U.S. DEPARTMENT OF COMMERCE National Oceanic and Atmospheric Administration National Ocean Service Center for Operational Oceanographic Products and Services Tidal Analysis and Prediction NOAA Special Publication NOS CO-OPS 3 NOAA Special Publication NOS CO-OPS 3 Tidal Analysis and Prediction Bruce B. Parker, Ph.D. Silver Spring, Maryland July 2007 Library of Congress Control Number: 2007925298 U.S. DEPARTMENT OF COMMERCE Carlos M. Gutierrez , Secretary National Oceanic and Atmospheric Administration Conrad C. Lautenbacher, Jr. Undersecretary of Commerce for Oceans and Atmosphere and NOAA Administrator National Ocean Service John H. Dunnigan, Assistant Administrator Center for Operational Oceanographic Products and Services Michael Szabados, Director NOTICE Mention of a commercial company or product does not constitute an endorsement by NOAA. Use for publicity or advertising purposes of information from this publication concerning proprietary products or the tests of such products is not authorized. Request for copies should be addressed to Center for Operational Oceanographic Products and Services 1305 East West Highway SSMC IV, Rm 6616 Silver Spring, Maryland 20910-3281 Phone (301) 713-2981ext. 125 http://tidesandcurrents.noaa.gov Table of Contents page 1. Introduction 1 1.1 Purpose of This Book 1 1.2 Relationship to Previous Tidal Analysis and Prediction Books and Manuals 3 1.3 The Uses For Tidal Analysis and Prediction 6 1.4 Brief Historical Background 7 1.5 The Importance of Hydrodynamics 10 1.6 Organization of This Book 11 2. Theory Behind Tidal Analysis and Prediction .15 2.1 Short Introductory Overview – the Big Picture 15 2.1.1 Some Definitions 15 2.1.2 The Generation of Tides 24 2.2 Astronomical Considerations 27 2.2.2 The Orbits of the Moon Around the Earth and the Earth Around the Sun 27 2.2.2 The Origins of the Largest Tidal Harmonic Constituents 31 2.2.3 Derivation of Tidal Frequencies from the Basic Astronomical Frequencies – Insights from Tidal Spectra 35 2.2.4 The Origin of Node Factors and Equilibrium Arguments 39 2.2.5 Definitions Revisited, With Relationship to Harmonic Constants 42 2.3 Hydrodynamic Considerations 48 2.3.1 Hydrodynamic Effects On Tide Ranges, Phase Lags and Tidal Current Speeds 49 2.3.2 Nonlinear Effects of Shallow Water – Overtides and Compound Tides 55 2.3.3 Nonlinear Tidal Interaction With River Flow and Storm Surge 62 2.3.4 Nonlinear Hydrodynamic Effects On Node Factors 67 2.3.5 Zero-frequency Effects From Nonlinear Tidal Interactions 69 2.3.6 Special Aspects of Tidal Currents 69 3. Methods of Tidal Analysis and Prediction .81 3.1 Introduction 81 3.2 Simple Demonstration of Extracting Tidal Constituents From a Data Time Series 82 iii Tidal Analysis and Prediction 3.3 The Synodic Period – Length of Time Series Needed To Separate Two Constituents 84 3.4 Harmonic Analysis 90 3.4.1 Introduction 90 3.4.2 The Harmonic Tide Prediction Equation 90 3.4.3 The Harmonic Tidal Current Prediction Equations 95 3.4.4 Fourier-based Harmonic Analysis 97 3.4.5 Least-squares-based Harmonic Analysis 99 3.5 Other Frequency Domain Methods 102 3.5.1 The Response Method 102 3.5.2 The Harmonic Shallow-Water Corrections Method 103 3.5.3 The Species Concordance Method 104 3.5.4 The Method for Superfine Resolution of Harmonic Constituents 105 3.5.5 Continuous Wavelet Transform Method 105 3.5.6 Harmonic Constants From a Long Series of High and Low Waters 106 3.6 Nonharmonic Comparison Methods 109 3.6.1 Overview and Short Historical Background 109 3.6.2 Types of Nonharmonic Analyses 111 3.6.3 Variations In Nonharmonically Determined Time and Height Differences 112 3.7 Analyses for determining Sa and Ssa 119 3.7.1 The Meteorological Origins of Sa and Ssa 119 3.7.2 Considerations In Whether To Use Sa and Ssa In a Prediction 120 3.7.3 Methods for Calculating Sa and Ssa 122 3.8 Tidal Filtering and Detiding 123 3.8.1 Introduction 123 3.8.2 Specially Designed Tidal Filters 127 3.9 Calculating Consistently Defined Maxima and Minima 129 3.9.1 Consistently Defined High and Low Waters 129 3.9.2 Consistently Defined Maximum Floods and Ebbs and Slacks (or Minimums) 131 3.10 Spectral Analysis and EOF Analysis 132 3.10.1 Spectral Analysis of Water Level Data Time Series 133 3.10.2 Tidal Cusps In Spectra 134 3.10.3 Spectral Analysis of Current Data Time Series – Rotary Spectra 135 3.10.4 Cross-spectral Analysis 136 3.10.5 Empirical Orthogonal Function (Principal Component) Analysis 137 4. Harmonic Analysis of Water Level Data .139 4.1 Considerations In Carrying Out the Analysis 139 4.1.1 Which Tidal Constituents Should Be Solved For 139 4.1.2 Keeping or Rejecting Small Constituents 142 4.1.3 Instrument Errors and Their Effects On Analysis Results 142 4.1.4 Assessing the Potential Effects of Nontidal Influences 146 4.1.5 Use of Node Factors and Equilibrium Arguments (or of Satellite Constituents) 148 4.2 Methods for Analyzing Short Time Series 150 iv Table of Contents 4.2.1 Schureman’s Inference and Elimination Technique 152 4.2.2 Use of a Reference Station To Improve the Results of a Short Analysis 153 4.2.3 Inference In the Frequency Domain 155 4.2.4 Other methods 156 4.3 Assessing the Quality of the Predicted Tide Series 158 4.3.1 Introduction 158 4.3.2 Comparison of High and Low Waters In the Predicted Time Series Versus In the Observed Time Series 160 4.3.3 Spectral Analysis of the Residual Time Series 161 4.3.4 Examination of the Residual Time Series For Periods With Transient Tidal Oscillations 162 4.3.5 The Use and Misuse of “Build-up Factors” 162 4.3.6 Looking For Changes in Bathymetry or Shoreline 163 4.4 Producing a Reference Station for Table 1 of a Tide Table 164 4.5 Summary Overview: Steps In Harmonically Analyzing Water Level Data 165 5. Harmonic Analysis of Current Data .169 5.1 Special Problems With Current Data 169 5.2 Tidal Current Harmonic Constituent Ellipses 174 5.3 Considerations In Carrying Out the Analysis 181 5.3.1 Which Tidal Current Constituents Should Be Solved For 181 5.3.2 Selection of Orthogonal Axes 181 5.3.3 Instrument Errors and Their Effects On Analysis Results 182 5.3.4 Assessing the Potential Effects of Nontidal Influences 186 5.3.5 Whether To Include the Calculated Mean In a Tidal Current Prediction 188 5.3.6 Use of Node Factors and Equilibrium Arguments (or of Satellite Constituents) 189 5.4 Methods for Analyzing Short Time Series 189 5.5 Assessing the Quality of the Predicted Tidal Current Series 190 5.5.1 Introduction 190 5.5.2 Comparison of Maximum Floods and Ebbs and Slacks In the Predicted Time Series Versus In the Observed Time Series 190 5.5.3. Spectral Analysis of the Residual Time Series 191 5.5.4 Examination of the Residual Time Series For Periods With Transient Tidal Current Oscillations 192 5.5.5 Looking For Changes in Bathymetry or Shoreline 192 5.6 Producing a Reference Station for Table 1 of a Tidal Current Table 193 5.7 Summary Overview: Steps In Harmonically Analyzing Current Data 194 6. Nonharmonic Analysis of Water Level and Current Data .195 6.1 The Need For Nonharmonic Tidal Analyses 196 6.2 Hydrodynamic Considerations 196 6.3 Monthly Mean Tide Analysis 197 6.3.1 Details of the Analysis 197 6.3.2 The Limitations of the Monthly Mean Tide Analysis 202 v Tidal Analysis and Prediction 6.4 Tide-by-Tide Analysis 202 6.4.1 Details of the Analysis 202 6.4.2 The Benefits and Limitations of the Tide-by-Tide Analysis 202 6.5 Reversing Reduction Analysis for Tidal Currents 205 6.5.1 Details of the Analysis 205 6.5.2 Advantages and Limitations of the Reversing Reduction Analysis 210 6.6 Rotary Reduction Analysis for Tidal Currents 210 6.6.1 The Benefits of Using the Rotary Analysis 210 6.6.2 The Limitations of the Rotary Analysis 212 6.6.3 Details of the Analysis 212 6.6.4 Special Consideration and Choices To Be Made For the Rotary Analysis 215 6.7 Selecting a Reference Station 217 6.8 Putting a Subordinate Station In a Tidal Current Table and On a Tidal Current Chart 218 6.9. Minimizing the Errors Due to the Variations Throughout the Month of Nonharmonically Determined Time and Height Differences 219 6.10 Summary Overview: Steps In Analyzing Water Level or Current Data With a Nonharmonic Comparison Method 220 7. Interpretation of Tidal Analysis Results Based On Hydrodynamics .223 7.1 Products for showing Tidal Analysis Results 223 7.1.1 Cotidal, Corange, and Co-amplitude Charts 223 7.1.2 Cospeed and Cophase Charts 226 7.1.3 Tidal Current Ellipses 226 7.1.4 Various Types of Graphs 229 7.2 Spatial Variation in Tidal and Tidal Current Parameters In a Waterway Due to Hydrodynamic Processes 229 7.3 The Equations of Motion as the Basis for Tidal Hydrodynamic Models 232 7.3.1 The Equation of Continuity 232 7.3.2 The Momentum Equations 233 7.3.3 Simplifying the Equations of Motion For Idealized Cases 236 7.4 Tidal Variations Along a Very Narrow Waterway 238 7.4.1 Tidal Variations In a Narrow Rectangular Waterway 238 7.4.2 Tidal Variations In a Narrowing Waterway 247 7.5.