Comparison of Gap Interpolation Methodologies for Water Level Time Series Using Perl/PDL
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Comparison of Gap Interpolation Methodologies for Water Level Time Series Using Perl/PDL
ByAimee Mostella, Alexey Sadovski, Scott Duff, Patrick Michaud, Philippe Tissot, Carl Steidley
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Necessity of Interpolation
of Nearshore Research (TAMUCC-DNR) and Texas Coastal Ocean Observation Network (TCOON) collect, archive and analyze various types of time series including water level time series
Gaps in time series limit the types of methods which may be used to study and further our understanding of water level patternsTexas A&M University-Corpus Christi Division
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lrwlfill
TAMUCC-DNR and TCOON have developed lrwlfill, a Perl script designed to interpolate gaps in water level time seriesEase and power with Perl
Efficient computation and data storage with the Perl Data Language Module (PDL)
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Basic Algorithm
Retrieve data according to user provided parametersSearch data for missing valuesPerform linear regression to obtain two sets of coefficientsCalculate missing values with coefficients
Combine two sets into oneInsert new values in place of missing data
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Retrieving the DataRetrieve water level values corresponding to these parameters: Time frame
Retrieve from one month before to one month after time provided by the user
Station identifier Number of coefficients Method of fitting the
resulting data
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Water LevelRWL = AWL - HWL RWL => Residual Water Level AWL => Actual Water Level HWL => Harmonic Water Level
Record the location of gaps in the AWLRecord the difference between AWL and HWL as the RWL
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Linear RegressionFor 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
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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
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Testing ConditionsFull sets of existing dataTwo station locations: Embayment station Open coast station
Three typical weather conditions Periods of calm weather Periods of frequent
frontal passages Extreme weather
Varying gap sizes and numbers of coefficients
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Testing Standards
United States National Ocean Service (NOS) standards: Root mean square error (RMSE) Central Frequency (CF)
Other statistical measures: Standard deviation (SD) Maximum error (ME)
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Results AnalyzedTiming and accuracy for varying numbers of coefficientsBest procedure to fit FLR and BLRTiming and accuracy as gap size increasedPrecision as weather conditions changedAccuracy for embayment stations in contrast to open coast stations
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Results
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Effect of Number of Coefficients
Timing was negligibleAccuracy peaked and then declined depending upon weather conditionsRMSE was used to determine the optimal number of coefficients
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Coefficients
Figure 1 displays our chosen coefficients according to weather conditionAlthough these coefficients are optimal, the accuracy of interpolation still declines as weather becomes more extreme
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Best Fit
Convex linear combination demonstrated the highest level of accuracy in fitting the data as shown in Figure 2.
2.74
4.36
9.61
3.88
5.88
14.35
3.56
5.82
12.36
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Root Mean Square Error in
cm
Convex LinearCombination
ConvexTrigonometricCombination
Combination atIntersection
Figure 2: Accuracy of Differing Methods of Fitting FLR and BLR
Calm Weather Frontal Passages Extreme Weather
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Effects of Gap Size and Weather Conditions
Generally, timing was negligiblePrecision decreases steadily as gap size increases as shown in figure 3Precision also decreases as the weather becomes more extreme
Figure 3: Effect of Increased Gap Size upon Accuracy of Interpolation Program
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Gap Size in Hours
RM
SE
Calm Weather Frontal Passages Extreme Weather
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Embayment vs. Open Coast
Overall, data from embayment stations produced better results than data from stations along the open coast (figure 4)
The flow of water into the bay dampens the change in water level
Figure 4: Embayment Data vs. Open Coast Data
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Gap Size in Hours
RM
SE
Embayment Station Open Coast Station
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Future Direction
Convert lrwlfill to a real-time web based implementationExperiment with the amount and orientation of previous data used to calculate coefficients Study the results of using more frequent time series values in the linear regression process as weather becomes more erratic to make up for rapid changes in weather
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BibliographyHigh Performance Computing Development Center, Texas A&M University-Corpus Christi.
http://www.sci.tamucc.edu/~hpcdc/
Mostella, Aimee; Duff, Scott; Michaud, Patrick R. (2001) HARMAN and HARMPRED: Web-based Software to Analyze Tidal Constituents and Tidal Forecasts for the Texas Coast
NOAA, 1994: NOAA Technical Memorandum NOS OES 8. National Oceanic and Atmospheric Administration, Silver Spring, Marilyn.
Sadovski, Alexey L.; Michaud, Patrick R.; Steidley, Carl; Tishmack, Jessica; Torres, Kelly & Mostella, Aimee L. (2003). Integration of Statistics and Harmonic Analysis to Predict Water Levels in Estuaries and Shallow Waters of the Gulf of Mexico. Presentation at the MATA International Conference (Cancun, Mexico), April 2003.
Sadovski, Alexey L.; Tissot, Philippe; Michaud, Patrick; Steidley, Carl (2004) Statistical and Neural Network Modeling and Predictions of Tides in the Shallow Waters of the Gulf of Mexico. In “WSEAS Transactions on Systems”, Issue 2, vol. 2, WSEAS Press, pp. 301-307.