ENSO has been rather active lately, with 1997-8 being one of the strongest El Ni?o events on record, and an extended El Ni?o event persisted from 1991-95. Several “indices” that provide a measure of the ENSO state are available. There is also some discussion (Trenberth and Hoar, 1995, Rajagopalan et al, 1995) as to whether the frequency of ENSO events is changing and whether it is related to anthropogenic climate change. Since this debate is inconclusive, we shall strive to use the indices related to these ?oscillations? as diagnostic tools for understanding oceanic teleconnections to Devils Lake. Here, we have used the Sea Surface Temperature (SST) in the NINO3 region (5N-5S, 150W-90W) in the Eastern tropical Pacific. A positive value of this index corresponds to El Ni?o conditions, and a negative value to La Ni?a conditions. The PDO is represented by the PDO Index develop by Nathan Mantua (Mantua et al., 1997). It corresponds to the leading spatial pattern from a Principal Component Analysis of N. Pacific (30N to 70N) SSTs. An NAO Index is also available, as the difference of normalized sea level pressures (SLP) between Lisbon, Portugal and Stykkisholmur, Iceland. The time series of the NINO3, PDO, and NAO indices have statistically significant cross-correlations at various lags, but have rather different time scales of variation based on spectral analysis.
The time-frequency variation in the three indices, and in the monthly change in the volume of Devils Lake and the Great Salt Lake is examined (Figure 14) through a wavelet analysis (Torrence and Compo, 1998) of their monthly time series. The spectra of these series are generally quite different in their overall character with different frequency bands emphasized. Note the intermittent nature of the narrow band oscillations (indicated by each global wavelet analysis and the corresponding time-frequency plot) in the series, over the period of record. The characteristic time scales of NINO3 are interannual, of the NAO, decadal, and the PDO interannual to interdecadal. The two lakes are more similar to each other than to the indicators, as would be expected. One expects the lakes to have a general long memory behavior. However, they are more organized in frequency bands than say the PDO, which despite preferred interannual and multi- decadal attributes is redder in
Figure 14. Wavelet spectra for NINO3, PDO, NAO, and monthly volume change of Devils Lake, and Great Salt Lake. The full record spectrum is shown to the right. The dashed red line in each such plot is the 95% significance level for red noise. The wavelet spectrum shows the corresponding variation in spectral power for each period as a function of time. For Devils Lake, most of the power at all frequencies is concentrated in the last decade, emphasizing the unusual nature of this period. Considerable time variation in the time-frequency structure as well as common time-frequency structures across the series are also notable.
character. The Devils Lake annual cycle is not nearly as clear as for the Great Salt Lake reflecting somewhat different basin/climate dynamics. The wavelet spectrum for the Great Salt Lake shows a much greater time-frequency commonality with the climate index spectra, than does the Devils Lake. The recent period is clearly the most anomalous for the Devils Lake record.
The role of the three low-frequency climate patterns in generating such an anomaly is explored through an analysis of the correlation between the three climate indices and summer and October continental precipitation in Figure 15. While the spatial patterns of correlation of each index with continental precipitation differ for each season, particularly for the summer, they (a) show strong spatially coherent response structures for each index, and (b) have statistically significant correlations with the general Devils Lake region that are consistent with the increased regional precipitation for these seasons shown in Figures 5 and 7. Note that NINO3 has been positive on average over this period with a protracted positive anomaly over 1991-95 and the large positive anomaly through 1997-8, and is positively correlated with the Devils Lake region precipitation for summer. Likewise PDO has been in its positive phase and is positively correlated while the NAO, which is negatively correlated with precipitation in the region, has been in its negative phase. The correlation patterns of the three indices with the summer 700 mb geopotential height are shown in Figure 16. The correlation of the Devils Lake region’s pressure with NINO3 is not significant. However, the correlations of the PDO and NAO with the regional pressure surface are consistent with the corresponding indications for precipitation. Thus it appears that the conditions in the Pacific and the Atlantic Ocean may jointly influence the atmospheric regime that leads to anomalous summer precipitation in the Devils Lake region. Indeed correlations between the indices and the atmospheric pressure over the U.S. for June through August are consistent with patterns that would steer storms to the Devils Lake area. Similar, but weaker correlations are noted for October.
The connections of the climate index teleconnections to Devils Lake volume changes were further explored through multi-taper spectral analysis. The spectral coherence (correlation at specific frequency bands) between the monthly indices and monthly changes in lake volume was estimated using the multi-taper method of spectral analysis (Thomson, 1982, Mann and Park, 1995). The analysis was performed for all three climate indices and the product of all possible pair-wise combinations (e.g., NAO*PDO) of these indices as a crude approximation to the physical interaction of the climate modes. Significance was assessed in terms of the spectra of the individual series correlated and the coherence at a given frequency. The results indicate that there are three dominant modes of coherent variability, with periods of 12-25 years, 3-5 years, and 2 years. PDO is coherent with the lake volume change with periods of 12-25 and 2 years, while NINO3, NAO, and the product of NINO3 and NAO are coherent with the lake with periods of 3-5 years. Note that the time-frequency analysis using wavelet methods shows considerable time variation in the frequency structure of the Devils Lake monthly volume changes and the climate indices, except for the recent period. Thus it is likely that these teleconnections are largely episodic. Our hypothesis is that these spectral peaks do not necessarily correspond to exactly periodic behavior and coupling at these frequencies. Rather, they represent organization in a space-time dynamical process that leads to preferred, narrow band, spectral signatures.
The significance of the interaction terms in the coherency analysis motivates the search for smaller-scale patterns of SST that may be more directly related to DL volume. This is accomplished by computing the lag cross-correlation between changes in DL volume and gridded SST values in the Pacific and Atlantic Oceans. The patterns observed in the Pacific Ocean were very similar to observed patterns already represented in other indices (PDO, ENSO). However, in the Atlantic Ocean five areas of SST (Figure 17) were identified as potential predictors. Preliminary forecasts of DL volume using all of these SST areas as predictors, as well as the indices for the large-scale climate patterns, consistently chose one climate index (PDO) and one SST area (SEC), as detailed in a later section. A lag cross-correlation analysis indicates that both ENSO and NAO are significantly correlated with SSTs in the SEC area.
Since the Devils Lake and basin hydrology act as integrators of the larger-scale atmospheric circulation and of regional precipitation, it is also useful to look at the time series of the Devils Lake volume in conjunction with running (cumulative) sums of the departures of these indices from their long-term mean. The perspective here is that the large-scale climate states suggested by these indices translate into preferred, concurrent local precipitation and temperature signals, that are modified in some way by the basin hydrology, and then ?added? up over time as lake volumes. Thus, the lake volume can be thought of as a random walk through time where the increments of the random walk are determined in some way by the underlying climate states. Since we don?t
Figure 15. Correlation of U.S. summer (June, July, August) rainfall with (a) NINO3, (b) PDO and (c) NAO, and of October precipitation with (d) NINO3, (e) PDO and (f) NAO Indices for the period 1958-1998. Correlations larger than 0.32 in absolute magnitude are significant at the 95% level. The summer correlations are stronger than those for October, with all 3 indices significantly correlated to the Devils Lake region in the summer, and only the NINO3 significant in October.
Figure 16. Areal averages of sea surface temperature in the Atlantic Ocean that are correlated with changes in Devils Lake volume. The acronyms indicate the approximate location of the area: South-East coast (SEC), Gulf of Mexico (GM), West African coast (WAC), European coast (EUR), and Labrador (LAB).
know precisely how these influences can be described, it is useful to compare the cumulative tendencies of the presumed causative factors (selected climate indices) with the resulting random walk (the Devils Lake volume). This comparison is presented below in Figure 18. Recall that the slope of a cumulative sum curve represents the ?local? mean of the process. In this respect, the NINO3 and the SEC index have recently been at rather different levels than their long-term mean (which is 0), and the PDO has recovered from a 1942-1976 period of an anomalously negative mean. If we assume that these indices are useful predictors of concurrent changes in monthly Devils Lake volume, then these rather long-term changes in state are important for understanding the potential changes in the long-run probabilities of lake volume.
Figure 18. Cumulative trends (obtained through a running sum of anomalies taken about the long-term mean) of NINO3, PDO and SEC climate indices compared to the Devils Lake volume. Note the ?turning points? in the climate series marked by the solid vertical lines around 1942 and 1976. The cumulative SEC and NINO3 indices are at extreme anomalous levels, like the Devils Lake volume at the end of the period.
In summary, evidence for correlation between summer precipitation and atmospheric pressure regimes in the Devils Lake region and indices of quasi-oscillatory climate modes related to the Pacific and Atlantic Oceans was found. The superposition of specific phases of these modes of natural climate variability may then be responsible for the recent, anomalous summer/fall wetness and rise of the lake. Since the indices used to diagnose these climate modes are themselves correlated, one has to be careful in making causal statements about their individual or joint role. Further, the analyses presented thus far consider concurrent variability of the climate indices and the regional hydroclimatic variable of interest. It is not clear that these 0-lag correlations are particularly useful for long lead forecasting of Devils Lake volumes. They merely establish that slowly varying sea surface temperature conditions that are known to be associated with interannual to interdecadal atmospheric circulation pattern changes are relevant for understanding the fluctuations of Devils Lake. No conclusive long run demonstrations of the ability to statistically or deterministically forecast these climate indices have so far emerged. Thus, at present, it is important to recognize that it is possible to establish such links and to think about what the low-frequency nature of the underlying climate state implies for operational and planning models. Clearly, one would need to develop paradigms for both the nature of dependence of the hydrologic variable (Devils Lake) on the climate state, and for the time variation of the underlying climate state. Formal investigations of this sort were not pursued here. The forecasting methods described in the next two sections approach the problem stated here empirically, in ways that have been useful in other related contexts.
Interannual Forecasts Using Nonlinear Time Series Analysis Methods
A nonlinear time series modeling framework that has its roots in the literature (see Diks, 1999; Kantz and Schreiber, 1997; Cutler and Kaplan, 1997; Abarbanel, 1995) on state space reconstruction of dynamical systems from time series data is considered here. The essence of this approach is that attributes of a multivariate dynamical system can be reconstructed from time series of one or more state variables through embedding, An embedding is defined by constructing a multivariate state space using appropriately time lagged copies of the observed time series. For instance if we had observations of only the Devils Lake volume, (denoted as xt, t=1?n), then Takens embedding theorem indicates that the dynamics of the lake volume can be reconstructed by forming the embedding Dt defined as (xt, xt-t, xt-2t,?. xt-(d-1) t), where t is an appropriate sampling frequency, d is the total number of lags considered and d (2p+1), and p is the number of state variables that interact to produce the fluctuations of the lake volume series, xt. Forecasts of future values of the state variable are then possible if the following relationship can be identified from the available data:
(1)
where T is the lead-time for the forecast, and f(.) is a function that describes the dynamical relationship between the past states and the future state.
In practice, we have a finite amount of data, and appropriate values of t and d, as well as the form of f(.) are not known a priori. If the function f(.) is linear in its arguments, this modeling framework is similar to traditional autoregressive time series models. Forecasts from such systems tend to dissipate to the long-term mean value of the time series from whatever condition they are in, unless special efforts are made to include oscillatory or other long-memory features. As was discussed earlier, one would be hard pressed to explain the fluctuations of Devils Lake or other similar processes under the traditional, linear, stationary system paradigm. However, if f(.) is nonlinear, there is a possibility of regime and oscillatory dynamics as well as chaotic dynamics. The system may stay in a particular regime for some time and then switch out to a different regime of behavior. Chaos or loss of predictability is often associated with the regime transitions, as exemplified in the famous work of Lorenz (1963). Oscillations that are the outcomes of positive and negative feedbacks within the system may occur within regimes or across regimes at a variety of time scales. These are the aspects of the climate system that were of interest in the preceding section.
Lall et al (1995) developed a forecasting model for the Great Salt Lake (GSL) using the above ideas, where a nonparametric, spline regression methodology was used to estimate f(.), and statistical criteria were used to choose t, d, and the subset of lagged coordinates used in building the model. They noted that certain regime transitions (e.g., the start of the 1983 rise of the lake) of the Great Salt Lake were not predictable even a few months in advance, but in general one could expect relatively accurate forecasts 1-4 years into the future, even during the extreme rise and fall of the lake. Data on the Great Salt Lake had been reconstructed back to 1847 for the analysis. This allowed some of the extreme fluctuations in the 19th century that are similar to the 1980’s GSL fluctuations to be represented in the data set available for model building. Unfortunately, for Devils Lake, the record could only be reconstructed back to 1905, limiting the ability to reconstruct the dynamics associated with the extreme recent fluctuations. Consequently, in our work here, we have used an extension (see Moon, 1995; Ames (1998)) of the Lall et al (1995) algorithm, that allows for the reconstruction of the dynamics of a target variable (xt) using time series of selected climate indicators. The general forecasting model is represented as :
(2)
where y1, y2,..ym, refer to m potential auxiliary predictors (e.g., climate indicators), with associated sampling frequencies t1, t2, ...tm, and embedding dimensions d1, d2, ?dm, and et is an error process that includes components due to measurement error and due to approximation error in estimating f(.). The approximation error may result from under-specification of the true state space (useful predictors are missing), or from limitations of the numerical scheme used to fit f(.).