world tides user manual

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WORLD TIDESUSER MANUAL

Version 1.03

January 25, 2007

By

John D. Boon, Ph.D.

John D. Boon Marine Consultant, LLC

P.O. Box 1042

Gloucester Point, VA 23062 USA

Website: www.worldtidesandcurrents.com

Email:[email protected]

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http://www.worldtidesandcurrents.com/mailto:[email protected]:[email protected]://www.worldtidesandcurrents.com/


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TABLE OF CONTENTS

1. INTRODUCTION . 3

2. PROGRAM REQUIREMENTS 3

3. GETTING STARTED 4

4. TIDE ANALYSIS ... 4

5. CHOOSING TIDAL CONSTITUENTS .... 6

6. TIDE PREDICTIONS .... 12

7. ANALYZING STORM SURGE AND STORM TIDES 13

8. ENTERING DATA WITH EXCEL 14

9. FREQUENTLY ASKED QUESTIONS . 16

10.APPENDIX A . 19

11.APPENDIX B . 21

12.REFERENCES ... 25

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1. INTRODUCTION

WORLD TIDES is a desktop computer program for the analysis and prediction of water

levels in tidal waterways. Designed to be extremely easy to use, its Graphical User

Interface (GUI) permits quick separation of a time series of water level measurements into

its tidal and non-tidal components using a selective least squares harmonic reductionemploying up to 35 tidal constituents. After saving the tidal constants for the constituents

selected during analysis, the user can generate predictions of the astronomical tide, the

water level that varies at known tidal frequencies attributable to gravitational interactionsbetween the earth, moon and sun.

Many software packages are available today that allow tide predictions to be made in tidalwaterways throughout the world. With few exceptions, these programs use tidal constants

determined by governmental agencies and the casual user of this software is generally

unaware of any of the details involved in the agencys analysis, not least the breakdown of

observed water level variation into its tidal and non-tidal parts.

If you are wondering why anyone should care about this distinction, consider the following

reason. It is easy to fault the astronomical prediction formula when predictions dont agreewith observations but, depending on the region, the astronomical tide may in fact be quite

small compared to wind-generated and other forms of locally induced water level change

that exists in the same record. In addition, little information is available to users, casual orotherwise, about the length of the water level time series used to estimate tidal constants or

about how old the data are. Predictions for coastal waterways that have undergone

significant hydrologic change (storms, dredging activity) may be subject to errors resultingfrom outdated measurements. For those with suitable data at hand, along with a personal

computer or laptop, there is no reason to accept the status quo.

WORLD TIDES is the ideal package with which to explore and develop preliminary to

finalized tidal predictions from serial records spanning several weeks to several months.Although its operating features are intuitive and can be quickly grasped by users familiar

with MS Windows terminology, it is important to have a general understanding of the

theory of tides before using WORLD TIDES. Comprehensive references such as

Cartwright (2000) and Pugh (2004) are highly recommended for this purpose, as is Boon(2004) for a practical introduction.

2. PROGRAM REQUIREMENTS

WORLD TIDES is an application of the MATLAB technical computing language, aproduct of The MathWorks, Inc. For product information, contact The MathWorks, Inc., 3

Apple Hill Drive, Natick, MA 01760-2098 USA (Email: [email protected]). The

present program is compatible with MATLAB Versions 7.0.4 through 7.3.0 (R2006b)

running on personal computers with the MS Windows operating system (Windows XP).

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3. GETTING STARTED

No installation is required. Simply download the zipped package of WORLD TIDES files

to a working directory on your computer. After unzipping the package and starting

MATLAB in the working directory, enter the single command worldtides to start the

main program. The window that will appear next on your screen offers two choices: TideAnalysis and Tide Prediction. The best way to learn how the program operates is to

conduct a water level analysis with the example data provided. Start this task after reading

Section 4. WATER LEVEL ANALYSIS, and continue with Section 5. CHOOSINGTIDAL CONSTITUENTS. Then use the results to generate tide predictions as described in

Section 6. TIDE PREDICTION.

At this time, WORLD TIDES operates only on personal computers with Microsoft

Windows (Windows XP) and Microsoft Excel 2000 or Microsoft Office Excel 2003. The

analysis portion of the software accepts only files of type .xls for data input and requires

serial date and time as defined in Excel worksheets to operate correctly. Acceptable data

input formats are presented in Section 8. ENTERING DATA WITH EXCEL.

CAUTION: Users are cautioned not to substitute worksheets ccreated with other operatingsystems. The author is aware of one instance in which an Excel worksheet constructed with

a Macintosh computer ran successfully but provided meaningless results because a

different time origin was used by the Mac version. To test for the correct time origin onyour worksheet, first format a test cell using Time category 3/14/01 13:30, and then

enter 1/1/1900. The cell should display 1/1/00 0:00 (1/1/1900 12:00:00 AM on the

toolbar). Next re-format the cell to the Number category; the number 1.00 shouldappear, the number of days since midnight beginning December 31, 1899 (WORLD

TIDES time origin). MATLAB serial time functions use yet another origin but conversionsare handled within the WORLD TIDES program and are transparent to the user.

4. TIDE ANALYSIS

The method used by WORLD TIDES to analyze a water level time series is commonly

known as Harmonic Analysis, Method of Least Squares (HAMELS). It achieves aprogressive reduction in variance (mean square deviation about the mean) by adding

harmonic terms with specific astronomical frequencies to a general least squares model of

the type used for multiple regression. It is not Fourier analysis, a procedure that employsonly the Fourier frequencies. A brief description of HAMELS is given in Appendix A. For

a complete description of the least squares harmonic analysis method employed here, the

reader is referred to Boon (2004).

We sometimes call this a water level analysis (rather than a tide analysis) because the

measured change in water level in coastal waterways varies at both tidal and non-tidal

frequencies, including frequencies so low they appear as a mean level or linear trend inshort series. The objective of the analysis is to separate these components so that a tidal

height predictioncan be made with the component that is predictable the water level that

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oscillates at tidal frequencies. So, if youve started WORLD TIDES and are on the main

page, you are ready to begin by clicking Tide Analysis.

Tide Analysis - Clicking Tide Analysis starts the GUI page that performs tide analysis

(An example of this page is shown in Figure 1 of Appendix B). Directing your attention to

the menu bar at the top of the Analysis page, please click on Disclaimer and read thedisclaimer message before proceeding. Click on Program Help immediately to the left of

the Disclaimer button to view information about input files, file analysis, selection of

tidal constituents and other topics. The analysis occurs in two steps in which the user hasthe following choices to make:

(1) Settings: Only two settings are required: theseries length(in days) and the water levelunitsemployed (meters or feet). Two example data sets appear in the listbox in the upper

right corner of the page: one from Ballyheige, a town at the entrance to the Shannon River

in western Ireland (bally20040607.xls) courtesy of the Irish Geological Survey, and one

from the Chesapeake Bay Bridge Tunnel, Chesapeake Bay entrance (cbbt20021101.xls)

courtesy of the U.S. National Oceanic and Atmospheric Administration (NOAA). Bothcontain a 29-day water level record in meters. Before double-clicking on an input file to

run, first go to the gray frame at left where the default value of 29 days is shown in blue1.

After setting the series length (29 days for the examples given), use the radiobuttons in the

frame to select the appropriate units.

When uncertain about the units (feet or meters), open the Excel file and examine the data

before you proceed. If your file isnt already annotated, its a good idea to set up a header

worksheet with station name, location, series length, measurement units, time zone andwater depth, placing it after the first worksheet containing the water level data. When

ready, double-click the file you want to analyze and wait a moment while the data are readin (the time required depends on the length of the series and the sampling rate employed).

(2) Analysis: After the data are read in, the message File ready for analysis will appear in

the databox directly below the listbox containing the file names. The second step beginswith a press of the large ANALYZE button on the right side of the page. The number of

days in the file selected will be briefly displayed in the databox, followed by the date and

time of the first record in the file. A graph will appear next showing the results of a leastsquares harmonic analysis fitting the five main tidal constituents, O1, K1, N2, M2 and S2,

to the water level data (Appendix B, Figure 2). A listbox at the bottom of the page displays

the tidalconstants(amplitude and phase) computed for all five constituents.

You have probably already noticed that the above five symbols appear next to some

checked boxes inside a wide blue frame containing thirty more boxes that arent checked.The first five are always selected but the thirty others should be thought of as potential

constituents that you can choose for inclusion in the next round of analysis. All you have to

do is check the ones you want to include in the tide model and press ANALYZE.

Constituent selection is explained in the next section.

1Numbers shown in blue may be changed by the user to make another selection or setting.

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5. CHOOSING TIDAL CONSTITUENTS

For a relatively short series of observations (29 days to 58 days), there are limits to the

number of constituents that can be used in a harmonic analysis of the tide. In general, the

difficulty caused by short series length arises in the resolution of certain constituents that

are close to others in frequency (consult the listbox at the top of the Analysis page for a listof available constituents and their frequency). The major solar semidiurnal constituent S2,

for example, has a frequency of exactly 2 cycles per mean solar day; the semidiurnal

constituents T2 (1.9973 cpd), R2 (2.0027 cpd) and K2 (2.0055 cpd) are all very close tothis frequency and can be difficult to resolve from a short series. Its not that these

constituents dont have a correct amplitude and phase its just that you cant learn what

they are if you dont have the data. You could try to analyze a 29-day record after checkingall 35 boxes in WORLD TIDES but you will probably get some strange results if you do.

How strange? If you check all 35 boxes you will see the amplitude of some of the main

constituents increase by a factor of 100 or more compared to a basic analysis with five

constituents. Even though this choice may explain a large portion of the variance in thepresent data, it is far from realistic as you would quickly see if you saved these constants

and tried to make future predictions with them.

The 3-day Plot - During analysis, the 3-day plot feature in the gray frame on the upper left

side of the page can be very helpful. Like the main plot that appears after pressing theANALYZE button, the 3-day plot uses Julian days to display time and select a time

interval for plotting (the corresponding calendar date is also displayed for convenience).

The 3-day plot of observed (red), predicted (blue) and residual (green) water level gives awave-by-wave view showing how well the tidal harmonic model fits the data. Obviously,

the fit is very good if the residual is almost a flat line. However, when it isnt and the bluecurve starts showing double peaks when the red curve has only single peaks, this may be a

further indication of the problem of trying too many tidal constituents with too little data.

Of course another reason to use the 3-day plot is to investigate errors; e.g., dropped datapoints, vertical datum shift, or a shift to incorrect times. The least squares algorithm used

in WORLD TIDES is not affected by small data gaps, provided the time remains correct.

This is one reason the data are entered in a multi-column spreadsheet you can check thatevery water level reading is associated with the correct serial date and time (see Section 8.

ENTERING DATA WITH EXCEL). Although a short gap may be acceptable, the

program will still issue a warning if the number of observations found is less than thenumber expected based on the series length specified and the calculated sampling rate.

WORLD TIDES determines sampling rate from the first two recorded sample times in the

data series (all times need to be correct but especially these two!).

Several tools are provided to assist the user in choosing constituents for inclusion in a

harmonic model of the astronomical tide. Rather than relying on any single one of these

tools, use them in combination while keeping the series length in mind. Following a briefdescription of the available tools listed below, two examples of the recommended tidal

analysis procedure are presented to illustrate their use.

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Using the Residual Periodogram - A special tool you will want to use in choosing each

constituent to include in a water level analysis is the residual periodogram. While notinfallible, the Fourier periodogram or line spectrum can often identify important tidal

constituents from energy peaks associated with specific frequencies representing

oscillations left out of the model left out but still present in the residual. For convenience,

both a high band periodogram(1 to 8 cpd) and a low band periodogram(0 to 3 cpd) areprovided. The high band periodogram is used most often for constituent identification; the

low band feature can be used to characterize subtidal oscillations that are usually

associated with meteorological forcing (wind stress, atmospheric pressure change). Thepoint to keep in mind is that having significant energy (variance) at subtidal frequencies

puts a cap on what you can achieve with an astronomical tide model no matter how many

constituents are used.

Using RMS Error and Percent Reduction in Variance - Two statistical parameters are

provided near the center of the Analysis page to assist the user in evaluating the degree of

success achieved by the model in representing the data. The RMS error, calculated as the

square root of the mean square difference between observed and predicted water levels, isa measure of the expected error associated with an individual water level prediction. The

Percent Reduction in Variance (%R_Var) is the percentage of the total variance in waterlevel explained by the astronomical tide model. Ideally, inclusion in the model of any one

constituent suggested by the periodogram should result in a noticeable decrease in RMS

error combined with an increase in %R_Var. Again, if the data are taken from a regionwith strong meteorological forcing in relation to the tidal regime, you may be unable to

achieve either a high %R_Var or a low RMS error.

Using Constituent Amplitude and Phase Estimates - After conducting an analysis with a

new tidal constituent added to the model, first check the amplitude found for thatconstituent in the listbox at the bottom of the page. It should exceed at least one percent of

the largest major constituent amplitude. More importantly for a short series, it should not

cause another constituent at an adjacent frequency to change either its amplitude or phase

by more than a few percent (K2 and S2 may be exceptions).

Analysis in Stages - To proceed with an analysis, work in stages starting with the high

band periodogram and the five major constituents (O1, K1, N2, M2, S2) as Stage I. Use theMATLAB data cursor to obtain the frequency of the highest peaks in the periodogram,

treating the y-coordinate, energy, as a relative measure2. Proceed with the following steps:

1. Use the listbox at the top of the page to identify the constituent(s) that come closestto matching the peak frequencies. In most cases, there wont be an exact match and

the Fourier frequencies of some peaks may fall midway between two tidalfrequencies nearby. In the latter case, analyze the constituents in separate stages.

2. Check the boxes of the constituents selected above as model candidates for StageII. Include constituents of different type classes (diurnal, semidiurnal, etc.) in onestage but do not include several constituents within the same class that are adjacent

2In the absence of band-averaging, frequency intervals are small but energy estimation errors are large.

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in frequency. Look at the tidal form numberto see which class is dominant (a low

form number means semidiurnal is dominant, high infers diurnal class dominance).

3. With the high band radiobutton on, press ANALYZE to begin Stage II.

4. Verify the constituent(s) selected as model candidates in the previous stage byconfirming 1) peak elimination in the residual periodogram, 2) appropriate size forthe resulting constituent amplitude, 3) decreased RMS error, increased %R_Var.

Uncheck the constituent if it clearly fails any of these tests. Otherwise, the newstage will be marked by a periodogram showing new residual peaks at a lower

energy level. To amplify the remaining peaks at each new stage, the y-coordinate

scale expands as the energy level drops.

5. Select constituent candidates as before for Stage III. Continue this process until allconstituents that can be successfully matched to a residual peak frequency are

found and included in the astronomical tide model.

When analyzing a short series (58 days or less), watch for signs of a failed resolution

between neighboring constituents on the frequency scale. This usually takes the form of alarge change in amplitude and phase for such constituents when analyzed jointly versus

separately. For tides of small range especially, avoid selecting a constituent that is very

close in frequency to one of the major constituents in a short series; e.g., T2 (1.9973 cpd)and R2 (2.0027 cpd) adjacent to S2 (2.0000 cpd).

Seasonal Constituents Note the four data boxes on the left side of the analysis page withzero values entered in blue. They allow you to manually enter an amplitude and phase for

the solar annual (Sa) and solar semiannual (Ssa) tide constituents if available for the tidestation supplying the data. These numbers are available for most primary tide stations in

the United States and can be applied at nearby stations as well. Otherwise, several years of

observations are required to determine Sa and Ssa, the so-calledseasonal tides.

Vertical Datums Manual entries can also be made for Highest Astronomical Tide (HAT)

and Lowest Astronomical Tide (LAT), two of the most commonly used vertical datums

outside the United States. The numbers entered will be saved with the tidal constants andused in making tidal predictions (see Section 6. TIDE PREDICTIONS). Enter a negative

number for LAT to signify its offset below Mean Sea Level (MSL). Otherwise if the

number is left at zero, tide predictions will be made relative to MSL. It is also possible tocheck the compute Datums box in the lower right corner of the page before pressing

ANALYZE for your final run. The program then performs 19 years of predictions

internally to find and display HAT and LAT relative to MSL. Caution: Select only theminimum number of constituents for the final run if analyzing a short series (


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Example 1 - To get a feel for the use of the features described above, the reader can run a

29-day analysis of input file bally20040607.xls, a file containing water levels (in meters)sampled at 5-minute intervals at a station near the entrance to the Shannon River, west

coast of Ireland3. After double clicking this file and getting the File ready for analysis

message in the databox, press the large ANALYZE button below the message. The

databox will then display the number of days in the file followed by the date and time ofthe first record. A few seconds later a window graph and a pop-up message will appear.

The window graph displays the observed water level(red), the astronomic tide(blue) andthe residual(green) or difference between observed and predicted for the analyzed 29-day

series. The pop-up message appears because there is a data gap somewhere in the file

(8,340 records were found, 8,353 were expected). When this happens, we need to find outwhy. Although the window graph displays the entire series, the gap is too small to be seen

there. Using the 3-day plot feature, the gap in question can be clearly seen mid-morning on

Julian day 173 (21-Jun-2004) at the point where the red curve abruptly shifts to the right

by one hour. This is a time shift rather than a simple data gap, an error that will have to be

corrected before proceeding with the analysis.

Although the 3-day plot confirms the error, the easiest way to locate it initially is to createa test file in the Excel workbook with a duplicate serial date and time column that you

construct (see Section 8. ENTERING DATA WITH EXCEL). Once this is done, you can

scroll down the worksheet looking for the row in which the dates and times no longeragree. An example of a test file can be found in the workbook bally20040607.xls. And

while you have the workbook open, drag the worksheet labeled Fixed_file left to the first

position in the workbook. This is a repaired file that allows us to proceed with the analysis.

After loading and reading the fixed file, run the analysis again with the high bandperiodogram turned on. This time you should not see a warning message but there will be a

new graph containing a Fourier periodogram displaying two prominent peaks close

together in frequency. Using the MATLAB data cursor, click on the left peak. An x-axisreading of 1.862 cpd should be visible. Clicking on the peak at right, 1.966 cpd should

appear (Appendix B, Figure 3). The first frequency falls midway between constituents 2N2

and MU2; The second midway between LAM2 and L2. For stage I of this file with the five

major constituents, RMS error starts at 0.138 m and %R_Var begins at 98.15%.

What we are dealing with now is a form of multivariate analysis.To monitor the variables

(constituent amplitude and phase) employed in a step-wise (stage) analysis you shouldconsider entering the amplitude and phase of all the constituents involved at each stage on

a separate worksheet. One has already been entered for you in the bally20040607 Excel

workbook just click on the tab labeled Stage Analysis. Here you will see that to includeall four of the constituents 2N2, MU2, LAM2, and L2 is not a good idea based on the large

changes in amplitude that result. Choosing one constituent for each peak in the

periodogram works much better. After selecting MU2 and LAM2, the RMS error is

reduced to 0.109 m and %R_Var is now 98.80 percent.

3Data supplied courtesy of the Irish Geological Survey, Dublin, Ireland.

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In the next stage of the analysis, the first dominant peak suggests constituents Q1 and

RHO1 and the second one matches the quarter-diurnal constituent, M4. Referring again tothe Stage Analysis worksheet, RHO1 and M4 emerge as the most reasonable choices,

reducing the RMS error to 0.105 m and increasing %R_Var to 98.88 percent. In the final

stage, MNS2, K2, M3, MN4, MS4 and M6 are added, reducing RMS error to 0.100 m

and increasing %R_Var to 98.98 percent. Noting the change in amplitude and phase for S2at this stage may surprise you, but in this instance it is clearly linked to relatively large

amplitude in the luni-solar semidiurnal constituent, K2. K2 is often one of the major

constituents in regions where the tide type is fully semidiurnal and the range is more thantwo meters (the mean range at Ballyheige is approximately three meters). Normally the K2

peak is not pronounced in a short series because of its proximity to S2 on the periodogram

frequency scale (S2 is included at the outset as one of the major constituents). If in doubt,it doesnt hurt to add K2 and conduct a trial analysis as part of a later stage.

SAVE - During the course of the above four stages, the peak energy shown in the

periodogram falls by two orders of magnitude from about 2 x 10-3

to about 3 x 10-5

m

2

/cpd. At this point we may enter a name and save the 15 constituents selected in a tidalconstants file (e.g.,Ballyheige_MSLindicating that no offset from MSL was used).

Example 2 This example involves a 29-day analysis of water levels from the Chesapeake

Bay Bridge Tunnel at the entrance to Chesapeake Bay on the east coast of the United

States. The file named cbbt20021101.xlswas obtained from the NOAA Tides & Currentsweb site and features processed 6-minute water level data reported at hourly intervals for

30 days. As with the previous example, a stage analysis for this file can be found on a

separate worksheet in the Excel workbook. Tidal type here is mixed, mainly semidiurnal.

Analysis in stages yields 13 constituents for this file (Q1, O1, M1, K1, J1, OO1, MNS2,MU2, N2, M2, L2, S2 and K2) but, unlike example 1, the resulting model accounts for

only 71.33 percent of the variance in water level. The reason why can be clearly seen in the

residual curve for the window plot of the 29-day series. During this particular month, a

subtidal oscillation was present at the bay entrance whose amplitude at times exceeded theamplitude of the astronomical tide. If you switch to the low band periodogram, you will

see a single large peak at a frequency of 0.2069 cpd or, taking the inverse, 4.83 days per

cycle the peak period of the subtidal oscillation shown in the window plot. Although alonger analysis would improve estimates of harmonic constituents such as S2 and K2 in

the astronomical tide model, its a drop in the bucket compared to the subtidal change.

Sa and Ssa The NOAA Tides & Currents web site, http://tidesandcurrents.noaa.gov/,

provides amplitude and phase information for the solar annual (Sa) and solar semiannual

(Ssa) constituents. These values can be found on the Harmonic Constants worksheet infile cbbt20021101.xls. Enter them in the data boxes indicated on the left side of the

Analysis page.

Print to file After checking the enable print to filecheck box, the next pressing of theANALYZE button will save a listing of the observed, predicted and residual water levels

in a text file with the same name as the input .xls file but with the .txt extension.

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SAVE When you are satisfied with your analysis and the set of tidal constants selected,

the results can be saved in a MATLAB data file4by entering a file name (without the .mat

extension) in the data entry box in the lower left corner and pressing SAVE.

Special Features Providing tidal constants for tide predictions (Section 6) is just one

application that the WORLD TIDES: Analysis program is well suited for. It is especiallyuseful for informative graphs displaying monthly water level histories at active monitoring

stations. It is now common to see a near real time plot from NOAA stations on the web;

plots typically covering the past three days and conveying the information that observedwater levels are trending above or below predicted levels. A monthly plot can add value to

this product by displaying monthly and weekly variations in tidal range in relation to non-

tidal events such as a transient storm surge (seen in the residual curve). Only after viewingthe phase relationships between the storm surge peak and the nearest peak and range of the

astronomical tide can one understand why the resultingstorm tide reached the height that it

did. An example history for Hurricane Isabel is shown in Figure 4, Appendix B.

Preset constituent selection - To avoid having to check multiple constituent symbol boxeseach time the user repeats a tidal analysis, a worksheet named tidecn can be added to the

Excel file after the first worksheet containing the water level data. This worksheet mustcontain the list of symbols for WORLD TIDES 37 tidal constituents (including Sa and

Ssa) in the first column of the tidecn worksheet in order of increasing frequency moving

downward. Enter the amplitude and phase of Sa and Ssa, if known, in the second and thirdcolumns; otherwise enter zeros. For the remaining constituents, enter 1 or 0 in the

second column to indicate symbols to be automatically checked or unchecked once the file

has been loaded; enter a zero in the third column. An example worksheet can be found inExcel file HRVA2006m.xls.

Tidal datums To display the position of vertical datums on monthly (30-day) water level

plots, enter datum elevations in a new worksheet named datums, placing this tab in any

position after the first worksheet in the Excel file. Two formats are allowed as follows:

U.S. format Non-U.S. formatDatum Elev.(ft) Datum Elev.(m)

plotmax 6.00 plotmax 3.00

HAT 3.53 HAT 2.47

MHHW 2.76 MSL 0.00

MSL 1.35 STND 0.00

MLLW 0.00 LAT -2.51

LAT -0.69 plotmin -3.00

plotmin -2.00

STND is the station datum or the zero point of the measurement scale in use (in feet or

meters). If MSL in relation to the station datum is unknown, it should be set to zero.

WEB services See USING WEB SERVICES WITH WORLDTIDES document

4The variables stored in this file may be examined using the MATLAB load command.

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6. TIDE PREDICTIONS

From the main page of WORLD TIDES, click on Tide Prediction. This will bring up the

Prediction page (Appendix B, Figure 5) with another listbox in the upper right corner with

the names of tidal constituent files created with the SAVE button on the Analysis page. As

examples, two MATLAB data files of type .mat should appear from the Ballyheige andChesapeake Bay analyses described in Section 5. Double click on either one to start.

After double clicking on a file, the analysis date (year and Julian day starting) is displayedin the databox. If you would like to see the symbols of the constituents used, click on

Tidal Constituents in the menu bar above. Your next choice is to set the month and year

of the predictions wanted along with the units desired units that can be different fromthose of the water level data analyzed to obtain the tidal constants. Other radio buttons turn

on a plot grid and allow a change from Local Standard Time (LST) to Local Daylight Time

(LDT) it is assumed that LST was used during analysis. Pressing the large PREDICT

button will then populate the calendar matrix at left with the days of the month and year

selected. A daily, weekly or monthly plot of the predicted tide is displayed after pressingthe appropriate button.

Printing to a file - The enable print to file check box below PREDICT allows the user to

print 12-minute tidal height predictions to a text file for the month and year selected. If the

year box is also checked, predictions are printed for the year selected using a 30-minuteprediction interval. Note: Excel serial date and time is printed for each height value. To

convert these to calendar date and time, copy the file to an Excel worksheet and use the

cell format command (see Section 8. ENTERING DATA WITH EXCEL).

Predicted tidal height distributions: Percentiles - To display a histogram of predicted tidalheights as a percentage of total time, check the histogram box before pressing PREDICT.

A cumulative curve will appear with percentile markings for the heights that are equaled or

exceeded 20%, 50% and 80% of the time for the month selected. If the year box is also

checked, the percentages refer to total time for the year selected.

Predicted tidal height distributions: LAT and HAT estimates If the tidal constants file

selected does not contain a vertical datum offset (i.e., MSL-LAT=0.0), the histogram ofpredicted tidal heights will display the lowest astronomical tide (LAT) and the highest

astronomical tide (HAT) for the month or year selected (Appendix B, Figure 6). Final

estimates of LAT and HAT for use as reference datums can be found using the Analysisprogram (see Vertical Datums, Section 5, page 8).

Predicted tidal height distributions: HAT estimate If the tidal constants file selected doescontain a vertical datum offset (i.e., MSL-LAT>0.0), the histogram of predicted tidal

heights will reference heights above LAT and display the highest astronomical tide (HAT)

for the month or year selected. Why HAT? Although generating most tidal height

predictions relative to HAT would be confusing due to largely negative numbers, HAT is agood vertical reference for comparing storm tide peaks (see Section 7). HAT marks the

extreme upper limit of the astronomical tide for a given time and place and its contour

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against the shore is often visibly marked (e.g., algal lines on rocks and piles). Normal tides

will reach this contour in most, but not all years. And by using HAT as a reference, we cancompare extratidal water levels between locations that have different tidal ranges.

7. ANALYZING STORM SURGE AND STORM TIDES

Storm tides are water levels made higher by the superposing of astronomical tides with

storm surge, the transient change in water level resulting from the effects of a storm. In the

United States, the term storm surge is used most often in connection with hurricanes andtropical storms, although tropical depressions and extra tropical storms or northeasters

produce damaging storm surge as well. WORLD TIDES is uniquely suited for conducting

post-storm investigations of storm surge it readily performs the task of separating thestorm surge from water level observations and shows the nature of its interaction with the

astronomical tide to produce the resulting water level extremes. An example from a NOAA

tide station at Yorktown, Virginia, is shown in Figure 7, Appendix B. It was created from a

WORLD TIDES 29-day analysis of Yorktown records following a visit by tropical

depression ERNESTO on 1 September, 2006.

Good timing this time - Figure 7 provides a good illustration of the importance of timingbetween the arrival of the storm surge peak and the stage of the astronomical tide. As luck

would have it, the surge peak arrived much closer to low tide than high tide at Yorktown

on the morning of September 1. Luck chance may be a better word is involved againas the surge happened to arrive duringtropic tidesevidenced by a strong diurnal inequality

in the daily highs, as Figure 7 clearly shows (tides are mixed, mainly semidiurnal in this

area). Thus the risk of an exceptional high storm tide was by no means spread evenly overa 24-hour period given the possibility of the storm surge peak arriving at another time.

Tracking sea level change - Figure 7 demonstrates the utility of the MATLAB figure editor

in changing features such as scaling and labeling of figure axes, figure legends, line

thickness and many others if you want to make a special point. Figure 8, for example, is

identical to Figure 7 but sheds the green storm surge curve while adding dotted lines tomark the MHHW vertical datum and monthly mean sea level (mmsl) for the 29-day lunar

month just analyzed. This allows another point to be made about the ERNESTO event: in

Figure 8 the current mmsl is not found midway between the MHHW and MLLW verticaldatums it is above MSL and located much closer to MHHW. This indicates that local sea

level change (the sea level anomaly), whether seasonal or long-term, is an important

factor in determining the risks we will face as future storms approach our coastline.

The case for MHHW, HAT - A final word about storm surge and storm tides. Many of us,

including the media, get these terms confused; e.g., reporting a 6-foot storm surge atYorktown, instead of the 6-foot storm tide that actually occurred. The storm surge from

ERNESTO, as shown in Figure 7, was approximately 4 feet. Part of the problem (the

author believes) is that the NOAA chart datum of MLLW is invariably used in referencing

water levels in the U.S. This certainly helps to avoid the confusion of dealing with negativenumbers but what if storm tides were reported in feet or meters above MHHW or highest

astronomical tide, HAT? As Figure 8 shows, the storm tide from ERNESTO would then

13


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be about 3.4 feet above MHHW if reported this way. Not only would storm surge and

storm tide be numerically more similar, we would acknowledge that MHHW (or HAT) isthe more relevant datum for comparing storm tides along the waterfront. HAT in particular

marks the level where normal flooding from the astronomical tide ends and extreme

flooding from tides plus surge begins. Choosing either MHHW or HAT as the reference

for storm tides also allows better comparisons between a location with high tidal range(Eastport, ME) and one with low tidal range (Gulfport, MS) by removing range as a factor.

8. ENTERING DATA WITH EXCEL

WORLD TIDE input files consist of an array of water level data entered on the firstworksheet of a Microsoft Excel workbook with file extension .xls. This is the ONLY file

type permitted in the file directory listbox shown in the upper right corner of the Analysis

page. Two separate formats are available for data entry:

A. Col. 1 - Record number, station number or Julian day (not used in calculations)Col. 2 - Date in Excel month-day-year-time format (3/14/01 13:30)

Col. 3 Water level in feet or meters (Columns > 3 must be empty)

B. Col. 1 - Record number, station number or Julian day (not used in calculations)

Col. 2 - Date in Excel month-day-year format (3/14/01)

Col. 3 - Local Standard Time in Excel 24-hour time format (13:30)

Col. 4 Water level in meters or feet

A set of non-numeric column labels may be inserted as the first row of either array. Note

that a 4th

column is not allowed when using format A.

One of the advantages of using Excel is that it can accept text as well as numeric data from

a variety of sources. For example, data that has been downloaded from an archive on theWorld Wide Web can be copied to a file or taken directly from your computer screen by

highlighting the data and using the Windows copy command (A process sometimes

known as web scraping). Then, after inserting the copied material into a new worksheetwith the paste command, you can use the Excel text-to-columns command to parse the

text into columns. Excels numerous tools can then be used to arrange the data so that it

conforms to one of the above formats.

Prior to 2006, the format used by the online tide and current archives of the U.S. National

Ocean Service, U.S. National Oceanographic and Atmospheric Administration (NOAA),

enabled immediate use of NOAA data after parsing; i.e., the calendar dates and times fellinto the columns shown in format B. above and conformed exactly to the Excel format. A

new format accompanying a new web page has unfortunately ruled out format B. for direct

entry of NOAA water level data. However, using format A is almost as fast as thefollowing paragraph explains:

Even if you have only the starting and ending dates and times for your data series, its easy

to construct a column of dates and times in a worksheet with the following steps:

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1. Highlight an empty column and convert it to a datecategory of type3/14/01 13:30using the Format Cells command.

2. Enter the starting date and time in, say, cell B2: e.g., 1/1/2006 0:00. Place the cursoron the cell and note the reading 1/1/2006 12:00:00 AM in the upper display line.

3. Enter the formula =B2+1/(10*24) in cell B3. The cell will display 1/1/2006 0:06as this particular formula specifies a time interval of 6 minutes assuming a sampling

rate of 10 readings per hour (1 reading per hour is the minimum that can be used).

4. Copy the formula in cell B3 to each of the cells below it until the ending date andtime is reached in the last row of the data. Splitting the screen into upper and lowerparts is helpful here. Use the small button at the top of the slider bar on the right.

Perhaps you already have date and time information downloaded with the other data on

your worksheet - but not in the form Excel requires. Inserting a second column with date &

time constructed as above is a good way to verify that the information is correct, removingthe downloaded column afterwards. WORLD TIDES counts the number of records in a file

and warns the user if the number is less than expected based on the sampling rate andseries length specified. While this warns of a data gap (not uncommon in water level data

sets), there can still be other problems, such as a time shift, which require closer inspection

of the file itself, aided by the 3-day plot feature described in Section 5.

After entering the water level data to be analyzed in the first worksheet of an Excel

workbook, insert header information about the data into a separate worksheet. If you copyinformation from a web source, be careful not to include email addresses or links in blue.

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9. FREQUENTLY ASKED QUESTIONS

Q.Why is the series length of 29 days given as the default on the Analysis page?

A. Asynodicmonth of 29 days is the average interval between corresponding phases of the

moon, an interval significant for analysis purposes because it approximates a time whenthe major tidal constituents each complete a whole number of cycles. More generally,

synodic period refers to the recurrence interval between successive conjunctions of like

phases of various lunar and solar constituents, yielding synodic periods of 14, 15, 29, 58,87, 105, 134, 163, 192, 221, 250, 279, 297, 326, 355, and 369 days. Performing an analysis

with some other number of days as the series length is thought to introduce bias due to the

sampling of fractional cycles. With longer series, the exact length is less important.

Q. If 14 days is one of the synodic periods, why should I be concerned with the other

periods that require more data for analysis?

A. Just as two points determine a straight line, three points determine a single harmonicterm. The problem is, we will never get the right three points because of insufficient

precision, measurement error and other harmonic terms requiring more points. Althoughthe method of least squares with an excess of constituents may provide a convincing fit to

a very short data series, it is likely to fail in terms of making reasonable future predictions.

Avoid selecting too many tidal harmonic constituents to fit too few data points.

Q. Why are your tidal amplitudes and phases different from those given by NOAA for its

tide stations?

A. Constituent amplitudevalues obtained with WORLD TIDES do not contain a buildupfactor. NOAA increases the amplitude of certain constituents by a fixed percentage to

obtain better agreement between predicted and observed tidal extremes. Organizations with

legal responsibilities in the area of marine safety have reason to avoid under-predicting

tidal extremes, circumstances that may justify the use of buildup factors. On the otherhand, scientists and engineers attempting to simulate the behaviour of coastal estuaries are

often unaware that buildup factors have been applied when calibrating or verifying

hydrodynamic models with NOAA constituent data.

Constituent phase values obtained with WORLD TIDES are not comparable with the

phase values obtained by NOAA and other governmental organizations. WORLD TIDESuses a single time origin (midnight beginning 31 December 1899) whereas conventional

tide and tidal current prediction systems change the time origin from one year to the next

using the equilibrium phase lagconcept (phase lag of a given tidal constituent relative toits hypothetical equilibrium phase). In an earlier age, it was necessity to generate lengthy

tables of equilibrium phase arguments for the Greenwich meridian, transferring these

arguments to other meridians through conversion formulas. WORLD TIDES simply

requires the user to adhere to a single time origin and use local time (Local Standard Timeis recommended) for analysis. It then calculates and applies corrections for the nodal

16


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(amplitude) factor f and the phase variation term u for the lunar constituents as these

parameters change from year to year during the 18.6-year cycle of the lunar nodes.

Q. Isnt the Response Method a better way to analyze and predict tides and tidal currents?

A. The response or admittance method described by W.H. Munk and D.E. Cartwright(1966) begins with a cross-spectral analysis between the computed tidal potential and a

corresponding series of observations. The sea level or current spectrum is then separated

into a part that is coherent with the generating potential and a part consisting of noise.The coherent part produces estimates of the oceanic admittance, which can be expressed as

an amplitude ratio and a phase lag. Comparisons between the response method and

harmonic methods have shown the former to give predictions that are more accurate butwith an improved reduction in variance that is small compared to variations in the

meteorological tide. Due to its complexity, it has never been used for routine tide table

and current table predictions5.

Q. Why should I use Lowest Astronomical tide (LAT) rather than Mean Lower Low Water(MLLW) as the vertical reference datum for predicted tides?

A. Both LAT and MLLW are offsets from Mean Sea Level (MSL). In the United Statesand its territories, federal law gives NOAA the right to define MSL, MLLW and several

other vertical datums used primarily on nautical charts which it does by using a specific

19-year period termed the National Tidal Datum Epoch (NTDE) to calculate the averages

involved. Traditionally, 19-year averages have been used to account for variations in tidalrange over the 18.6-year lunar node cycle, rounding to 19 years to include a complete

annual cycle. NOAA then periodically revises the NTDE in response to sea level change

perceived at its various tide stations.

But is all this averaging really necessary? Using a 19-year average to account for sea level

change is arbitrary at best because no relationship is known to exist between tidal rangeand mean sea level: sea level change within a range of lengthy periods related to ocean-

atmosphere interactions is arguably more important. Other organizations outside the U.S.

have chosen other means to track sea level and most now recognize that, however meansea level is defined, it is much more practical to determine a datum offset above or below it

using tidal harmonic analysis. A datum so defined is LAT, with the result that the predicted

tide never falls below it, yet it is not so low that unduly shallow depths result when it is

used as a chart datum. Similarly, Highest Astronomical Tide (HAT) is determined as thevertical datum which the predicted tide by itself cannot exceed. And you dont have to wait

19 years to find it!

Q. How do you justify using time local tide predictions in lieu of the official tide

predictions by NOAA and other agencies to determine storm surge as the difference in

elevation between observed water levels and predicted tides?

5D.E. Cartwright, 2000. Tides: A scientific history, Cambridge University Press, pp. 195-198.

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A. The tide predictions that NOAA generates are normally based on a 369-day analysis of

water levels measured at its primary tide stations. While NOAA makes the tidal harmonicconstants from these analyses available to the public, it does not routinely publish the dates

for the measurement series used in the analysis. In the United Kingdom Hydrographic

Department, neither the tidal harmonic constants nor the dates of their determination for

major ports are public information. Even when a series is relatively recent, NOAA tidepredictions refer to the MSL tidal datum or one of the vertical datum offsets determined for

the current National Tidal Datum Epoch (NTDE), a 19-year series whose median age is

never less than a decade when the new values are released to the public. Thus a surgedetermination using NOAA tide predictions requires yet another offset to account for sea

level change over a decade or more (the sea level anomaly) and the user generally does

not know the magnitude of the offset employed at any one time.

Conversely, time local tide predictions used in WORLD TIDES are based on tidal

behavior as it occurs during the same lunar month in which the storm surge of interest

occurs. This process can be thought of as eliminating the maximum variance (in the least

squares sense) that is present at tidal frequencies in the water level record (low pass filtersremove these same tides but much else besides not all storm surge energy winds up in the

subtidal band). The vertical reference for these predictions is monthly mean sea leveldetermined relative to the measurement datum (normally MLLW) for either a lunar month

(m29) or a calendar month (m30) according to the series length selected. Given the

elevation of the MSL datum above the measurement datum, WORLD TIDES clearlydisplays thesea level anomalyfor each monthly record as either the difference m29-MSL

or the difference m30-MSL.

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10. APPENDIX A

The Harmonic Model for the prediction of tides and tidal currents assumes that tidal

motion can be represented by the sum of a series of simple harmonic terms (tidal

constituents), each term being represented by an oscillation at a known frequency of

astronomical origin. While the astronomical frequencies associated with celestial motionsof the earth-moon-sun system are well known, the success of a harmonic model depends

entirely on the oceans response to extraterrestrial (gravitational) forcing at these same

frequencies, recognizing that oceans are free to respond at the same time to local(meteorological) forcing. The equation for the harmonic model in this instance is:

( )1)(cos)( *1

0 jjj

m

j

jj utHfhth ++= =

where t= time in serial hours6, h(t)= predicted water level (water current) at t, h0= mean

water level (water current),fj= lunar node factor for jth constituent, Hj= mean amplitude

for jth constituent over 18.6-year lunar node cycle, j= frequency of jth constituent, uj =nodal phase for jth constituent, j

* = phase of jth constituent for the time origin in use

(midnight beginning December 31, 1899) and m= number of constituents. For purely solar

constituents,fj= 1 and uj = 0. Others are obtained by formula (see Doodson and Warburg,1941, reprinted 1980; Schureman, 1958).

Harmonic analysis by the method of least squares (HAMELS) is a simple but powerfulmeans of obtaining tidal constituent amplitude (Hj) and phase (j

*), the so-called tidal

harmonic constants needed for tidal predictions using equation (1). The least squares

criterion requires a solution for the harmonic constants that will produce the minimum

possible sum of squared differences for a series of observations ht of length n

=

=n

t

t thh1

2)]([ minimum

For this purpose, we rewrite equation (1) in the equivalent form

= =

++=m

j

m

j

jjjj tBtAAth1 1

0 )2(sincos)(

whereA0= h0,22

jjj BAR += =fjHj and

=

j

j

jA

B1

tan = . The unknownsA0,Aj,

Bj in equation (2) are obtained by solving the general matrix equation for least squares

approximations:

jj u*

[ ] [ ] [ ] ( )31 SXYSSXC =

6Serial time is the fractional number of hours past a specified time origin as opposed to 24-hour solar time.


20/25

In the above, is a 2m+1x 1vector of unknowns,[ ]C [ ] [ ]'22110 .. mmBABABAAC = ,

with and [ ][ ] [ ] [ ]XXSSX '= [ ] [ ]YXSXY '= where

[ ]

=

nmnmnn

mm

mm

mm

tttt

tttt

tttttttt

X

sincos..sincos1

............

sincos..sincos1

sincos..sincos1sincos..sincos1

11

333131

222121

111111

and is a vector containing n observations. The prime symbol used in

these equations indicates the transpose of a matrix or vector whereas the unit negative

exponent indicates theinverseof the 2m+1x 2m+1square matrix,

[ ] [ ]'321 .. nhhhhY =

[ ]SSX . Note also that,

while there is a term representing the mean in equation (1), there is no term representing alinear trend (or portion of a long-period oscillation that appears as a trend in the data). It is

therefore advisable to de-trend the observations prior to analysis.

MATLAB requires only five lines of m-code to perform least squares harmonic analysis

for m constituents given a data vector Y consisting of nobservations7:

X = ones(n,1);for j = 1:m

X = [X cos(w(j)*t) sin(w(j)*t)];

end

A = (X*X)\(X*Y);

See Boon (2004, pp. 153-166) for more information concerning computational methodsand the application of matrix algebra in tide and tidal current analysis.

7The backslash operator \ in line 5 performs left-matrix division in MATLAB.

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11.APPENDIX B

Figure 1. WORLD TIDES Analysis Page.

155 160 165 170 175 180 185 190-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2bally20040607.xls: 29-day analysis

Julian Day 2004

WaterLevel(meters)

-0.206

O1 K1 N2 M2 S2 observed

predicted

residual

Figure 2. Analyzed water level at Ballyheige (series mean level: -0.206m).

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0 1 2 3 4 5 6 7 80

0.5

1

1.5

2

2.5x 10

-3

X: 1.966

Y: 0.00186

Frequency(cycles per day)

Energy

bally20040607.xls - Residual Periodogram

2003/08/28 2003/09/02 2003/09/07 2003/09/12 2003/09/17 2003/09/22 2003/09/27 2003/10/02-2

-1

0

1

2

3

4

5

6

7

8

Hampton Roads (Sewells Point), VA

WaterLevel(feet)

2.27 m30

Water level s from U.S. National Oceanic and Atmospheric Admini stration (NOAA)

3.53 HAT

2.76 MHHW

1.35 MSL

0.00 MLLW

-0.69 LAT

XHW = 4.36observed

astronomic

residual

Figure 3. High Band Periodogram.

18 September 2003

Hurricane Isabel

Figure 4. Water level History for Hampton Roads, VA, September, 2003.

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Figure 5. WORLD TIDES Prediction Page.

-3 -2.4 -1.8 -1.2 -0.6 0 0.6 1.2 1.8 2.4 30

10

20

30

40

50

60

70

80

90

100

Height above MSL (meters)

Frequency(percentoftotaltime)

Ballyheige MSL 2006

LAT = -2.47

HAT = 2.44

>-1.1

>-0.082

>0.957

Figure 6. WORLD TIDES Height-Frequency Histogram for Ballyheige, Ireland.

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8/31 9/1 9/2 9/30

1

2

3

4

5

6

Yorktown, VA

storm tid e

astronomical tide

storm surge

mmsl (lunar)

Water

Level

(feet

above

MLLW)

Figure 7. Storm tide and storm surge at Yorktown, VA,

Tropical depression ERNESTO, 1-Sep-2006.

8/31 9/1 9/2 9/30

1

2

3

4

5

6

Yorktown, VA

storm tid e

astronomical tide

Water

Level

(feet

above

ML

LW)

mmsl

MHHW

MLLW

3.38 ft

Figure 8. Yorktown storm tide referred to 1983-2001 MHHWTropical depression ERNESTO, 1-Sep-2006.

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12.REFERENCES

Bloomfield, P., 1976.Fourier Analysis of Time Series: An Introduction. John Wiley &

Sons, New York, 258 pp.

Boon, J.D., 2004. Secrets of the Tide: Tide and Tidal Current analysis and Predictions,

Storm surges and Sea Level Trends. Horwood Publishing, Chichester, U.K.

212 pp.

Cartwright, D.E., 2000. Tides: A scientific history. Cambridge University Press, 292pp.

Doodson, A.T. and H.D. Warburg, 1944.Admiralty Manual of Tides. Admiralty

Charts and Publications, London, England, 270 pp.

Munk, W.H. and D.E. Cartwright, 1966. Tidal Spectroscopy and Prediction. Phil.

Trans. Royal Society of London. Series A, Mathematical and PhysicalSciences, Vol. 259, No. 1105, pp. 533-581.

Pugh, David T., 2004. Changing Sea Levels: Effects of Tides, Weather and Climate.

Cambridge University Press, 265 pp.

Schureman, P., 1958. Manual of Harmonic Analysis and Prediction of Tides. U.S.

Dept of Commerce, Coast and Geodetic Survey. Special Publication No. 98,

Washington, D.C., 317 pp.

world tides user manual

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