Introduction

Several tools and techniques have been developed for measuring climatic elements in order to know the current weather (^{Gutiérrez, García, Magaña, & Escalante, 2007}) and provide information for short-term forecasting, statistical analysis, farm support, air and maritime navigation safety, and characterization and classification of the weather and atmospheric state at a particular time and place. A meteorological instrument is the combination of a sensitive element and a system able to convert measurements of this element into a numerical value, which represents the meteorological variable being evaluated. The representative area of a region having meteorological equipment, at which climatic elements can be observed, measured, recorded, concentrated and processed, is commonly referred to as a weather station (^{Nájera & Arteaga, 1998}).

Often weather station records for a certain period are incomplete due to the absence or replacement of the operator, recording device failures or operational negligence (^{Aparicio, 2007}; ^{Campos-Aranda, 1998}), which has limited the conducting of agro-climatic studies whose results enable increasing productivity, optimizing resources, reducing the risk of crop loss, planning irrigation and drainage infrastructure in an integrated manner and making weather forecasts (^{Toro-Trujillo, Arteaga-Ramírez, Vázquez-Peña, & Ibáñez- Castillo, 2015}). According to the World Meteorological Organization (^{OMM, 2011}), the planning and implementation of user projects cannot be deferred until there are sufficient meteorological or climatological observations; therefore, estimation is used to expand, fill in or complete the information. It also recommends that the study period be at least 30 years.

Among the most common methods for estimating missing data series are: normal ratio (NR), which is used when the mean annual precipitation of any of the neighboring stations differs by more than 10 % (^{Aparicio, 2007}), and when it differs by less than 10 % it is estimated with the arithmetic average of the records of the neighboring stations (^{Alfaro & Pacheco, 2000}; ^{McCuen, 1998}). ^{Aparicio (2007)} recommends using at least three auxiliary stations in both cases.

The WMO (2011), ^{Campos-Aranda (2015)}, ^{Allen, Pereira, Raes, and Smith (2006)} and ^{DeGaetano, Eggleston, and Knapp (1995)} propose using linear regression (LR) and multiple regression (MR). The latter consists of obtaining a mathematical equation that expresses the relationship between the dependent variable (Y) and the independent one (X), or explanatory variables (X1, X2... Xn), although rarely is a perfect linear relationship observed because the phenomena studied in climatology are usually nonlinear, so both records must be homogeneous; that is, they need to represent the same conditions. When there are no stations surrounding the station with missing information, the missing values are deduced with the deductive rational (DR) method proposed by ^{Campos-Aranda (1998)}, which allows estimating them with the information provided by the complete years of the same series (^{Puertas- Orozco, Carvajal-Escobar, & Quintero-Angel, 2011}).

The method most used in hydrological and geographical studies is the U.S. National Weather Service’s inverse square distance (ISD) method (^{Ramírez-Cruz, López- Velasco, & Ibáñez-Castillo, 2015}). In this case, the influence of rain at a station for the calculation thereof at any point is inversely proportional to the distance between the station and the auxiliary stations (^{OMM, 2011}). The most significant advantage of the ISD method is that it uses daily data, grouped into periods of five or ten days, monthly or yearly (^{Teegavarapu & Chandramouli, 2005}; ^{Toro-Trujillo et al., 2015}).

Other procedures include moving averages (Campos- Aranda, 1998) and autoregressive, logarithmic, and triangulation methods, all based on normality assumptions. According to the WMO (2011), there are also estimation methods known as geostatistics based on theoretical foundations, which are statistical assumptions. The most important are the Kriging and optimal interpolation methods. The former is a spatial interpolation method based on showing the proportion at which the variance changes between points in space, and is expressed in a variogram. These procedures are limited by requiring a certain amount of data to produce a reliable and appropriate variogram (^{Toro-Trujillo et al., 2015}). In addition, ^{Wagner, Fiener, Wilken, Kumar, and Schneider (2012)} suggest that these procedures do not represent an improvement over the ISD method.

To evaluate and select the method that best fits the variable analyzed, there are various tests and procedures known as statistical indices, which consist of estimating the error or bias; some of them are: the mean standard error or root mean square error (RMSE), the Willmott index of agreement (d) (^{Willmott, 1981}), the coefficient of determination (R2) and the relative error (RE) (^{Cervantes-Osornio, Arteaga-Ramírez, Vázquez- Peña, Ojeda-Bustamante, & Quevedo-Nolasco, 2013}; ^{Kelso-Bucio, Bâ, Sánchez-Morales, & Reyes-López, 2012}; ^{Wright, 2001}; ^{Cai, Liu, Lei, & Pereira, 2007}).

The aim of this study was to compare the deductive rational (DR), normal ratio (NR) and United States Weather Services’ inverse square distance (ISD) methods, and then select the best one to estimate the missing daily precipitation and maximum and minimum temperature data at weather stations located in San Luis Potosí, Mexico.

Materials and methods

Prior analysis of information from 108 weather stations

Since not all the stations contain records of the climatological variables under study and many lack information in different periods, those stations whose precipitation and maximum and minimum temperature records were more than 85 % complete were selected. Years with no data or few records were removed. In addition, the data series were updated to 2010, with the values obtained from the National Meteorological Service (SMN) database, called KMZ, an extension that allows downloading up-to-date weather records. Finally, 27 years were considered as the study period for precipitation and 16 years for maximum and minimum temperature.

Of the total of 190 weather stations, 81 located in San Luis Potosi were examined, plus some neighboring ones in Tamaulipas, Zacatecas, Nuevo León, Querétaro, Guanajuato, Hidalgo and Veracruz, of which 27 were selected for analysis.

Figure 1 shows the percentage of daily information obtained from the precipitation and maximum and minimum temperature records of the 108 weather stations. In the case of precipitation, two neighboring stations with a lesser amount of information were taken to correctly delimit San Luis Potosí. Similarly, for maximum temperature, four stations with less than 85 % information were taken, while the minimum temperature records have the required information. For all three cases the same selected stations were taken.

Figure 1 Percentage of historical information by weather station. 

Figure 2 shows the location and distribution of the 108 weather stations. It also shows the analyzed and auxiliary stations, which served to define the best method for deducing the missing data.

Figure 2 Location of the study area and the network of weather stations. 

Results and discussion

Table 2 shows a summary of the statistical precipitation indices, and a better fit with the ISD method with an RMSE average of 38.7 mm is observed. The d values on average were 0.85, 0.68 and 0.68 for ISD, DR and NR, respectively, which have no values greater than 0.95 as recommended by ^{Cai et al. (2007)} and ^{Ruiz-Álvarez et al. (2012)}; however, the ISD method shows a better fit.

On the other hand, stations 11050, 24027, 24038, 24049 and 24093 show a lower RMSE value with ISD, compared with that obtained with DR and NR, while the data analyzed at station 24094, according to the RMSE statistic, indicate a lower error with the DR method.

In a study conducted in Colombia, ^{Toro-Trujillo et al. (2015)} report similar values among the statistical indices when analyzing the precipitation variable, but a better Willmott index of agreement (d) result with the ISD method is highlighted; therefore, they propose it as the best method for estimating missing data, a situation also presented in this study, since the best estimate was obtained with the ISD method. Moreover, considering that R2 averaged 0.231 and 0.197 for ISD and NR, respectively, it can be deduced that there is little strength in the linear relationship between the estimated and observed value, mainly because precipitation is a discrete variable with a high frequency of values close to zero.

Figure 3 illustrates the relationship between observed and estimated precipitation values, with the ISD and NR methods at weather stations 24027 El Salto, 24038 Los Pilares and 24049 Papagayos. The ISD method has, on average, a better fit of the estimated values with respect to the observed ones, which is consistent with the results obtained by ^{Hubbard and You (2005)}, ^{Eischeid, Pasteris, Diaz, Plantico, and Lott (2000)} and ^{You, Hubbard, and Goddard (2008)}. Moreover, ^{Teegavarapu and Chandramouli (2005)} mention that the ISD method better estimates the observed data, plus it can be used in any time period, i.e., in daily, monthly and annual data.

Figure 3 Comparison between the estimated and observed precipitation value. 

Figure 4 shows the relationship between the observed and estimated maximum temperature with the ISD and NR methods; similar behavior can be observed in the two methods, possibly because temperature is a more localized variable (^{Toro-Trujillo et al., 2015}; ^{You et al., 2008}).

Figure 4 Comparison between the estimated and observed maximum temperature value. 

Table 3 shows the statistical indices calculated with the three described methods to determine which one estimates more accurately the missing maximum temperature data records. It also shows that the values are similar to one other, although the RMSE results are higher than those with the ISD method, followed by the DR and finally the NR methods. According to the Willmott index, the NR method has a better fit because, in general, it has a higher value at the stations, followed by the DR and ISD methods, respectively.

At stations 11050, 24038, 24049 and 24093, the coefficient of determination R2 is greater with the ISD method, and with the others it is greater with NR, although the R2 are very similar. ^{Campos-Aranda (2005)} recommends against using low linear relationship methods if the coefficients of determination do not have values greater than 0.8; therefore, because the present study presents a low linear relationship, it is advisable to use either the ISD or NR methods, since they have similar statistics.

Figure 5 compares the observed and estimated minimum temperature with the ISD and NR methods, at the stations studied.

Figure 5 Comparison between the estimated and observed minimum temperature. 

Table 4 shows the statistics that allow identifying the method that has the best fit to the minimum temperature data. The Willmott index is higher with DR at stations 24027, 24049 and 24094, with NR at 11050 and 24038, and with ISD at 24093. It also shows that the d value varies in a very similar way in the three methods. Stations 24027, 24049 and 24094 reflected lower RMSE values with the DR method, whereas 24038 and 24093 did so with the ISD method, and 11050 with the NR method.

Conclusions

The methods used to estimate the missing precipitation data records allowed identifying the one which best estimates the results, and according to the statistical indicators it is the ISD method; therefore, for precipitation, this is the best alternative when data are available for the same period at auxiliary stations. The results obtained by analyzing the maximum and minimum temperature data indicate little variability among methods, as the calculated data are similar to those observed; therefore, both the behavior and errors are very similar with all three methods. Accordingly, the U.S. National Weather Service’s inverse square distance (ISF) method is an efficient alternative for completing missing data records in a continuous series; therefore, it was the method used to estimate missing data at 108 weather stations in the state of San Luis Potosí.