the quasi-periodic fluctuations in the drought indices over the north american great plains

16
Natural Hazards 2: 1-16, 1989. 1 © 1989 Kluwer Academic Publishers. Printed in the Netherlands. The Quasi-Periodic Fluctuations in the Drought Indices over the North American Great Plains E. OLUKAYODE OLADIPO Department of Geography, Ahmadu BelIo University, Zaria, Nigeria (Received: 24 February 1988; revised: 15 December 1988) Abstract. The maximum entropy spectral technique is used to search for periodicities in drought indices over the Great Plains of North America, covering a period of at least 70 years. The results show that in general, quasi-biennial, quasi-triennial and quasi-five-year cycles dominate the spectra of drought indices of many stations in the region. These significant periodicities in the 2-6 year waveband are, however, sporadic in their spatial distribution; they can be evident at one station while being absent at another a few hundred kilometres away. From the results of the spectral analysis, it is concluded that significant peaks obtained in the drought indices are only short-lived time variabilities in drought as recorded by individual stations and do not represent any persistent drought characteristics over a broad geographical region. In general, the drought series display only what appears to be a random variation, with the possible exception of a systematic quasi-biennial oscillation. Key Words: Drought, maximum entropy spectral analysis, quasi-periodicities. 1. Introduction Drought, as a natural hazard, has been a scourge of mankind since antiquity. Its devastating effects in Africa in recent years have been extensively documented (Glantz, 1987a, 1987b; Glantz and Katz, 1985, 1987). Drought, however, does not only occur in Africa. It is an inevitable part of climate and it occurs in high as well as in low rainfall areas. In the North American Great Plains, droughts have occurred throughout available historical records (Hope, 1938; Chakravarti, 1976; Bark, 1978) and will recur in the future. However, the time between consecutive droughts is uncertain and searches for significant periodicities in drought are common in climatological investigations as a first step towards drought prediction. There are different viewpoints on periodic fluctuations of drought/flood condi- tions in the North American Great Plains. Mitchell et al. (1979) presented evidence of drought in the western two-thirds of the United States varying in a pulse-like manner on the same 22-year double sunspot cycle timescale as the Hale magnetic cycle on the Sun. This drought rhythm was, however, later modified to claim a mixed model with terms near 22 and 19 years and a general ill-defined recurrence intervals of from 15 to 25 years (Stockton and Meko, 1983). Currie and others (Currie, 1981, 1984a,

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Page 1: The quasi-periodic fluctuations in the drought indices over the North American Great Plains

Natural Hazards 2: 1-16, 1989. 1 © 1989 Kluwer Academic Publishers. Printed in the Netherlands.

The Quasi-Periodic Fluctuations in the Drought Indices over the North American Great Plains

E. O L U K A Y O D E O L A D I P O Department of Geography, Ahmadu BelIo University, Zaria, Nigeria

(Received: 24 February 1988; revised: 15 December 1988)

Abstract. The maximum entropy spectral technique is used to search for periodicities in drought indices over the Great Plains of North America, covering a period of at least 70 years. The results show that in general, quasi-biennial, quasi-triennial and quasi-five-year cycles dominate the spectra of drought indices of many stations in the region. These significant periodicities in the 2-6 year waveband are, however, sporadic in their spatial distribution; they can be evident at one station while being absent at another a few hundred kilometres away. From the results of the spectral analysis, it is concluded that significant peaks obtained in the drought indices are only short-lived time variabilities in drought as recorded by individual stations and do not represent any persistent drought characteristics over a broad geographical region. In general, the drought series display only what appears to be a random variation, with the possible exception of a systematic quasi-biennial oscillation.

Key Words: Drought, maximum entropy spectral analysis, quasi-periodicities.

1. Introduction

Drought, as a natural hazard, has been a scourge of mankind since antiquity. Its devastating effects in Africa in recent years have been extensively documented (Glantz, 1987a, 1987b; Glantz and Katz, 1985, 1987). Drought, however, does not only occur in Africa. It is an inevitable part of climate and it occurs in high as well as in low rainfall areas. In the North American Great Plains, droughts have occurred throughout available historical records (Hope, 1938; Chakravarti, 1976; Bark, 1978) and will recur in the future. However, the time between consecutive droughts is uncertain and searches for significant periodicities in drought are common in climatological investigations as a first step towards drought prediction.

There are different viewpoints on periodic fluctuations of drought/flood condi- tions in the North American Great Plains. Mitchell et al. (1979) presented evidence of drought in the western two-thirds of the United States varying in a pulse-like manner on the same 22-year double sunspot cycle timescale as the Hale magnetic cycle on the Sun. This drought rhythm was, however, later modified to claim a mixed model with terms near 22 and 19 years and a general ill-defined recurrence intervals of from 15 to 25 years (Stockton and Meko, 1983). Currie and others (Currie, 1981, 1984a,

Page 2: The quasi-periodic fluctuations in the drought indices over the North American Great Plains

2 E. OLUKAYODE OLADIPO

1984b, Hameed and Currie, 1986) suggested the existence of the 18-19 and 10-11 year cycles in drought/flood index series for North America. The cyclic variation of nearly 19 years is attributed to the lunar tide in the atmosphere whose forcing function has a period of 18.613 years, while 10-11 year cycle is presumably related to the well-known solar cycle. Vines (1982, 1984) also presented evidence of a pronoun- ced rainfall cycle of about 22 years in addition to the 18.6 year lunar cycle in the Western United States, and another cycle closer to 16 years in the north-eastern coast and including southern Quebec and the Canadian maritime provinces. In a similar manner, Bradley (1976a) noted a significant quasi-biennial signal in the summer precipitation series of many stations in the Western United States. On the other hand, some authors (Karl and Koscielny, 1982; Diaz, 1983) have found no significant periods in drought index series over the contiguous United States. None of these studies covered the whole of the North American Great Plains and there appears to be no significant attempt at comparing the periodicities of drought between the Canadian Prairies and their counterpart plains of the United States. In this paper, a recently developed spectral technique is used to describe the temporal fluctuations of drought indices over the whole of the Great Plains.

2. Data and Procedures

The data base for the study consists of monthly precipitation totals for 407 stations during the period 1875-1978. The lengths of the periods of observation range from 48 to 103 years with an average station length, for all stations, of 56 years. Interpolation for the missing data, which constituted only about 0.5% of the observations, was carried out according to the median ratio method of Bradley (1976b). In this study, only the time series of the monthly precipitation totals for the growing season months of April to September were used. These are the months during which most of the stations in the North American Great Plains receive about 75 % of their annual totals. Thus, it is expected that the occurrence of drought during any growing season would strongly indicate drought conditions for the year as a whole.

Statistical properties of the 407 precipitation data series were examined by some statistical procedures enumerated in Mitchell et al. (1966) and discussed in Oladipo (1987). In general, the precipitation series were found to be largely homogeneous and normally distributed with no significant evidence of long-term trends.

The growing season precipitation series were used to derive a Bhalme and Mooley (1980)-type drought index for the region. The Bhalme and Mooley drought index (BMDI) is an empirical drought severity index method that uses only monthly precipitation as the sole climatological input. Its derivation is based on the assumption that agriculture in a region adjusts to the average precipitation variability. In spite of its simple approach, BMDI has been shown to perform comparatively as well as the more popular but more complex Palmer severity drought index (PSDI) in detecting periods and intensity of drought (Oladipo, 1985).

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QUASI-PERIODIC FLUCTUATIONS IN DROUGHT INDICES 3

For the North American Great Plains, the growing season BMDI is given by

Ik = 0.548 lk-I + Mk/42.67, (1)

where Ik and Ik-1 are drought intensities for the kth and ( k - 1)th months, respectively, and M k is the moisture index for the kth month. For each month considered (April-September), the moisture index M k is

M k = 100(p - p)/s. (2)

In Equation (2), p is the monthly precipitation, ~ is the long-term mean monthly precipitation, and s is the corresponding standard deviation. For the initial month (k = I) under consideration, we assume in Equation (1)

11 = M1/42.67. (3)

The sequences of monthly drought intensities (I's) for each growing season month and for each station were generated from Equations (1) and (3). From these monthly indices, the mean index series for the seven growing season months were obtained for each of the recorded years and for each of the 407 stations. In the following sections, however, the results of detailed spectral analysis of drought indices for only 105 stations thought to be a really representative of the region, and with a minimum record length of 78 years, are presented. The network of stations selected for analysis is shown in Figure 1. While the transformation of the precipitation data to obtain mean growing season drought indices would have some smoothing effect, the writer strongly believes that this will not seriously damage the signals in the spectra of the drought series.

Two classical techniques of spectral analysis, namely the periodogram (Jones, 1965) and the Blackman and Tukey (1959) methods, are mostly used to estimate the amplitude, frequency and phase components of drought indices or rainfall series. Limited frequency resolution and leakage in the spectral domain are the two dominant limitations of these conventional techniques of spectral analysis. For these reasons, we have used the Maximum Entropy Method (MEM) which has been found to estimate spectra with increased frequency resolution (Currie, 1981, 1984a, 1984b; Hameed and Currie, 1986).

The maximum entropy method (MEM) is based on the transformation of an autoregressive process which neither adds information to nor subtracts information from the data (the concept of maximum entropy). Since its development by Burg (1967) and Parzen (1969), detailed and refined arguments about MEM from the standpoint of theory and practice have been given in Childers (1978) and Kay and Marple (1981), to mention a few. For a discrete equally spaced series of values with zero mean and length N, which is a segment of an infinite stationary series, the power spectrum gives the power at frequencyfas

PM + 1/W (4) s ( f ) = ~ ~ ,

1 + ~ aM, r exp(-- 2njfk At) k=l

Page 4: The quasi-periodic fluctuations in the drought indices over the North American Great Plains

4 E. OLUKAYODE OLADIPO

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Page 5: The quasi-periodic fluctuations in the drought indices over the North American Great Plains

QUASI-PERIODIC FLUCTUATIONS IN DROUGHT INDICES 5

where aM. K are a set of prediction error filter (PEF) coefficients (ak) of order M, PM÷I is the mean output power of the filtered series, At is the sampling interval which is equal to unity in our case and, thus, 14I, the Nyquist interval -½At ~< W~< ½At, is also equal to unity. The coefficients of the filter are then applied to the time- series as the weightings in a running mean, and this ideally removes all the predic- table portions of the signal, thus leaving only the noise.

We used the optimal least squares algorithm of Barrodale and Erickson (1980) to obtain the filter coefficients and the output of Equation (4). The algorithm minimizes the sum of squares of the forward and backward one-step prediction errors. There is some difficulty in obtaining a correct estimate of the required length (M) of the PEF. In general, small M values yield a highly smoothed spectrum, whereas too many produce spurious detail. Many criteria (Akaike, 1969, 1974; Parzen, 1969; Kashyap, 1977; Newton, 1982; Rovelli, 1982) have been suggested for estimating M. The MEM also suffers from the difficulty of assessing the statistical significance of its spectra and from a lack of clearly defined variance estimates.

The few procedures proposed for the statistical significance testing of MEM spectra (Baggeror, 1976; Goers and Koscielny, 1977; Reid, 1979) are very com- plicated and computationally expensive. Their usefulness was therefore limited in this study because of the large amount of data that were involved. A simple procedure, using many values of M, was therefore adopted to determine the significance and stability of the spectral peaks in the drought indices obtained by MEM. Akaike's (1969) final predication error (FPE) criterion was initially used as a guide for the determination of M. Then many values of M were tried experimentally to examine the range for M which yielded similar spectral results.

This approach is somewhat subjective but it ensures that only significant spectra are taken into consideration. The spectra for various M values were graphed and examined for consistency and stability. By this approach, we subscribe to the view of Prof. R. G. Currie (1986, private communication) that practical experience and judgement are very important in interpreting MEM spectra, because none of the criteria for finding the PEF work in the real world.

3. Results and Discussion

The graphs of power spectral density generated by MEM using many M values were examined for amplitude modulation, production or reduction of sidebands and spectra consistency and stability. Because spectral lines resolved by MEM at different M values are liable to peak shifting and splitting, only spectral peaks that remained resolved without significant shifting with frequency or splitting into multi- plets with a range of filter orders and that account for at least 2% of the total spectral variance, were accepted to be real. Whenever there were multiple adjacent peaks in a frequency band range (A f), a single peak that accounted for the highest variance is taken to be the prominent of the multiplets.

In all cases, a low-order autoregressive process, usually AR(1), was chosen, with an

Page 6: The quasi-periodic fluctuations in the drought indices over the North American Great Plains

E. OLUKAYODE OLADIPO

order of 2, AR(2) chosen for only 18 stations, while AR(3) and AR(4) were chosen in one and two cases, respectively, when the FPE criterion was used to select the optimum M values. These results are in good agreement with the work of Katz and Skaggs (1981), but the M coefficients selected by Akaike's criterion did not give much spectral resolution of the drought indices and produced over-smoothed spectra.

Since the Akaike FPE criterion is designed for autoregressive series, its inability to resolve the spectral behaviour of the drought indices is an indication that drought occurrence in the Great Plains, as designated by the BMDI, may not be a purely autoregressive process as modelled by Katz and Skaggs (1981). It may also be an indication of the failure of the FPE criterion to exhibit a clear minimum as required. In any case, the FPE is of limited value in the spectral resolution. When the MEM algorithm was programmed to use many values of M over the range 0.2N ~< M ~< 0.5N, the drought indices spectra remain stable for a range of N/ 3 ~< M ~< 2N/ln 2N orders. The individual spectra are too voluminous to be fully presented. Therefore, the spectral estimates with M = 2N/ln 2N (Berryman's 1978 criterion) are used in the following discussion.

Results of the power spectral analysis for 20 stations (one for each state/province) are illustrated in Figure 2. A scan of all the spectra of the drought indices showed that most were dominated by three peaks associated with the quasi-bennial, quasi- triennial and quasi-five-year oscillations. The geographical distribution of significant peaks in the seasonal spectra of the drought indices in each of the three preferred frequency ranges is shown in Figures 3 to 5.

The quasi-biennial oscillation can be particularly seen in the drought indices spectra of the stations in the Canadian prairies, Illinois, Louisiana, Minnesota, Montana, Nebraska, South Dakota and Texas (Figure 3). This quasi-biennial behaviour is known to exist in several meteorological phenomena, but the fun- damental physical causes of this approximately two-year periodicity is still unclear. Its relationship to the quasi-biennial oscillation in tropical stratospheric winds (Reed et al., 1962) is uncertain because there is still insufficient data available for the latter phenomenon to enable proper comparison to be made. Moreover, these strong peaks near the two-year period should be interpreted with caution because they are very close to the Nyquist frequency and may, therefore, have resulted from energy at higher frequencies folded back into the spectrum at the Nyquist frequency.

Most of the stations with significant quasi-triennial periodicity are concentrated in Iowa, Kansas, Minnesota and South Dakota (Figure 4). The exact significance of the quasi-three-year oscillation in the drought series is not known, although similar periodicity has been observed by Neil (1981) for some precipitation series in some sub-areas of the study region. This periodicity however, is, close to one of the two modal peaks of the Southern Oscillation and it may suggest that the oscillations of the drought indices may be teleconnected with the large-scale atmospheric pressure in the Indo-Pacific region.

Quasi-periodic variations in the range of 5.0 to 5.9 years are strongest and most common in Arkansas, Illinois, Montana and Southern Texas (Figure 5). This

Page 7: The quasi-periodic fluctuations in the drought indices over the North American Great Plains

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Page 8: The quasi-periodic fluctuations in the drought indices over the North American Great Plains

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Page 9: The quasi-periodic fluctuations in the drought indices over the North American Great Plains

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Page 10: The quasi-periodic fluctuations in the drought indices over the North American Great Plains

10 E. OLUKAYODE OLADIPO

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approximately 5-6 year period in climatological time series has received little attention, although similar evidence for its occurrence can be found in the Indian, Indonesian, African and South American rainfall series (Rao et al., 1973; Rhode and Virji, 1976; Fleer, 1981).

The number of stations with significant periodicities in various frequency bands are shown in Figure 6. Only periods of 2 to 25 years are shown in the figure because periods shorter than 2 years are not resolvable with seasonal data, and also because periods greater than one-third of the record length are generally not reliable. Periods greater than 25 years, which are found to be significant in the spectra of drought for some stations, are not therefore taken to be statistically significant, because of the short length of the drought data. Figure 6 shows the repeated occurrence of peaks in three preferred ranges corresponding to about 2-2.9, 3-3.9 and 5-5.9 year frequency bands. In general, periodicities greater than the quasi-five-year frequency bands, while present, are insignificant in terms of the number of cases. The 11.3 and 22.6 year sunspot cycles, which are often regarded as important in the region of study (Thompson, 1973; Hancock and Yanger, 1979; Mitchell et al., 1979; Currie, 1984b), are found to be statistically significant in only about 5 and 7% of the drought indices, respectively. In a similar manner, the periodic 18.6 year signal obtained for the western North American drought index series (Currie, 1984a, 1984b), and associated with the long-term lunar (MN) tidal potential generated by gravitational interactions on the Earth-Moon-Sun system, is present in the spectra of only three of the analysed drought index series. Whether those cycles found in the drought series

Page 11: The quasi-periodic fluctuations in the drought indices over the North American Great Plains

QUASI-PERIODIC FLUCTUATIONS IN DROUGHT INDICES 11

120 ° 115 o 110 ° 105o 100 ° 95 ° 8 0 ° 85 °

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110 ° 105 ° 100 ° 9 5 ° 90 o

Fig. 3. Spatial distribution of stations with significant 2-2.9 year periodicities.

Page 12: The quasi-periodic fluctuations in the drought indices over the North American Great Plains

12 E. OLUKAYODE OLADIPO

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Page 13: The quasi-periodic fluctuations in the drought indices over the North American Great Plains

QUASI-PERIODIC FLUCTUATIONS IN DROUGHT INDICES 13

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Page 14: The quasi-periodic fluctuations in the drought indices over the North American Great Plains

14 E. OLUKAYODE OLADIPO

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by other researchers are really persistent features of drought in the Great Plains is not very clear. On the other hand, instrumental records are generally too short for detecting changes in the amplitude of these periodicities. It is therefore possible that the lack of evidence about the solar-cyclic (11.3 and 22.6 year sunspot cycles) and the lunar nodal tidal (18.6 year) modulations of the drought indices studied here is a result of the sample size used and the period covered.

4. Conc lus ions

The year-to-year fluctuations in the drought indices over selected climatic stations of the North American Great Plains are largely dominated by three spectral peaks associated with the quasi-biennial, quasi-triennial and quasi-five-year oscillations.

Spectral analysis, of course, is unable to confirm physical relationships and, in this case, it is. not very clear whether the significant spectral peaks in the 2-6 year period will persist in the future. Moreover, an examination of the areal distribution of stations with significant quasi-biennial, quasi-triennial and quasi-five-year oscilla- tions (Figures 3-5) does not indicate significant regional preferences that can be used to identify possible causes of drought in the region of study. Consequently, there is no claim of a definite identification of all periods in the drought indices with any predictive power. The present exploratory study does not provide more than a spectral description of preferred timescales in drought occurrence in the Great Plains of North America. However, many of the spectral peaks in the 2-6 year periodicities are coxnmon for many other geophysical and meteorological time series.

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QUASI-PERIODIC FLUCTUATIONS IN DROUGHT INDICES 15

In conclusion, the results of the maximum entropy spectral method suggest that, for all. practical purposes, the mean growing season drought index sequences in the Great Plains of North America are largely time independent. The spectral behaviour indicates that drought occurrence in the region has a large random component with only short-lived time variabilities such as the quasi-biennial, quasi-triennial and quasi-five-year periodicities. These oscillations are not similar for stations located within relatively short distances of each other, suggesting a spatial incoherence in the temporal behaviour of drought occurrence in the region. This apparent lack of regional°.preferences in the temPoral patterns of the drought indices in the Great Plains of North America will complicate forecasting and modelling of the causal mechanisms of drought in the region.

References

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Bark, L. D.: 1978, History of American drought, in N. Rosenberg (ed.) North American Drought, Westview Press, Boulder, pp. 9-24.

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Be~ryman, J. G.: 1978, Choice of operation length for maximum entropy spectral analysis, Geophysics 43, 1384-1391.

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Blackman. R. B. and Tukey, J. W.: 1959, The Measurement of Power Spectra, Dover, New York. Bradley, R. S.: 1976a, Precipitation History of the Rocky Mountain States, Westview Press, Boulder. Bradley, R. S.: 1976b, Secular changes of precipitation in the Rocky Mountain States, Mon. Wea. Rev. 104,

513-523. Burg, J. P : 1967, Maximum entropy spectral analysis, in D. G. Childers (ed.), Modern Spectrum Analysis,

IEEE Press, New York, pp. 34-41. Chakravarti, A. K.: 1976, Precipitation deficiency pattern in the Canadian Prairies, 1921 to 1970, J.

Canad. Plains Res. Center 1, 95-110. Childers, D. G.: 1978, Modern Spectrum Analysis, IEEE Press, New York. Currie, R. G: 1981, Evidence of 18.6 year M~ signal in temperature and drought conditions in North

America since 1800 AD, J. Geophys. Res. 86, 11055, 11064. Currie, R. G.: 1984a, Evidence for 18.6 - year lunar nodal drought in western North America during the

past millenium, J. Geophys. Res. 89, 1295-1308. Currie, R. G.: 1984b, Periodic (18.6 - year) and cyclic (11 year) induced drought and flood in western

North America, J. Geophys. Res. 89, 7215-7230. Diaz, H. F.: 1983, Some aspects of major dry and wet periods in the contiguous United States, 1895-1981,

J. Climate Appl. Meteor. 22, 3-16. Fleer, H.: 1981, Large-scale tropical rainfall anomalies, Bonner Meteor. Abhand. 26. Glantz, M. H.: 1987a, Drought and Economic Development in Sub-Saharan Africa, in M. H. Glantz (ed.),

Drought and Hunger in Africa: Denying famine a future, Cambridge University Press, Cambridge (in press).

Glantz, M. H.: 1987b, Drought, famine, and the seasons in sub-Saharan Africa, in R. Huss-Ashmore and S. Katz (ed0, Anthropoligical Perspectives on the African Famine, Gordon and Breach, New York (in press).

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16 E. OLUKAYODE OLADIPO

Glantz, M. H. and Katz, R. W.: 1985, Drought as a constraint in Sub-Saharan Africa, Ambio 6 334-339. Glantz, M. H. and Katz, R. W.: 1987, African drought and its impacts: revived interest in a recurrent

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Hamced, S. and Currie R. G.: 1986, Cyclic variations in Canadian and United States drought, Paper presented at the Canadian Hydroloffy Symposium (CHS:86), Regina, Saskatchewan, pp. 113-122.

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