WORLD TIDES USER 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] Email:
[email protected]
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TABLE OF CONTENTS
1. INTRODUCTION …………………………………………………… ……………………………………………………………. ……….
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2. PROGRAM REQUIREMENTS ………………………… ……………………………………………… ……………………
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3. GETTING STARTED ………………………………………………… ………………………………………………………… ……… 4 4. TIDE ANALYSIS …………………………… ……………………………..…………………………… ..………………………………. …. 4 5. CHOOSING TIDAL CONSTITUENTS ………………...………… ………………...……………………. …………. 6 6. TIDE PREDICTIONS …………………………… ……………………………..………………………….. ..………………………….. 12 7. ANALYZING STORM SURGE AND STORM TIDES ……………………
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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 reduction employing 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 interactions between the earth, moon and sun. Many software packages are available today that allow tide predictions to be made in tidal waterways 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 agency’s 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 don’t agree with 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 or otherwise, 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 resulting from 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, a product 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: ‘Tide Analysis’ 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. CHOOSING TIDAL 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 operating systems. 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 on your 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’ should appear, the number of days since midnight beginning December 31, 1899 (WORLD TIDES time origin). MATLAB serial time functions use yet another origin but conversions are 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 a progressive 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 employs only 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 in short series. The objective of the analysis is to separate these components so that a tidal height prediction can be made with the component that is predictable – the water level that
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oscillates at tidal frequencies. So, if you’ve 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 the disclaimer 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 has the following choices to make: (1) Settings: Only two settings are required: the series length (in days) and the water level units employed (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). Both contain a 29-day water level record in meters. Before double-clicking on an input file to 1 run, first go to the gray frame at left where the default value of 29 days is shown in blue . 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 isn’t already a nnotated, it’s a good idea to set up a ‘header’ worksheet with station name, location, series length, measurement units, time zone and water 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 read in (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 begins with 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 least squares 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 tidal constants (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 aren’t 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.
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Numbers 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 list of 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 to this frequency and can be difficult to resolve from a short series. It’s not that these constituents don’t have a correct amplitude and phase – it’s just that you can’t learn what they are if you don’t have the data. You could try to analyze a 29-day record after checking all 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 the present 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 the ANALYZE 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 a wave-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 isn’t and the blue curve 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 data points, 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 that every 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 the number 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 brief description 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 not infallible, 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) are provided. 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). The point 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, is a 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 water level 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 region with 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 that constituent 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 the MATLAB data cursor to obtain the frequency of the highest peaks in the periodogram, 2 treating the y-coordinate, energy, as a relative measure . Proceed with the following steps: 1. Use the listbox at the top of the page to identify the constituent(s) that come closest to matching the peak frequencies. In most cases, there won’t be an exact match and the Fourier frequencies of some peaks may fall midway between two tidal frequencies nearby. In the latter case, analyze the constituents in separate stages. 2. Check the boxes of the constituents selected above as model candidates for Stage II. Include constituents of different type classes (diurnal, semidiurnal, etc.) in one stage but do not include several constituents within the same class that are adjacent 2
In 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 number to 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 by confirming 1) peak elimination in the residual periodogram, 2) appropriate size for the resulting constituent amplitude, 3) decreased RMS error, increased %R_Var. Uncheck the constituent if it clearly fails any of these tests. Otherwise, the new stage 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 all constituents 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 a large 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 with zero 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 tide station 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-called seasonal 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 and used 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 to check 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 the minimum number of constituents for the final run if analyzing a short series (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 a good 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 can compare “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 and tropical 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 the storm 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 ‘timing’ between 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 again as the surge happened to arrive during tropic tides evidenced 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 over a 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 to mark 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 vertical datums – 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 at Yorktown, 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 negative numbers 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
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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) is the 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 first worksheet 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 th that a 4 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 the World 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 worksheet with the ‘paste’ command, you can use the Excel ‘text-to-columns’ command to parse the text into columns. Excel’s 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 fell into 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 the following paragraph explains: Even if you have only the starting and ending dates and times for your data series, it’s easy to construct a column of dates and times in a worksheet with the following steps: 14
1. Highlight an empty column and convert it to a date category of type ‘3/14/01 13:30’ using the ‘Format Cells’ command. 2. Enter the starting date and time in, say, cell B2: e.g., ‘1/1/2006 0:00’. Place the cursor on 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 0:06’ as 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 and time is reached in the last row of the data. ”Splitting” the screen into upper and lower parts 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, removing the 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 and series 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 copy information 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. A synodic month of 29 days is the average interval between corresponding phases of the moon, an interval significant for analysis purposes because it approximates a time when the 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 harmonic term. The problem is, we will never get the right three points because of insufficient precision, measurement error and other harmonic terms requiring more points. Although the 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 amplitude values obtained with WORLD TIDES do not contain a “buildup” factor. 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 other hand, 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 TIDES uses 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 lag concept (phase lag of a given tidal constituent relative to its 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 Time is recommended) for analysis. It then calculates and applies corrections for the nodal
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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. Isn’t 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 but with 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 5 and current table predictions . 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 States and 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 tidal range 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 range and mean sea level: sea level change within a range of lengthy periods related to oceanatmosphere 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 mean sea level is defined, it is much more practical to determine a datum offset above or below it using tidal harmonic analysis. A datum so d efined 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 the vertical datum which the predicted tide by itself cannot exceed. And you don’t 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?
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D.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 harmonic constants 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 tide predictions 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 surge determination 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 filters remove 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 level determined 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 clearly displays the sea level anomaly for 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 motions of the earth-moon-sun system are well known, the success of a harmonic model depends entirely on the ocean’s 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: h(t ) = h0 +
m
∑ f H cos(ω t + u j
j
j
j
− κ j* ) (1)
j =1
6
where t = time in serial hours , h(t) = predicted water level (water current) at t , h0 = mean water level (water current), f j = lunar node factor for jth constituent, H j = mean amplitude for jth constituent over 18.6-year lunar node cycle, ω j = frequency of jth constituent, u j = * nodal phase for jth constituent, κ 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, f j = 1 and u j = 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 powerful * means of obtaining tidal constituent amplitude ( H j) and phase (κ 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
∑ [h
t
− h(t )]2 = minimum
t =1
For this purpose, we rewrite equation (1) in the equivalent form h(t ) = A0 +
m
∑ A
j
j =1
cos ω j t +
m
∑ B
j
sin ω j t (2)
j =1
⎛ B j ⎞ −1 ⎟ = κ j* − u j . The unknowns A0, A j, where A0 = h0, R j = A j2 + B j2 = f j H j and φ j = tan ⎜ ⎜ A j ⎟ ⎝ ⎠ B j in equation (2) are obtained by solving the general matrix equation for least squares approximations:
[C ] = [SSX ]−1[SXY ] (3) 6
Serial time is the fractional number of hours past a specified time origin as opposed to 24-hour solar time.
In the above, [C ] is a 2m+1 x 1 vector of unknowns, [C ] = [ A0 A1 B1 A2 B2 .. Am Bm ] , '
with [SSX ] = [ X ] [ X ] and [SXY ] = [ X ] [Y ] where '
'
⎡1 ⎢1 ⎢ [ X ] = ⎢1 ⎢ ⎢.. ⎢⎣1
cos ω 1t 1
sin ω 1t 1
..
cos ω 1t 2
sin ω 1t 2
.. cos ω m t 2
cos ω 1t 3
sin ω 1t 3
.. cos ω m t 3
..
..
cos ω 1t n
sin ω 1t n
..
cos ω m t 1
..
.. cos ω m t n
sin ω m t 1 ⎤
⎥ ⎥ sin ω m t 3 ⎥ ⎥ .. ⎥ sin ω m t n ⎥⎦ sin ω m t 2
and [Y ] = [h1 h2 h3 .. hn ] 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 the inverse of the 2m+1 x 2m+1 square matrix, [SSX ] . Note also that, while there is a term representing the mean in equation (1), there is no term representing a linear 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 7 for m constituents given a data vector Y consisting of n observations : 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 methods and the application of matrix algebra in tide and tidal current analysis.
7
The backslash operator ‘ \ ‘ in line 5 performs left-matrix division in MATLAB.
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11. APPENDIX B
Figure 1. WORLD TIDES Analysis Page. bally20040607.xls: 29-day analysis 2
O1 K1 N2 M2 S2
observed
1.5
predicted residual
1 0.5 ) s r e t 0 e m ( l e -0.5 v e L r e -1 t a W -1.5
-0.206
-2 -2.5 -3 155
160
165
170 175 Julian Day 2004
180
185
190
Figure 2. Analyzed water level at Ballyheige (series mean level: -0.206m ).
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2.5
x 10
-3
bally20040607.xls - Residual Periodogram
X: 1.966 Y: 0.00186
2
1.5 y g r e n E
1
0.5
0
0
1
2
3 4 5 Frequency(cy cles per day)
6
7
8
Figure 3. High Band Periodogram. Hampton Roads (Sewells Point), VA observed astronomic residual
8
7
XHW = 4.36
Hurricane Isabel 18 September 2003
6
5
) t e e 4 f ( l e v e L 3 r e t a W 2
3.53 HAT 2.76 MHHW 2.27 m30 1.35 MSL
1
0
0.00 MLLW -0.69 LAT
-1 Water level s from U.S. National Oceanic and Atmospheric Admini stration (NOAA)
-2 2003/08/28
2003/09/02
2003/09/07
2003/09/12
2003/09/17
2003/09/22
2003/09/27
Figure 4. Water level History for Hampton Roads, VA, September, 2003.
22
2003/10/02
Figure 5. WORLD TIDES Prediction Page. Ballyheige MSL 2006 100 LAT = -2.47 90 ) e m i t l a t o t f o t n e c r e p ( y c n e u q e r F
80
>-1.1
70 60 50
>-0.082
40 30 20
>0.957
10 0 -3
HAT = 2.44 -2.4
-1.8
-1.2 -0.6 0 0.6 1.2 Height above MSL (meters)
1.8
2.4
3
Figure 6. WORLD TIDES Height-Frequency Histogram for Ballyheige, Ireland.
23
6 storm tid e
Yorktow n, VA
astronomical tide storm surge
)5 W L L M
mmsl (lunar)
e v4 o b a t e e3 f ( l e v e2 L r e t a W1
0 8/31
9 /1
9 /2
9 /3
Figure 7. Storm tide and storm surge at Yorktown, VA, Tropical depression ERNESTO, 1-Sep-2006.
6 storm tid e
Yorktow n, VA
astronomical tide
)5 W L L M e v4 o b a
3.38 ft
t e e3 f (
MHHW
l e v e2 L
mmsl
r e t a W1
0 8/31
MLLW
9 /1
9 /2
9 /3
Figure 8. Yorktown storm tide referred to 1983-2001 MHHW Tropical 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 Physical Sciences, 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.
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