1. Sea Level in month i =α 2 sin t i + α 3 cos t i + α 4 sin2 t i + α 5 cos2 t i seasonality + α 6 SOI i climate indices + α 1 + α 7 time i + (deviations from linearity ; df)spline time trend + ε i autocorrelated errors Sea Level in month i =α 2 sin t i + α 3 cos t i + α 4 sin2 t i + α 5 cos2 t i seasonality + α 6 SOI i climate indices + α 1 + α 7 time i + (deviations from linearity ; df)spline time trend + ε i autocorrelated errors Methods for Estimating Trends in Sea Level Methods for Estimating Trends in Sea Level Terry Koen and David Hanslow Cowra Research Centre, Office of Environment & Heritage, P.O. Box 445, Cowra NSW 2794. Australia. Terry.Koen@environment.nsw.gov.au Introduction Many consider trend analysis to be a straight forward affair, possibly resorting to simple linear regression if one wishes to report a single summary statistic such as a linear rate of change. However, data in the form of a time series are likely to exhibit serial correlation, meaning the assumption of independence of errors may be violated. The trend is unlikely to be purely linear in nature, so alternative approaches such as flexible regression techniques (e.g. Generalized Additive Models) may be worth considering. Also, covariates may need to be added to the model, such as the Southern Oscillation Index (SOI) when examining Sea Level (SL) variability. This paper explores methods for the modelling of trends in monthly Sea Level (SL) time series data (1985 to 2014) originating from 14 NSW tidal stations. Various methods for trend analysis are compared including results based on monthly and annual means derived from the same data. Neil J. White, Ivan D. Haigh, John A. Church, Terry Koen, Christopher S. Watson, Tim R. Pritchard, Phil J. Watson, Reed J. Burgette, Kathleen L. McInnes, Zai-Jin You, Xuebin Zhang, Paul Tregoning. (2014). Australian Sea Levels – Trends, Regional Variability and Influencing Factors. Earth-Science Reviews 136, 155-174. Results When assuming iid errors, se(β)’s are under-estimated by a factor of 0.4-0.7 using monthly data; by 0.6-0.9 when using annual data; Series of differing record length, as well as start and end dates, add to the variability in estimated trend; Fitting a spline may suggest evidence of a nonlinear trend and/or effects from longer period climate indices (eg IPO) and processes; Inclusion of a meaningful covariate is likely to influence trend estimate; Trends (mm/year) were similar whether using monthly or annual data; Variation between sites requires further investigation; The records analysed here are short compared with climate indices that have been shown to influence sea level, thereby limiting the ability to differentiate between longer term sea level trends and longer term variability. Site 1Site 2Site 3Site 4Site 5Site 6Site 7Site 8 DataModelN β±seN β±seN β±seN β±seN β±seN β±seN β±seN β±se Monthlylm iid2348.7±1.02380.6±1.62591.9±0.5324-0.1±0.52941.5±0.51983.8±0.93233.0±0.53333.0±0.4 gls AR18.6±2.20.5±2.61.8±0.9-0.1±0.91.4±1.03.7±1.43.0±0.83.0±0.7 spl AR16.9±1.51.8±1.21.6±0.4-0.8±0.50.4±0.52.8±0.62.5±0.51.9±0.4 Annuallm iid 20 8.7±1.9200.1±2.4241.8±0.529-0.1±0.8261.6±0.9174.0±1.1282.6±0.7283.0±0.8 gls AR1 8.4±2.80.1±2.21.8±0.5-0.1±1.11.6±1.24.1±1.32.6±0.73.2±1.2 spl AR1 6.6±1.4-0.2±2.31.3±0.4-0.9±0.60.3±0.73.0±0.81.6±0.61.8±0.8 Site 9Site 10Site 11Site 12Site 13Site 14 DataModelN β±seN β±seN β±seN β±seN β±seN β±se Monthlylm iid3221.4±0.43571.7±0.32832.4±0.5826.5±2.93061.9±0.43161.2±0.4 gls AR11.4±0.71.7±0.52.3±0.86.4±4.31.9±0.71.2±0.7 spl AR10.8±0.41.2±0.31.4±0.58.1±2.21.1±0.40.3±0.5 Annuallm iid271.3±0.7301.8±0.5242.5±0.976.8±2.9261.5±0.7281.1±0.9 gls AR11.3±0.91.8±0.72.3±1.28.1±1.51.2±1.11.2±1.0 spl AR10.8±0.61.7±0.81.4±0.87.6±1.10.7±0.50.0±0.7 Annual cycle Semi-annual Sea Level in month i =α 2 sin t i + α 3 cos t i + α 4 sin 2t i + α 5 cos 2t i inter-annual seasonality + α 6 SOI i climate indices + α 1 + β time i + (deviations from linearity ; df)spline time trend + ε i autocorrelated errors Annual cycle Semi-annual Methods Following White et al. (2014), the general model needed to include terms for: seasonality (annual and semi-annual variability), a climate ‘noise’ covariate (eg SOI), nonlinear trend, modelled as a low order spline, and serial correlation between the residual elements. The R commands lm ( stats package), gls ( nlme ) and gam ( gam ) were used to fit models with iid errors, AR(1) errors, and smoothing splines of degree 2, respectively. The approach was to fit lm (SL ~ Time) assuming iid errors fit gls (SL ~ Time) assuming AR(1) errors adjust SL for effects of seasonality and SOI fit lm (adjSL ~ Time) assuming iid errors fit gam (adjSL ~ s(Time; 2)) assuming iid errors derive ‘deviations from linearity’: DevLin = s(Time;2) - lm fit gls (adjSL ~ Time + DevLin) assuming AR(1) errors