1. Introduction
The elevation of the Great Salt Lake (GSL), Utah, reflects variability on the order of periodic, quasi-periodic, and aperiodic time scales. Such features have challenged numerous attempts to forecast GSL volume variations; for example, by traditional Box–Jenkins methods (James et al. 1984), nonparametric forecasting (Abarbanel and Lall 1996; Lall et al. 1996; Sangoyomi et al. 1996), singular spectral analysis (Moon and Lall 1996), and locally weighted polynomial regression (Lall et al. 2006). Furthermore, based upon the fact that surface hydrologic systems are ultimately forced by the atmospheric circulations, Moon and Lall (1996) and Moon et al. (2008) have included such indices as the monthly Southern Oscillation index (SOI) and the Pacific–North America (PNA) index as predictor variables. Such a combination has resulted in a noticeable improvement in short-term (1–2 year) forecasts for the GSL volume.
In other work, Wang et al. (2010b) discerned that the GSL elevation is highly coherent with, yet opposite to, sea surface temperature (SST) variations in the equatorial Pacific within the 10–15-yr time scale; known as the Pacific quasi-decadal oscillation (QDO) (e.g., White and Liu 2008; Meehl et al. 2009; Wang et al. 2010a). The coherence exists because a trans-Pacific teleconnection wave train develops during the alternating periods between the extreme warm and cool phases of the Pacific QDO and modulates the atmospheric circulations over the Gulf of Alaska; these in turn affect the synoptic patterns and thus the precipitation over the GSL watershed. The result of these processes is manifest in the precipitation maxima–minima being reached 3 yr (on average) after the extreme phases of the Pacific QDO. It then takes another 3 yr for the GSL to fully integrate such precipitation changes, leading to an overall 6-yr lag between the extremes of the GSL elevation and the Pacific QDO. The entire process is illustrated in Fig. 1a. Given such circumstances, forecasting the GSL elevation out to 6 yr in advance becomes a possibility.
(a) Adopted from Wang et al. (2010b): 10–15-yr bandpassed ΔSST(Niño-4) (black curve; domain 5°N–5°S, 160°E–150°W), precipitation in the GSL watershed (blue dotted curve), and the GSL elevation (red dashed curve). The GSL elevation lags the precipitation by 3 yr, while the precipitation lags the Pacific QDO by another 3 yr. (b) Annual GSL elevation tendency (ΔGSL; black dashed curve) superimposed with the PCLag model fit (red line) and forecast (green line). (c) As in (b), but for the annual GSL elevation; the forecast includes the 95% confidence interval (green dotted curves). Observations of 2008–10 are presented as black dots.
Citation: Journal of Hydrometeorology 12, 3; 10.1175/2010JHM1352.1
Inspired by the aforementioned linkage between the GSL elevation and the Pacific QDO, this paper investigates the possibility of a longer-term (≥6 yr) forecast of the GSL elevation by utilizing both SST and precipitation time series as predictor variables. Following Wang et al. (2010b), we utilized the Kaplan Extended SST (Kaplan et al. 1998) and the National Weather Service Cooperative Observer Program (COOP) station precipitation (denoted as P) for the period of 1900–2007. The COOP station records were provided by the Utah Climate Center (http://climate.usurf.usu.edu/reports/dynamic.php); only the active COOP stations within the GSL watershed were selected for the precipitation analysis (see Wang et al. 2010b for geographical domain and station information). The structure of the paper is as follows: rationale for the proposed models is discussed in section 2, the models are discussed and evaluated in section 3, and conclusions as to the effectiveness of the prediction are given in section 4.
2. Rationale
The water budget equation for a closed-basin lake indicates that any lake volume change is balanced by precipitation, runoff, and evaporation (Shanahan et al. 2007). As a result, any GSL volume change tends to be highly coherent with the precipitation, especially at low-frequency spectrums (Lall and Mann 1995). Thus, it was decided to select the GSL elevation tendency (ΔGSL), rather than the GSL volume, to forecast and reconstruct the GSL elevation. Given our aim toward a longer-term forecast, the seasonal cycle of the GSL elevation, which is on the order of 2 feet, is not of immediate concern. Therefore, the analysis that follows focuses on the annual mean; that is, the ΔGSL was defined as the difference of GSL(t) − GSL(t − 1), where t is the calendar year. Variables SST and P were annual means throughout the water year (August–July) and the calendar year, respectively. The advantage of the described method is twofold: 1) the ΔGSL time series becomes more stationary because of differencing, and 2) ΔGSL is reflective of the GSL’s hydrologic forcing, which, as previously noted, directly responds to the Pacific forcings of climate. As shown in Figs. 1b,c (dashed lines), the ΔGSL time series exhibits substantially better stationarity than the GSL elevation, despite a discernible quasi-decadal variability.
3. Models—Construction and evaluation
a. Precipitation-lagged (PLag) regression model
Spectral coherence between ΔGSL and (a) P, (b) ΔSST(Niño-4), (c) PC1, and (d) PC2. Blue dotted curves indicate the 95% confidence interval. Red solid (dashed) line is the rejection criteria at the confidence level of 0.01 (0.05). Significant spectral coherences are shaded in light yellow.
Citation: Journal of Hydrometeorology 12, 3; 10.1175/2010JHM1352.1
The GSL elevation was then reconstructed by integrating the modeled–forecasted ΔGSL. Because of the maximum lag t − 42 in Eq. (1), model fitting and integration are only applicable from 1943 onward (the cause of this 42-yr lag will be discussed later). The root mean square error (RMSE) between the fitted ΔGSL and the observed ΔGSL was 0.904 feet (ft, where 1 ft = 0.305 m; Table 1). The RMSE between the fitted and the observed GSL elevation was 1.21 ft; this is less than the GSL elevation’s average seasonal variability.
The RMSE between fitted and observed values.
b. ΔSST(Niño-4)-lagged (SSTLag) regression model
For the sake of completeness, we also constructed an SSTLag regression model, following the same approach as for the PLag model. The best SSTLag model features the same lags as the PLag model, though the coefficient betas did not exceed the 0.05 significance level. However, its large RMSE indicates that the SSTLag model is not superior to the PLag model (Table 1); this was expected given the spectral coherence analysis outlined in Fig. 2b.
c. Principal component-lagged (PCLag) regression model
Projection time series of (top) PC1 and (bottom) PC2 for the PCLag model.
Citation: Journal of Hydrometeorology 12, 3; 10.1175/2010JHM1352.1
As shown in Fig. 1b, the PCLag model (red line) successfully captured the characteristic quasi-decadal variability in ΔGSL, though certain aspects of the interannual variability were not captured; this is not unexpected as the bias is a compromise given our decision to omit short-term lags, and this likely resulted in the underestimation in the fitted ΔGSL around the peaks centered at 1984 and 1998. Although such a bias can be reduced by adding more lags for PC1 between t − 2 and t − 5 (not shown), the added lags would increase the uncertainty of the forecast by at least a factor of 2. The reconstructed GSL elevation (Fig. 1c; red line) depicts the correct rhythm in the observed rises and declines but still inherits the bias of underestimated ΔGSL from 1983 onward. Nevertheless, the 8-yr forecast of the GSL elevation depicts a rising trend that intersects the observed elevation in 2009 and 20101. According to the observations in Fig. 1a, the years around 2009 coincide with a warm-to-cold transition of ΔSST(NINO4) and a projected decline in the precipitation. Thus, the GSL elevation is expected to be in the process of integrating the increased precipitation around 2006 and should be on the rise. The PCLag model forecast is consistent with such a process.
A comparison of the RMSE′s for GSL elevation (Table 1) to first order would suggest that the PLag model is marginally better. To further compare the performance between the PLag and PCLag model, both models were used to make out-of-sample predictions. In regard to longer-term forecasting, predictions up to eight years ahead were made every year after 1985. All model coefficient betas were estimated exclusively on the in-sample dataset. The forecasting performance of the models was assessed on the out-of-sample dataset. Numerically, we have data Xt, t in years, and two predictors XP (PLag) and XPC (PCLag). These predictors depend on the final year tf used for parameter estimation XP (tf) and XPC (tf), where tf < t, and they were used to predict the next 8 years ahead. The out-of-sample prediction errors were then computed. The result is shown in Fig. 4. The RMSEs of the PCLag model are consistently smaller than those of the PLag model, except in four prediction periods starting 1992, 1993, 1997, and 1998. These years happened to be at the extreme phases of ENSO, which may distort or overpower the signal of the Pacific QDO (Wang et al. 2010a). The mean of the RMSEs of the PCLag model is 3.02—about one foot smaller than that of the PLag model, suggesting that the PCLag model performs better in predicting eight years ahead. It is also interesting to note that the PCLag model performs particularly well during the GSL “overturn” periods, such as in the periods of 1991–98 and 2000–07, when the GSL level bottomed out and began to rise (cf. Fig. 1c).
The RMSEs of the PCLag model (red line) and the PLag model (blue line) for predictions up to eight years ahead made every year after 1985.
Citation: Journal of Hydrometeorology 12, 3; 10.1175/2010JHM1352.1
Three periods of the GSL elevation overturn time were further explored, which include 1) an upward trend turning downward (1986–93), 2) a downward trend turning upward (1991–98), and 3) the start of a downward trend (2000–07). During the period of 1986–93 (Fig. 5a), both the PLag and PCLag models predicted the GSL elevation to drop after the rising trend, but in each case the turning point is delayed by 2 yr. The PCLag model forecasted a continuous downward trend while the PLag model forecasted an upward trend after an initial downward trend, which disagreed with reality. For the period 1991–98 (Fig. 5b), both models forecasted the GSL elevation to rise after a few years of a downtrend trend, but the PCLag model aligned more closely with the observation. Finally for the 2000–07 period (Fig. 5c), the PLag model forecasted the 1999 peak but overestimated the 2004 trough, while the PCLag model underestimated the 1999 peak but captured the 2004 trough. The RMSEs of these forecasts are listed in Table 2.
The GSL elevation (black dashed curve) and model fits from the PLag model (red line) and the PCLag model (green line), along with the corresponding forecasts for the periods of (a) 1986–93, (b) 1991–98, and (c) 2000–07. Vertical lines indicate starting year of the forecast. The legend is provided alongside (a). The Shapiro–Wilk normality test results (W) for each model are provided.
Citation: Journal of Hydrometeorology 12, 3; 10.1175/2010JHM1352.1
The RMSE between forecasts and observations.
4. Conclusions
A principal component analysis and a lagged regression model were combined to predict the ΔGSL and, subsequently, the GSL elevation. This approach is somewhat different from previous nonparametric, nonlinear time series forecasting for the GSL volume, since the ΔGSL has a strong hydrological linkage with local rainfall. Based on the lagged relationship between precipitation in the GSL watershed and the Pacific QDO observed previously, we used the annual means of P and the water year means of ΔSST(NINO4) as the predictor variables transformed with PC analysis. The results show that the PCLag model was able to forecast the characteristic turnarounds of the GSL elevation, in which a prolonged increasing trend is followed by a persistent decrease or vice versa. Although previous short-term forecasts were able to predict the GSL volume out to four years in advance during extreme hydrologic events, those short-term forecasts were generally unable to capture the turnarounds of the GSL volume (e.g., Fig. 6 of Lall et al. 2006; Fig. 6 of Moon et al. 2008). Thus, the ability of the PCLag model to estimate the GSL elevation turnarounds is a noteworthy improvement. However, forecasts of the PCLag model exhibit biases in the GSL elevation due in part to the lack of shorter-term (interannual) variability—a trade-off given a desire for longer-term forecast capability. One remedy to this situation might be to couple the proposed longer-term forecast with the established short-term forecast methods.
Acknowledgments
We are indebted to David Stoffer for providing useful R-code that facilitated our statistical analyses. This study was supported by the USDA CSREES–funded Drought Management, Utah Project, and approved as journal number 8230 by the Utah Agricultural Experiment Station, Utah State University.
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The 2010 GSL elevation was measured up to 30 June; both the 2010 and 2009 data shown here are provisional.
