widrow and Lehr 1990):

$$ {\varepsilon}^2=\sum \limits_{i=1}^{no}{\varepsilon_i}^2 $$

(9)

where

ε _i :

d_i − y_i;

d _i :

desired output of the ith element of the last layer;

y :

output of the ith element of the last layer; and

no :

number of neurons of the last layer.

Considering the ith network neuron and using the descent gradient method, the weight adjustments can be formulated by (Widrow and Lehr 1990):

$$ {\boldsymbol{\varGamma}}_i\ \left(r+1\right)={\boldsymbol{\varGamma}}_i\ (r)+{\theta}_i\ (r) $$

(10)

where

θ_i (r):

− γ [∇_i (r)];

γ :

stability control parameter or training rate;

∇_i (r):

quadratic error gradient related to neuron i weights; and

Γ _i :

vector containing neuron i weights

=:

[w_0i, w_1i, w_2i… w_ni]^T.

The adopted direction in Eq. (1012) to minimize the objective function of the quadratic error corresponds to the gradient opposite direction. The γ parameter determines the vector length ϕ_i(r). The sigmoid function is defined by (Widrow and Lehr 1990):

$$ {y}_i\underset{\_}{\varDelta }{y}_i\left(\lambda, {\vartheta}_i\right)=\left\{\right(1+\mathit{\exp}\left(-\lambda {\vartheta}_i\right)\left\}/\right\{\left(1+\mathit{\exp}\left(-\lambda {\vartheta}_i\right)\right\} $$

(11)

or

$$ {y}_i\underset{\_}{\varDelta }{y}_i\left(\lambda, {\vartheta}_i\right)=1/\left\{\right(1+\mathit{\exp}\left(-\lambda {\vartheta}_i\right)\Big\} $$

(12)

where

λ :

constant that determines the slope of the function y_i.

The variation of Eqs. (11) and (12) are (− 1,+ 1) and (0,+ 1), respectively.

Next, calculating the gradient as shown in Eq. (1012) and considering the sigmoid function defined by Eq. (11) or (12) and the momentum term, the following adaptation weight scheme is obtained (Lopes et al. 2003):

$$ {\varPi}_{ij}\ \left(r+1\right)={\varPi}_{ij}\ (r)+{\varDelta \varPi}_{ij}\ (r) $$

(13)

where

$$ \varDelta {\varPi}_{ij}(r)=2\gamma \left(1-\eta \right){\beta}_j{x}_i+\eta \varDelta {\varPi}_{ij}\left(r-1\right); $$

(14)

Π_ij weight corresponding to the connection with the ith and the jth neuron;

γtraining rate; and

ηmomentum constant (0 ≤ η < 1).

If the jth element is in the last layer, then:

$$ {\beta}_j={\sigma}_j\ {\varepsilon}_j $$

(15)

where

σ _j :

Δ sigmoid function derivative, given by Eq. (11) or (12), respectively, related to ϑ_j

$$ =\lambda /2\left(1-{y_j}^2\right). $$

(16)

$$ =\lambda {y}_j\left(1-{y_j}^2\right). $$

(17)

If the jth element is in other layers, we have:

$$ {\beta}_j={\sigma}_j\sum \limits_{k\in \varGamma (j)}{w}_{jk}{\beta}_k $$

(18)

Γ(j):

set of the element indices that are in the next layer to the jth element layer and are interconnected to the jth element.

The γ parameter that is used as a stability control for the iterative process is dependent on λ. The network weights are randomly initialized from the interval [0,1]. For convenience, the parameter γ (training rate) can be redefined by the following (Lopes et al. 2003):

$$ \gamma ={\gamma}^{\ast }/\lambda $$

(19)

Replacing Eq. (19) in Eq. (12) will “cancel” the amplitude dependency of σ related to λ. The σ amplitude will be maintained constant to every λ. This alternative is important considering that λ will only actuate in the left and right tails of σ. Equation (14) can then be written as the following:

$$ \varDelta {\varPi}_{ij}(r)=\left\{2{\gamma}^{\ast}\left(1-\eta \right){\beta}_j/\lambda \right\}{x}_i+\eta \varDelta {\varPi}_{ij}\left(r-1\right). $$

(20)

The BP algorithm executes as follows (Widrow and Lehr 1990):

• Step 1. Present a pattern X to the network, which provides an output Y.

• Step 2. Calculate the error (difference with the desired value and the output) for each output.

• Step 3. Determine the backpropagated error by the network associated with the partial derivative of the quadratic error.

• Step 4. Adjust the weights of each element.

• Step 5. Present a new pattern to the network and repeat the process until the convergence is attained (according to a predefined tolerance).