A Seven-Parameter BRDF Model with Double-Peak Characteristic Suitable for Sandy Soil

A seven-parameter BRDF model with double-peak characteristic was proposed in this paper, which can fit double-peak data. The global genetic algorithm was used to model the BRDF experimental data of sandy soil, and the parameter values and relative mean square error of the seven-parameter BRDF model were obtained. The results proved the correctness of the model, and the relative mean square errors of this model are, respectively, 0.30%, 0.22%, 0.26%, and 0.25% corresponding to the incident angles of 15, 30, 45, and 60. Additionally, we also combined data from four incident angles to derive seven parameters, which do not depend on incident angle, and the overall error is 1.79%. Finally, in order to intuitively show the BRDF of sandy soil, 3D BRDF graph of sandy soil with different incident angles is, respectively, given. It will be of great significance in practical project applications.


Introduction
Bidirectional Reflectance Distribution Function (BRDF) has been widely used in the fields of agricultural remote sensing, atmospheric radiation transmission, spectral scattering characteristics of rough surface targets, and material diagnosis [1][2][3].In the fields of agricultural remote sensing and digital agriculture, BRDF is used to study the spatial distribution characteristics and spectral characteristics of the reflected light of object, as shown in Figure 1.Through measuring BRDF of crop leaves, crop growth parameters can be accurately characterized and quantitatively analyzed.So it has great advantages for researches and applications in the fields of vegetation remote sensing, agriculture, etc. [4,5].In recent years, the measurement and reproduction of the entire appearance of an object are one of the most important research directions in the field of color vision, and they are also one of the main application directions of BRDF.The use of BRDF to measure and model the surface appearance of reflective objects is a new method for entire appearance description, which has a very good development prospect [6].In two-dimensional imaging, by method of BRDF the spectral and geometric characteristics of objects in a three-dimensional space can be captured, and the texture and color of the object surface are acquired, then under the required conditions the objects can be faithfully reproduced.Compared with traditional digital imaging, twodimensional BRDF imaging can better reflect the interaction among observers, artworks, lighting, etc., in which it has its unique advantages [7].In addition, environmental issues have become increasingly prominent in society.BRDF is also commonly used to measure the atmospheric environmental quality in a certain area.According to GOCIL1B data and solar zenith angle, combined with dark pixel method and 6S atmospheric radiation transmission model, area surface reflectance in typical weather can be inverted and accurately verified by using MODIS data [8,9].Since the reflection of visible light is different on the different materials surface, the use of BRDF to derive an effective light model is also the basis for the three-dimensional reconstruction [10,11].So it can be seen that BRDF has been widely used in aspects of society and the researches on it has become increasingly important.
In 2012, Yang Yufeng et al. combined a Minnaert model with a five-parameter semiempirical model to propose a sixparameter semiempirical model that satisfies the principle of conservation of energy and reciprocity under certain conditions [12][13][14].The model has a good effect on the target with weak scattering characteristics.In the same year, Bai Lu However, these models are only for the specific practical application scenarios to innovate and optimize.But so far, most BRDF models can only fit the sample data which has single-peak, and it is difficult to accurately fit sample data with double peaks.
The study of soil bidirectional reflection characteristics has important significance for the development of quantitative remote sensing and soil remote sensing technology, which is the core of soil water content survey, surface temperature, surface albedo, etc.At the same time, it is also an important factor that must be considered in the global ground remote sensing research [19][20][21][22][23][24][25].However, the BRDF data of

BRDF Seven-Parameter Statistical Model
The sandy soil sample is shown as Figure 2, whose particle size is less than 120m, and the BRDF measurement experiments were entrusted to the 802 Institute in Shanghai that is specialized in BRDF data measurement for many years, which provided BRDF data.During the experiment, REFLET system, as shown in Figure 3, was used for BRDF measurement, which is a compact mechanized optical system produced by Light Tech of France for the detection of scattering properties of all materials and objects.It can quickly and easily measure the light distribution of illuminants or the spectral composition of the scattering lobes.And a complete description of the scattering properties in 3D space after light enters the surface can be fully described.The completion time of this experiment is June 5, 2017, and the quality of the measured data meets the experimental requirements.
Sandy soil BRDF usually has double peaks, as shown in Figure 4.The single-peak BRDF data only has a maximum value at a certain scattering angle, and the double-peak data has two peaks whose scattering angles are almost symmetrical.Due to the coincidence of the laser transmitter and receiver at a certain angle, the receiver does not receive a signal at this angle, the data show a few individual minimums at the certain angle, as shown in Figure 4.
Figure 5 shows the 3D data plot of the sandy soil sample with incident angles of 15 ∘ , 30 ∘ , 45 ∘ , and 60 ∘ (when incident angle is equal to 0 ∘ , there is only one peak at the scattering zenith angle of 0 ∘ ; its form is simple.So the 3D graph of this angle is not given out).From Figure 5, it can be seen that, as the incident angle increases, the sandy soil BRDF data gradually shows the second peak.When the incident angle is 60 ∘ , two peaks can be clearly seen from Figure 5(d).
Based on above, we proposes a seven-parameter empirical BRDF model with double-peak characteristic suitable for sandy soil: where In ( 1

Seven-Parameter BRDF Model Validation
In order to verify the correctness of the model, sevenparameter BRDF model is verified by BRDF measured data of some sandy soil sample, in which the global genetic algorithm is used.Then we use MATLAB to fit the model to the data and perform error analysis on the model based on the relative mean square error.Because the BRDF value is small, the absolute error does not make sense, so we choose the relative mean square error here.The relative mean square error formula is as follows:

𝐸
In (3), N represents the total number of data, y data represents the measured data, and y model represents the value of the model fitting curve.
As shown in Table 1, we use a set of parameters to fit the data for each incident zenith angle.
In the following, we use the BRDF model to analyze BRDF experimental data of the sandy soil sample with the parameters in Table 1.There are four sets of data corresponding to four scattering zenith angles.We conduct error analysis on four sets of data, respectively.For computational convenience here, we remove individual minimums that occur at a certain angle due to coincidence of laser emitters and receivers.
Figure 6 shows BRDF experimental data of sandy soil sample and the fitting effect of the model.The wavelength is equal to 650nm and the incident zenith angles are, respectively, 15 ∘ , 30 ∘ , 45 ∘ , and 60 ∘ .As Figure 6 shows, the fitting effect of this model is very good, especially for BRDF experimental data with double peaks as shown in Figures 6(a), 6(b), 6(c), and 6(d), in which the error is only 0.30%, 0.22%, 0.26%, and 0.25%, respectively.It can be seen from Table 1 that the error in fitting the four groups of data is very small.But it can also be seen from Figure 6, as the incident angle increases, the sandy soil BRDF data gradually shows the second peak, and the value of the peaks are, respectively, 0.0965, 0.1104, 0.1406, 0.2495, and 0.2406.
According to the BRDF experimental data of four sets of scattering angle distributions at different incident angles in the incident surface, a large parameter range is set for the initial value of the parameter.When a set of better solutions is obtained, the solution is centered on this set of solutions.By narrowing down the original set of parameters  and repeating the calculation 2 to 3 times, it is basically possible to determine a set of general parameters suitable for different incident angles, as shown in Table 2, and the overall error is 1.79%.As shown in Figure 7, this set of data also has a good fitting effect, especially for BRDF experimental data with doublepeak characteristic; the seven-parameter BRDF model proposed in this paper can fit closer to the ideal near the peak.
Using the seven parameters derived, we can calculate the BRDF at any angle of incidence.In order to more clearly and intuitively represent the sandy soil BRDF, as shown in Figure 8, we draw out the fitting 3D graphs of sandy soil BRDF at 20 ∘ , 35 ∘ , 50 ∘ , and 65 ∘ using the seven general parameters we derived, which shows the fitting 3D graphs of sandy soil BRDF at different zenith angles.We can see that the seven general parameters have a good fitting effect at other angles, which shows that this set of data has wide adaptability.

Conclusion
In this paper a seven-parameter BRDF model is proposed that can fit double-peak BRDF data.With a sandy soil sample as an example, the correctness of the model is verified.As the increase of incident zenith angle, the sandy soil BRDF gradually shows double peaks.This model can overcome a common limitation of most previous models which cannot accurately fit double-peak data.In this paper, the global genetic algorithm is used to perform BRDF fitting analysis for the sandy soil BRDF data, and the corresponding parameter values and relative mean square error of the model are obtained.And the seven-parameter BRDF model can overcome the incapability of most of the current BRDF models, such as Walthall empirical model, Hapke model, and SOMILSPECT model, which also can fit very accurately.Finally, we combine data from four incident angles to derive seven parameters, which can give BRDF data of any angle by this seven-parameter BRDF model, not depending on incident angle.
), it is mainly divided into two parts.The first part represents the specular reflection component, and k a , k b are the mirror reflection coefficient.The second part represents the diffuse reflection component, and k c is the diffuse reflection coefficient.exp[ 1 (1 − cos  1 )  ] and exp [ 2 (1 − cos  2 )  ]represent reflection function,   and   , respectively, represent incident zenith angle and scattering zenith angle,   and   , respectively, represent incident azimuth angle and scattering azimuth angle, and k a , k 1 , a, k b , k 2 , b, and k c are the parameters to be determined.

Figure 7 :
Figure 7: The fitting effects of the seven-parameter model with a fixed set of parameters for different incident zenith angles.(a) 15 ∘ incident zenith angle.(b) 30 ∘ incident zenith angle.(c) 45 ∘ incident zenith angle.(d) 60 ∘ incident zenith angle.

Table 1 :
Model parameters and error for Figure6.

Table 2 :
Model parameters and error for Figure7.