Evaluation of Interpolation Methods for Refractivity Mitigation †

: Refraction has to be eliminated or mitigated for medium-or long-range applications requiring high accuracy (such as deformation monitoring). The high variability of meteorological parameters, and thus refraction, makes mitigation complicated. This study explores the possibility of direct refractivity interpolation with different algorithms for a nine-station meteorological sensor network in Cortes de Pall á s (Spain). Our Multiple Linear Regression (MLR) model can potentially contribute to improving the refraction correction of the monitored area. MLR provides, on average, an RMSE of 0.6 (dimensionless) compared to 1.5 obtained with Inverse Distance Weighting (IDW). For future improvements, the previous smoothing of meteorological data will be considered, and the possibility of using GNSS for vertical atmospheric information will be studied.


Introduction
Signals traveling through the atmosphere are affected in different ways, such as the effect of refraction along the signal path.Atmospheric refraction affects all geomatic techniques (electronic distance measurement (EDM), total station (TS), terrestrial laser scanner (TLS), image, levelling, etc.) [1,2].Assuming measurements are made with a relative precision of 1ppm, this effect can be only neglected for short distances, i.e., lower than 200 m.However, for medium or long distances, where accuracy is critical (as in the case of deformation monitoring), it is crucial to eliminate or mitigate the effect of refraction.In these cases, neglecting the effect of atmospheric refraction can lead to erroneous results, for example, when comparing two point clouds acquired by TLS, coordinate differences up to decimeters caused by refraction under different atmospheric conditions can be erroneously detected as deformations [2].
Different approaches to mitigate refraction can be found in the literature [3,4].For this study, a meteorological sensor network deployed in the geodetic network of Cortes de Pallás [5,6] is used to explore the possibility of interpolating refractivity to obtain a 3D model that can be applied to different geomatic techniques.
Refractivity depends on meteorological parameters, including temperature, pressure, and humidity [7].Several studies have tested methods for temperature interpolation [8,9], but other parameters such as humidity are more complicated to model due to their high local variability.For this reason, interpolating refractivity instead of meteorological parameters is a more convenient approach.
In this paper, Inverse Distance Weighting (IDW) interpolation (both 2D and 3D) and Multiple Linear Regression (MLR) are applied to a nine-station network in Cortes de Pallás (Spain) for a period of about 8 hours.The results of the models are analyzed in terms of root-mean-square error (RMSE).The next section of this paper shows the theoretical basis of the refractivity calculation, interpolation methods, and model evaluation.The data used are also described.Then, the results are presented, and finally, some conclusions are drawn in the last section.

Refractivity Calculation
The refractive index for a specific medium (n) is defined as the ratio of the speed of the propagation of an electromagnetic wave in vacuum to that in medium.Its value for air is very close to one, so, usually, the refractivity N defined as is used instead.Refractivity is calculated from meteorological parameters, and its expression depends on the signal wavelength.Visible and infrared, for EDM to within 1 ppm, is calculated as [7]: where T is the temperature (in K), P is the total pressure (in hPa), e is the partial water pressure (in hPa), and N gr is the group refractivity.The group refractivity in Equation ( 1) is obtained as: N gr = 287.6155+ 4.5566 with λ being the carrier wavelength of the EDM in µm.In this study, the value of the Leica TM-30 TS is used (0.658 µm) as it has been used to monitor the zone in the different field campaigns.

Interpolation Methods
There exist a great number of interpolation algorithms for different purposes.In fields such as geography or meteorology, some commonly used are Kriging, co-kriging, and IDW [8,9].Some of them have been tested and discarded for the data of this study, for instance, the Kriging method [8].This method is based on the premise that the spatial variation continues with the same homogeneous pattern, so a function can be fitted to the semivariogram.However, this does not occur in our data.The semivariogram obtained cannot be adjusted because the premise of homogeneous spatial variation is not true.On the other hand, the Inverse Distance Weighting (IDW) method and Multiple Linear Regression (MLR) can be used as follows:

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Inverse Distance Weighting: This estimates values by averaging the ones of the sample data points.The closer a point is to the point where the value is being estimated, the more influence or weight it will have in the averaging process.
Several types of distances can be used (Euclidean, Manhattan, etc.).Despite some other options having been tested, in this study, only the Euclidean distance (both 2D and 3D) is used.

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Multiple Linear Regression: This allows for the generation of a linear model in which the value of the dependent variable is determined from a set of independent variables.In this case, the variable to be determined is refractivity N, and the input variables are 3D point coordinates.
The general expression for MLR is [9]:

Model Evaluation
The different models have been evaluated by means of the root-mean-square error (RMSE) obtained using cross-validation.For cross-validation, the model is generated for n-1 stations, and the excluded station is then used to calculate the error.The RMSE is then obtained as: where n is the number of stations, V ic the calculated value, and V io is the obtained value by the model.

Data
The data used in this study were acquired during a field campaign in Cortes the Pallás.We installed nine data loggers in different stations, so meteorological parameters (temperature, pressure, and relative humidity) were automatically stored every minute.Table 1 shows the coordinates of the stations used in this study (Easting (E), Northing (N) corresponds to UTM coordinates zone 30, and H is the orthometric height).These stations are periodically monitored by Universitat Politècnica de València (UPV) [5].

Model Correctness
Three different models (IDW 2D, IDW 3D, and MLR) were obtained for each epoch (that is every minute).The RMSE with cross-validation was also obtained epoch by epoch.
Figure 1a shows the evolution of the RMSE of refractivity obtained with each model during the observation period.Figure 1b shows the coefficient of determination R 2 , for the MLR model, as this is the one that performs better in terms of RMSE.

Model Evaluation
The different models have been evaluated by means of the root-mean-square e (RMSE) obtained using cross-validation.For cross-validation, the model is generated n-1 stations, and the excluded station is then used to calculate the error.The RMSE is obtained as: where  is the number of stations,  the calculated value, and  is the obtained v by the model.

Data
The data used in this study were acquired during a field campaign in Cortes the lás.We installed nine data loggers in different stations, so meteorological parameters (t perature, pressure, and relative humidity) were automatically stored every minute.T 1 shows the coordinates of the stations used in this study (Easting (E), Northing (N) co sponds to UTM coordinates zone 30, and H is the orthometric height).These stations periodically monitored by Universitat Politècnica de València (UPV) [5].

Model Correctness
Three different models (IDW 2D, IDW 3D, and MLR) were obtained for each ep (that is every minute).The RMSE with cross-validation was also obtained epoch by ep Figure 1a shows the evolution of the RMSE of refractivity obtained with each m during the observation period.Figure 1b shows the coefficient of determination R 2 , fo MLR model, as this is the one that performs better in terms of RMSE.

Interpolated Refractivity
Following cross-validation with a model generated by excluding one station, both real and calculated refractivity can be graphically displayed.As an example, Figure 2 shows the evolution of refractivity N at station 8006.Refractivity tends to decrease during observation hours.This change is mainly due to temperature changes during the day.
Around 16:00 p.m., refractivity suddenly decreases with a peak in the obtained RMSE and R 2 (Figure 1).

Interpolated Refractivity
Following cross-validation with a model generated by excluding one station, both real and calculated refractivity can be graphically displayed.As an example, Figure 2 shows the evolution of refractivity N at station 8006.Refractivity tends to decrease during observation hours.This change is mainly due to temperature changes during the day.Around 16:00 p.m., refractivity suddenly decreases with a peak in the obtained RMSE and R 2 (Figure 1).

Conclusions
IDW interpolation shows similar behavior using 2D or 3D coordinates, and the value obtained for the refractivity RMSE is higher than the one obtained using MLR.MLR performs better in terms of RMSE, with a value under one (dimensionless) at most epochs.In terms of R 2 , MLR models have values above 0.85 in most epochs.
In the worst case, the model shows an RMSE of around 2 and R 2 of 0.7.When raw meteorological data are used, no smoothing is applied, so an error in any of the stations can lead to outliers.Meteorological data smoothing could be studied in future works.
The evolution of real refractivity and refractivity calculated with MLR shows good agreement.This, along with the RMSE and R 2 of the MLR models, indicates that this method is highly applicable to potentially mitigate atmospheric refraction in mountainous areas such as Cortes de Pallás.
Due to the harsh orography of the area, the local variability of meteorological parameters, and the critical precision required to prevent rockfalls, more research seems necessary.For future works, the model is pretended to be refined by adding gradient information or using additional data as the atmospheric information provided by GNSS techniques.

Figure 2 .
Figure 2. Evolution of real and interpolated refractivity at station 8006.