A Global Analysis of the Fluctuation of Total Ozone. I. Application of the Optimum Interpolation to the Network Data with Random and Systematic Errors

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1 April 1980 F. Hasebe 95 A Global Analysis of the Fluctuation of Total Ozone I. Application of the Optimum Interpolation to the Network Data with Rom Systematic Errors By Fumio Hasebe Geophysical Institute, Kyoto University, Kyoto, 606, Japan (Manuscript received 29 October 1979, in revised form 4 March 1980) Abstract Statistical characteristics of the observational errors in total ozone measurements by Dobson spectrophotometers M-83 filter ozonometers are examined by the use of the structure functions. In order to explain the seasonal variation of the functions, it is necessary to consider not only rom errors but also systematic errors which are correlated with both the total ozone amount the errors at other stations. The optimum interpolation scheme is modified in case of the existence of systematic errors. By this method, grid point values of total ozone fluctuations are obtained together with the estimates of errors. Spatial mean values are computed by the average of grid point values. When the interpolation networks overlap, estimates of errors in spatial averages are calculated as variances of statistical samples which are not independent each other. By these procedures, an objective treatment is possible for observational data by Dobson spectrophotometers M-83 filter ozonometers with the estimates of errors both in grid point values in spatial averages. The interpolation errors thus obtained are found to be small enough to discuss the fluctuation of total ozone on a global basis. 1. Introduction A half century after the proposal by Chapman (1930) of the photochemical theory of ozone in a pure oxygen atmosphere, it has become clear that HOx, NOx C1X play an important role in the stratospheric ozone balance (e.g., Johnston, 1975; Nicolet, 1975). Attentions have also been paid to the destruction of stratospheric ozone by human activities (e.g., Crutzen, 1974; Molina Rowl, 1974; Cicerone et al., 1974). Some efforts have been made to detect the possible depletion of the observed total ozone values on a global basis (e.g., London Kelley, 1974; Angell Korshover, 1976; London Oltmans, 1978/79). Although some of these studies analyzed global distribution of total ozone using their own criteria for reliability, there still remains arbitrariness in their estimation. Observations of total ozone are commonly based on two types of instruments: the Dobson spectrophotometer the M-83 filter ozonometer. However, previous analyses of the fluctuation of total ozone have referred mainly to the Dobson spectrophotometer observations, because the M-83 filter ozonometer gives errors as large as 30% under the different conditions of solar zenith angle or visibility, as shown by Bojkov (1969a, b). The filter ozonometer is used mainly in the Soviet Union, covering widely high latitudes where temporal variations in total ozone are large, while there are few Dobson stations near this region. The use of filter ozonometer data is, therefore, inevitable for more detailed analysis of the global distribution of total ozone. For this purpose, it is necessary to establish a method of objective treatment of the M-83 filter ozonometer data in combination with the Dobson data. It is well known that the optimum interpolation proposed by Gin (1963) is useful to give the reliability of the analysis when there are only homogeneous rom errors (Alaka Elver, 1972). In the case of total ozone considered here,

2 96 Journal of the Meteorological Society of Japan Vol. 58, No. 2 however, the observational errors are considered to be neither homogeneous nor rom, but they include some systematic errors* due to the characteristics of the instruments. In the present study, it is attempted to examine the systematic errors to modify the optimum interpolation scheme in case of the existence of such errors. where *ij is the correlation coefficient between f i' fj', (8),* is the dimensionless variance of rom error given by 2. The principle of the optimum interpolation The principle of the optimum interpolation is outlined here in such case that there are only homogeneous rom errors (Gin, 1963; Alaka Elver, 1972). Its modification for the total ozone analysis is discussed in the following section. The optimum interpolation expresses the anomaly of a physical quantity f at the point of interest, fo', as a linear combination of the observed anomalies f ( at i-th station (i =1, 2, n): *, (1) (2) where the upper bar indicates time averaging, pi is a weighting factor, Io is the interpolation error, f i' includes observational errors local irregularities. The weight pi is determined from the condition that the mean square interpolation error E2 be minimum; (3) (4) When only rom errors are involved, the deviation of the observed value from the time average, fi'. is expressed as a sum of the deviation of the true value fi' the rom error *i; The rom error ~ti is characterized by (6) where * ij denotes Kronecker's delta, *2 is the variance of the rom error. Under the assumption of homogeneity for the variances both homogeneity isotropy for the covariances, Eq. (4) can be expressed in the non-dimensional form: (5) (7) * In this article, the word "systematic error(s)" is used to signify all the part of errors that cannot be considered to be rom. which is independent of the location of stations f'2 is the mean variance. We now replace i by the correlation function *(*), where *0* is the distance between i-th station the point of interest. Under the condition of Eq. (4), the mean square interpolation error E2 (=I 02) is reduced to (9) (10) If we obtain * *ij, f0' is calculated from Eqs. (2) (7). With the confidence of about 70%, (11) Substituting Eq. (5) into Eq. (11) taking Eq. (6) into account, we obtain (12) It is expected from Eq. (12) that bij is a monotonously increasing function with the distance between i-th j-th station pij, when * is small enough compared with the characteristic scale of the spatial fluctuation of f. When *ij tends to zero, the first term in the right h side of Eq. (12) vanishes. Thus, (13) The actual analysis is usually performed on the diagram in which the logarithmic value of bij is plotted against the distance of the two stations. First, we draw a regression line that fits the distribution of bij, the intersection of the line with the ordinate (b0) would give the value of 2*2. Then, * is readily calculated from Eq. (9). 3. Statistical characteristics of the observational errors In order to examine the statistical characteristics of the observational errors, structure functions of total ozone for March September are plotted in logarithmic values in Fig. 1. For detailed description of the data, see part II

3 April 1980 F. Hasebe 97 In general, the observed value f i is expressed as a sum of the true value fi, the rom error Fig. 1 The structure functions of total ozone for March September. Thick lines indicate the regression lines of crosses (DD), thin lines those of triangles (DM), brolen lines those of circles (MM). (Hasebe, 1980). Hereafter the observations by Dobson spectrophotometers are indicated by "D", those by M-83 filter ozonometers are by "M". In Fig. 1, crosses denote the values for the combination of two Dobson stations (DD), triangles for a Dobson an M-83 (DM), circles for two M-83's (MM). Regression lines are separately drawn for each combination of instruments. From Fig. 1, the following two features are obviously seen: (i) b0 of DD is smaller by about an order of magnitude than that of MM. This feature can also be found for the other months not shown here. (ii) b0 of MM is larger than that of DM in March, but the reverse is the case in September. We also found that b0 of MM was smaller than that of DM from June to September, that the reverse was true for the other months. These features are not expected from homogeneous rom errors as described in section 2. Therefore, the optimum interpolation cannot be applied straightforward to the total ozone analysis. Thus, we try to determine the statistical characteristics of the observational errors which may explain those two features. (14) (15) First we assume that the both errors of D M would be rom, i.e., there are no systematic errors (*i=0), the variance of the rom error of D M is *D2 *M2, respectively. In this case the structure functions are schematically illustrated in Fig. 2. The disposition of three curves is characterized by the fact that b0 of DM is the average of those of DD MM. This configuration is not consistent with the fact (ii) mentioned above, it is suggested that the observed values must contain some systematic errors originated in the instruments. As to Dobson spectrophotometers, observational uncertainties of the direct sun measurements are believed to be about 1 % (Dobson Norm, 1958); it is thought that the observational errors are rom at most 2% (Krueger Pressman, 1975). The comparative study by Bojkov (1969a, b) shows that M-83 filter ozonometers have errors as large as 30% suggests that the errors are correlated with the amount of total ozone. So we next assume that the systematic errors would be included in the data of M-83 filter ozonometers only that the rom errors do not depend on the type of instruments (* D2 = *M2 = *2). If systematic errors might exist only in Dobson spectrophotometer's data, the following considerations are also valid if the notations DD MM are exchanged. Spatially correlated errors were treated by Bergman Bonner (1976) concerning the satellite-derived temperature data. According to Fig. 2 The schematic diagram of the structure functions expected when there would be only rom errors.

4 98 Journal of the Meteorological Society of Japan Vol. 58, No. 2 [D], the regression line of DM always lies between those of DD MM, in apparent dis- the suggestion by Bojkov (1969a, b), the systematic agreement with the fact (ii). In case [C], DM is error at one station may be correlated situated above MM if *2 is larger than 2* as not only with errors at other stations (*i'*j') but seen in the figure, whereas DM is below MM if also with the amount of total ozone (fi'*j'). So 2 is smaller than 2 *. Among the four possible * systematic errors are introduced in a simplified cases, therefore, only case [C] can explain the form as described below. seasonal variation of the disposition of the structure First, we assume that there is no correlation between rom errors systematic ones, i.e., functions of DD, DM, MM. In addition, (16) We further assume that fi'*j' *i'*j' have either the following two properties: (a) they are constant with respect to *ij, (b) they vanish except for i = j, which give the following four possible cases: (17) (18) (19) (20) where *2 is the variance of the systematic errors, * is the covariance between total ozone the systematic error. In each case, the expected structure functions are shown in Fig. 3. In case [A], the effect of systematic errors does not appear in the structure function of MM. This does not agree with the fact (i). In cases [B] the fact (i) indicates that systematic errors exist in the data of M-83 filter ozonometer rather than in those of Dobson spectrophotometer insofar as they may exist only in either of the two types of the instrument. Note that the values of *2, *2 * can be uniquely determined by b0's of the regression lines. Thus, it may reasonably be concluded that the rom error does not depend on the instrument that the systematic error exists only in M-83 filter ozonometer observations. 4. Correlation coefficients The correlation coefficient *ij between the true values f i' f j' defined by Eq. (8) should have some relationship with the coefficient *ij between the observed values f i' fj'. We now derive the relation between *ij, *ij, when the errors have the statistical characteristics as shown in the preceding section. Under the assumption of homogeneity of variances, the Eqs. (6), (19), (16) for rom errors systematic errors of type [C] can be expressed in dimensionless forms, respectively, Fig. 3 The schematic diagram of the structure functions for the four possible cases as indicated by Eqs. (17), (18), (19), (20).

5 April 1980 E. Hasebe 99 (MM) (29) (21) If the three limits of the dimensionless structure functions are obtained, Eqs. (27), (28), (29) are solved simultaneously the values of i, T2 2 * are obtained. The results are tabulated in Table 1. Incorpolation of these values into Eqs. (23), (24), (25) would give the correlation coefficients *ij. (22) where T2 is the dimensionless variance of the systematic error, * is the dimensionless covariance between a physical quantity the systematic error. With these parameters, *ij is related to the true correlation coefficient *ij for the instrument combination of DD, DM, MM as follows: (DD) (DM) (MM) (23) (i*j), (24) (25) Similarly, the dimensionless structure function ij between fi' fj' is given by * (26) When the distance *ij between stations i j tends to zero, *ij tends to unity. Thus the following relations are derived from Eqs. (23), (24), (25), (26): 5. Optimums interpolation using data with rom systematic errors The optimum interpolation scheme with rom systematic errors of type [C] is presented. Let us consider the interpolation network consisting of m Dobson stations (i =1,..., m) n - m M-83 filter ozonometer stations (i = m +1,..., n). In this case, basic equations (1), (2), (3), (4) are still valid, but Eq. (7) which determines the weighting factor pi is replaced by (30) The expression of the mean square interpolation error (Eq. (10)) is also valid in this case. The correlation function which is used to infer i is assumed to take the form, allowing more *0 degree of freedom than those by Gin (1963) Alaka Elver (1972): (31) where * is the distance between the grid point the i-th station. Coefficients A1, A2, A3, Table 1. Calculated values for *, 2* T2. (DD) (27) (DM) (28) ** The dimensionless structure function *ij is different from that by Gin (1963) by a factor of two.

6 100 Journal of the Meteorological Society of Japan Vol. 58, No. 2 A4 are determined by iterative calculations are listed in Table 2, where * is measured in kilometers. For constructing Eq. (31), twenty representative values of the correlation coefficients, similar to those by Alaka Elver (1972), have been used. These representative values the obtained correlation functions are shown for each month in Fig. 4. For November December, however, iterations are not achieved successfully, as is suspected from large Table 2. Coefficients of the correlation functions as expressed in the form of Eq. (31). The numbers in the parentheses indicate the exponential part of the coefficients. scatter in Fig. 4; therefore, the first guess values from initial condition are used for these two months. This large scatter is probably due to the less reliable estimates of * T2 owing to the small number of bij for the MM pair. 6. Estimates of errors in the spatially averaged values The interpolation networks for individual grid point are usually determined so that they may not overlap with one another. In this case, because the interpolation error at each grid point can be considered to be independent, the errors in the spatially averaged values are easily obtained. Since the observational stations of total ozone are not so abundant, it is usually necessary to use observational data at the same station to calculate the interpolated values at two or more grid points. In this case the estimation of errors in the spatial averages are rather complicated because of the interdependency of errors. Let's consider the k-th grid point with the interpolated value f0k' the mean square interpolation error Ek2, f0k' is expressed as the linear combination of the observed anomalies fik' with the weighting factor Pik in the k-th interpolation network, (32) Fig. 4 Representatives of the correlation coefficients correlation functions iteratively fitted to them.

7 April 1980 F. Hasebe 101 (33) where f0k' is the true value at the k-th grid point. The spatial mean value of a physical quantity, denoted by [ ], is given by the average of its grid point values; (34) where Wk denotes the weight which expresses the contribution of the grid point to the spatial mean value. For the cross latitudinal averaging, it is a function of latitude * such as (35) The spatial mean of the interpolation error is represented by the variance of some statistical samples which are not independent each other (e.g., Okugawa, 1958); work determination, appear in part II (Hasebe, 1980). It should be noted that if there is no station in the interpolation network of the grid point the deviation f 0' is taken to be zero that the interpolation error is set to the square root of the mean variance. Interpolations are made for the grid points with 10* latitude intervals (5*, 15*,..., 85*N S), 20* longitude intervals in middle low latitudes. Along the 65* 75* latitude circles, the longitudinal spacing is taken to be 40*, for the 85*N S, taken to be 90*. The fluctuations of the monthly mean values of total ozone for every grid point from 1962 to 1976 are calculated with the estimates of interpolation errors. The northern midlatitude values (area weighted mean of 35*N, 45*N, 55*N) at 180* longitude 20*E are shown as time series in the upper two illustrations (1) (2) of Fig. 5, respectively. The third illustration (3) represents the zonal mean values for (36) The second summation of Eq. (36) appears due to the dependency of interpolation errors. The first term under the second summation, i.e., f 0k'f 0l' consists of the interpolated values, the second term, i.e., f 0k'fol'( is deduced from the correlation function. Using Eq. (32), the third term, equivalently the fourth, is further decomposed to (37) (37) The first term in Eq. (37) is again obtained from the correlation function, the last two terms vanish by the statistical characteristics of errors as mentioned in sections 2 3 (Eqs. (6) (19)). As a result, [E2] can be obtained from Eqs. (36) (37). 7. Examples of the result In order to illustrate the usefulness of the method of the present analysis, the result is shown for some examples in this section. More detailed results, together with the locations of the observational stations the procedure of net- Fig. 5 Examples of the fluctuation of total ozone. (1) northern hemisphere midlatitude mean values along 180* longitude meridian, (2) the same as (1) but for 20*E, (3) zonal mean values of 45*N, (4) the global means. Vertical bars indicate the estimates of errors as the confidence limits of about 70%.

8 102 Journal of the Meteorological Society of Japan Vol. 58, No. 2 45*N, whereas the bottom illustration (4) indicates taken into account objectively. the global mean values. The vertical bars (ii) Estimates of interpolation errors in both grid represent the interpolation errors indicating about point values spatial averages can easily the 70% confidence limits. The comparison of (1) (2) in Fig. 5 shows be obtained. (iii) The results of interpolation obtained here that the interpolation error of (1) is evidently depend on *i' only, but not on the ensemble larger than that of (2). This is because of the mean value of the systematic observational fact that the observational stations are sparse for error (*i), which is estimated as large as the Northern Pacific whereas they are relatively 30% of the observed values (Bojkov, 1969a, dense for the Central Europe. It is seen, however, b). Therefore, we can avoid the influence of that statistically significant deviations occur the large value of *i. even for the Northern Pacific as well as for the It is now possible to analyze the fluctuation Central Europe. of the observed total ozone on a global basis If the errors in the grid point values are with the estimates of errors. The results of such mutually independent, the errors in spatial mean an analysis are presented in part II (Hasebe, values would be inversely proportional to N1/, 1980). where N is the number of grid points used for the calculation, i.e., 3, 3, for the four Acknowledgements illustrations (1)-(4) of Fig. 5, respectively. For The author wishes to express his hearty example the errors for the global mean ((4)) are gratitude to Professor R. Yamamoto of the Geophysical Institute of Kyoto University for con- expected to be (18/260)1/2 times less than those for the zonal mean ((3)). In the present study, tinuing guidance, encouragement critical however, the interpolation errors are actually reading of the manuscript. He would like to dependent with each other, so that they should thank Professor I. Hirota for helpful comments. be calculated from Eqs. (36) (37). As seen Many thanks are also due to Dr. K. Fukuyama in Fig. 5, therefore, the errors for the global mean for many suggestions critical reading of the are not so small as expected from the N-1/2 original manuscript. He is pleased to acknowledge law. However the errors are small enough for the discussions of the members of the the discussion of the fluctuations of total ozone Meteorological Laboratory of Kyoto University, on a global basis. especially Mr. H. Kobayashi in his discussion on the estimation of interpolation errors. Finally 8. Conclusion but not least, the author would like to express Our primary concern in the present study is his sincere thanks to the referee whose careful the rational treatment of the errors involved in comments contributed considerably to the improvement of the present manuscript. the total ozone measurements. From the examination of the observational errors in section 3, it is The computations were made by the use of reasonably assumed that values of Dobson the computer at the Data Processing Center of spectrophotometer contain only rom errors, Kyoto University. whereas those of M-83 filter ozonometer have systematic errors characterized by Eq. (19) as References well as rom errors of the same magnitude as Alaka, M. A., R. C. Elver, 1972: Optimum in the Dobson's data. For the first time these interpolation from observations of mixed quality. errors are taken into account objectively for Mon. Wea. Rev., 100, calculating the grid point values by the modified Angell, J. K., J. Korshover, 1976: Global analysis scheme of the optimum interpolation proposed of recent total ozone fluctuations. Mon. Wea. here. In spatial averaging of the interpolated Rev., 104, grid point values, the dependency of interpolation Bergman, K. H., W. D. Bonner, 1976: Analysis errors arising from the overlapping of the interpolation network is exactly considered. error as a function of observation density for satellite temperature soundings with spatially correlated errors. Mon. Wea. Rev., 104, The advantages of the procedures described in Bojkov, R. D., 1969a: Differences in Dobson spectrophotometer filter ozonometer measurements the present work may be summarized as follows: (i) The difference in the accuracy of the two of total ozone. J. Appl. Meteor., 8, types of instruments (Dobson spectrophotometer M-83 filter ozonometer) can be ozone deduced from 1969b: Some characteristics of the total Dobson-spectrophotometer

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