Review of Mathematical and Statistical Models to Predict (Extrapolate) Water Levels and to Fill Gaps (Interpolate) of the TCOON Data Texas A&M University - Corpus Christi 6300 Ocean Dr. Corpus Christi, Texas 78412, USA This year team: Alexey Sadovski Scott Duff Garry Jeffress Carl Steidley Philippe Tissot Beate Zimmer Zack Bowles David Beck Jeremy Flores Aimee Mostella Kelly Knott (Torres)

Project Goals Develop effective & reliable prediction tools Developed methods: – –Harmonic analysis – –Numerical methods based equations of hydrodynamics – –Statistical models – –Neural networks

Primary Water Level Water Temperature Wind Speed Wind Gust Wind Direction Typical TCOON station web page Typical TCOON station web page

Texas Coastal Ocean Observation Network Monitors water levels and other coastal parameters along the Texas coast

Tide Charts In general this is the first choice Astronomical forcing –Earth, Sun, Moon motions Limitations –Areas such as the Gulf of Mexico where the dominant forcing is meteorological in nature

Harmonic Analysis Standard method for tide predictions Represented by constituent cosine waves with known frequencies based on gravitational (periodic) forces Elevation of water is modeled as h(t) = H 0 +  H c f y,c cos(a c t + e y,c – k c ) h(t) = elevation of water at time t a c = frequency (speed) of constituent c f y,c e y,c = node factors/equilibrium arg-s H 0 = datum offset H c = amplitude of constituent c k c = phase offset for constituent c

Prediction vs. Observation It’s nice when it works…

Prediction vs. Observation …but it often doesn’t work in Texas

Water Levels Tides In Texas, meteorological factors have significant effect on water elevations

Two Reliable Statistical Models –Both are linear multi-regression models –Both deal with combinations of previous water levels only –Difference in models Between 4 and 8 variables in one kind of model, which takes into account first and second differences of water levels All 12 to 48 variables in the other models, in which only previous water levels are used

Statistical Models

One possible future application –Occasional losses of data Regression models, using forward and backward regression, evaluate lost data as a linear combination of forward and backward predictions with weights proportional to the distances from the edges of the gap

Factor Analysis Question: Why do models with only previous water levels work better than models with all data provided by TCOON stations? No more than 5 factors explain over 90% of variance for water levels

Bob Hall Pier (014)

Flower Garden (028)

Factor Analysis –In off-shore deep waters, the first two or three components are periodical –In coastal shallow waters and estuaries the major or the first component is not periodical –Our conclusion is that the prime factor is “weather” –Linear regression models for different locations have different coefficients for the same variables –This difference may be explained by the geography where the data is collected

Improved Predictions Model differences between the observed water levels and the harmonic predictions by using multiple regression (so-called marriage of harmonic and regression analysis) Build a model based on past observations; use that to make a model to predict differences in future observations

Statistical Models

Predicted Levels

Station 005: Packery Channel

Evaluation Criteria Criteria for the evaluation of water level forecasts –Different criteria were developed mostly by the U.S. National Oceanic and Atmospheric Administration (NOAA) to address the different priorities of coastal users

Evaluation Criteria Average error will address the possible bias of a model The absolute error will give information on the overall accuracy of the model Standard deviation will give information on the variability of the forecasts

Evaluation Criteria Specialized criteria,e.g., positive and negative outlier frequencies, will be useful to characterize model performance for unusually high or low water level situations Some forecasting methodologies will be better suited for some criteria and worse for others, e.g., predictions based on harmonic analysis are very good when evaluated by the standard deviation criteria and not as good when using the absolute error criteria.

Rockport (015) Training Set - March 2003 Prediction for 48 hours

Bob Hall Pier (014) Training Set - March 2003 Prediction for 24 hours

Basic Algorithm Retrieve data according to user provided parameters Search data for missing values Perform linear regression to obtain two sets of coefficients Calculate missing values with coefficients Combine two sets into one Insert new values in place of missing data

Water Level RWL = AWL - HWL –RWL => Residual Water Level –AWL => Actual Water Level –HWL => Harmonic Water Level Record the location of gaps in the AWL Record the difference between AWL and HWL as the RWL

Linear Regression For each gap in the data –Perform forward and backward linear regression (FLR & BLR, respectively) using hourly data to obtain coefficients –Calculate the missing data points with these coefficients

Methods of Combination Combine the results of FLR & BLR using one of the following methods: –Convex linear combination Based on weighted proportion –Convex trigonometric combination Based on trigonometrically weighted proportion –Combination at intersection Fuse together at the intersection

Results

Effect of Number of Coefficients Timing was negligible Accuracy peaked and then declined depending upon weather conditions RMSE was used to determine the optimal number of coefficients

Coefficients Figure 1 displays our chosen coefficients according to weather condition Although these coefficients are optimal, the accuracy of interpolation still declines as weather becomes more extreme

Artificial Neural Network Modeling Started in the 60’s Key innovation in the late 80’s: backpropagation learning algorithms Number of applications has grown rapidly in the 90’s especially financial applications Growing number of publications presenting environmental applications

Why ANN’s? Modeled after human brain Neurons compute outputs (forecasts) based on inputs, weights and biases Able to model non-linear systems

Neural Network Features Non-linear modeling capability as well as generic modeling capability Robustness to noisy data Ability for dynamic learning Limitation: Requires availability of high density of data

Artificial Neural Network Setup ANN models developed within the Matlab and Matlab NN Toolbox environment Found simple ANNs are optimum Use of ‘tansig’ and ‘purelin’ functions Use of Levenberg-Marquardt training algorithm ANN trained over 1 year of hourly data (8760 observations)

Transform Functions Tansig Purelin y = x y =(e x – e -x )/(e x + e -x )

Optimum ANN Structure Simple ANNs work best: 1 hidden neuron and 1 output neuron Optimum number of previous water level inputs varies between 3 and 24 hours Optimum number of previous wind measurement inputs varies between 1 and 12 hours Actual number of inputs chosen does not significantly change model performance

ANN schematic Philippe Tissot H (t+i) Output LayerHidden Layer Wind Squared History Water Level History Input Layer Water Level Forecast  (a 1,i x i ) b1b1 b2b2  (X 1 +b 1 ) b3b3  (X 2 +b 2 )  (X 3 +b 3 )  (a 2,i x i )  (a 3,i x i ) Tidal Forecasts

Model Assessment Based on five 1-year data sets: ‘97, ‘98, ’99, ’00, ‘01 including observed water levels and winds, and tide forecasts Based on five 1-year data sets: ‘97, ‘98, ’99, ’00, ‘01 including observed water levels and winds, and tide forecasts Train the ANN model using one data set e.g. ‘97 for each hourly forecast target, e.g. 12 hours Train the ANN model using one data set e.g. ‘97 for each hourly forecast target, e.g. 12 hours Apply the ANN model to the other four data sets, Apply the ANN model to the other four data sets, Repeat the performance analysis for each training year and forecast target and compute the model performance and variability Repeat the performance analysis for each training year and forecast target and compute the model performance and variability

Artificial Neural Network forecasting of water levels Use historical time series of previous water levels, winds, barometric pressure as input Train neural network to associate changes in inputs and future water level changes Create water level forecasts using a static neural network model

ANN Inputs Tested model with input from different locations: –Rockport only –Rockport with Port Aransas (Ship channel) –Rockport with Bob Hall Pier (Coastal station) Tested model with different meteorological time series: –Water Level only –Water Level and Previous Wind measurements –Water Level, Previous Wind, and Wind Forecasts

Training with one set (X = 15cm) Morgan’s Point

Tropical Storms Tropical storms are a challenge for any predictive model They are relatively infrequent and unique As storms are often destructive, improved predictions are very useful to emergency management

Performance applied to 1998 Hours (1998) Water level (m)

Tropical Storm Frances - September 7-17, 1998 Frances Trajectory Landfall on Sept. 11

Forecasts in storm events Rockport ANN 24-hour Forecasts During 1998 Tropical Storm Frances (ANN trained over 1997 Data Set)

New Directions and Problems Statistics on ANN structures. New feedback functions and optimal criteria for ANN to improve prediction quality. Evaluating quality and quantity of information produced by ANN during training and predictions using ergodic and entropy analysis. Spatial-temporal analysis (statistics) of data from stations in and around Corpus Christi Bay and Galveston Bay. Dynamic models of the water levels (Integral-differential equations with boundary values).

New Directions and Problems Visualization and simulation of water levels. Predictions during tropical storms using as inputs the first, the second, and the third differences of water levels. Optimal control in multi-species models for fisheries and fish harvesting. Spectral analysis of stochastic processes based on TCOON data. Developing of the theory of quasi-periodic functions. Development of mathematics for autonomous boat.

Acknowledgments The presented work is funded in part by the following federal and state agencies of the USA: –National Aeronautic and Space Agency (NASA Grant #NCC5-517) –National Oceanic and Atmospheric Administration (NOAA) –Texas General Land Office –Coastal Management Program (CMP)

Resources Division of Nearshore Research Website TCOON Data Query Page Web-based Predictions Development Page