Estimating the First-year Corrosion Losses of Structural Metals for Continental Regions of the World

The knowledge of the first-year corrosion losses of metals (K1) in various regions of the world is of great importance in engineering applications. The K1 values are used to determine the categories of atmospheric corrosivity, and K1 is also the main parameter in models for the prediction of long-term corrosion losses of metals. In the absence of experimental values of K1, their values can be predicted on the basis of meteorological and aerochemical parameters of the atmosphere using the dose-response functions (DRF). Currently, the DRFs presented in ISO 9223:2012(E) /1/ standard are used for predicting K1 in any region of the world, along with the unified DRFs /2/ and the new DRFs /3/. The predicted values of corrosion losses (K1 ) of carbon steel, zinc, copper and aluminum obtained by various DRFs for various continental regions of the world are presented. In this work we used the atmosphere corrosivity parameters and experimental data on the corrosion losses of metals for the first year of exposure (K1 ) for the locations of the tests performed under the international UN/ECE program, the MICAT project, and the Russian program. For the first time, a comparative assessment of the reliability of various DRFs is given by comparing the values of K1 pr and K1 ex using graphical and statistical methods. The statistical indicators of reliability of predicting the corrosion losses of metals are calculated for various categories of atmosphere corrosivity. It is shown that the new dose-response functions offer the highest reliability for all categories of atmosphere corrosivity.


Introduction
The corrosion losses of metals for the first year of exposure (K 1 ) are used to determine the category of atmosphere corrosivity toward each metal [1] and for long-term predictions of corrosion mass losses of metals based on models, including the power and power-linear functions. The variability of climatic and aerochemical atmosphere parameters over the years leads to different K 1 values in each test location. Repeated one-year corrosion tests of metals are required to obtain the K 1 values corresponding to the average long-term parameters and the level of atmosphere pollution at a given time. To avoid this, dose-response functions (DRFs) have been developed for estimating K 1 at any location. They are based on long-term average annual atmosphere corrosivity parameters.
The fact that metal corrosion depends on numerous climatic and aerochemical factors of the environment creates great difficulties for the quantitative estimation of the coefficients at the parameters used in DRFs. The coefficients for the main parameters can be obtained from the regularities found by statistical analysis of outdoor or laboratory tests. However, the use of the coefficients obtained for all dependences in DRFs does not ensure the prediction reliability.
 Use of those DRFs which give the most reliable K 1 pr for a given category. This requires preliminary information on the DRFs which are the most trustworthy for territories with a certain corrosivity category.
The DRFs presented in the Standard [1], unified DRFs [2] and new DRFs [3,4] have been developed for territories with a wide variation range of atmosphere corrosivity parameters. The K 1 pr values were calculated using various DRFs in previous studies [3,4]. However, those studies gave no quantitative estimates of the reliability of K 1 pr values obtained by different DRFs. This does not allow one to decide which DRF is the best. A quantitative estimation of the reliability of K 1 pr is required to make the right choice of the evaluation criterion. The standard determination coefficient R st 2 is used to determine the reliability of predicting the corrosion losses by a DRF. It is calculated from a data array presented in the y = K 1 pr , x = K 1 exp coordinates [5][6][7]. However, it was shown that the R st 2 coefficient is not recommended for determining the DRF reliability [8]. A number of statistical methods were suggested both for developing DRFs and for estimating the reliability of K 1 pr [8][9][10].
Based on the above considerations, this work includes:  Analysis of statistical indicators that are used to estimate the properties of DRFs;  Selection of a statistical indicator for estimating the reliability of the K 1 pr which would be most applicable in engineering practice;  Calculation of K 1 pr values for structural metals by all DRFs;  Assessment of the reliability of K 1 pr values of metals by the selected indicator for each category of atmosphere corrosivity;  Selection of the most reliable DRFs that provide the more reliable K 1 pr for each category of atmosphere corrosivity.
The purpose of this work is to estimate the reliability of K 1 pr values of these standard metals as calculated by different DRFs for each category of atmosphere corrosivity.

Procedure
This study was performed using the climate parameters of atmosphere corrosivity and the results on the first-year corrosion losses of metals in continental test locations K 1 exp reported previously [3,4]. Data for CS, Zn and Cu were taken from the UN/ECE international program [11,12], data for Cu and Al only were taken from the MICAT project [13,14], and data for all the metals were taken from the Russian Federation program (RF) [15]. The K 1 pr values were calculated for all the metals using the new DRFs [3,4,8] (hereinafter DRF N ) and the DRF presented in the standard [1] (hereinafter DRF S ). Calculations by the unified DRFs [2] (hereinafter DRF U ) are given only for carbon steel (CS) and Zn due to the lack of required atmosphere corrosivity parameters used in the equations.
Errors made in the previous study [3,4] should be noted: It should also be reminded that the K 1 pr values obtained by DRF U for steel were increased 7.8-fold like in the previous study [3]. It was based on the assumption that the DRF U for steel were developed for corrosion losses in μm rather than in g/m 2 according to the assumption made by Tidblad et al. [2].
It should also be noted that DRF N developed for carbon steel and zinc [3] and for copper [4] were improved using statistical methods of analysis for zinc [8], and in the development of coastal atmospheres DRF for carbon steel and copper [17]. The improvement was achieved by changing the coefficients at T>10 °C (carbon steel, zinc, copper) and Рrec (carbon steel, zinc). These changes are justified by the fact that the continental places according to the UN/ECE and RF programs and the MICAT project used for the development of the DRF N almost never feature high temperatures (T> 10 °C) and have a large amount of precipitation, as was observed in coastal places under the MICAT project. DRF N remained unchanged for aluminum.

Statistical Methods for Estimating the DRF Reliability and the K 1 pr Accuracy
State standards or international standards exist for any corrosion studies. The use of standards allows the testing, acquisition of meteorological and aerochemical data, processing of samples, and comparison of the results to be performed consistently. There are no established standards for processing large amounts of data with application of statistical tests and for graphical demonstration of results. This leads to difficulties in comparing the results obtained for different experimental data sets or within a single data set. In view of this, it is necessary to consider the statistical methods used by different authors for estimating the DRF reliability and K 1 pr accuracy and select at least one of them which is most suitable for practical use.
The experimental first-year corrosion losses (K 1 exp ), the atmosphere corrosivity parameters and the K 1 pr values obtained from DRFs for test locations form a large data array. The choice of atmosphere corrosivity parameters and the estimation of their effects on corrosion are performed using statistical methods. Statistical methods are also used to assess the accuracy of K 1 pr .
Where x i and y i are the observed values, x av and y av are their average arithmetic values, and n is the number of observations. To prevent negative R st values, R st 2 is used as the standard coefficient of determination. It is believed that the closer the value of R st 2 to 1, the more reliable the K 1 pr values.
It was shown [8] that absolute accuracy of K 1 pr was reached if the K 1 exp values equaled the K 1 pr values. Absolute accuracy of K 1 pr for all test locations is graphically represented by the K 1 pr = K 1 exp line, i.e., the y = x line. Therefore, it is necessary to consider the scatter of points (x i ; y i ) with respect to the y = x line with a = 1, b = 0 (hereinafter, the bisectrix of the angle between the axes if the x and y axes are plotted using the same scale) [8,10].
Taking this into account, various DRFs were estimated not by R st 2 but by the generalized coefficient of determination R new 2 indicating the deviation of points (x i ; y i ) from the bisectrix [8]. The generalized coefficient of determination is calculated by Equation 6: , .
Hence, the standard coefficient of determination R st 2 cannot be recommended for the comparative estimation of DRF reliability [8,10].  (9) This indicator is used to estimate the mean error expressed in g/m 2 or μm for all test locations. This error should not be large for locations with weakly corrosive atmospheres, but it can be substantially larger for locations with a corrosive environment.
Furthermore, the confidence angle Δ of the scatter corresponding to the angle of deviation of points (x i ;y i ) from the bisectrix whose slope is π/4 was considered, Equation 10 [8]: (10) In the study by Surnam and Oleti (2012), not only R st 2 , but also the maximum Mahalanobis distance was used to choose the DRF for long-term predictions [28]. The smaller the latter parameter is, the more reliable the model. The model that had both a large R st 2 value and a small maximum Mahalanobis distance was selected.
It is desirable to use all the indicators presented above, except for the standard coefficient of determination R st 2 , in the development of DRF. These indicators allow one to estimate the proximity of points (K 1 pr ; K 1 exp ) to the K 1 pr = K 1 exp line, excludes the points with significant deviations, and estimate the absolute and relative errors in K 1 pr calculations. In DRF development, analysis of the statistical indicators obtained will make it possible to estimate the need to improve them by using various mathematical functions and/or atmosphere corrosivity parameters and by changing the values of DRF coefficients.
The distribution of points (K 1 exp ; K 1 pr ) along the straight line K 1 = K 1 exp is unimportant for practical purposes. It is important that the error in the calculation of K 1 pr values should not exceed certain MAPE relative measurement errors, that is, δ av , Equation 7. Taking into account that a logarithmic distribution is characteristic of corrosion losses [1], it is advisable to calculate the mean relative errors for y i > x i (+δ) and for y i < x i (-δ): (11) where Where m is the number of points with δ i > 0.
The values of δ av , +δ and -δ can be used to estimate the safety margin of a metal or to determine the choice of protective means.
ISO 9223:2012(E) standard gives the uncertainty intervals (Table 1). For engineering applications, these uncertainty intervals can be taken as the intervals of permissible relative deviations of K 1 pr from the actual corrosion losses (K 1 exp ). The presented intervals are rather wide, while the interval for +δ is larger than that for -δ. The total uncertainty consists of two components: the uncertainty due to the imperfection of a DRF and the uncertainty in the measurements of environmental parameters and K 1 exp values. The uncertainty due to DRF imperfection is predominant. The SMAPE indicator takes the possible errors in the K 1 exp values into account. However, assuming in the first approximation that the K 1 exp are true values, only the δ av indicator, Equation 7, and the values +δ and -δ, Equation 11, were used to analyze the reliability of K 1 pr .

Selection of x and y Variables
Statistical indicators have been developed and are now used in various cases to identify the relationship between random variables that characterize some real processes observed in nature. If R st 2 is used to estimate the reliability of DRF and the accuracy of the K 1 pr values obtained, the choice of K 1 pr and K 1 exp like x and y variables does not matter. This is due to the fact that R st 2 characterizes only the proximity of the location of (x i ; y i ) points. Therefore, for example, in some publications [11,19,20,27] K 1 pr and K 1 exp are taken as the x and y variables, respectively, which is quite acceptable.
If the δ av , +δ and -δ indicators are used, it is necessary to provide a justified choice of x and y variables. For this purpose, let us compare K 1 pr and K 1 exp depending on the choice of x and y variables, using an abstract example. Let K 1 exp and K 1 pr in test locations A, B, C and D have the values listed in Table 2. The comparison of K 1 pr with K 1 exp is graphically represented for two cases: x = K ex , y = K 1 pr and x = K pr , y = K ex .
If x = K ex and y = K 1 pr , the points (K 1 exp ; K 1 pr ) corresponding to the test locations are arranged in the order of increasing K 1 exp values ( Figure 1a). The bisectrix corresponds to the K 1 pr = K 1 exp line. The absolute error can be estimated by the length of the vertical segment from a point to the bisectrix. One can see that K 1 pr have overestimated values in comparison with K 1 exp at locations A and C and underestimated values at locations B and D. The relative errors calculated by Equation 11 range from -63.6% to 166.7%, Table 2. The thin lines in Figure 1a indicate the prediction errors δ i corresponding to -33% and + 50% in Table 1, as calculated by the equations K 1 pr = 0.67·K 1 exp and K 1 pr = 1.50·K 1 exp , respectively. If x = K pr , y = K exp , the points (K 1 pr ; K 1 exp ) are arranged in the order of increasing K 1 pr values (Figure 1b), which is different from their order in Figure 1a. In this case, the lengths of the vertical segments of point deviations from the K 1 exp = K 1 pr line correspond to the absolute errors of K ex rather than K 1 pr , i.e. the K exp -K 1 pr differences. The vertical segments for each location are the same in both figures, that is, like in Figure 1a, K 1 pr have overestimated values at locations A and C and underestimated values at locations B and D. Hence, the K 1 pr values are overestimated for the points located below the bisectrix and underestimated for the points above the bisectix. To draw the lines with relative errors of -33% and + 50%, one has to make recalculations by the equations K 1 exp = 1.49·K 1 pr and K 1 exp = 0.67·K 1 pr , respectively. The relative errors calculated by Eq. (12) for x = K pr , y = K exp differ significantly from the δ i values calculated for x = K exp , y = K 1 pr , both in magnitude and in sign ( Table 2). This is due to the fact that in Eq. (12), while the numerators are equal in absolute value, the denominators change from x = K exp to x = K 1 pr .
The results obtained indicate that the choice of x and y variables significantly affects the calculation of the relative error obtained from Equation 12. In addition, if x = K 1 pr is used, the graphical images of the results become inconvenient for the visual assessment of the relative errors.

Calculation of K 1 pr using Different DRF s
The first-year corrosion losses of metals K 1 pr were calculated using DRF S , DRF U and DRF N for the test locations under the UN/ECE, RF programs and the MICAT project [3,4,8]. The K 1 pr results obtained were presented only in graphical form. Tables 3-6 show the numerical values of K 1 pr . These data allow one to estimate the relative error using MAPE (δ av ), -δ and +δ indicators.

Estimating the reliability of K 1 pr for all test locations
The relative errors δ av , -δ and +δ calculated by Equations 7 and 11 based on the K 1 exp and K 1 pr data, Tables 3-6, are presented in Table 7.  Table 7. Comparison of the results on -δ, + δ with the uncertainty intervals ±δ according to the Standard, Table 1, indicates that the -δ and +δ values obtained for all the DRFs are within the uncertainty intervals, except for +δ for steel based on DRF U (103.5%), for Cu (58.0%) and Al (85.7%) based on DRF N , as well as -δ for Cu (-61.4%) based on DRF S . It can be assumed that DRF N , DRF S and DRF U can be used for K 1 predictions, except for the exceptions indicated above. No rejection of test locations was carried out, so significant outliers (K 1 exp ; K 1 pr ) for certain test locations due to various reasons can significantly affect the -δ and +δ values.

Estimation of K 1 pr Reliability for Test Locations with Different Categories of Atmosphere Corrosivity
It is stated in ISO 9223:2012(E) standard that the uncertainty of K 1 predictions is the smallest in the medium range of K 1 exp values corresponding to atmosphere corrosivity category C3 but it is much higher for territories with atmosphere corrosivity of categories C1 and C5. For category CX, the uncertainties of K 1 prediction are the highest, therefore, predictions of K 1 using the developed DRFs may be unreliable for areas with this category of atmosphere corrosivity.
The C2 category is divided into three additional subcategories in Russia: C2-1, C2-2 and C2-3 (Table 8) [31]. In view of this, let us consider the values of +δ, -δ and δ av for each category, including the additional C2 subcategories (Table 9). To compare the reliability of K 1 pr values for locations with different corrosivity, the coordinate field is divided into categories of atmosphere corrosivity determined from the K 1 exp values. One can see that for all the metals, the equality K 1 pr = K 1 exp is observed only in a small number of test locations for each DRF. In most cases, the K 1 pr values are over/underestimated relative to K 1 exp , which corresponds to the location of points above/below the line of absolute match, K 1 pr = K 1 exp . Some of the points fall outside the limit lines corresponding to the relative errors of -33% and +50% for steel, zinc and copper, as well as -50% and +100% for aluminum.   Table 9 gives an estimate of the relative errors -δ, +δ, and δ av of K 1 pr values for each corrosivity category.   Carbon steel. The atmosphere corrosivity toward carbon steel at test locations as determined from K 1 exp falls within categories C1-C4. For the K 1 pr values calculated by DRF N , the values of -δ and +δ fall within the specified interval for all categories except for categories C1 and C2-2: +δ = 124.1% and + 63.2% for one and three test sites, respectively. For SRF S , the -δ values fall beyond the lower limit of the range for categories Ρ1 and Ρ2, but for categories Ρ3 and Ρ4, the -δ and +δ values do not fall beyond the specified range, as one can clearly see in Figure 2b. The K 1 pr values based on DRF U are more unreliable: they are extremely overestimated for categories C1, C2-1 and C2-2 (the +δ values range from 51.6 to 462.7%) but underestimated for category C4 (-δ = -55.3%). The most reliable K 1 pr values are only provided for subcategory C2-3, Figure 2c.
For zinc, the atmosphere corrosivity at the test location ranges from C2-2 to C4. For all the DRFs, the deviation of points from the K 1 pr = K 1 exp line is observed for a large number of locations, Figure 3. Some of the points fall outside the relative error range from -33% to +50%. Outside this range, the points for DRF N are located symmetrically relative to the K 1 pr = K 1 exp line, mostly below the line for the DRF S and mostly above it for DRF U , taking into account that for certain test locations, the hydrogen ion concentration in precipitation, Rain[H + ], was not taken into account in the K 1 pr calculations. However, despite the scatter of points, the mean values of -δ and +δ correspond to the relative error range from -33% to +50% in all the corrosivity categories for DRF N and DRF S , and for DRF U only in categories C3 and C4.
The corrosion testing of copper under the UN/ECE program and the MICAT project was carried out in a significantly smaller number of locations than that of steel and zinc. Considering that the corrosivity categories range from C1 to C5, the number of test locations in each of them is small. Therefore, considerable deviations of individual points can significantly affect the -δ and +δ values, for example, in category C1 for DRF N and DRF S , Table 5. In the other categories, the values of -δ and +δ for DRF N are within the relative error interval from -33% to +50%, except for categories C2-3, where the +δ is +76.13%. In contrast to DRF N where the scatter of points is symmetrical with respect to the K 1 pr = K 1 exp line, for DRF S the points are mainly located below this line, Figure 4. As a result, the -δ values are well below the -33% limit for all the categories.
The corrosion tests of aluminum were carried out only under the MICAT project in a small number of locations and under the RF program, therefore the total number of locations is as small as 52. For certain test locations under the MICAT project, the K 1 exp values were only 0.027, 0.054 and 0.081 g/m 2 , Table 6. Due to the absence of the upper limit for category C1 in the Standard [1], category C2-1 was assigned for those places as the smallest one where K 1 exp ≤ 0.2 g/m 2 in accordance with the Standard. According to K 1 exp , the corrosivity category for all the locations was only C2 with additional subcategories (Table 8) or C3. For DRF N , the arrangement of points relative to the K 1 pr = K 1 exp line is rather symmetrical, but for DRF S , the values of K 1 pr generally have underestimated values in comparison with K 1 exp , Figure 5. It can also be seen from Figure 5 that the point with the (K 1 exp = 1.24 g/m 2 ; K 1 pr = 0.10 g/m 2 ) coordinates is apparently an outlier. Despite the scatter of points, the -δ and +δ values for DRF N correspond to the relative error range from -50% to +100% in all the categories, even taking into account the assumed outlier point, except for category C2-1 where +δ = 111.8% that is not much higher than the upper relative error value. The overestimated K 1 pr values are primarily due to the complexity of accurate simulation of extremely small K 1 exp values for which +δ = +111.8% is not so significant. Unlike DRF N , the K 1 values based on DRF S for category C2-1 and C2-2 are quite reliable, which is, in general, due to the underestimation of K 1 pr values for all the categories ( Figure 5); -δ and +δ correspond to the relative error range of -50% to +100%, respectively. However, for categories C2-3 and C3 there are only underestimated K 1 pr values compared to K 1 exp with -δ = -63.4 and -61.9%, respectively.
The value of the relative error δ av for each category of atmosphere corrosivity (Table 5) provides incomplete information on how much the K 1 pr values differ from K 1 exp : whether K 1 pr values are overestimated or underestimated, and what measures should be taken to protect structures from corrosion in practice based on DRF results.
The results obtained from comparing the K 1 pr values calculated by different DRFs with K 1 exp indicate that an absolute equality of K 1 pr and K 1 exp is unlikely for any test site. The relative errors -δ and +δ for each category are averages of scattered points, and the smaller their values, the greater the probability that K 1 pr can be obtained for any test site within the relative errors provided. The -δ and +δ values indicate that the most reliable K 1 pr can be obtained:  For carbon steel, using DRF N for locations with all corrosivity categories; using DRF Sfor places with corrosivity categories C2-3, C3 and C4; and using DRF Uonly for categories C2-3 and C3;  For zinc, using DRF N and DRF S for test locations with corrosivity C2-2, C2-3, C3 and C4, and using DRF Uonly for categories C3 and C4;  For copper, only using DRF N for test locations with corrosivity from C1 to C5, although this model can give overestimated K 1 pr values for C2-2 and C2-3 categories, whereas DRF S will mostly give significantly underestimated K 1 pr values for all categories;  For aluminum, only using DRF N for test sites with corrosivity C2-1, C2-2, C2-3 and C3, and using DRF Sonly for categories C2-1 and C2-2.

Conclusions
 Various statistical indicators characterizing the properties of dose-response functions and the need to use these indicators to estimate the reliability of predicting the corrosion losses of metals have been considered. It has been shown that to estimate the reliability of K 1 pr values calculated by DRFs, it is advisable to use the MAPE indicator to find the average relative errors and the -δ and +δ values that characterize underestimated and overestimated K 1 pr values in each category of atmosphere corrosivity, taking into account the uncertainty range of corrosion loss predictions in accordance with ISO 9223:2012(E).
 The values of first-year corrosion losses of standard metals were first calculated by three types of dose-response functions (DRF S , DRF U and DRF N ) for various continental regions of the world. Based on the large data array of K1pr and experimental K1exp values thus obtained, the MAPE index, -δ and + δ values were calculated.
It has been found that reliable K 1 pr can be obtained with higher probability: o For carbon steel: using DRF N for all corrosivity categories, using DRF S -for locations with corrosivity categories C2-3, C3 and C4, and using DRF U -only for categories C2-3 and C3; o For zinc: using DRF N and DRF S for all the atmosphere corrosivity categories listed where tests were carried out, i.e., C2-2, C2-3, C3 and C4, and using DRF U , only for categories C3 and C4; o For copper: only using DRF N ; o For aluminum: using DRF N for test sites with corrosivity C2-1, C2-2, C2-3 and C3, and using DRF S , only for categories C2-1 and C2-2.

Conflicts of Interest
The authors declare no conflict of interest.