Forecast of Tropical SSTs using Linear Inverse Modeling(LIM)
Cecile Penland, Ludmila Matrosova, Klaus Weickmann and Catherine Smith
NOAA-CIRES/Climate Diagnostics Center, Boulder, Colorado
Using the methods previously described in issues of the Experimental Long-Lead Forecast Bulletin, in Penland and Magorian (1993), and in Penland and Matrosova (1998), the pattern of IndoPacific sea-surface temperature anomalies (SSTA; Fig. 1) as well as SSTA in the Nino 3.4 region (6N-6S, 170W-120W; Fig. 2) the tropical North Atlantic (Figs. 4 and 5), and the Caribbean (Figs. 4 and 6) are predicted. A prediction at lead time tau is made by applying a statistically-estimated Green function G(tau) to an observed initial condition consisting of SSTA in an appropriate domain. Although the parameters of the model are obtained statistically, the dynamical assumption of stable linearity implicit in the method (an assumption that in the case of tropical SSTA is largely corroborated by data) requires a fixed-point attractor in phase space. The technique, therefore, cannot be considered a purely statistical prediction method (Penland 1989; Penland and Sardeshmukh 1995). SST data were provided by NCEP and consolidated into COADS-compatible monthly statistics at CDC. Two sets of predictors/predictands are used, one for the IndoPacific and one for the tropical Atlantic. In both cases, three-month running means of the temperature anomalies are used, the seasonal cycle has been removed, and the data have been projected onto the 20 leading empirical orthogonal functions (EOFs).
The prediction of IndoPacific SSTA uses tropical SSTA in the region (30N-30S, 30E-70W) as predictors. The COADS 1950-1979 climatological annual cycle has been removed, and the leading 20 EOFs explain about 70% of the remaining variance. The Nino 3.4 region has had an RMS temperature anomaly of about 0.7C; the inverse modeling prediction method has an RMS error of about 0.5C at a lead time of nine months and approaches the RMS Nino 3.4 value at lead times of about 15-18 months. The predicted IndoPacific SSTA patterns based on the JJA 2002 initial condition for the following SON, DJF, MAM and JJA are shown in Fig. 1. Fig. 2 shows the predictions (light solid lines) of the Nino 3.4 anomaly for FMA, MAM, AMJ, MJJ and JJA 2002 initial conditions. Light dotted lines indicate the one-standard-deviation confidence interval for predictions based on DJF and MAM 2002. Verifications including the truncation error (heavy dashed line) and omitting the truncation error (heavy solid line) are also shown. Confidence intervals include estimations of the uncertainty due to seasonally-varying stochastic forcing (Penland and Sardeshmukh 1995; Penland 1996, Penland and Matrosova 2001), as well as uncertainties in the initial condition and in the empirically--estimated Green function.
The short-lead (< 6 months) forecasts shown in Fig. 2 may be called many things, but "successful" probably isn't one of them. Fig. 3 shows the time series of Nino 3.4 forecast errors standardized to the one-standard-deviation confidence interval estimated at the time of forecast (see dotted lines in Fig. 2). The horizontal lines at +/- 1.96 indicate the 95% confidence interval for a Gaussian process. The 9-month forecast, which is arguably the most accurate, has now climbed out of this interval.
The prediction of tropical Atlantic SSTA is confined to the north tropical Atlantic (NTA) and Caribbean (CAR) sectors (Fig. 4) since persistence on the timescales shown is a remarkably good predictor of SSTA in the equatorial and south tropical Atlantic (Penland and Matrosova 1998). The added predictability in the northern tropical Atlantic is primarily due to the effect of the Pacific, so SSTA in the global tropical strip (30N-30S) are used as predictors. The leading 20 EOFs in this case contain about 67% of the variance. Forecast skill is discussed in the March 1997 issue of this Bulletin. SSTA in NTA recently took a dive (Fig. 5) so that verification in that region is now within or even cooler than the one-sigma confidence intervals. However, given the large unpredicted anomalies in the recent past, we are reluctant to claim the current forecast as a success. Cooler than observed anomalies continue to be forecast in CAR (Fig. 6).
References:
Penland, C., 1989: Random forcing and forecasting using Principal Oscillation Pattern analysis. Mon. Wea. Rev., 117, 2165-2185.
Penland, C., and T. Magorian, 1993: Prediction of Nino 3 sea surface temperatures using Linear Inverse Modeling. J. Climate, 6, 1067-1076.
Penland, C., and P. D. Sardeshmukh, 1995: The optimal growth of tropical sea surface temperature anomalies. J. Climate, 8, 1999-2024.
Penland, C., 1996: A stochastic model of IndoPacific sea surface temperature anomalies. Physica D, 98, 534-558.
Penland, C., and L. Matrosova, 1998: Prediction of tropical Atlantic sea surface temperatures using Linear Inverse Modeling. J. Climate, 11, 483-496.
Penland, C., and Matrosova, 2001: Expected and Actual Errors of Linear Inverse Model Forecasts. Mon. Wea. Rev., 129, 1740-1745.
Figure captions:
Fig. 1: Forecasts of IndoPacific SST anomalies projected onto 20 leading EOFs, based on JJA 2002 initial conditions. Anomalies were calculated relative to the 1950-1979 COADS climatology. SST data were provided by NCEP and summarized onto COADS-compatible monthly statistics at CDC. The contour interval is 0.3C.
Fig. 2: Predictions (light blue solid lines) of the Nino 3.4 SSTA for initial conditions FMA, MAM, AMJ, MJJ and JJA 2002. Light black dotted lines indicate the one-standard-deviation confidence interval appropriate to a forecast based on FMA 2002 initial conditions. Verifications including the truncation error (heavy red dashed line) and omitting the truncation error (heavy red solid line) are also shown.
Fig. 3: Nino 3.4 forecast error time series for lead times of 3, 6, 9 and 12 months, normalized to the one-standard-deviation confidence interval estimated at the time of forecast (see dotted lines in Fig. 2).
Fig. 4: Map showing the North Tropical Atlantic (NTA) and Caribbean (CAR) regions within which average SSTA is predicted.
Fig. 5: Time series of linear inverse modeling (LIM) predictions (blue solid line) of NTA SSTA for lead times of 3, 6, 9 and 12 months. Anomalies are calculated relative to the 1950-1993 climatology. Also shown are the verification series (red solid line) and the one-standard-deviation confidence interval appropriate to the LIM forecast (black dotted lines).
