Empirical Orthogonal Functions and the Statistical Predictability of Sea Ice Extent

Transcription

1 Empirical Orthogonal Functions and the Statistical Predictability of Sea Ice Extent JOHN E.WALSH ABSTRACT The predictability of Alaskan sea ice extent over time scales of several months is evaluated quantitatively. The predictors include year-month means of the surface pressure and temperature fields as well as prior anomalies of sea ice extent. The problem of artificial predictability is described in order to motivate the representation of the predictors in terms of empirical orthogonal functions. Included in the results are statistics of the dominant orthogonal functions and preliminary estimates of the decay of the forecast skill with the length of the forecast interval. The skill at a lag of one month is highest for persistence. The skill scores obtained from the pressure and temperature fields are small but statistically significant at lags of 1-2 months. Little significant skill is found at longer lags. INTRODUCTION Substantial year-to-year variations in the extent of sea ice have been observed in nearly every geographical sector of the Arctic. Schule and Wittmann (1958) examine cases in which the September position of the ice edge north of Alaska differed by several hundred kilometers in successive years. Haupt and Kant (1976) and Volkov and Sleptsov-Shevlevich (1972) describe similar variations in the North Atlantic and in the Soviet Arctic, respectively. These variations can seriously affect the navigability of the peripheral Arctic seas and undoubtedly contribute to local variations in the surface energy budget. While the large-scale behavior of sea ice is complex (Untersteiner, 1974), it is known that the ice responds both thermodynamically and dynamically to the atmospheric forcing. The importance of the thermodynamic response is apparent in the results of Washington et al. (1976). The dynamical response is attributable to the fact that the air stress is the primary motive force for sea ice. One might 373

2 374 JOHN E. WALSH therefore ask whether year-to-year fluctuations in sea ice extent are predictable from corresponding fluctuations in the surface temperature and pressure (geostrophic wind) fields. Indeed, the Soviets (Sancevich, 1976) contend that summer navigability in the Barents and southwestern Kara seas is dependent on earlywinter thermal anomalies. In the East Siberian and Chukchi seas, however, the surface wind (pressure) patterns are claimed by the Soviets to be the controlling factor. Markham (1975) reports that summer ice conditions ("good, fair, poor") in the southeastern Beaufort Sea appear to be correlated with January-March pressure gradients over the Arctic ice pack. Barnett (1976), on the other hand, finds that an April pressure sum correlates with the severity of the shipping season in the Barrow-Prudhoe corridor. Lilly and Stewart (1977) have recently claimed to be able to predict the severity of the Alaskan shipping season six months in advance. Details of the Lilly and Stewart methodology have not been published, however. The present work is directed at the need for a more rigorous quantification of longer-term sea ice predictability. Empirical orthogonal functions are used to represent the meteorological and sea ice data. The formulation of and the motivation for such data representations are described in the following section. Some statistics of the empirical orthogonal functions and preliminary predictability results are then presented. METHODOLOGY Linear statistical predictors are used in this study to quantify the predictability of sea ice extent in terms of the prior fields of sea ice and atmospheric (surface) pressure and temperature. The pressure data were obtained from the northern hemisphere grids archived at the National Center for Atmospheric Research, while the temperature data are a subset of the objectively analyzed Arctic surface temperature grids described by Walsh (1977). The sea ice data are in the form of concentration grids constructed from the charts of the U.S. Naval Océanographie Office ( ) and the U.S. Navy Fleet Weather Facility (1976). In the general case of a set of predictors, x,, x 2,..., x, w and a predictand y, a prediction of y is }' = M 2* a m x m m \ The root mean square error in the prediction is minimized by taking M a, = S (x x m ) 1 (x,j ) n = \ where the brackets indicate a sample covariance and the negative exponent denotes a matrix inverse. The skill, S, of the prediction can be taken as the fraction of the variance that is successfully predicted:

3 Empirical Orthogonal Functions and Statistical Predictability 375 Ç = 1 _ \V y) ) _ Y y \ x ny I \ x n x m ) V^m)' / /n <>,2 > m = \ n = \ <J' 2 ) S is the square of the correlation between the predicted and observed quantities. As shown by Davis (1976), however, computed values of S will consist of the true skill, S t, and an artificial component attributable to the differences between the sample and population covariances: 1 M - Mr S = S t + L Y ^ = S, + (2) N mt x At N At where N is the sample size, At is the sampling interval, and T is a mean of the "correlation time scales" T, : T = {y(t)y(t+iat)) (x m (t)x m (t+iat)) ^ i = ~oo (y 2 > <x m 2 ) The artificial skill, Mr IN (At), is the amount by which the computed skill exceeds the skill to be expected in the case of an infinite sample drawn from the same statistical system. If a particular prediction formula is applied to a data sample that is independent of the data used in deriving the formula, the skill will be less than the true skill by an amount equal to the artificial skill. In order to minimize the artificial skill, it is necessary not only to maximize the sample size but also to minimize M, which must be viewed as the number of a priori chosen predictors. In this work, N is maximized by utilizing the data on a monthly basis. N is therefore equal to 235< 12 months rather than 23 seasons; the penalty is a slight increase in r/at to about 1.5. M is minimized by introducing empirical orthogonal function representations of the spatial structure of the various quantities. Kutzbach (1967) outlines the construction of empirical orthogonal functions, which are also referred to as eigenvectors or principal components. The functions and the associated eigenvalues are obtained by solving an eigenvector equation formulated in terms of the sample covariance matrix. Empirical orthogonal functions have the advantage that they are the most efficient possible data representations in the sense that the amplitudes of the dominant functions account for more variance than any other combination of the same number of parameters or functions. One can therefore maximize the variance contained in a specified number of variables by representing a time series of spatial fields in terms of empirical orthogonal functions and retaining as variables the coefficients of only the most dominant functions or components. The advantage of such a truncation is evident from the linear M-dependence in (2). Other useful properties of the functions are their orthogonality, which ensures the statistical independence of the predictors, and the fact that the eigenvalue of a particular eigenvector is a convenient measure of the fraction of variance described by that eigenvector.

4 376 JOHN E. WALSH In the present work the predictand is the extent of sea ice in the twelve areal subsets of the ice concentration grid shown in Figure 1. The predictors are the coefficients of the dominant pressure, temperature, and sea ice eigenvectors of preceding months. Both the predictor and predictand data are expressed as departures from the monthly means ( ) prior to the eigenvector construction, thereby isolating the nonseasonal variability. Figures 2 and 3 show the three dominant eigenvectors of the sea-level pressure and surface temperature, respectively. The eigenvectors were constructed from the 276 months of data for the 30 points shown in the figures. In each case the first eigenvector corresponds to a general excess (mass or temperature) over the region, while the second and third eigenvectors correspond to gradients in approximately perpendicular directions. Eigenvectors P 2 andp 3 would be expected to be most significant for ice drift anomalies, while T x should correspond to broad anomalies in ice growth and decay. The first 3 of the 12 ice eigenvectors are shown in Figure 4. l x evidently represents a summer (Beaufort/Chukchi Sea) mode of variability, while I 2 represents a winter (Bering Sea) mode. I 3 is a summer mode representing an east-west gradient in the departure from normal sea ice extent along the northern Alaskan coast. Figure 5 shows the cumulative fractions of variance described by the eigenvectors of pressure, temperature, and sea ice. It is evident that most of the variability is contained in the first few modes of each variable, especially in the cases of pressure and temperature. The frequency spectra of the coefficients of the dominant eigenvectors are shown in Figure 6. The spectra of the pressure and temperature coefficients are essentially white noise. The corresponding spectra for the ice eigenvectors are red, implying that the time scales of the ice anomalies are longer than those of the Figure 1. Ice concentration grid. Sea ice predictors and predictands are constructed from mean concentrations along the 12 numbered lines.

5 Empirical Orthogonal Functions and Statistical Predictability 377 p 3 \y6 y^ y>0.1 / /^L \ / xarh.3 / y A / \ ^/\/ 1/1/ ' <.';.-*V. ; '. '>*- 1 \\ \ \\ i? ^V? "^ 1 WA l WV^'"^ \ li-aa / "^^ '^T^^^^' *~~^T-./. \ Figure 2. First three sea-level pressure eigenvectors plotted in units of normalized departures from the monthly means. "H" and "L" indicate maxima (high) and minima (low) in the eigenvectors.

6 378 JOHN E. WALSH T 2 T ; -0.2 / \ IK / n /* y \JM T T \, \ Figure 3. First three surface temperature eigenvectors plotted in units of normalized departures from the monthly means. "W" and "C" indicate maxima (warm) and minima (cold) in the eigenvectors.

7 Empirical Orthogonal Functions and Statistical Predictability 379 l.or Iz \ \ \ \ \ / II / A -A ^^ ^^- K * o y' / _ A Bering Sea i 1 1 1! 1 1! 1 1 I 4 Chukchi Sea 6 8 Pt. Barrow 10 Beaufort Sea 12 Banks Island Figure 4. First three eigenvectors of sea ice extent plotted in units of normalized departures from the monthly means. The abscissa is the coastal location given by the identification scheme of Fig. 1. atmospheric pressure and temperature anomalies. This fundamental difference in the spectra does not bode well for the skill of pressure- and temperature-based ice predictions. RESULTS It is evident from (1) that the covariances between the predictors and predictands are at the core of the computed predictability statistics. Examples of the covariances, expressed as correlations, are shown in Figure 7 as functions of the lags between the two variables. The plots are based on the entire 276 months of data. It should be noted that the ice charts used in this work depict ice conditions at the end of the month for which the corresponding pressure and temperature data are means. Lag "n" could therefore be labelled as lag "n + 1/2." The decay of the autocorrelation curves is more rapid for the meteorological eigenvectors than for the ice eigenvectors, in agreement with Figure 6. The pressure/temperature correlations are in most cases modest ( ) at lag 0 and negligible at other lags. The pressure/ice and temperature/ice correlations are modest at lag 0 in some cases (e.g., (TJ 2 ), (T S I 2 ), (P 3 I 2 )) but negligible at lag 0 in other cases. The decay of the modest values is essentially complete by lag 2-3. As discussed later, the smallness of the cross-correlations involving the ice vectors is partially attributable to the absence of ice variability at certain times of the year.

8 380 JOHN E. WALSH Pressure Figure 5. Cumulative fractions of variance described by the eigenvectors of pressure (P), temperature (T), and sea ice (/). Total numbers of eigenvectors are 30 (P), 30 (7"), and 12 The computed values of the skill, S, are shown in Figure 8 for the summer and winter. The skill is plotted for cases in which the pressure, temperature, and ice eigenvector coefficients were used separately as predictors, and for cases in which a combination of the three was used. The number of predictors appears parenthetically next to each curve. The values of S are plotted as functions of the forecast interval, which again should be interpreted as n + 1/2 months when pressure and temperature data are the predictors. Also included in Figure 8 are the values (dashed lines) of the artificial skill computed from the unapproximated form of (2). The summer curves of Figure 8 imply that persistence is the best one-month predictor, showing a skill of The correlation between the predicted and observed concentration anomalies is therefore V(0.35)~ 0.6. The pressure and temperature skill values, while certainly not impressive, do exceed the artificial skill for lags 0-2 months. A small rise in the temperature skill at 6-8 months is apparent, although it would seem premature to attach much significance to the winter temperatures on the basis of these preliminary results. The rise in the composite (P, T, I) values of 5 at 6-8 months is evidently attributable to the temperature skill. As shown in Figure 8, the composite skill suffers from a larger artificial skill because it is based on a larger number of predictors. The corresponding plots for winter (November-April) show essentially no

9 Empirical Orthogonal Functions and Statistical Predictability 381 Frequency (cycles per year) Figure 6. Frequency spectra of dominant pressure (P), temperature (T), and sea ice (/) eigenvectors. (Note that the spectral density scale is not continuous between upper and lower sets of curves).

10 382 JOHN E. WALSH Lag of y in <xy> Figure 7. Examples of correlations between the coefficients of the dominant pressure, temperature and sea ice eigenvectors. Correlations are plotted as functions of the number of months by which y lags x in (xy ). significant skill beyond one month. In the first two months (lags 0-1), however, the skill values based on the temperature data are significantly higher than those computed from the summer data. These skill values represent correlations of approximately 0.6 and 0.4 at lags 0 and 1, respectively, between the predicted and observed fields of ice extent. A correlation of only V (0.075) ~ 0.27 would be expected in the case of no true skill. CONCLUSIONS The work described here represents an attempt to quantify the extended-range statistical predictability of sea ice extent and to distinguish the true and artificial components of such predictability. While the results are preliminary, they do imply that the pressure and temperature fields of the previous 1-2 months contain a small amount of true predictive value. The only hint of longer-range predictability is in the temperature-based predictions of summer ice extent, although this predictability may well be too small to be useful. The quantitative results of the preceding section must be considered prelimi-

11 Empirical Orthogonal Functions and Statistical Predictability 383 May-October Lag (months) November-April Figure 8. Values of the skill, S, computed from (1) for cases in which sea ice extent is the predictand and the coefficients of the pressure (P), temperature (T), and sea ice (/) eigenvectors are the predictors. Results are plotted for summer (May-October) and winter (November-April) as functions of the number of months by which predictands lag predictors. Numbers of predictors are indicated parenthetically. Dashed lines are computed values of the artificial skill. nary because of an inherent negative bias in the formulation of the skill, S. The formulation does not isolate the forecast errors of interest, since it includes as errors the pressure- and temperature-based forecasts of ice variability for cases in which there is obviously no variability of interest, e.g., the Bering Sea in summer and the northern Alaskan coast in winter. The construction of an unbiased measure of forecast skill for this statistical system is therefore in order, although such a construction may not be trivial. In a further application of the approach described here, predictors representing means for periods longer than a single month may be used. In addition, eigenvectors may be constructed from more than one variable to further reduce the artificial skill. The work can also be extended to other geographical areas for which ice data are available. ACKNOWLEDGMENTS This work was supported in part by the National Science Foundation, Division of Polar Programs, through Grant DPP Computing facility support was

12 384 JOHN E. WALSH provided by the National Center for Atmospheric Research. The U.S. Navy Fleet Weather Facility in Suitland, Maryland, provided a substantial portion of the sea ice data. REFERENCES Barnett, D. G A practical method of long-range ice forecasting for the north coast of Alaska, Part I, Fleet Weather Facility Technical Report TR-1, Suitland, Md. Davis, R. E Predictability of sea surface temperature and sea level pressure anomalies over the North Pacific Ocean. Journal of Physical Oceanography, 6, Haupt, I., and V. Kant Satellite ice surveillance studies in the Arctic in relationship to the general circulation. In Proceedings of the Symposium on Meteorological Observations, Their Contribution to the First GARP Global Experiment, pp Kutzbach, J. E Empirical eigenvectors of sea-level pressure, surface temperature and precipitation complexes over North America. Journal of Applied Meteorology, 6, Lilly, K. E., and J. Stewart Arctic Alaska North Slope resupply operations. Mariners Weather Log, 21, Markham, W. E Ice climatology of the Beaufort Sea. Beaufort Sea Technical Report No. 26, Dept. of the Environment, Victoria, B.C. Sancevich, T. I Methods of long-term hydrometeorological forecasts for the Arctic. In Ice Forecasting Techniques for the Arctic Seas (ed. B. A. Krutskih, Z. M. Gudkovic, and A. L. Sokolov), published for the Office of Polar Programs (NSF) by Amerind Publishing Co. Pvt. Ltd., New Delhi. Schule, J. J., Jr., and W. I. Wittmann Comparative ice conditions in the North American Arctic, 1953 to 1971, inclusive. Transactions, American Geophysical Union, 39, U.S. Navy Fleet Weather Facility Western Arctic Sea Ice Analyses, , ADA , Suitland, Md. U.S. Naval Océanographie Office Report(s) of the Arctic Ice Observing and Forecasting Program, Tech. Reports TR-49 through TR-52, TR-66, TR-69; Special Pubs. SP-70 through SP-81, Washington, D.C. Untersteiner, N The Arctic Ice Dynamics Joint Experiment. Arctic Bulletin, 1, Volkov, N. A., and B. A. Sleptsov-Shevlevich Cyclic variations in the ice cover coefficient of the Arctic Seas (English translation). AIDJEX Bulletin, 16, Walsh, J. E The incorporation of ice station data into a study of recent Arctic temperature fluctuations. Monthly Weather Reveiw, 105, Washington, W. M., A. J. Semtner, Jr., C. Parkinson, and L. Morrison On the development of a seasonal change sea ice model. Journal of Physical Oceanography, 6,