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The Integral Equation Model with multiple scattering (IEMM) represents a well-established method that provides a theoretical framework for the scattering of electromagnetic waves from rough surfaces. A critical aspect is the long computational time required to run such a complex model. To deal with this problem, a neural network technique is proposed in this work. In particular, we have adopted neural networks to reproduce the backscattering coefficients predicted by IEMM at L- and C-bands, thus making reference to presently operative satellite radar sensors,

Modeling the electromagnetic wave scattering from random rough surfaces is an important issue for remotely sensing both land (e.g., soil moisture and roughness) and ocean (speed and direction of the wind blowing over the sea surface) geophysical parameters from satellite microwave sensors. As a consequence, a number of theoretical models were developed to deal with this problem. These models necessarily made some simplifying assumptions because of the great complexity of realistic scattering problems [

If a very rough surface is considered, the phenomenon of multiple scattering should be accounted for [

An alternative to complex theoretical models is represented by semiempirical techniques. These techniques were widely adopted in the literature, for instance to predict the backscattering coefficient measured by a microwave radar (e.g., a Synthetic Aperture Radar: SAR) aboard satellites or aircrafts (e.g., [

From the previous discussion, the need to join the simplicity and the efficiency of the semiempirical backscattering models to the precision of physical ones clearly emerges. To succeed in combining these two key features, a neural network approach can be attempted. Since a multilayer feed-forward neural network (NN), having at least one hidden layer, can approximate any nonlinear function relating inputs to outputs [

In this work, a neural network approach to the problem of reproducing the behavior of the IEMM is proposed. We have considered only the backscattering case, because the radar sensors presently operative are monostatic systems, although bistatic experiments have been recently envisaged (e.g., [_{I}_{i}_{v}^{0}). The incidence angles previously mentioned (hereafter denoted also as nominal incidence angles) have been considered in this case. Successively, a second exercise has been carried out in which the incidence angle has been assumed as additional input parameter in order to make the NN-based model applicable for simulating observations of terrains with complex topography. Other four databases (both training and test sets for the two frequencies) have been set up for this purpose. The validation of our method has been carried out by comparing, for the test databases, the IEMM-derived ^{0} with the NN-derived ones.

In Section 2, a summary of the IEMM is provided, while Section 3 introduces the algorithm that has been selected to train the networks, gives some details about the various databases we have built to train and test the behavior of the networks, and describes the design of the NNs architecture. In Section 4, the results are discussed by assessing the simulations of the backscattering coefficients obtained by running the trained NNs against the IEMM outputs. Section 5 draws the main conclusions.

The IEMM can be considered as an extension of the Integral Equation based surface scattering model (IEM). With respect to the latter, the IEMM removes the assumption on the phase factor exp(

IEMM expresses the total scattered field as the sum of a term derived from the Kirchhoff tangent plane approximation [_{0} is the incident field amplitude, superscripts

In (3), _{0} is the electromagnetic wavenumber and _{s}_{i}_{s}_{i}_{qp}_{qp}_{p}^{k}_{p}^{k}_{p}^{c}_{p}^{k}_{i}_{0}^{−jk0}^{ki·r}_{qp}

Several approximations were accomplished to make _{qp}_{qp}_{qp}_{qp}_{qp}_{qp}_{qp,up}_{qp,down}

In [

We can conclude this brief summary by pointing out that the IEMM model leads to a more accurate calculation of the multiple scattering contribution with respect to IEM [

An artificial NN is a nonlinear parameterized mapping from an input vector

In the architecture of a NN, all nodes are interconnected to each other, and this interconnection is characterized by weights and biases. The hidden and output nodes are characterized by an activation function, which is generally assumed to be differentiable and nonlinear. Here, we have chosen the tan-sigmoid function, which is characterized by the node gain and the node bias.

The network is trained by a supervised learning using a training database _{r}_{r}_{D}

The minimization with respect to

It is worth considering that, when attempting to emulate rough surface scattering models, the NN training set can hardly encompass all the possible model inputs, because of the high variability of surface characteristics (e.g., roughness and soil moisture in our case). The capability of a neural network to properly respond to unexpected inputs is called generalization. The procedure to improve generalization, called regularization, usually adds an additional term to the error objective function. Such a modified function, denoted as _{R}_{W}_{i}^{T}_{D}_{W}_{D}_{W}_{D}_{D}_{W}

A critical issue of the regularization procedure is the evaluation the optimal values for the regularization parameters α_{D}_{W}

For each of the two frequency bands considered here, two cases have been considered, so that four NNs have been designed. All the training and test databases have been built by means of the IEMM, whose runs have been performed by adopting an exponential autocorrelation function (ACF). In the first case, for which the incidence angle has been maintained constant (C-band: 23°; L-band: 34°), a training set consisting of _{r}_{v}_{v}_{0}_{0}_{0} = 2π/

In the second case, the incidence angle has been assumed as an additional parameter to allow the NN-based model to be suitable for terrains with complex topography too. While the soil parameters have been randomly generated with the same limits as those listed above (so that an exponential ACF has been chosen again to run the IEMM), for _{i}_{r}

For the soil permittivity, a model proposed in the literature that relates it to soil moisture, temperature and composition has been selected [

The same procedure followed to produce the training databases has been employed to build the test sets. 500 input/output pairs have been generated (both at C- and L-bands and independently of those produced for the training database) considering the nominal _{i}_{i}

The architectures of the NNs designed to reproduce the ^{0} predicted by the IEMM model at C- and L-bands consist of three (corresponding to _{v}, s_{v}, s, l_{i}_{i}_{o}_{h}_{h}

From _{R}_{h}_{h}_{h}_{E}_{h}

Considering that a two-hidden-layer MLP network can approximate any function to any degree of nonlinearity (see Section 3) and taking into account the high degree of nonlinearity of a very complex scattering model such as the IEMM, we have also designed an architecture consisting of two hidden layers of _{h}_{1} and _{h}_{2} neurons, respectively. The results are reported on _{R}_{E}_{R}_{h}_{1} = 15 and _{h}_{2} = 10.

The same exercise has been accomplished for the case in which _{i}_{i}_{R}_{h}_{R}_{h}_{1} and _{h}_{2}. By looking at _{i}_{R}_{h}_{1} = 30, _{h}_{2} = 25. An architecture defined by: _{i}_{h}_{1} = 15, _{h}_{2} = 10 and _{o}_{i}_{i}_{i}_{h}_{1} = 30, _{h}_{2} = 25 and _{o}

Since the activation functions of the hidden layers are tan-sigmoid, and those of the output neurons are linear, these architectures imply that the NN outputs can be expressed by the following relationships:
_{ji}_{j}_{hj}_{h}_{kh}_{k}_{i}

Before ending this section, it is worth underlining that the generation of the backscattering coefficients by means of the IEMM has taken about 24 hours per 1,000 records (by employing a personal computer with a 3.2 GHz Pentium 4 processor and 2 GB of RAM). The two networks whose architecture is shown in the upper panel of

To assess the proposed approach, we have compared the backscattering coefficients produced by the IEMM model and belonging to the test sets with those generated by the NNs for the same inputs. Note that, as pointed out at the end of Section 3, the trained NNs generate outputs in the interval [−1,1], so that an inverse normalization has been accomplished to restore the nominal range for the backscattering coefficients.

_{i}

The performances can be also evaluated by looking at ^{0} predicted by NNs and those estimated by IEMM) and root mean square error

Other details on the behavior of the NN-emulators designed for the nominal incidence angle can be inferred by looking at _{0}_{0}_{0}

It can be expected that the NNs designed to include the incidence angle in the input parameters (_{i}_{i}

Comments on _{0}_{0}_{i}_{i}

A final test has been accomplished to assess the reliability of the proposed approach. Considering separately the test sets built for the nominal incidence angle (L-band: 34°; C-band: 23°), we have added to the IEMM-based ^{0} a random Gaussian noise of zero mean and 1 dB of standard deviation, in order to simulate both the model and the instrument errors. Then, we have tried to retrieve the corresponding input _{v}

In (13), ^{0} to which the noise was added thus simulating a radar sensor measurement, while with subscript NN we indicate the outputs of the trained network. The minimization of (13) has been performed by applying a simulated annealing technique (e.g., [

_{v}

A neural network approach to approximate the behavior of the Integral Eqution Model with multiple scattering (IEMM) has been proposed to deal with the problem of the IEMM computational efficiency. The backscattering coefficients evaluated at C-band, considering an observation angle of 23° and at L-band, assuming an observation angle of 34°, have been initially considered to evaluate the reliability of the methodology. The approach has been also extended by considering the incidence angle as an additional input parameter to make the derived model applicable for terrains with complex topography. The use of neural networks has considerably decreased the computational time required by the IEMM.

It has been proved that the neural networks we have designed reproduce the behavior of IEMM fairly well both for flat and for tilted surfaces. The correlation between IEMM- and NN-derived backscattering coefficients has turned out to be close to the unit and we have also found an almost null mean error and a root mean square error not exceeding 1.3 dB at L-band and 0.7 dB at C-band (considering a variable incidence angle). A simple theoretical exercise has also indicated that the use of the trained networks within an iterative retrieval algorithm can be suitable for a typical problem such as retrieving soil moisture from radar data.

It is worth noting that the proposed approach can also be used to train the network on a database merging both model outputs and real measurements, thus providing a way to deal with the uncertainties of any theoretical model.

The Neural Network architectures. Upper panel: NN topology for the nominal incidence angle (_{i}_{h}_{1} = 15, _{h}_{2} = 10, _{o}_{i}_{h}_{1} = 30, _{h}_{2} = 25, _{o}

Comparison between IEMM- and NN-derived ^{0} (test sets of 500 records). Left panels: vertical polarization; right panels: horizontal polarization. Upper panels: L-band; lower panels: C-band. Dotted lines represent perfect agreement.

Trend of the _{0}

Same as _{i}

Same as _{i}

Number of epochs (_{E}_{r}_{R}_{h}_{i}_{i}

_{h} |
_{E} |
_{R} |
---|---|---|

5 | 79 | 10.5 |

10 | 134 | 5.4 |

15 | 286 | 3.1 |

20 | 343 | 2.9 |

25 | 545 | 2.5 |

Same as _{h}_{1} and _{h}_{2} neurons, respectively.

_{h}_{1} |
_{h}_{2} |
_{E} |
_{R} |
---|---|---|---|

8 | 7 | 380 | 0.56 |

12 | 7 | 420 | 0.29 |

12 | 10 | 440 | 0.17 |

15 | 7 | 645 | 0.10 |

15 | 10 | 920 | 0.04 |

17 | 12 | 1,350 | 0.03 |

Same as _{i}_{i}_{r}

_{h}_{1} |
_{h}_{2} |
_{E} |
_{R} |
---|---|---|---|

15 | 10 | 780 | 2.81 |

20 | 15 | 962 | 0.77 |

25 | 20 | 1,202 | 0.32 |

30 | 25 | 1,580 | 0.06 |

Comparison between IEMM- and NN-derived ^{0} in terms of correlation coefficient

| ||||
---|---|---|---|---|

L-band | C-band | L-band | C-band | |

0.99 | 1.00 | 0.99 | 1.00 | |

−0.08 | −0.01 | −0.02 | −0.01 | |

0.66 | 0.31 | 0.78 | 0.27 |

Same as _{i}

| ||||
---|---|---|---|---|

L-band | C-band | L-band | C-band | |

0.95 | 0.99 | 0.98 | 1.00 | |

0.05 | −0.03 | 0.05 | 0.01 | |

1.21 | 0.66 | 1.26 | 0.52 |

Results of the comparison between the soil moistures belonging to the test sets built for fixed _{i}

0.88 | 0.74 | |

^{3}/m^{3}] |
0.01 | 0.01 |

^{3}/m^{3}] |
0.05 | 0.07 |