International Journal

on Marine Navigation

and Safety of Sea Transportation

Volume 2

Number 2

June 2008

143

A Required Data Span to Detect Sea Level Rise

T. Niedzielski

Space Research Centre, Polish Academy of Sciences, Warsaw

Poland Institute of Geography and Regional Development, University of Wrocław,

Wrocław, Poland

W. Kosek

Space Research Centre, Polish Academy of Sciences, Warsaw, Poland

ABSTRACT: Altimetric measurements indicate that the global sea level rises about 3 mm/year, however, in

various papers different data spans are adopted to estimate this value. The minimum time span of

TOPEX/Poseidon (T/P) and Jason-1 (J-1) global sea level anomalies (SLA) data required to detect a statisti-

cally significant trend in sea level change was estimated. Seeking the trend in the global SLA data was per-

formed by means of the Cox-Stuart statistical test. This test was supported by the stepwise procedure to make

the results independent of the starting data epoch. The probabilities of detecting a statistically significant trend

within SLA data were computed in the relation with data spans and significance levels of the above-

mentioned test. It is shown that for the standard significance level of 0.05 approximately 5.5 years of the SLA

data are required to detect a trend with the probability close to 1. If the seasonal oscillations are removed from

the combined T/P and J-1 SLA data, 4.3 years are required to detect a statistically significant trend with a

probability close to 1. The estimated minimum time spans required to detect a trend in sea level rise are ad-

dressed to the problem of SLA data predictions. In what follows, the above-mentioned estimate is assumed to

be minimum data span to compute the representative sample of SLA data predictions. The forecasts of global

mean SLA data are shown and their mean prediction errors are discussed.

1 INTRODUCTION

Climate change studies are usually associated with

seeking variation rates of various elements of the

environment. Among others, the sea level rise

reflects current global climatic changes as it is

caused by complex interactions between the solid

Earth, atmosphere, oceans, hydrosphere and

cryosphere. Thus, the rate of sea level rise may be

used as an indicator of the global environmental

changes and can be extrapolated in order to build

future scenarios.

Sea level rises as a result of several natural

processes acting in the global environment, which

can be classified into three main groups: geological,

eustatic and steric effects (e.g. Dobrovolski 2000).

The first group concerns the processes, which make

the ocean basins and the coasts change their

parameters. The main changes of this type are

associated with orogenic movements, spreading,

sedimentation, tectonics, subsidence of the sea floor

and the post-glacial rebound. The second group of

processes is connected to the climate itself as being

forced by the increase of water mass of the oceans.

In fact, the eustatic changes are mainly derived from

melting the ice-sheets and glaciers. Finally, the steric

effect is connected to the increase in the water

volume without the change in its mass. This is

largely caused by the thermal expansion of the water

in the oceans as a result of the increase in the global

sea surface temperature.

The considerable number studies focus on the

determination of the rate of sea level change and these

estimates differ due to the wide spectrum of methods

and data sets applied. The trends are being usually

fitted by the least-squares or the robust techniques.

144

The data on the sea surface topography may be

measured both relatively to the Earth crust (tide

gauges) or absolutely (satellite altimetry). For

instance, Douglas (1991) analysed the tide gauge

data and found that a good approximation of the rate

in question is of 1.8 ± 0.38 mm/year. In contrast,

Leuliette et al. (2004) used the recent and precise sea

level anomaly (SLA) time series obtained from the

satellite altimetry TOPEX/Poseidon (T/P) and Jason-1

(J-1) and argued that the rate of sea level change was

of 2.8 ±0.4 mm/year. If no Jason-1 date is

considered, the discussed trend computed by a robust

procedure is equal to 1.46 mm/year (Kosek 2001).

Various data sets on the sea level variation are

usually of dissimilar lengths and hence may be sparse.

The vast majority of the estimates in question is based

upon fitting a trend without much concern whether it

is statistically significant. Hence, there is a need to

reverse the problem and not to estimate the trend itself

but, in turn, to estimate the data span which is

required to detect a statistically significant trend. The

practical usage of such estimates follows from the

SLA prediction studies. In what follows, in order to

construct the representative sample of SLA forecasts,

one needs to fix arbitrarily the first starting prediction

epoch. If one knows the minimum time span of the

SLA data to detect a statistically significant trend in

them (which is the main and the most straightforward

component for extrapolation), it is assumed to be the

first starting prediction epoch.

The method for seeking the above-mentioned es-

timates was proposed by Niedzielski & Kosek

(2006) and presented first at the General Assembly

of the European Geosciences Union in Vienna in

April 2006. The results gained using this simulation-

based statistical technique (Niedzielski & Kosek,

submitted) are applied in this article to support the

evaluation of the prediction results obtained by dif-

ferent forecasting techniques. Thus, this paper aims

to combine the SLA predictions with the detailed

analysis of the rate of sea level rise.

2 METHODS

2.1 Estimation of minimum data span for prediction

According to Niedzielski & Kosek (submitted), the

minimum time span of SLA data required to detect a

statistically significant trend in sea level rise can be

estimated using the statistical simulation based upon

the Cox-Stuart test (McCuen 2003). This statistical

test is designed to test for the existence of an upward

and/or downward trend within the time series. For

the analysis of sea level change it is straightforward

to focus only on upward trends. If the latter applies,

the null hypothesis assumes that there does not exist

a trend in the time series, whereas the alternative hy-

pothesis assumes an upward trend in the underlying

data. In general, the idea behind the Cox-Stuart tech-

nique is simple. It is based on subdividing the time

series x

of size n (even number) into two smaller

data sets. The first one is comprised of the first n/2

data and the second one consists of the remaining

n/2 elements of the initial time series. If n is an odd

number, the middle data point is excluded from the

study and hence n should be replaced by n-1. The

objective of the subsequent statistical analysis is to

compare these two data sets using the 0-1 random

variable defined by







),()(

ibiaif

(1)

where a

=(x

,…, x

), bt =(x

,…, x

) and k = n/2 and

m =(n/2) +1 (if n is an even number); k =(n – 1)/2

and m =(n + 3)/2 (if n is an odd number). Hence, the

random variable

∑

(2)

counts the number of elements of the second time

series b

being greater than the corresponding ele-

ments in the first data set a

. The probability law of T

is binomial b(l,p), where l is a number of a

(or

equivalently b

) elements. The null hypothesis stated

before may be expressed in terms of N

as the equal

amount of zeros and ones. Thus, under the null hy-

pothesis the probability distribution of T is b(l,1/2).

Testing the hypothesis of no trend in sea level rise is

based upon T values and hence – as a result of the al-

ternative hypothesis definition (upward trend) – the

upper tail of the probability distribution is consid-

ered.

In order to make the analysis independent of the

specific starting data epoch it is convenient to apply

the simulation (Niedzielski & Kosek, submitted). In

what follows, one ought to test the above-mentioned

hypothesis for various subsets of a given SLA time

series. To do this, one fixes the small positive inte-

ger t and defines the data block of size t. Subse-

quently, one moves the block forward and applies

the Cox-Stuart test for the new subset of data of size

t. The procedure should be repeated N–t + 1 times.

This allows the computation of the probability of de-

tecting the trend after the time t as

{ }

)(:

)(

1t-N

sjpj

(3)

where p

(j) is a p-value of the Cox-Stuart test for the

j-th location of the block of size t within the entire

time series and s is a significance level. The subse-

quent analysis is based on the stepwise algorithm

145

that performs the above-mentioned analysis by in-

creasing t in each step.

Thus, the required time to detect a trend in sea

level rise is a minimum t for which the probability

given by the equation (3) is close to 1. This time may

support forecasting SLA data as t can be assumed to

be a minimum number of data points.

2.2 Prediction techniques

The variety of forecasting methods is big and hence

one may consider both uni- and multivariate predic-

tion techniques of linear and non-linear structures. In

this paper we apply the most straightforward time

series tools, i.e. fitting and extrapolating the

harmonic-polynomial deterministic model (LS),

autoregressive stochastic modelling and prediction

(AR) and multivariate autoregressive stochastic

modelling and prediction (MAR). The LS approach

is used as a preprocessing tool and thus it estimates

and subsequently extrapolates well-known oscillations

and trends. The residuals from the fitted LS models

are being modelled and predicted using the above-

mentioned stochastic approaches.

The LS model can be denoted as

∑

γ++ϕ+ω=

kkkt

BttAf

)sin(

(4)

where A

,…, A

, B and φ

,…, φ

, γ have to be esti-

mated by least-squares algorithm and ω

,…, ω

are

known frequencies. There are two objectives of LS

modelling. First, the LS model can be extrapolated

to obtain the deterministic prediction. Second, the

LS model may be applied to calculate the residuals,

which can be subsequently modelled and predicted

by stochastic methods.

The AR technique is a simple stochastic method

used to build a model for stationary residuals. The

AR approach is based upon the following equation

tptpt

YcYcY

ε+++=

−−



111

(5)

where Y

is the residual time series obtained as the

difference between the data and the LS model; p is

the order of the autoregressive process; c

,…, c

are

the autoregressive coefficients; and ε

is the white

noise (e.g. Brockwell & Davis 1996). The order p is

being usually chosen by the Akaike Information Cri-

terion (AIC) and the autoregressive coefficients are

estimated using the combination of Yule-Walker and

maximum likelihood methods.

The MAR method is a multivariate extension of

the AR technique. The MAR process is defined by

the following equation

= M

t–1

+…+ M

t–p

+ E

, (6)

where Y

is a vector of stationary residuals computed

at each axis as the difference between the data and

the corresponding LS model; p is the order of the

multivariate autoregressive process; M

,…M

are

the coefficient matrices for multivariate autoregres-

sion; and E

is the white noise vector with mean 0

and covariance matrix C (e.g. Reinsel 1997). The

common method for order selection is a Schwarz

Bayesian Criterion (SBC). The estimation of the co-

efficient matrices is performed by the stepwise LS

algorithm (Neumaier & Schneider 2001).

For the prediction equations we relate to

Brockwell & Davis (1996) and Reinsel (1997) or

more specifically in the field of satellite geodesy to

Niedzielski & Kosek (2005). Three prediction

approaches are used, i.e. LS – extrapolation of the

LS deterministic polynomial-harmonic model;

LS+AR – combination of the LS extrapolation and

the AR prediction of residuals; LS+MAR –

combination of the LS extrapolation and the MAR

prediction of vector residuals.

The verification of the computed predictions is

performed by the analysis of root mean square error

(RMSE) defined as

( )

,)(

)()(













−=

∑

LitLit

XPX

LRMSE

(7)

where X

t(i)+L

are the data at the time t(i)+L, PX

t(i)+L

the L-step prediction of the data.

3 DATA

The modelling and prediction is based upon the two

data sets. The first one is the global mean SLA data

and hence corresponds to the sea level change. The

second time series is a mean global sea surface tem-

perature (SST) and corresponds to the physical de-

scription of the steric effect.

The SLA data are obtained from T/P and J-1

satellite altimetry. In fact, the SLA itself is the

difference between sea surface height (SSH)

computed in respect to the reference ellipsoid and

the mean sea level computed in respect to the geoid

JGM-3. Altimetric measurements are absolute, i.e.

the sea level fluctuations are not mixed with vertical

land movements. The T/P and J-1 satellites are

providing the data on SSH every 1 cycle which is

equal to 9.9140625 days. In this study we use the

T/P global mean SLA time series measured in the

period 01.01.1993 – 01.08.2002, which corresponds

to the T/P cycles No 12-364. As regards J-1 global

mean SLA data, the period 04.02.2002 – 14.07.2003

146

is chosen, i.e. the J-1 cycles No 3-56 are considered.

In this study, both 10-day and monthly global mean

SLA are analysed. As the T/P and J-1 time series

overlap in time, both data sets are combined and bias

correction between the two is introduced. The com-

bined T/P and J-1 data set exhibits an upward trend.

The most energetic oscillations in the data are annual

and semi-annual seasonal components.

The SST data are NOAA OI.v2 SST monthly

fields derived by the weakly optimum interpolation.

These gridded fields are averaged over the entire

ocean in order to obtain the global mean SST data.

The analysed time period coincides with the time pe-

riod of the analysed SLA data, i.e. 01.01.1993 –

14.07.2003. The most energetic oscillations in the

global mean SST data are – similarly to the global

mean SLA data – annual and semi-annual compo-

nents. However, the strength of the semi-annual os-

cillation in global mean SST data is relatively greater

than the strength of the semi-annual seasonality in

the global mean SLA data. Besides, there is no trend

within global mean SST time series.

4 RESULTS

4.1 Estimates of the minimum data time span

Following Niedzielski & Kosek (submitted), seeking

the minimum time span to detect the statistically

significant trend in sea level rise may be subdivided

into two parts. First, the global mean SLA data are

being processed. Second, the non-seasonal global

mean SLA data are considered. If the latter applies,

the global mean SLA time series should be pre-

processed by removing annual and semi-annual

components. The removal of these oscillations al-

lows the analysis of the linear trend itself and sto-

chastic fluctuations. Figure 1 shows the probability

of detecting the statistically significant trend in both

seasonal and non-seasonal global mean SLA data.

The probability is dependent on time. Hence, the

time for which the probability reaches 1 is assumed

to be the minimum required data span. For the stan-

dard significance level of 0.05 the estimates are

equal to 5.5 years (the analysis for seasonal global

mean SLA data) or 4.3 years (for non-seasonal

global mean SLA time series). As noted earlier, the

trend belongs to the key deterministic components

within the studied data and hence – in order to ex-

trapolate it – one needs to know the data span which

guarantees the statistical significance of the model.

Thus, considering the minimum data time span of

4.39 years we assume the cycles No 162 and No 53

(for SLA data with 1 cycle and 1 month sampling in-

terval, respectively) to be the first starting prediction

points.

Fig. 1. The probability of detecting the trend in sea level rise as

a function of time (T/P and J-1 cycles) for a standard signifi-

cance level of 0.05

4.2 Prediction of global mean sea level anomalies

In accordance with the above-mentioned prediction

methods we apply LS, LS+AR and LS+MAR predic-

tion procedures. In the LS and LS+AR cases, the

predictions are based on the past of global mean

SLA data. However, the MAR approach is applied to

combine both global mean SLA data with global

mean SST data in order to consider the contribution

from the steric effect as the explanatory variable.

The deterministic LS modelling of the data is based

on the equation (4) in the following way: for the

global mean SLA data we model annual, semiannual

oscillations and the trend, whereas for the global

mean SST data we consider annual and semiannual

oscillations. As the SST data are monthly, the

LS+MAR analysis is only performed for the data

with the sampling interval of 1-month.

Table 1. Basic statistics (in cm) for the LS predictions of global

mean SLA data.

Statistics

Length of prediction

2-month

6-month

1-year

1.5-year

For data in cycles

Maximum

1.829

1.752

1.803

2.000

RMSE

0.485

0.503

0.589

0.704

For data in months

Maximum

0.960

1.179

1.123

1.118

RMSE

0.338

0.403

0.459

0.577

Tables 1-3 present both (1) maximum absolute

values of the difference between global mean SLA

data and their predictions, (2) values of RMSE. In

general, the comparison of the values of these statis-

tics shows that the predictions of monthly SLA data

are significantly more accurate than the predictions

of the SLA data with the sampling interval of 1 cy-

cle. The interpretation is straightforward and follows

form smoothing of time series. Indeed, the monthly

data are essentially smoothed relatively to the data in

cycles due to the time-averaging procedure. Thus,

the extremes which exist within the time series re-

147

corded in cycles are eliminated (or lessened) in the

process of smoothing. In fact, predicting the ex-

tremes is usually difficult and introduces the consid-

erable error. In the case of absence of the extremes,

the predictions often work well.

Table 2. Basic statistics (in cm) for the LS+AR predictions of

global mean SLA data.

Statistics

Length of prediction

2-month

6-month

1-year

1.5-year

For data in cycles

Maximum

1.953

1.827

1.807

2.002

RMSE

0.496

0.524

0.608

0.707

For data in

months

Maximum

0.901

1.229

1.030

1.121

RMSE

0.340

0.441

0.476

0.580

It is difficult to address the issue of comparison of

the calculated predictions. In fact, the analysis of

maximum absolute values of the difference between

the data and their predictions and RMSE indicate

that all selected procedures lead to the forecasts of

similar accuracy (Tab. 1-3).

Table 3. Basic statistics (in cm) for the LS+MAR predictions of

global mean SLA data.

Statistics

Length of prediction

2-month

6-month

1-year

1.5-year

For data in months

Maximum 1.147 1.122 1.079

1.044

RMSE

0.371

0.411

0.451

0.531

One should suspect that the application of multi-

variate time series analysis would improve the pre-

dictions. This is true only for the long-term (1.5year)

forecasts. In this case the improvement is of order

0.5 mm RMSE and hence is rather insignificant.

5 CONCLUSIONS

The required time span of global mean SLA data to

detect a statistically significant trend in them is

found to be 4.3 years. This estimate is utilized in this

paper to find the minimum data span for forecasting

these data. The comparison results in the conclusion

that the performances of these three approaches are

similar.

ACKNOWLEDGEMENTS

This paper was supported by the Polish Ministry of

Education and Science under the project No 4 T12E

039 29.The authors thank the Center for Space Re-

search, University of Texas at Austin, USA for pro-

viding T/P and J-1 data. The National Centers for

Environmental Prediction, USA are also acknowl-

edged for NOAA OI.v2 SST monthly data provided

in ASCII format by the Data Support Section of the

Scientific Computing Division at the National Cen-

ter for Atmospheric Research. NCAR is supported

by grants from the National Science Foundation. The

authors of R 2.1.1., a Language and Environment

2004 (GNU General Public License) and the addi-

tional packages are also acknowledged.

REFERENCES

Brockwell, P.J. & Davis, R.A., 1996. Introduction to time se-

ries and forecasting. New York: Springer.

Dobrovolski, S.G., 2000. Stochastic Climate Theory: Models

and Applications. Berlin – Heidelberg: Springer.

Douglas, B.C., 1991. Global sea level rise. Journal of Geo-

physical Research 96: 6981-6992.

Kosek, W., 2001. Long-term and short period global sea level

changes from TOPEX/Poseidon altimetry. Artificial Satel-

lites 36: 71-84.

Leuliette, E.W., Nerem, R.S. & Mitchum, G.T., 2004. Calibra-

tion of TOPEX/Poseidon and Jason Altimeter Data to Con-

struct a Continuous Record of Mean Sea Level Change.

Marine Geodesy 27: 79-94.

McCuen, R.H., 2003. Modeling Hydrologic Change: Statistical

Methods. Boca Raton, London, New York, Washington,

D.C.: Lewis Publishers.

Neumaier, A. & Schneider, T., 2001. Estimation of parameters

and eigenmodes of multivariate autoregressive models,

ACM Transactions on Mathematical Software 27: 27-57.

Niedzielski, T. & Kosek, W., 2005. Multivariate stochastic pre-

diction of the global mean sea level anomalies based on

TOPEX/Poseidon satellite altimetry. Artificial Satellites 40:

185-198.

Niedzielski, T. & Kosek, W., 2006. A minimum time span of

TOPEX/Poseidon and Jason-1 global sea level anomalies

data required for trend determination and the multivariate

autoregressive forecast of these data. Geophysical Research

Abstracts 8, 2006, European Geosciences Union, abstract

EGU-06-A-04198.

Niedzielski, T. & Kosek, W. Minimum time span of

TOPEX/Poseidon and Jason-1 global altimeter data to de-

tect the significant trends in sea level change. Submitted to

Advances in Space Research, under review.

Reinsel, G.C., 1997. Elements of multivariate time series

analysis. Berlin, Heidelberg, New York: Springer.