Accurate interpretation of geological structures inverted from gravity data is highly dependent on the coverage of the recorded gravity data. In this work, Artificial Neural Networks (ANNs) are implemented using Levenberg-Marquardt algorithm (LMA) to construct a background density model for predicting gravity data across Northern Cameroon and its surroundings. This approach yields statistical predictions of gravity values (low values of errors) with 97.48%, 0.10, and 0.89, respectively, for correlation, Mean Bias Error, and Root Mean Square Error for two inputs (latitude, longitude) and 97.08%, 0.13, and 1.14 for three inputs (latitude, longitude, and elevation) for a set of anomalies as output. The model validation is obtained by comparing the results to other classical approaches and to the computed Bouguer, lineaments, and Euler maps obtained from measured gravity data. The depth of most of the deep faults and their orientation are in agreement with those obtained from other studies. The results achieved in this study establish the possibility of enhancing the quality of the analysis, interpretation, and modeling of gravity data collected on sparse grid of recording stations.
The Northern Cameroon and its surroundings, the subject of this study, have prompted many researchers and prospectors to identify superficial and deep structures and to indicate their geodynamics and tectonic implications [
Haykin [
Spatial distribution in the study area.
The data used in this study cover three main domains: northern extension of Benue trough in the East of Nigeria, Northern Cameroon, and West of Chad (Doba basin). Fairhead and Okereke [
Geological map of the study area:
The measurement of gravity fields can be used to calculate relative and absolute values regarding variations of the fields across earth surface. It is linked to frameworks like Global Positioning System and Digital Terrain Model. The data are treated with respect to the equipment used (for example, Scintrex CG3/CG5 relative gravity meters or Micro-g Lacoste and Romberg as shown in Figure
Lacoste and Romberg G/D gravity meters (Micro-g Lacoste, USA).
The data is processed to remove undesirable influences from the surroundings in order to isolate Bouguer anomaly (BA). It represents the difference between the measured and calculated gravity (formula (
We aim to develop a connectionist model (Neural Network) to correlate Bouguer anomalies with the geographical variables. In general, an array of Artificial Neural Networks is a juxtaposition of unitary, functional, and interconnected elements [
Details of a neuron (with
The inputs
The output y of the MLP network is given by equation below:
From (
From an arbitrary weight (random value) defined at the beginning, the weights are adjusted by backpropagating the error according to the expression:
The data set is composed of 2922 samples which comprise latitude, longitude, elevation, and the corresponding Bouguer anomalies. These data are extracted from a database computed for the whole Cameroon by Collignon [
Multilayered Perceptron (MLP) network for (a) 2 inputs and (b) 3 inputs.
60% of these data (1754 samples) are used for training, 20% of the data (584 samples) are used for the validation, and 20% (584 samples) for testing the models. Data conditioning processes are conducted to speed up the training ANNs, which includes interpolating missing data, normalizing the data, and then randomizing them. Usually, the missing data are calculated imprecisely by averaging the neighboring values. In this study, the missing values are forecasted by ANN.
The diagram (Figure
General layout of the neural model.
To quantitatively evaluate the ANN and verify its trend, we conduct statistical analysis involving the coefficient of determination (R2), the Root Mean Square Error (RMSE), and the Mean Bias Error (MBE). The network structure identification is 2-190-1 and 3-370-1, respectively, for 2 and 3 inputs where the first number indicates number of neurons in the input layer, the last number represents neurons in the output layer, and the numbers in between represent neurons in the hidden layers. We present the best achieved results for the MLP ANN models (Figures
Performance (regression) of ANN model for prediction of Bouguer anomalies around Benue trough with 2 inputs.
Performance of ANN model for prediction of Bouguer anomalies around Benue trough with 3 inputs.
As shown in Figures
Comparison of present ANN model (blue color) and measured anomalies (red color) for two inputs (
MBE and RMSE yield very low values as shown below: For model 1 (model inputs L and l) the network structure is 2-190-1 for 0.95027, 0.10, and 0.89 representing, respectively, R2, RMSE and MBE. For model 2 (model input L-l-h), the network structure is 3-370-1 and the values of statistical errors R2, RMSE, and MBE are, respectively, 0.94254, 0.13, and 1.14.
The results indicate that, for the test base, there is a very good correlation (Figures
Comparison of present ANN model (blue color) and measured anomalies (red color) for three inputs.
Comparing observed and simulated data, we can see as shown in Figures
Comparing interpolation methods.
Methods | Correlation factor | Mean Bias Error | Root Mean Square Error |
---|---|---|---|
2 inputs | |||
| |||
ANN2 | 0,95 | 0,89 | 0,10 |
| |||
Kriging | 0,06114 | -0,00142648 | 12,2701515 |
| |||
Minimum Curvature | 0,06114 | -0,00142648 | 12,2701515 |
| |||
Radial Basis Function | 0,06114 | -0,00142648 | 12,2701515 |
| |||
Polynomial Regression | 0,06099 | 0,00476981 | 12,2701501 |
| |||
Multiple linear regression | 0,0610 | 5,2595e-11 | 12,2680 |
| |||
Inverse distance to a Power | 0,06099 | 0,00476981 | 12,2701501 |
| |||
3 inputs | |||
| |||
Multiple linear regression analysis | 0,2273 | -9,5263e-11 | 11,1289 |
| |||
ANN3 | 0,942 | 1,14 | 0,13 |
For three inputs, we use a classical multiple linear regression implemented through a matrix approach programmed [
For Multiple Linear Regression Analysis, the following constants were obtained: Multiple Linear Regression Analysis with 2 inputs (MLRA2): Multiple Linear Regression Analysis with 3 inputs (MLRA3):
In Figure
Errors against the number of iterations and neurons in the hidden layer for 2 inputs.
Iterations | NN | RMSE | MBE |
---|---|---|---|
34 | 10 | 6,33 | -0,25 |
| |||
97 | 30 | 3,68 | -0,17 |
| |||
37 | 50 | 2,92 | 0,24 |
| |||
92 | 70 | 2,03 | 0,039 |
| |||
47 | 90 | 1,77 | 0,049 |
| |||
28 | 120 | 1,27 | 0,05 |
| |||
63 | 150 | 1,15 | -0,165 |
| |||
21 | 170 | 1,14 | 0,13 |
| |||
64 | 180 | 0,88 | 0,0456 |
| |||
36 | 190 | 0,88 | 0,0456 |
| |||
15 | 200 | 1,27 | -0,15 |
MLRA3 (red color) versus Bouguer anomalies measured (blue color).
Root Mean Square Error (RMSE) versus number of neurons in the hidden layer (NNHL) for two inputs.
Plotting errors against iterations to us did not have any mathematical explanation; instead we plot errors against number of neurons in the hidden layer where it is obvious that one has low value of RMSE with increasing number of neurons in the hidden layer, unless the situation (above 190) where there is an overfitting (increasing RMSE) exists. The same conclusion arises for three inputs (Figure
Root Mean Square Error (RMSE) versus number of neurons in the hidden layer (NNHL) for three inputs.
The results obtained show very good precision for Neural Network compared to classical approaches. Though ANNs can approximate any function, regardless of its linearity, they have some limitations such as their “black box” nature, greater computational burden, increasing accuracy by a few percent which can bump up the scale by several magnitudes (proneness to overfitting), and the empirical nature of model development (needs a lot of data for the training and cases for validation and test).
Bouguer, Bouguer 2 and 3 entries maps (Figures
Bouguer (a), lineaments (b), and Euler maps (c).
Bouguer (a), lineaments (b), and Euler maps (c) for two entries.
Bouguer (a), lineaments (b), and Euler (c) maps obtained for three inputs.
By using and comparing also Bouguer, Bouguer 2 entries, and Bouguer 3 entries, there is also a similarity between the lineament maps (Figures
The lineaments obtained are later compared with existing results from other works. From Figures
The ANN based model for gravity anomalies is accurate for the prediction of these anomalies in Northern Cameroon and its surroundings.
In this paper, an Artificial Neural Network (ANN) model was estimated for the prediction of gravity anomalies using, respectively, two (longitude and latitude) and three (longitude, latitude, and elevation) inputs in Northern Cameroon and its surroundings along with the corresponding anomaly. Existing gravity data were used for training, validation, and testing of the Neural Network. With each of these inputs, we obtained a good correlation on the plot for regression for all data in both networks, where R2 has values of 0.95027 and 0.94254, respectively. In order to validate the model, results were compared to those from classical interpolation approaches; in addition Bouguer, Euler, and lineaments maps were compared to our prediction. Low values of MBE and RMSE indicate the effectiveness of the approach. We provide in this work new deep faults for the studied area. The model is promising for evaluating the gravity anomaly at a specific point where there is no measured value. This method can therefore be recommended in geophysics to improve the resolution of geological features for uneven coverage of recorded gravity data and also to reduce the cost of geophysical surveys.
The data used to support the findings of this study are available from the corresponding author upon request.
The authors declare that there are no conflicts of interest regarding the publication of this paper.
The authors are indebted to IRD (Institut de Recherche pour le Developpement) for providing them with the data used in this work.