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

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

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)

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

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

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

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

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

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

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)

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

Results

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

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

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

0

5

10

15

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

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

0

2

4

6

8

10

12

0 12 24 36 48 60 72 84

Gap Size in Hours

RM

SE

Calm Weather Frontal Passages Extreme Weather

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

0

1

2

3

4

5

6

0 12 24 36 48 60 72

Gap Size in Hours

RM

SE

Embayment Station Open Coast Station

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

                      

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.