R ECONSTRUCTION OF REGIONAL MEAN SEA LEVEL ANOMALIES FROM TIDE GAUGES USING THE NEURAL NETWORK APPROACH

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R ECONSTRUCTION OF REGIONAL MEAN SEA LEVEL ANOMALIES FROM TIDE GAUGES USING THE NEURAL NETWORK APPROACH

M. Wenzel and J. Schr¨oter

Alfred-Wegener-Institute for Polar and Marine Research, Bremerhaven, Germany

The Neural Network

Regional mean sea level anomalies (RMSLA) are estimated from tide gauge values directly using the neural network approach. A neural network is an artificial neural system, a computational model inspired by the notion of neurophysical processes. It consists of several processing elements called neurons, which are interconnected with each other exchanging information (see e.g. Freeman and Skapura, 1991) . In this presentation a backpro- pagation network (BPN) is used. In this type of network the neurons are ordered into layers: an input layer on the top, one or more hidden layers below and an output layer at the bottom.

Figure 1 shows a BPN that is enriched by direct connections between the input and the output layer. Its output~y in dependence to the input~x can be desribed by the equation:

~y =

O

₍_~_b_o ₊_WIO·~x+WHO·

H

₍_b_~_h ₊_WIH·~x))

where

H

₍₎ _and

O

₍₎ desribe the transfer functions of the hidden and the output neurons, respectively. The matrices of the connection strength between the neurons from the different layers (WIO, WIH and WHO) as well as the bias terms ~b_h and ~b_o are estimated in a training phase, i.e. the BNP learns from given examples. This leads to a costfunction that is minimized by gradient descent.

input layer

hidden layer

output layer

bias

bias 1.0

1.0

equal to input layer

neural network layout

x(1) x(2) x(3) x(N-2) x(N-1) x(N)

y(1) y(2) y(K-1) y(K)

Fig. 1: [above] Layout of a backpropagation network (BPN) enriched by direct connections between the input and the output layer (as indicated by the blue lines). [right top] Scetch of a hidden/output layer neuron [right bot- tom] Three possible transfer functions for the hidden/output layer neurons.

The final choice is H ₍_~_{x) =} _tanh(_~_x) for the hidden and O ₍_~_{x) =}_~_{x for the}

output layer.

z_m

input section

transfer/output section

a_m = ^N_n=1 W_m,n x_n

z_m = a_m + b_m x_n

0 1

most common neuron transfer functions

y = 1/(1+exp(-x))

-1 0

1 y = tanh(x)

-1 0

1 y = x

The Data

30 60 90 120 150 180 210 240 270 300 330 360

-80 -60 -40 -20 0 20 40 60 80

number of avail. data

= 1286

selected tide gauges

1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000 0

5 10 15 20 25 30 35 40 45 50 55

tide gauge

tide gauge data

0.5 0.7 0.9 0 5 10 15 20 25 30 35 40 45 50 55 normalized data

per tide gauge

0.0 0.2 0.4 0.6 0.8 1.0

N

normalized data per timestep

Fig. 2: In the plate on the left the positions of the selected tide gauges and the corresponding amount of data are given by the red circles and the vertical bars, respectively. The data availability is demonstrated in more detail in the graphs on the right.

For our purpose 56 tide gauges are selected from the PSMSL monthly data that comply with the following conditions:

• there are more than 11 annual mean values given in [1993,2005]

• more than 50 annual mean values are given in [1900,2007] and

• the tide gauge is neighboured by at least one ocean point on a 1^◦ x 1^◦grid.

The selected tide gauges are GIA corrected using the Peltier ICE5G_VM4_L90 dataset also available on the PSMSL web side.

1993 1995 1997 1999 2001 2003 2005 2007

-0.6 -0.4 -0.2 0.0 0.2 0.4

cm/month

TP/JS (CSIRO) TOPEX (GFZ)

Trop. Pacific (TPAC) network training (target values)

regional mean SLA - monthly differences

1993 1995 1997 1999 2001 2003 2005 2007

-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3

cm/month

TP/JS (CSIRO) TOPEX (GFZ)

Global Ocean network training (target values)

regional mean SLA - monthly differences

Fig. 3: Comparing the RMSLA’s (monthly differences) from the CSIRO and the GFZ dataset for the tropical Pacific (left) and the global ocean (right).

A single BPN is trained to compute all regional mean SLA’s (trop. Indian Ocean, ... South Atlantic Ocean to Global Ocean) at once from the tide gauge values. To avoid possible problems with the different local reference frames all computations are done in the space of temporal derivatives. Beyond that, this makes the data more suitable for the BPN because it better limits the possible range of the numerical values.

To train the BPN known regional mean target values are needed. These values are derived either from the TOPEX/Poseidon data processed by GFZ Potsdam (T.Sch¨one, S.Esselborn pers. communication) and/or from the combined TOPEX/Poseidon and Jason-1 sea level fields available at the CSIRO sea level webpage (www.cmar.csiro.au/sealevel/sl_data_cmar.html). Differences between these datasets appear mainly in the tropical belt (15^◦N-15^◦S, e.g. the tropical Pacific, Fig. 3 left) and are also visible in the global mean (Fig. 3 right).

Filling Tide Gauge Data Gaps

Although every tide gauge has more then 50 years of data, many values are missing, especially prior to 1950 (see Fig. 2). To fill these data gaps at the input layer of the BPN several alternatives (see Table on the right) are tested. This includes a reconstruction using an EOF basis estimated from all timesteps that have a complete tide gauge dataset. Furthermore a fore- cast network is build, that is trained to compute the values at all tide gauge positions for timestep (n+1) from all values at the steps (n) and (n-1). Ad- ditionally an equivalent backcast network is constructed that computes the values for step (n-1) from the steps (n) and (n+1). Each of these networks has the following dimension: 112 input, 224 hidden and 56 output neurons.

The best estimate, i.e. with minimal error at known data points, is achieved at most timesteps by the backcast network with input gaps filled by EOF reconstruction (case 8, ∼40%) and by the forecast network with EOF filling (case 5, ∼34%).

1918 1919 1920 1921 1922 1923 1924 1925 1926 1927 1928 -20

-15 -10 -5 0 5 10 15 20

cm/month

fc/recurr fc/mac fill fc/eof fill

bc/recurr bc/mac fill bc/eof fill

fc/bc mean fc/bc best org data

gaps filled by Prince Rupert 130.33W 54.32N

tide gauge SLA / monthly differences

1938 1940 1942 1944 1946 1948 1950 1952 1954 -10

-5 0 5 10

cm/month

fc/recurr fc/mac fill fc/eof fill

bc/recurr bc/mac fill bc/eof fill

fc/bc mean fc/bc best org data

gaps filled by

Lewes (Breakwater Harbor) 75.10W 38.78N

tide gauge SLA / monthly differences

Fig. 4: Data gap filling given for two example the tide gauges: Prince Rupert (left) and Lewes, Breakwater Harbor (right)

acronym method

1: mac mean annual cycle (MAC) 2: eof EOF reconstruction (EOFR)

3: fc/recurr forecast network, recurrent, reset input to known values 4: fc/mac fill forecast network, input gaps filled by MAC

5: fc/eof fill forecast network, input gaps filled by EOFR

6: bc/recurr backcast network, recurrent, reset input to known values 7: bc/mac fill backcast network, input gaps filled by MAC

8: bc/eof fill backcast network, input gaps filled by EOFR

9: fc/bc best best of 3 to 8 (minimal fore-/backcast error at known values) 10: fc/bc mean error weighted mean of 3 to 8

Regional Sea Level Anomaly

1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000 -30

-20 -10 0 10

cm

TOPEX Jevrejeva etal Church+White

mac eof fc/recurr

bc/recurr fc/mac fill bc/mac fill fc/eof fill bc/eof fill fc/bc mean fc/bc best all mean

Global

1.60

estimated mean trend [mm/year]

0.37

target values estimated from:

CSIRO TOPEX/Jason data and

GFZ Potsdam TOPEX data

reconstructed MSLA

0 20 40 60 80 100

%

[ 100% 56 gaps ] number of input data gaps

1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000 -60

-50 -40 -30 -20 -10 0 10

cm

TOPEX mac eof

fc/recurr bc/recurr fc/mac fill bc/mac fill fc/eof fill bc/eof fill fc/bc mean fc/bc best all mean

North Pacific

2.73

1.35

1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000 -60

-40 -20 0 20

cm TOPEX mac eof

North Atlantic

2.40

1.75

1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000 -40

-20 0 20 40

cm

TOPEX mac eof

trop. Indian

0.33

1.83

1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000 -60

-50 -40 -30 -20 -10 0 10

cm

TOPEX mac eof

trop. Pacific

2.48

0.81

1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000 -50

-40 -30 -20 -10 0 10

cm ^TOPEX ^mac ^eof

trop. Atlantic

2.02

1.04

1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000 -10

0 10 20 30

cm

TOPEX mac eof

South Indian

-1.04

0.83

1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000 -50

-40 -30 -20 -10 0 10

cm

TOPEX mac eof

South Pacific

2.45

0.97

1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000 -20

-10 0 10 20

cm

TOPEX mac eof

South Atlantic

0.03

0.72

⇐= Fig. 5: RMSLA for the different ocean regions (color

shaded areas in Fig. 2,left). The solutions from all filling cases (see Table above) are shown. The thick red curves (all mean) give the ensemble mean and the grey shading the cor- responding standard deviation. For the global ocean (top of left column) the results from Church and White (2006) and from Jevrejeva et al (2006) are included.

NOTE: All curves are smoothed before plotting to eliminate the annual cycle!

regional mean sea level trend [mm/year]

target values for training taken from

region CSIRO only GFZ only CSIRO+GFZ TIND 1.01±1.36 0.58±1.61 0.33±1.83 SIND -0.12±0.51 -0.17±0.38 -1.04±0.83 NPAC 1.27±1.24 2.07±1.49 2.73±1.35 TPAC 2.56±0.99 1.79±0.59 2.48±0.81 SPAC 1.95±0.67 2.64±0.97 2.45±0.97 NATL 1.57±1.54 1.53±1.73 2.40±1.75 TATL 1.59±0.77 0.34±0.93 2.02±1.04 SATL 0.57±0.80 -0.75±1.12 0.03±0.72 global=

Σregion 1.42±0.39 1.29±0.31 1.60±0.37

A single BPN (56 input, 112 hidden and 8 output neurons) is trained to compute the monthly differences for the eight RMSLA’s from the tide gauge values. The training includes the constraint that the area weighted sum of the regional must coincide with the given global. The corresponding target values for the training are derived from the CSIRO or/and the GFZ data.

Finally a recall is done for all data gap filling alternatives (see above). The resulting RMSLA’s for the

"CSIRO+GFZ" training are displayed in Fig. 5. The results for the single ensemble members are relatively in- sensitive to what is filled into the tide gauge data gaps as long as the number of gaps does not exceed 20%, i.e.

beyond 1950. Before this date the sensitivity is higher for the regional than for the global mean sea level.

The linear trends are summarized in the table (ensemble mean and standard deviation). For the regional trends we find a strong dependence to the target data chosen for training, while the global trend shows less dependence and fits well to the estimates given by Church and White (2006) or Jevrejeva et al (2006): 1.7±0.3 mm/year and 1.8 mm/year, respectively.

References:

Freeman and Skapura (1991) Neural Networks – Algorithms, Applications and Programming Techniques, Addison-Wesley Publishing Company, Reading, MA Church et al (2004) Estimates of the Regional Distribution of Sea Level Rise over the 1950 to 2000 Period. Journal of Climate, 17, 2609-2625.

Church and White (2006) A 20th century acceleration in global sea-level rise, Geophys. Res. Lett., 33, L01602, doi: 10.1029/2005GL024826 Jevrejeva et al (2006) Nonlinear trends and multiyear cycles in sea level records, J. Geophys. Res., 111, C09012, doi:10.1029/2005JC003229

Corresponding e-mail adresses:

Manfred.Wenzel@awi.de Jens.Schroeter@awi.de