3.1

Select Experimental Environment

The experimental road is the Caiban Valley Tunnel on the S229 Yipi Road in Shandong Province. The tunnel is 402m long on one side and has a symmetric layout of lighting fixtures on both sides. The speed limit on the road is 80 km/h, and the tunnel has an asphalt road surface. The sidewalls are coated with fire-resistant material, and the luminance ratio is between 0.7 and 0.8.

3.2

Experimental environment set-up

To avoid daylight interference and obtain better data on road surface illuminance of the tunnel environment, the road surface illuminance collection experiment was conducted during the night. Fig.2(a) shows a photo of the experimental site for road surface illuminance collection in the tunnel, and Fig.2(b) provides a schematic diagram of the experimental setup, along with the following explanation: The lighting fixtures are installed on one side of the tunnel at a 45° angle with respect to the horizontal plane. The experiment includes both single lighting fixture road surface illuminance collection and multiple lighting fixtures road surface illuminance collection in the tunnel.

Fig.2

Experimental environment

3.3

Experimental data acquisition

The tunnel road surface illuminance was measured using the TES 1355 illuminance meter by TES, as shown in Fig.3(a). The meter is capable of measuring illuminance values ranging from 0 to 400K Lux and has an incident light cosine angle that meets the requirements of the experiment. Fig.3(b) illustrates the setup of the road surface illuminance collection points in the tunnel, with each point measuring 0.5m × 0.5m. The X-axis is parallel to the direction of travel, and the Y-axis is perpendicular to the direction of travel. A plane coordinate system was established based on this setup for data collection and storage.

Fig.3

Road surface illumination data collection

The illuminance data at each point of the road surface were collected using a grid. A data collection threshold of 10lx was set, and data points with illuminance values lower than 10lx were considered as boundaries of effective road surface illuminance and were not included in the dataset. A portion of the collected dataset is shown in Table.1. The collected discrete data was visualized in three dimensions using Matlab, as shown in Fig.4.

Table.1

Tunnel road illumination data set

Fig.4

Experimental data visualization

4.1

Inverse Distance Weighting Interpolation Method

The principle of geospatial information statistics followed by Inverse Distance Weighting (IDW) interpolation is that objects closer to each other are more similar than objects that are farther apart^[6].

As shown in Fig.6, the illuminance value of a luminaire at a point P in space can be calculated using Equation (2), while the illuminance value of an adjacent point Q in the same plane can be calculated using Equation (3).

Fig.6

Calculation of illuminance of adjacent points in the spatial plane

where E_P,E_Q are the illuminance values of the luminaire at point Р, Q respectively, I₁,I₂ are the luminous intensity of the luminaire in the direction of θ₁,θ₂. r₁,r₂ are the linear distance of the luminaire in space from point Р, Q and can be calculated by equation (4).

where θ₂ = θ₁ + α, which is substituted into equations (2), (3) and (4) to obtain equation (5)：

When point Р and point Q are in a similar circle,I₁ ≈ I₂, and the values of E_P and E_Q are similar The similarity is higher, while when the point P and the point Q are in a farther circle, does not hold and the similarity between E_P and E_Q is worse. In summary, the distribution of luminaire illuminance values in the space plane is in accordance with the principle of inverse distance weight (IDW) interpolation, and it is theoretically feasible to use the IDW interpolation method to interpolate the luminaire plane illuminance.

The mathematical description of the IDW interpolation method is shown in equation (6)^[7].

Where Z*(S₀) is the predicted value at point S₀, N is the number of surrounding sample points selected for interpolation at point S₀, Z(S_i) is the measured value at sample point S_i;, and λ_i is the weight of the influence of the ith sample point on the predicted value at point S₀. λ_i can be calculated by equation (7).

Where d_i0 is the distance between the prediction point S₀ and each sampling point, and p is the power of the distance. As the distance between the sampling point and the prediction point increases, the influence of the sampling point on the weight of the prediction point decreases exponentially, but the sum of the weights of all the sampling points used to predict the S₀ value is 1.

In this paper, the experimental data were processed and statistically analyzed using EXCEL software, and the initial experimental data were interpolated by inverse distance weights (IDW) through Matlab, and the iso-illumination curves and 3D plots were made.

First make p = 2, whereupon we have, and interpolate according to this interpolation formula to obtain Fig.7(a), (b), (C).

Fig.7

Interpolation results of inverse distance weight (IDW)

Letting p = 4 again, we have , and interpolation according to this interpolation formula yields Fig.7(d), (e), (f).

Figures 7(a), (b), (c) show the 3D plots, the iso-illumination curve plots and the general overview of the interpolation results obtained byinterpolation, respectively. From Fig.7(a).p = 2interpolation of the 3D plot of the interpolated surface and the fit of the original data points, it can be seen that when p = 2, the inverse distance weight interpolation method is less effective in the regions with higher and lower illuminance, but good in the regions where the illuminance value transitions from high to low, and Fig.7(b).p = 2 interpolation of the iso-illuminance plot shows that there are more More anomalous data points.

Fig.7(d), (e), (f) show the 3D plots, the iso-illumination curve plots and the general overview of the interpolation results obtained by interpolation, respectively. As can be seen from Fig.7(d).p = 4 interpolated 3D plots, the interpolated surfaces of the inverse distance weight interpolation method in all interpolated regions fit the original data better at p = 4 than at p = 2, and the iso-illuminance plots drawn from the interpolated data are smoother and free of anomalous data points.

The interpolation results for p = 2 and p = 4 were analyzed in terms of the interpolation loss time, the smoothness of the interpolation results, and the fit of the interpolated surface to the original data for each algorithm with the same interpolation grid, and are shown in Table.2.

Table.2

Comparison of interpolation results for different distance powers

Analysis of the interpolation results for different powers of the inverse distance weights shows that the smoothness of the interpolated surface is better when p = 4 than when p = 2, that the boundary transition of the iso-illumination curve drawn from the interpolated data in the selected region is smoother, and that the interpolated surface fits the original data to a greater extent.

4.2

Kriging interpolation method

The Kriging interpolation method, also known as the spatial self-covariance best interpolation method^[8], takes full account of the spatial relationship between known and unknown points, and its estimation formula is:

Where Z*(S₀) is the predicted value at point S₀, N is the number of surrounding sample points selected for interpolation at point S₀, Z(S_i) is the measured value at sample point S_i;, and λ_i is the weight of the influence of the ith sample point on the predicted value at point S₀. Unlike the inverse distance weight interpolation, the entry is not a function of the inverse distance correlation, but rather an optimal set of coefficients that can satisfy the minimum difference between the predicted value Z*(S₀)and Z(S₀) at point S₀, The mathematical assumptions of the Kriging interpolation method are that the spatial property Z (x, y) is homogeneous, that the values of the spatial coordinates are unknown but have a constant value E[Z(x,y)] = μ and the same variance σ², and that the unbiased estimation condition E(Z*(S₀) – Z(S₀)) = 0. To ensure the unbiased estimability of the valuation to be made, there also exists .

When using kriging interpolation to predict unknown points based on planar finite data points^[9], the covariance functions of Eqs. (9), (10), and (11) are used.

where:(x_i,y_i) and (x_j,y_j) are the plane coordinates of the i-th and j-th illuminance measurement points, and C_i,j· is the covariance of the illuminance values of the i-th and j-th illuminance measurement points. The illuminance values at different spatial locations are measured as a function of d_i,j and 𝛾(d_i,j). Spherical functions, exponential functions, linear equations and Gaussian functions are commonly used to fit the relationship between d_i,j and 𝛾(d_i,j), and this topic chose to interpolate the illuminance plane through exponential functions as well as Gaussian functions, and the interpolation results are shown in Fig.8.

Fig.8

Graph of interpolation results of Kriging

Figures 8(a), (b) and (e) show the 3D plots, iso-illumination plots and overview plots of the interpolation results obtained by fitting the d_i,j and 𝛾(d_i,j) functions to the exponential function, respectively. As can be seen from the fit of the interpolated surface to the original data points in Fig.8(a) for the exponential function interpolation 3D plot, the kriging interpolation method fits the illuminance throughout the region as expected when fitting the d_i,j and 𝛾(d_i,j) function relationships by the exponential function, and there are no data outliers in the smooth transition from high to low illuminance regions.

Fig.8(d), (e) and (f) show the interpolated three-dimensional plots, the iso-illumination curve plots and the general overview of the interpolation results when fitting the relationship between the d_i,j and 𝛾(d_i,j) functions by Gaussian functions, respectively. As can be seen in Fig.8(d) for the Gaussian function interpolation, when fitting the relationship between the d_i,j and 𝛾(d_i,j) functions by the Gaussian function, the Kriging interpolation method produces multiple data anomalies pooling at the edges of the interpolated region, while the interpolated surfaces in other regions fit the original data points more closely.

The mean square error of the kriging interpolation results for the exponential function and the Gaussian function fitted to the d_i,j and 𝛾(d_i,j) function relationship were analyzed and the results are shown in Fig.9.

Fig.9

Mean square error analysis of interpolation results

Fig.9(a) shows a plot of the mean square error of the kriging interpolation results by fitting the d_i,j,and 𝛾(d_i,j) function relationships through the exponential function. It can be seen through Fig.9(a) that the mean square error of the interpolation results of the illuminance values remains within a small range in the illuminance effective interval; Fig.9(b) shows a plot of the mean square error of the kriging interpolation results by fitting the d_i,j and 𝛾(d_i,j) function relationships through the Gaussian function. At the edges of the illumination effective interval, the interpolation results produce a large mean square error, but inside the interpolation interval the interpolation results have a smaller mean square error. Therefore, the relationship between the d_i,j and 𝛾(d_i,j) functions fitted by the exponential function is better than the Gaussian function in terms of the stability of the overall interpolation results.

The interpolation effect of fitting the d_i,j and 𝛾(d_i,j) function relationships by exponential and Gaussian functions was analyzed in terms of the interpolation loss time, the smoothness of the interpolation results, the fit of the interpolation plane to the original data, and the overall stability of the interpolation results for each algorithm on the same interpolation grid, and is given in Table.3.

Table.3

Analysis of interpolation results of different fitting functions

The analysis of the kriging interpolation results with different fitting functions shows that when the exponential function is used to fit the relationship between the d_i,j and 𝛾(d_i,j) functions, the smoothness of the interpolated surface is better than that of the kriging interpolation results with the Gaussian function, the boundary transition of the iso-illumination curve drawn from the interpolated data in the selected region is smoother, and the interpolated surface fits the original data to a greater extent, and the stability of the interpolation results is better.