Statistical and Neural Network Modeling and Predictions of Tides
in the Shallow Waters of the Gulf of Mexico
ALEXEY L. SADOVSKI, PHILIPPE TISSOT,
PATRICK MICHAUD, CARL STEIDLEY
Department of Computing and Mathematical Sciences
Department of Physical and Life Sciences
Conrad Blucher Institute for Surveying and Science
Texas A&M University - Corpus Christi
6300 Ocean Dr. Corpus Christi, Texas 78412,
USA
Sadovski@falcon.tamucc.edu http://www.cbi.tamucc.edu
Abstract: - This paper presents a preliminary report of statistical modeling for filling gaps in the data
collected to make predictions of tides in the shallow waters of the Gulf of Mexico. The approach
discussed is based on data gathered by the Texas Coastal Ocean Observation Network (TCOON). By
applying multiple regression and factor analysis to different kinds of data (water level, wind speed and
direction, water temperature) we were able to make quite reliable predictions from 6 to 72 hours. Results
of this investigation were compared with predictions based on the usage of Neural Networks, and
integration of these two approaches looks very promising.
Key-Words: - Predictions, Statistical Modeling, Regression, Factor Analysis, Neural Networks.
1 Introduction
The goal of this on going research is to
develop effective and reliable tools for
predicting water levels in the shallow waters
of the Gulf of Mexico. Different
methodologies for the prediction of water
levels include: statistical models [1],
harmonic analysis, numerical methods based
on finite elements/finite differences, neural
networks [2], etc. Here we would like to
discuss a statistical based model of
prediction (SMP) of tides and compare it
with neural network predictions (NNP).
Both of these two approaches are under
development at the Conrad Blucher Institute
in cooperation with the Department of
Computing and Mathematical Sciences both
of Texas A&M University-Corpus Christi
(A&M-CC). Many stations of the Texas
Coastal Ocean Observation Network
(TCOON) located in the coastal waters of
the Gulf of Mexico provide data for such
predictions [3].
TCOON consists of approximately 50 data
gathering stations located along the Texas
Gulf coast from the Louisiana to Mexico
borders. Data sampled at these stations
include: precise water levels, wind speed
and direction, atmospheric and water
temperatures, barometric pressure, and water
currents. The measurements collected at
these stations are often used in legal
proceedings such as littoral boundary
determinations; therefore data are collected
according to National Ocean Service
standards. Some stations of TCOON collect
parameters such as turbidity, salinity, and
other water quality parameters. All data are
transmitted back to A&M-CC at multiples of
six minutes via line-of-sight packet radio,
cellular phone, or GOES satellite, where
they are then processed and stored in a realtime, web-enabled database. TCOON has
been in operation since 1988.
The second approach is another multiregression model in which two-hour
The general idea is to predict water levels
predictions of water level are based on the
for the next two hours by using a multilevels of water during the previous 48 hours,
regression model. Then step by step - using
using 2-hour steps. Here we now believe
these predicted levels as the given levels that information about weather (pressure,
predict water level for 4, 6,…,48 hours. We
wind, temperature, etc.) is hidden in the
have considered three different models for
previous levels of water. This model worked
two-hour predictions, and two of these
remarkably well: R squared for all stations
produced quite reliable predictions. The first
was greater than 0.95. To make further
of these models is a multi-regression model
predictions we used the previously
in which two-hour prediction is based on the
determined levels of water. Such a step by
levels of water, speeds and directions of
step approach produced quite good
wind for the previous 48 hours with the step
predictions. The table below presents
of 2 hours. This model did not produce
statistical data for differences between
expected results, because R square for such
predicted and real levels of water for 6, 12,
a prediction was less than 0.5
18, 24, 30, 36, 42, and 48 hours:
Table 1. Statistical characteristics of prediction errors (in meters)
Mean
Median
Std.
Min.
Max.
Deviation range
range
Error 6hr
0.0124
0.0121
0.310
-0.858
0.796
2 Statistical Model
Error 12hr
0.0129
0.0117
0.105
-0.421
0.442
Error 18hr
0.0155
0.0108
0.313
-0.951
0.866
Error 24hr
0.00924
0.0023
0.177
-0.580
0.622
Error 30hr
0.0176
0.0062
0.297
-0.748
0.803
Error 36hr
0.0140
0.0198
0.184
-0.653
0.641
Error 42hr
0.0156
- 0.0034
0.293
-0.746
0.828
Error 48hr
0.0265
0.0289
0.193
-0.568
0.593
The third approach is based on linear multiregression of the levels of water, first
differences, and second differences for such
levels for the previous 48 hours with the step
equal to two hours. This approach produces
the same quality of water level prediction as
the second approach. These results are quite
understandable, because in both cases we
have to deal with linear combinations of
previous water levels. The difference in
these two models is the following: third
approach has between four (4) and eight (8)
significant variables in a linear regression
while in the second model of linear
regression we use all twenty four (24)
variables.
To fill gaps in water level data, we can use
the following procedure. First, find
backward and forward linear regressions for
predicted water levels, and then evaluate lost
data as a linear combination of forward and
backward predictions with weights
proportional distances from the edges of the
gap.
3 Factor Analysis
To determine why our regression models
that do not include wind and atmospheric
pressure data provide us with a better
prediction than the models that include such
data we preformed a Factor Analysis. The
analysis of the major components has shown
that 5 factors explain 95% of variance for
water levels. In deep waters the first three
components are periodical while in the
shallow waters the major component is not
periodical, while the other components are
periodical. Our conclusion is that the prime
factor is weather. It is well known that the
weather affects tides much more in shallow
waters than in deep waters [1], [2]. Linear
regression models for different locations
have different coefficients for the same
variables. This difference may be explained
by the geography of the location where the
data are collected.
4 ANN Modeling and
Predictions
The application of Artificial Neural
Networks (ANN) to a number of fields
including environmental modeling started
shortly after the development of the
backpropagation algorithm by Rumelheart et
al. [4]. During the past five to ten years
ANNs have been successfully applied to a
growing number of applications such as
coastal and riverine cases including the
forecasting of physical or water quality
parameters [5], [6], [7], [8], [9], the
forecasting of flooding along rivers [10],
[11] and the forecasting of water levels
along the coasts of the Gulf of Mexico [2],
[12]. Back propagation neural networks use
the repeated comparison between the output
of an ANN and an associated set of target
vectors to optimize the weights of the
neurons and biases of the model. The
learning process consists more specifically
in backpropagating a function of the error
through the network. The main advantages
and key characteristics of ANNs for water
level forecasting are their non-linear
modeling capability, their generic modeling
capacity, their robustness to noisy data, and
their ability to deal with high dimensional
data [13]. Forecasting water levels with
ANN consists of finding weights and biases
by training the model using historical
measurements. Our model’s inputs consist
of time series of previous water level and
wind measurements as well as tidal data.
All measurements and tidal forecasts for this
work were extracted from the TCOON [3]
database. The typical structure of the neural
networks used in this work is illustrated in
figure 1 and consists of one hidden layer
with 1 to a few neurons and one output layer
consisting of one neuron when predicting
individually each water level. The tidal
forcing is included in the model by using
water level differences between the
measured and hindcasted water levels and
the water levels predicted by the tide tables
published by NOAA. The water level
differences are then a direct function of the
meteorological forcing. Finally the model
predicts changes in water level differences
rather than absolute water level differences.
This methodology allows for a more direct
relationship between short-term forcing and
changes in water levels and also allows for
the inclusion of long-term effects such as
steric effects as part of the input to each
short-term forecast. The models were tested
with and without wind hindcasts. All the
ANNs discussed in this work were trained
using the Levenberg-Marquardt
backpropagation algorithm and implemented
within version 4.0 of the Matlab Neural
Network Toolbox and the MATLAB 6.0
Release 12 computational environment [14]
running on a Pentium PC.
To test the performance of ANNs for the
prediction of water levels at Bob Hall Pier,
Texas, the model was trained and tested
using three data sets composed of 3600
hourly measurements of water levels, wind
speeds and wind directions. The data set
covered the Spring seasons of 1998, 2000,
and 2001 from Julian day 21 to Julian day
182. This procedure provided a set of six
time series of predicted water levels to be
used for validation. For each time series the
average absolute error between predicted
and measured water levels was computed.
Averages and standard deviations were then
computed for the results of the six validation
time series for these two parameters. The
standard deviation gives an overall measure
of the variability due to the differences
between training sets as well as the
differences resulting from the training
process. The inputs to the model were
selected as the previous12 hourly water level
and wind measurements based on
experience gathered during the modeling for
other locations [12]. One model was trained
without wind predictions while for the
second case wind measurements were used
to simulate wind forecasts. These wind
forecasts consisted of future wind
measurements at 3 hour intervals up to 36
hours. A database of wind forecasts is
presently being constructed and models
based on wind forecasts are expected to be
more representative of future model
performance [15]. Figure 2 displays a
comparison between a 36-hour water level
hindcast, the tide tables, and TCOON
measurements. As can be observed in the
figure, the ANN model captures a large
fraction of the water anomaly and improves
significantly on the tide tables. The
performance of the models with and without
wind forecasts is compared with the
performance of the tide tables in Figure 3 for
forecasting times ranging from 6 to 36
hours. Both ANN models improve
significantly on the tide tables for
forecasting times up to 24 hours.
Improvements for 30-hours and 36-hours
predictions are still measurable. The
addition of wind hindcasts improves the
model performance although not
significantly as compared to the
improvement over the tide tables. Based on
these results an ANN model will be
implemented in the near future as a real-time
water level forecasting tool integrated within
the Texas Coastal Ocean Observation
Network.
Water Level
Time Series
Tidal Data
East-West Wind
Stress
North-South Wind
Stress
East-West Wind
Stress Forecasts

(a1,ixi)
 (X1+b1)

(a3,ixi)
b1

(a2,ixi)
 (X2+b2)
Forecasted
water level
variation
 (X3+b3)
X
b3
b2
North-South Wind
Stress Forecasts
Figure1. Schematic of the type of neural network applied to the problem of water level
forecasting including outputs, inputs, and neural network topology.
Fig. 2. Comparison of a 36-hour ANN hindcast, tide table readings and TCOON measurements of water
levels for the Bob Hall Pier Station located near Corpus Christi, Texas during the Spring of 2001.
0.14
Absolute Average Error [m]
0.12
0.10
0.08
0.06
0.04
Tide Tables
ANN without Wind Hindcasts
0.02
ANN with Wind Hindcasts
0.00
0
5
10
15
20
25
30
35
40
Forecasting Time [hrs]
Fig. 3. Comparison of the performance of the ANN model and the tide tables for the forecasting of water
levels at the Bob Hall Pier Station.
5 Conclusions
Comparing these two approaches we have
found that neural networks are more flexible
and give better predictions of the water
levels. On the other hand, statistical methods
are more simple to implement and can be
applied with only the knowledge of the
water levels. We are presently pursuing both
approaches with the goal of combining the
different approaches and improving the
overall quality of the forecasts.
6 Acknowledgements
The work presented in this paper is funded
in part by the following federal and state
agencies of the USA:
- National Aeronautic and Space Agency
(NASA Grant #NCC5-517)
- Texas General Land Office
-
National Oceanic and Atmospheric
Administration (NOAA)
- Coastal Management Program (CMP).
The views expressed herein are those of the
authors and do not necessarily reflect the
views of NASA, TGLO, NOAA, CMP or
any of their sub-agencies.
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