1. Motivation
The variability and change of future sea surface height (SSH, denoted η) is the center of much of the concern about the ongoing global warming. Understanding and predicting these key values, globally and regionally, involves projection and space–time integration of the numerous factors that influence SSH. These factors include the wind field, atmospheric pressure, tides, ice melt, river runoff, heat and freshwater exchange, and the shifting ocean circulation itself (Parker 1992; Church et al. 2013). The diverse physics spans a large range of time scales for oceanic response (e.g., Wunsch 2015).
Compared to the atmosphere, most relevant oceanic time scales are very long, ranging from months to thousands of years. The presence of that long time scale (long memory) and the observed small perturbations in the oceanic state suggest that many of the major components determining future values of η can be predicted from a knowledge of the present and past states of the ocean. The expected prediction error (PE) growth of η is not well established. Attempting to estimate the PE via ensembles of climate model simulations reveals large ensemble spread [e.g., the Intergovernmental Panel on Climate Change (IPCC)] (Church et al. 2013; Stainforth et al. 2005; Palmer 2012).
The goal here is to assess quantitatively the extent to which η variability is predictable using linear methods, describing both the deterministic (seasonal) changes as well as the underlying continuum treated as a wide-sense stationary linear process.1 As discussed, for example, by Wunsch (2013), such an approach provides a baseline against which predictions made with considerably more complex methods (nonlinear, nonstationary, extended to spatial structure) can be compared. The general case involves much more complex computations and raises the purely practical issue of whether the linear, univariate, stationary approach is adequate for SSH, and for how long.
An extensive body of literature explores the variability of η with varying degrees of complication, ranging from elementary statistics to the application of hierarchies of general circulation models (GCMs). These methods have varying degrees of regional success (Gille 1994; Chowdhury et al. 2007; Melillo et al. 2014) or global application (Rahmstorf et al. 2012; Church et al. 2013). The purely statistical approach is less concerned with capturing the underlying physics, while the GCM approach treats the η field as the deterministic integrated sum of ocean and atmospheric physics. This present study uses the simplest statistical approach to present a benchmark for more complex studies.
Treating oceanic change as linear may be counterintuitive. However, the modern observational record shows no major shifts in the large-scale baroclinic structure of the ocean (e.g., Roemmich et al. 2012). Apart from small regions of sea ice or convection, well-understood theory also supports the inference of only perturbation changes over periods from decades to centuries (Hirschi et al. 2013).
Interpretation of statistics from short records is difficult (see, e.g., Wunsch 1999; Percival et al. 2001; Ocaña et al. 2016). The methods that underlie much of what is presented here rely on the assumptions that η changes from the superposition of deterministic seasonal components and from a wide-sense stationary stochastic process. Of most relevance to the latter are general red noise processes and the extreme of white noise, which is, by definition, linearly unpredictable. Detection of true nonstationarity is not possible with the short records at hand. Similarly, an infinite number of generalizations to nonlinear representations are possible, but unless the linear assumption can be excluded, it remains an important reference point.
Local and global predictability are in many ways distinct; for example, regional variability in η has been attributed to shifts in wind features, tropical modes, and features such as the North Atlantic Oscillation (Yin and Goddard 2013; Roberts et al. 2016). Here, the approach is that of a univariate “black box,” with the underlying mechanisms (e.g., determining the changing global mean of η) having been discussed in many published papers (Parker 1992; Piecuch and Ponte 2011; Forget and Ponte 2015; Ocaña et al. 2016). The oceans store large portions of the added heat from global warming, and land ice is retreating, along with other external forcings, but discussion of these specific physical contributions as functions of time and position is postponed.
The methods are detailed in section 2, and the results are presented in section 3, where the seasonal and nonseasonal contributions to the variance of η are presented. As defined in this paper, the seasonal component is perfectly predictable, and the nonseasonal portion involves stochastic forecasting. A set of four reestimates is presented: 1) using monthly or annual means of
2. Numerical and ARMA models
Predictability of η is studied using the ECCOv4 global bidecadal state estimate, as described by Wunsch and Heimbach (2013), Forget et al. (2015), and others (see also ECCO Consortium 2017a,b). The state estimate is global, with latitudinal 1° resolution with tropical mesh refinement. A least squares with Lagrange multipliers approach is used to obtain the state estimate. The result is an adjusted, yet free-running, version of the MIT general circulation model (MITgcm; Adcroft et al. 2004). In contrast to most “reanalysis” products, the ECCO oceanic state satisfies basic conservation laws for enthalpy, salt, volume, and momentum, while remaining largely within error estimates of a diverse set of global data (Wunsch and Heimbach 2007, 2013; Stammer et al. 2016). Regions without data are filled in a dynamically consistent way, avoiding the use of untested statistical hypotheses (e.g., Reynolds et al. 2013).
At each point of latitude and longitude (
Example of the process of removing the seasonal signal at 67°S, 149°W. The mean (1992–2011) is removed, and the green line illustrates the fitted seasonal model
Citation: Journal of Climate 31, 7; 10.1175/JCLI-D-17-0142.1
To the extent that
Linear predictability of wide-sense stationary stochastic processes, not distinguishable from Gaussian, is well understood, with a very large literature including standard textbooks (e.g., Box and Jenkins 1970). Here, the formalism is discussed only insofar as it develops the notation to be applied. Most linear methods are based on the autoregressive (AR) process of order n [AR(n)], the moving average (MA) process of order
















The performance of the ARMA(






In the following discussion of the SSH time series, two cases are considered: one where only a time mean has been removed (η), and one where a best-fitting linear trend has been subtracted as well (
3. Results
a. Seasonal variance
Figure 2a shows the total variance
From ECCOv4 of monthly averages from 1992 to 2011. (a) The variance of η (m2)—note the high variance associated with equatorial, western boundary currents, and monsoonal regions. (b) The percentage of variance contained in
Citation: Journal of Climate 31, 7; 10.1175/JCLI-D-17-0142.1
Figure 2b illustrates the percentage of the total variance included in the
b. Seasonal prediction
The seasonal component
c. Nonseasonal variance
Figure 2c shows the percentage of the total variance accounted for by the nonseasonal background process. Here, a visually striking signal appears across the Pacific Ocean along the equator, probably associated with the El Niño–Southern Oscillation (ENSO) climate mode. Further zonal bands appear to the north of the equator. A strong signal occurs in the Bering Strait region. In the Atlantic Ocean, the subtropical regions show active areas, as well as along the paths of the Labrador and southern tip of the East Greenland Currents. In the South Atlantic, the nonseasonal component of variance accounts for a large portion of the variance in the Brazil Current and the Zapiola region of the Argentine Basin, as well as in zonal bands. In the Indian Ocean, the nonseasonal component of variance also accounts for a large portion of the total variance to the west, in the Mascarene Basin region, as well as along the western Indonesian coast associated with the propagation of the throughflow. The Southern Ocean produces a very large signal associated with regions of deep mixed layers and possible mode water formation, as well as in areas where the Antarctic Circumpolar Current (ACC) is directed southward. Predictability associated with this nonseasonal variance is addressed in the remainder of this paper.
d. Predictability after trend removal
A linear trend is now removed from the SSH values, meaning that a possibly perfectly predictable component is eliminated. The
The chosen order of ARMA(
Citation: Journal of Climate 31, 7; 10.1175/JCLI-D-17-0142.1
The simplest linear theory assumes that the underlying values are Gaussian, or close to it, an assumption tested in the ECCO estimated SSH in Fig. 4a using the Shapiro–Wilk test for normality (Shapiro and Wilk 1965). Large areas associated with features such as the ENSO signal appear to deviate from normality (i.e., p values close to 0). This result has implications for the predictability because these departures are important when interpreting the PE.
The Shapiro–Wilk test for normality for (a)
Citation: Journal of Climate 31, 7; 10.1175/JCLI-D-17-0142.1
Figure 5 shows the error growth asymptoting to its upper bound: the full variance of the background residual time series. Large differences as a function of region appear in the PE, as well as in their asymptotic rate of growth. The expected error e-folding structure is shown to illustrate the rate of predictability decay, independent of magnitude.
The prediction error as defined in Eq. (5) for
Citation: Journal of Climate 31, 7; 10.1175/JCLI-D-17-0142.1
The ARMA(
The ARMA(
Citation: Journal of Climate 31, 7; 10.1175/JCLI-D-17-0142.1
High predictability is seen in the North Pacific Ocean. Although the physics here are beyond the intended scope of this paper, these areas are associated with the Kuroshio crossing the Pacific and wave–eddy interactions stretching from Hawaii to the coast of California in a banana shape. Areas in the equatorial Pacific also show better performance of PE growth over time but are clearly nonnormal. Features associated with the Pacific–Antarctic Ridge in the Southern Ocean also produce better performance in terms of the ARMA(
e. Predictability with apparent trends
Tests of predictability are now made with the linear trend left in the time series. Including the trend treats it as an unresolved component of a red noise process. The inclusion of the trend is expected to increase the performance of the ARMA(
The normality of the stochastic background process with the trend is retested in Fig. 4b, illustrating that most of the ocean remains indistinguishable from having a normal distribution in
f. Predictability with annual averages
Interannual and monthly physics are distinct. Assessing the annually averaged
Figure 7 shows the chosen ARMA(
Chosen order of ARMA(
Citation: Journal of Climate 31, 7; 10.1175/JCLI-D-17-0142.1
Figure 8a assesses the extent to which the 20-yr time series can be viewed as coming from a normally distributed population when using annual averages. Most of the ocean passes this test for annually averaged
The Shapiro–Wilk test for normality for the annually averaged (a)
Citation: Journal of Climate 31, 7; 10.1175/JCLI-D-17-0142.1
The associated prediction performance based on the ARMA(
The prediction performance based on the ARMA(
Citation: Journal of Climate 31, 7; 10.1175/JCLI-D-17-0142.1
As expected, the role of the linear trend is also important in the annually averaged
Figure 8b, using annually averaged
4. Discussion and conclusions
In this paper, linear univariate predictability of SSH η is discussed, and benchmarks for more elaborate prediction methods are presented. In general, more complex models and prediction methods (e.g., GCM projections) would need to exceed this PE performance over time to be proven worthwhile. Prediction performance is presented here in terms of the time it takes the expected PE to grow beyond 1 cm. More complex models should necessarily do better, and their use may well be justified, particularly in specific, physically identifiable regions. This approach is supported by work such as Goddard et al. (2015), where certain events in
In the ECCOv4 state estimate, the seasonal cycle
The percentage of 20-yr variance in the stochastic, nonseasonal component
An important next step is to distinguish the physical mechanisms, whether atmospheric or oceanic, in regions where the ARMA prediction procedure does well and those where it works poorly. Stationary linear prediction methods can be of limited utility for a variety of reasons. These include dominance by unpredictable white noise (e.g., from atmospheric forcing), non-Gaussian forcing functions, strong nonlinearities in ocean physics, and nonstationary behavior from external forcing and lack of equilibrium in the ocean.
Investigating the contribution of specific mechanisms to the predictability structures of η is outside the scope of this study, but results presented suggest such analysis is merited. For example, work with linear models of
Predictability from annually averaged data, as in Figs. 9a and 9b, proves generally different. With the trend included, a region in the western South Pacific Ocean has striking performance. Paradoxically, an increase in prediction performance is seen in a band extending northeast of Hawaii when the trend is removed. This is likely a stochastic artifact. Given the long time scales controlling oceanic physics, the records remain far too short to infer statistically stable results. In this context, the continuing difficulties, generally experienced in distinguishing the lowest frequencies present between a general red noise process and a true secular trend of multidecadal applicability, remain a major issue. Whether unconstrained models, such as the CMIP5 ones used by Lyu et al. (2014), have true prediction skill remains unknown. Note, too, that the univariate approach used here is readily extended to accommodate multivariate predictive models employing correlated spatial structures of many different types, which may work much more effectively in some areas.
In brief summary, the present study produces a benchmark of univariate linear skill in predicting η. Figure 2b illustrates that up to 50% of the ocean η variability is accounted for >80% using only the seasonal signal over the 20 years of the ECCO state estimate. The remaining ocean η variability has a significant stochastic component, with expected prediction error growth largely taking over 2 months to exceed 1 cm. Figure 6b shows that treating the linear trend as part of the continuum enhances the predictive performance, as expected. In areas in the Southern and Pacific Oceans, the stochastic continuum
Acknowledgments
This work was funded by the U.S. National Aeronautics and Space Administration Sea Level Change Team (contract NNX14AJ51G) and through the ECCO Consortium funding via the Jet Propulsion Laboratory.
APPENDIX
Influence of Chosen Information Criteria


As discussed by Priestley (1981) and Yang (2005), the AIC tends to overestimate the true order, and the BIC tends to underestimate it. The AIC results showed better predictive power. The two criteria give different weights to penalizing the number of regression coefficients, with the BIC having a larger penalty term.
Figure A1 illustrates the orders chosen by the BIC for
Chosen order of ARMA(
Citation: Journal of Climate 31, 7; 10.1175/JCLI-D-17-0142.1
The predictive potential associated with the
The ARMA(
Citation: Journal of Climate 31, 7; 10.1175/JCLI-D-17-0142.1
The prediction performance difference with AIC − BIC using monthly averaged (a)
Citation: Journal of Climate 31, 7; 10.1175/JCLI-D-17-0142.1
For annually averaged
The difference in predictability with the AIC and BIC informing the choice of the order suggests that the AIC has the highest utility. The BIC underestimates the order, and the AIC is found to be more suitable. This is illustrated in detail, looking at the predictability with the apparent trend in
REFERENCES
Adcroft, A., C. Hill, J. M. Campin, J. Marshall, and P. Heimbach, 2004: Overview of the formulation and numerics of the MIT GCM. Proc. ECMWF Seminar Series on Numerical Methods, Recent Developments in Numerical Methods for Atmosphere and Ocean Modelling, Reading, United Kingdom, ECMWF, 139–150.
Aho, K., D. Derryberry, and T. Peterson, 2014: Model selection for ecologists: The worldviews of AIC and BIC. Ecology, 95, 631–636, https://doi.org/10.1890/13-1452.1.
Akaike, H., 1973: Information theory and an extension of the maximum likelihood principle. Proc. Second Int. Symp. on Information Theory, Budapest, Hungary, Institute of Electrical and Electronics Engineers, 267–281.
Box, G., and G. Jenkins, 1970: Time Series Analysis: Forecasting and Control. Holden-Day, 553 pp.
Burnham, K. P., and D. R. Anderson, 2002: Model Selection and Multimodel Inference: A Practical Information-Theoretic Approach. 2nd ed. Springer-Verlag, 488 pp., https://doi.org/10.1007/b97636.
Burnham, K. P., and D. R. Anderson, 2004: Multimodel inference: Understanding AIC and BIC in model selection. Sociol. Methods Res., 33, 261–304, https://doi.org/10.1177/0049124104268644.
Chowdhury, M. R., P.-S. Chu, T. Schroeder, and N. Colasacco, 2007: Seasonal sea-level forecasts by canonical correlation analysis—An operational scheme for the U.S.-affiliated Pacific Islands. Int. J. Climatol., 27, 1389–1402, https://doi.org/10.1002/joc.1474.
Church, J. A., and Coauthors, 2013: Sea level change. Climate Change 2013: The Physical Science Basis, T. F. Stocker et al., Eds., Cambridge University Press, 1137–1216.
ECCO Consortium, 2017a: A twenty-year dynamical oceanic climatology: 1994–2013. Part 1: Active scalar fields: Temperature, salinity, dynamic topography, mixed-layer depth, bottom pressure. ECCO Consortium Rep., 54 pp.
ECCO Consortium, 2017b: A twenty-year dynamical oceanic climatology: 1994–2013. Part 2: Velocities, property transports, meteorological variables, mixing coefficients. ECCO Consortium Rep., 45 pp.
Forget, G., and R. M. Ponte, 2015: The partition of regional sea level variability. Prog. Oceanogr., 137, 173–195, https://doi.org/10.1016/j.pocean.2015.06.002.
Forget, G., J.-M. Campin, P. Heimbach, C. N. Hill, R. M. Ponte, and C. Wunsch, 2015: ECCO version 4: An integrated framework for non-linear inverse modeling and global ocean state estimation. Geosci. Model Dev., 8, 3071–3104, https://doi.org/10.5194/gmd-8-3071-2015.
Gille, S. T., 1994: Mean sea surface height of the Antarctic Circumpolar Current from Geosat data: Method and application. J. Geophys. Res., 99, 18 255–18 273, https://doi.org/10.1029/94JC01172.
Goddard, P. B., J. Yin, S. M. Griffies, and S. Zhang, 2015: An extreme event of sea-level rise along the northeast coast of North America in 2009–2010. Nat. Commun., 6, 6346, https://doi.org/10.1038/ncomms7346.
Hirschi, J. J.-M., A. T. Blaker, B. Sinha, A. Coward, B. de Cuevas, S. Alderson, and G. Madec, 2013: Chaotic variability of the meridional overturning circulation on subannual to interannual timescales. Ocean Sci., 9, 805–823, https://doi.org/10.5194/os-9-805-2013.
Hughes, C. W., and S. D. P. Williams, 2010: The color of sea level: Importance of spatial variations in spectral shape for assessing the significance of trends. J. Geophys. Res., 115, C10048, https://doi.org/10.1029/2010JC006102.
Lyu, K., X. Zhang, J. A. Church, A. B. A. Slangen, and J. Hu, 2014: Time of emergence for regional sea-level change. Nat. Climate Change, 4, 1006–1010, https://doi.org/10.1038/nclimate2397.
Melillo, J. M., T. C. Richmond, and G. W. Yohe, Eds., 2014: Climate change impacts in the United States: The Third National Climate Assessment. U.S. Global Change Research Program Rep., 841 pp., https://doi.org/10.7930/J0Z31WJ2.
Ocaña, V., E. Zorita, and P. Heimbach, 2016: Stochastic secular trends in sea level rise. J. Geophys. Res. Oceans, 121, 2183–2202, https://doi.org/10.1002/2015JC011301.
Palmer, T. N., 2012: Towards the probabilistic Earth-system simulator: A vision for the future of climate and weather prediction. Quart. J. Roy. Meteor. Soc., 138, 841–861, https://doi.org/10.1002/qj.1923.
Parker, B., 1992: Sea level as an indicator of climate and global change. Mar. Technol. Soc. J., 25, 13–24.
Pattullo, J., W. Munk, R. Revelle, and E. Strong, 1955: The seasonal oscillation in sea level. J. Mar. Res., 14, 88–155.
Percival, D. B., J. E. Overland, and H. O. Mofjeld, 2001: Interpretation of North Pacific variability as a short- and long-memory process. J. Climate, 14, 4545–4559, https://doi.org/10.1175/1520-0442(2001)014<4545:IONPVA>2.0.CO;2.
Piecuch, C. G., and R. M. Ponte, 2011: Mechanisms of interannual steric sea level variability. Geophys. Res. Lett., 38, L15605, https://doi.org/10.1029/2011GL048440.
Ponte, R. M., and C. G. Piecuch, 2014: Interannual bottom pressure signals in the Australian–Antarctic and Bellingshausen Basins. J. Phys. Oceanogr., 44, 1456–1465, https://doi.org/10.1175/JPO-D-13-0223.1.
Priestley, M. B., 1981: Spectral Analysis and Time Series, Volumes 1–2. Academic Press, 890 pp.
Rahmstorf, S., M. Perrette, and M. Vermeer, 2012: Testing the robustness of semi-empirical sea level projections. Climate Dyn., 39, 861–875, https://doi.org/10.1007/s00382-011-1226-7.
Reynolds, R. W., D. B. Chelton, J. Roberts-Jones, M. J. Martin, D. Menemenlis, and C. J. Merchant, 2013: Objective determination of feature resolution in two sea surface temperature analyses. J. Climate, 26, 2514–2533, https://doi.org/10.1175/JCLI-D-12-00787.1.
Roberts, C. D., D. Calvert, N. Dunstone, L. Hermanson, M. D. Palmer, and D. Smith, 2016: On the drivers and predictability of seasonal-to-interannual variations in regional sea level. J. Climate, 29, 7565–7585, https://doi.org/10.1175/JCLI-D-15-0886.1.
Roemmich, D., W. J. Gould, and J. Gilson, 2012: 135 years of global ocean warming between the Challenger expedition and the Argo Programme. Nat. Climate Change, 2, 425–428, https://doi.org/10.1038/nclimate1461.
Schott, A., P. Friedrich, and J. McCreary, 2001: The monsoon circulation of the Indian Ocean. Prog. Oceanogr., 51, 1–123, https://doi.org/10.1016/S0079-6611(01)00083-0.
Shapiro, S. S. and Wilk, M. B., 1965: An analysis of variance test for normality (complete samples). Biometrika, 52, 591–611, https://doi.org/10.1093/biomet/52.3-4.591.
Stainforth, D., and Coauthors, 2005: Uncertainty in predictions of the climate response to rising levels of greenhouse gases. Nature, 433, 403–406, https://doi.org/10.1038/nature03301.
Stammer, D., M. Balmaseda, P. Heimbach, A. Köhl, and A. Weaver, 2016: Ocean data assimilation in support of climate applications: Status and perspectives. Annu. Rev. Mar. Sci., 8, 491–518, https://doi.org/10.1146/annurev-marine-122414-034113.
Wunsch, C., 1999: The interpretation of short climate records, with comments on the North Atlantic and Southern Oscillations. Bull. Amer. Meteor. Soc., 80, 245–255, https://doi.org/10.1175/1520-0477(1999)080<0245:TIOSCR>2.0.CO;2.
Wunsch, C., 2013: Covariances and linear predictability of the Atlantic Ocean. Deep-Sea Res. II, 85, 228–243, https://doi.org/10.1016/j.dsr2.2012.07.015.
Wunsch, C., 2015: Modern Observational Physical Oceanography: Understanding the Global Ocean. Princeton University Press, 511 pp.
Wunsch, C., and P. Heimbach, 2007: Practical global oceanic state estimation. Physica D, 230, 197–208, https://doi.org/10.1016/j.physd.2006.09.040.
Wunsch, C., and P. Heimbach, 2013, Dynamically and kinematically consistent global ocean circulation and ice state estimates. Ocean Circulation and Climate: A 21 Century Perspective, G. Siedler et al., Eds., International Geophysics Series, Vol. 103, Academic Press, 553–579.
Yang, Y., 2005: Can the strengths of AIC and BIC be shared? Biometrika, 92, 937–950, https://doi.org/10.1093/biomet/92.4.937.
Yin, J., and P. B. Goddard, 2013: Oceanic control of sea level rise patterns along the East Coast of the United States. Geophys. Res. Lett., 40, 5514–5520, https://doi.org/10.1002/2013GL057992.
“Wide sense” stationarity is the terminology of electrical engineering; mathematicians call it “weakly” stationary, and in both cases, only the first two moments (mean and variance) are assumed time independent (Priestley 1981).