Forecast of Tropical SSTs using Linear Inverse Modeling(LIM)
contributed by 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 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 value at lead times of about 18 months. The predicted IndoPacific SSTA patterns based on the MAM 2002 initial condition for the following JJA, SON, DJF and MAM are shown in Fig. 1. Fig. 2 shows the predictions (light solid lines) of the Nino 3.4 anomaly for DJF, JFM, FMA and MAM 2002 initial conditions. Light dotted lines indicate the one-standard-deviation (67%) 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.
Fig. 3a shows a time series of the Nino 3.4 SSTA (heavy solid line)along with the time series of observed Nino 3.4 SSTA and that of the spatial correlation between the optimal initial structure for variance growth (Penland and Sardeshmukh 1995) and the SST field (heavy dashed line). The second time series is shifted forward by eight months, and represents a forecast of El Nino from the optimal structure alone at a lead of eight months. If the linear relation between these two time series is used as a predictor (thin solid line), we expect the Nino 3.4 anomaly to be about 1.0C warmer than normal. This compares with LIM's "most prob-able prediction" of 0.3C. One wonders, then, whether or not the projection of the optimal structure might be a better eight-month predictor of the Nino 3.4 anomaly than the LIM most probable prediction. Fig. 3b shows that this is not the case in general, even though the LIM most probable fore-cast has to predict the entire pattern, not just the SSTA in Nino 3.4. The correlation between prediction and verification is 0.65 using the optimal structure projection as predictor and 0.71 using the LIM most probable forecast. It is clear from Fig. 3 that this difference in correlation, evaluated between 1991 and the present, arises from the excellent success of the LIM prediction model during spells of cold and moderate SSTA. However, the model is only moderately successful at predicting the magnitude and duration of warm SSTevents.
The prediction of tropical Atlantic SSTA is confined to the north tropical Atlantic (NTA) and Caribbean (CAR) sectors (Fig. 3a, 3b) 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 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.
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 MAM 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 DJF, JFM, FMA and MAM 2002. Light black dotted lines indicate the one-standard-deviation (67%) confidence interval appropriate to a forecast based on DJF and MAM 2002 initial conditions. That is, about one in three predictions could be expected to lie outside this interval even with a perfect model. Verifications including the truncation error (heavy red dashed line) and omitting the truncation error (heavy red solid line) are also shown
Fig. 3: a) Time series of Nino 3.4 SSTA (heavy solid line), the projection onto the optimal structure shifted forward by eight months (heavy dashed line), Nino 3.4 SSTA forecast based on best fit line through a graph of Nino 3.4 SSTA vs. optimal structure projection eight months earlier (light solid line), and one standard deviation expected error of that forecast (light dotted lines with crosses). b) Nino 3.4 SSTA (heavy solid line), corresponding most probable forecast from LIM at a lead time of eight months (heavy dashed line), and the one standard deviation confidence interval appropriate to the LIM forecast (light dotted lines).
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 (67%) confidence interval appropriate to the LIM forecast (black dotted lines).
