U.S. DEPARTMENT OF COMMERCE

National Oceanic and Atmospheric Administration

National Ocean Service

Center for Operational Oceanographic Products and Services

COMPUTATIONAL TECHNIQUES 

FOR 

TIDAL DATUMS HANDBOOK

NOAA Special Publication NOS CO-OPS 2
Center for Operational Oceanographic Products and Services

The National Ocean Service (NOS) Center for Operational Oceanographic Products and

Services (CO-OPS) collects and distributes observations and predictions of water levels

and currents to ensure safe, efficient and environmentally sound maritime commerce. The

Center provides the set of water level and coastal current products required to support NOS'

Strategic Plan mission requirements, and to assist in providing operational oceanographic

data/products required by NOAA's other Strategic Plan themes. For example, CO-OPS

provides data and products required by the National Weather Service to meet its flood and

tsunami warning responsibilities. The Center manages the National Water Level

Observation Network (NWLON), and a national network of Physical Oceanographic RealTime Systems (PORTS®) in major U.S. harbors. The Center: establishes standards for the

collection and processing of water level and current data; collects and documents user

requirements which serve as the foundation for all resulting program activities; designs

new and/or improved oceanographic observing systems; designs software to improve COOPS' data processing capabilities; maintains and operates oceanographic observing

systems; performs operational data analysis/quality control; and produces/disseminates

oceanographic products.
NOAA Special Publication NOS CO-OPS 2

C0MPUTATIONAL TECHNIQUES FOR TIDAL

DATUMS HANDBOOK

Silver Spring, Maryland

September 2003

U.S. DEPARTMENT OF COMMERCE

Don Evans, Secretary

National Oceanic and Atmospheric Administration

Conrad C. Lautenbacher, Jr. 

Undersecretary of Commerce for Oceans and Atmosphere and NOAA Administrator

National Ocean Service

Dr. Richard W. Spinrad

 Assistant Administrator Ocean Services and Coastal Zone Management

Center for Operational Oceanographic Products and Services

Michael Szabados, Director
iv

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.
v

TABLE OF CONTENTS

page

1. INTRODUCTION ....................................................... 1

1.1 Intended Audience . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Statement of Philosophy ........................................... 1

1.3 Prerequisite Knowledge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2. BACKGROUND ....................................................... 3

2.1 Characteristics of the Tides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.2 Lunitidal Interviews .............................................. 7

2.3 Other Signals in Water Level Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.4 Tide Station Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.5 Bench Marks and Differential Leveling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.6 Data Processing Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.7 The National Tidal Datum Epoch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3. GENERAL TIDAL DATUM COMPUTATION PROCEDURES

3.1 Datum Computation Procedures Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.2 Other Vertical Datums and Their Relationship to Tidal Datums . . . . . . . . . . . . 20

3.3 Steps required to Compute Tidal Datums at Short-term Stations . . . . . . . . . . . . 25

3.4 Datum Computation Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

Standard Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

Modified Range Ratio Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

Direct Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.5 Accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4. WORKED EXAMPLES OF TIDAL DATUMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.1 Procedural Steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.2 Comparison of Monthly Means . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.2.1 Modified Range Ratio Method - Semidiurnal Tides . . . . . . . . . . . . . . . . 31

 Definition, MTL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

Definition, DTL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

Definition, Mn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

Definition, Gt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

Definition, DHQ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

Definition, DLQ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.2.2 Modified Range Ratio Method - Diurnal Tides . . . . . . . . . . . . . . . . . . . . 40

Corrected MTL for Panama City Beach . . . . . . . . . . . . . . . . . . . . . . . . . 43

Corrected DTL for Panama City Beach . . . . . . . . . . . . . . . . . . . . . . . . . . 44

Corrected Mn for Panama City Beach . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

Corrected Gt at Panama City Beach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.2.3 Standard Method - Mixed Tides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

Corrected MTL at Alameda . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
vi

TABLE OF CONTENTS (cont.)

Corrected Mn at Alameda . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

Definition, DHQ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

Definition, DLQ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.3 Comparison of Simultaneous High and Low Waters . . . . . . . . . . . . . . . . . . . . . 54

4.3.1 Modified Range Ratio Method - Semidiurnal Tides . . . . . . . . . . . . . . . . 54

Type or Designation Conversion at Subordinate Station . . . . . . . . . . . . . 58

Tidal Datums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

Lunitidal Intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.3.2 Modified Range Ratio Method - Diurnal Tides . . . . . . . . . . . . . . . . . . . . 68

Tidal Datums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

Lunitidal Intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.3.3 Standard Method - Mixed Tides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

Tidal Datums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

Lunitidal Intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

4.4 Direct Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

5. SUMMARY and ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

6. REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

Appendix 1: Example spreadsheets for semidiurnal and mixed tides . . . . . . . . . . . . . . . . . . A-1

Appendix 2: Relationships of tide and geodetic datums . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-1
vii

LIST OF FIGURES

page

Figure 1. A depiction of the three primary kinds of tides. From the top panel downward they are

semidiurnal, mixed, and diurnal. Standard tidal terminology is used to describe the

various aspects of the tides. The zero on these graphs is illustrative of the relationship

of the tides to Mean Sea Level (MSL). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

Figure 2. Comparison of types of tide over one calendar month . . . . . . . . . . . . . . . . . . . . . . 5

Figure 3. An illustration of the spatial variability of the type of tide in the Gulf of Mexico . 5

Figure 4. An illustration of the effect of the regression of the moon’s nodes on the water levels at

Puget Sound, WA. The heavy black curve is the annual mean range, or the difference

in height between mean high water and mean low water. The time elapsed between

trough to trough or peak to peak, is the period of oscillation of the regression, and is

about 18.6 years. The more rapidly varying curve is the monthly mean range. Changes

in the monthly mean range are due to rapidly changing meteorological and

oceanographic conditions ............................................ 6

Figure 5. Locations of U.S. NWLON water level stations. . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

Figure 6. Illustration of the tide station network hierarchy. . . . . . . . . . . . . . . . . . . . . . . . . . 11

Figure 7. An schematic diagram of extending vertical control inland from the tidal datum by the

method of differential leveling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

Figure 8a and 8b. Observed 6-minute data for one-month and resulting tabulation of the tide 14

Figure 9. Example of a monthly tabulation of the tide. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

Figure 10. Relative sea level change at several locations in the U.S. . . . . . . . . . . . . . . . . . . . . 17

Figure 11. Illustration of the long term changes in sea level causing the need to update tidal datums

such as mean tide level (MTL). The graph shows the annual values of mean tide level

with horizontal lines drawn to show the 19-year MTL values for the 1941-59 and 1960-

78 NTDE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

Figure 12. Tidal datums (1960-1978 NTDE), extreme water levels and geodetic elevations at

station 941 4290 San Francisco, CA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

Figure 13a. Locations of stations for plots shown in Figure 13b . . . . . . . . . . . . . . . . . . . . . . . 23
viii

LIST OF FIGURES (cont.)

page

Figure 13b. The relationship of the geodetic datums to tidal datums in the St. John’s River, FL. The

perspective is going upstream from left to right. The graph was constructed by

assigning NAVD88 to a relative zero value at each site, and adjusting the other datums

accordingly. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

Figure 14. The estimated error in tidal datums from Swanson (1974) . . . . . . . . . . . . . . . . . . 27

Figure 15. Let x be the horizontal distance inland, and y the vertical rise of the land. By definition,

tan(")=y/x. Likewise, the cotangent of ", denoted by cot("), is given by cot(") = x/y

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

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

Figure 16. NWLON station locations along the South Atlantic Bight . . . . . . . . . . . . . . . . . . 32

Figure 17a. Time series plots of monthly mean tide level (MTL) for Fort Pulaski, GA and

Charleston, SC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

Figure 17b. Time series plots of monthly mean range of tide (Mn) for Fort Pulaski, GA and

Charleston, SC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

Figure 18. A map showing NWLON stations in the Gulf of Mexico . . . . . . . . . . . . . . . . . . . 40

Figure 19a. Time series plots of monthly mean diurnal tide level (DTL) for Pensacola, FL and

Panama City, FL. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

Figure 19b. Time series plots of monthly mean diurnal range of tide (Gt) for Pensacola, FL and

Panama City, FL. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

Figure 20. A map of California showing the locations of NWLON stations . . . . . . . . . . . . . . 47

Figure 21a. Time series plots of monthly mean tide level (MTL) for Alameda, CA and San

Francisco, CA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

Figure 21b. Time series plots of monthly mean range of tide (Mn) for Alameda, CA and San

Francisco, CA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

Figure 22. The menu driven interface to the CO-OPS internet database. The interface is

completed for the Fort Pulaski data of the tide by tide comparison between Charleston

and Fort Pulaski. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
ix

LIST OF FIGURES (cont.)

page

Figure 23. The data returned from the CO-OPS database for Fort Pulaski, GA. Referring to Figure

31, the selection of “W3-Highs/Lows”, returns the data denoted by the diamond

symbols. Their values are also labeled. The rest of the curve is obtained by selecting

“W2-Hourly Heights”, and resubmitting the data request. . . . . . . . . . . . . . . . . . . 57

Figure 24. The data returned from the CO-OPS database for Charleston, SC . . . . . . . . . . . . 58

Figure 25. A graph of Charleston and Fort Pulaski data. The high water and low water values

required for the tidal datum calculations are printed to illustrate the tide by tide

comparison. Refer to this graph when considering the organization of Table 26 below.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

Figure 26. A comparison of corresponding highs and lows for diurnal tides . . . . . . . . . . . . . 70

Figure 27. The water level time series for Alameda with the highs and lows plotted. The data are

referenced to station datum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

Figure 28. The water level time series for San Francisco with the highs and lows plotted. The data

are referenced to station datum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

Figure 29. A composite graph of Alameda and San Francisco. The time series at Alameda is

represented by the dashed curve. The data are referenced to station datum at each site.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

Figure 30. Comparison of data for a typical 7-day time period showing low waters cut-off at the

subordinate station . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
x

LIST OF TABLES

page

Table 1. Generalized accuracy of tidal datums for East, Gulf, and West Coasts when determined

from short series of record and based on +/- sigma. From Swanson (1974) . . . . . . 26

Table 2. Error in position of marine boundary as a function of the slope of the land (12 month

series). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

Table 3. NOS monthly mean water levels for Fort Pulaski, GA. These data are verified, monthly

mean values. STND refers to station datum, an arbitrary, vertical reference point at a

given location. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

Table 4. NOS monthly mean water levels for Charleston, SC. These data are verified, monthly

mean values. The data fields have the same meaning as in Table 3. . . . . . . . . . . . . 34

Table 5. NOS tidal datums for Charleston. Data are accepted. These are the 19-year values of the

tidal datums at Charleston, SC, and have been computed by primary reduction. . . . 34

Table 6. Worksheet for calculating the corrected MTL for the subordinate station. The purpose

of this worksheet is to solve Eq. (2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

Table 7. Worksheet for calculating the corrected DTL for a subordinate station. This worksheet

solves Eq. 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

Table 8. Worksheet for correcting the Mn at a subordinate station. This worksheet solves Eq. 6

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

Table 9. Worksheet for calculating the corrected Gt at a subordinate station. The purpose of this

worksheet is to solve Eq. 8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

Table 10. NOS monthly mean water levels for Panama City Beach, FL. . . . . . . . . . . . . . . . . 42

Table 11. NOS monthly mean water levels for Pensacola, FL. . . . . . . . . . . . . . . . . . . . . . . . . 42

Table 12. NOS accepted tidal datums for Pensacola. These are the 19-year, accepted, official values

of the tidal datums at Pensacola, FL, and have been computed by primary reduction. Note

that HWI -- Greenwich Mean High Water Interval in Hours, and LWI -- Greenwich Mean

Low Water Interval in Hours, are not calculated for this station because the type of tide

at Pensacola is diurnal. The data are in meters above station datum. . . . . . . . . . . . 43

Table 13. Worksheet for calculating the corrected MTL for the subordinate station. The purpose

of this worksheet is to solve Eq. (2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
xi

LIST OF TABLES (cont.)

Table 14. Worksheet for calculating the corrected DTL for Panama City Beach. The purpose of this

Table is to solve Eq. 4 for Panama City Beach. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

Table 15. Worksheet for correcting the Mn at a subordinate station . . . . . . . . . . . . . . . . . . . . 44

Table 16. Worksheet for calculating the corrected Gt at a subordinate station. . . . . . . . . . . . . 45

Table 17. NOS monthly mean water levels for Alameda, CA. The data are verified. . . . . . . 49

Table 18. NOS monthly mean water levels for San Francisco, CA. The data are verified. . . 49

Table 19. NOS tidal datums for San Francisco. Data are accepted. These are the 19-year values of

the tidal datums at San Francisco, CA, and have been computed by primary reduction.

The data are in meters above station datum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

Table 20. Worksheet for calculating the corrected MTL for Alameda. The purpose of this

worksheet is to solve Eq. (2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

Table 21. Worksheet for correcting the Mn at a subordinate station. The purpose of this worksheet

is to solve Eq. 6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

Table 22. Worksheet for calculating the corrected DHQ at a subordinate station. This worksheet

solves Eq. 14. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

Table 23. Worksheet for calculating the corrected DLQ at a subordinate station. This worksheet

solves Eq. 16. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

Table 24. NOS High and Low Waters for 8670870 Fort Pulaski, Savannah River, GA, USA from

19960304 to 19960308. Data are verified. Data are in meters above Station Datum

(STND). Times are on UTC (GMT). The fields in the table have the following meanings.

Station--Unique seven character identifier for the station. Date Time--Date and time the

data were collected by the DCP. WL_Value--Water level height. Type --Designation of

Water level height. LL = lower low water, L = low water, H = High water, HH Higher

High water. Infer--A flag that indicates that the water level value has been inferred.

Limit--A flag that when set to 1 indicates that either the maximum or minimum expected

water level height limit was exceeded. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

Table 25. NOS High and Low Waters for 8665530 Charleston, Cooper River Entrance, SC, USA

from 19960304 to 19960308. Data are verified. Data are in meters above Station Datum

(STND). Times are on UTC (GMT). The fields in the table have the same meanings as

Table 24. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
xii

LIST OF TABLES (cont.)

Table 26. Comparison of simultaneous observations for 96-3-4 TO 96-3-8. Datums computed for

the 1960-1978 tidal epoch. (A) designates the subordinate station, 8670870, Fort Pulaski,

Savannah River, GA. (B) designates the control or, standard station 8665530,

Charleston, Cooper River Entrance, SC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

Table 27. Calculation of the mean time differences between the subordinate and control stations.

The means computed below are entered into Equations 48 and 49 to estimate the

Greenwich lunitidal intervals at the subordinate station. The times entered below are in

GMT. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

Table 28. NOS high and low waters for 8729210 Panama City Beach, FL, USA from 1996-03-10

to 1996-03-16. Data are verified. Data are in meters above STND. Times are on UTC

(GMT). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

Table 29. NOS high and low waters for 8729840 Pensacola, Pensacola Bay, FL, USA from 1996-

03-10 to 1996-03-16. Data are verified. Data are in meters above STND. Times are on

UTC (GMT). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

Table 30. Worksheet for computing a tide by tide comparison of high and low waters, diurnal

example. Refer to Figure 31 to see how the high and low waters are matched. . . . 70

Table 31. Computation of the mean time differences between Panama City Beach and Pensacola,

FL.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

Table 32. NOS high and low waters for 9414750 Alameda, San Francisco Bay, CA, USA from

1997-03-01 to 1997-03-07. Data are verified. Data are in meters above STND. Times

are on UTC (GMT). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

Table 33. NOS high and low waters for 9414290 San Francisco, San Francisco Bay, CA, USA from

1997-03-01 to 1997-03-07. Data are verified. Data are in meters above STND. Times

are on UTC (GMT). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

Table 34. Worksheet for calculating a tide by tide analysis for a mixed tide case. . . . . . . . . . 77

Table 35. Computation of the mean time differences between Alameda and San Francisco, CA

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

Table 36. Worksheet for computing MHW at subordinate station using the Direct Method. . 89

Table 37. Worksheet for computing MHHW at subordinate station using the Direct Method . 89
xiii

LIST OF ACRONYMS

CO-OPS Center for Operational Oceanographic Products and Services

DQA Data Quality Assurance

DHQ Mean Diurnal High Water Inequality

DLQ Mean Diurnal Low Water Inequality

DTL Diurnal Tide Level

GMT Greenwich Mean Time

GPS Global Positioning System

Gt Great Tropic Range

IGLD 85 International Great Lakes Datum of 1985

HHW Higher High Water

HLW Higher Low Water

HWI High Water Lunitidal Interval

HWL Highest Water Level

LHW Lower High Water

LLW Lower Low Water

LW Low Water

LWI Low Water Lunitidal Interval

LWL Lowest Water Level

MHW Mean High Water

MHHW Mean Higher High Water

MLW Mean Low Water 

MLLW Mean Lower Low Water

Mn Mean Range of Tide

MSL Mean Sea Level

MTL Mean Tide Level

NAVD 88 North American Vertical Datum of 1988

NGWLMS Next Generation Water Level Measurement System

NGVD 29 National Geodetic Vertical Datum of 1929

NIST National Institutes of Standards and Technology

NOAA National Oceanic and Atmospheric Administration

NOS National Ocean Service

NTBMS National Tidal Benchmark System

NTDE National Tidal Datum Epoch

NWLON National Water Level Observation Network

NWLP National Water Level Program 

PBM Primary Bench Mark

QA Quality Assurance

QC Quality Control

SOP Standard Operating Procedure

STND Station Datum

USACE U.S. Army Corps of Engineers

USCG U.S. Coast Guard

UTC Universal Time
xiv
1

1. INTRODUCTION

1.1 Intended Audience

This handbook is intended to provide education and training for both internal and external

audiences to NOAA. It presents the National Ocean Service (NOS) methodology for the

computation of tidal datums and explains how to use the Center for Operational Oceanographic

Products and Services (CO-OPS) water level data and bench mark information available on the

internet for tidal datum computations. Fundamental background for tide measurement and data

processing is also reviewed. Detailed descriptions of tidal datum procedures, the background

mathematical formulas, and example spreadsheets are interwoven in the various sections.

The handbook is designed to be both a technical reference and a guidance document for the

practical determination of tidal datums using tide gauge measurements. It does not present methods

for surveying, or address the problems associated with instrument installation, calibration, data

collection, or quality assurance of water level data. Nor does it present specific algorithms for

computation, or recommend what software packages should be used. However, a knowledgeable

coastal engineer or scientist should be able to follow the key steps and arrive at the same results

posted on the CO-OPS website (http://www.tidesandcurrents.noaa.gov). 

1.2 Statement of Philosophy

The philosophy of this handbook is that fairly simple, straight-forward examples should be

presented. CO-OPS is confident that coastal engineers will be able to compute datums similar to

these “straight-forward” examples using this manual. The emphasis is on education and training,

illustrated by clear real-world examples of tidal datum calculations. By reading this material, coastal

engineers and surveyors will gain an understanding of how to reduce the data that they may have

collected themselves, and gain necessary skills to handle more difficult cases. The datum

computational methods described in this handbook produce valid datums where the tidal conditions

and tide station locations for datum determination are straightforward. Difficult cases should be

referred to CO-OPS for consultation. These cases might include project areas of rapidly changing

tidal characteristics either temporally or geographically, measurements collected during extreme

events, cases of poor data, data records with too many gaps, or poor station coverage. Additional

special cases that may render the methods not applicable include situations where the astronomic

tide is frequently masked by non-tidal effects (such as areas where wind-driven water level

variations dominate and areas affected by river runoff); and where man-made structures (such as

locks or water gates) affect the water level variations.

1.3 Prerequisite Knowledge

The reader will need to possess a mathematical understanding of means, standard deviations,

differences, and errors. The reader should possess knowledge of suitable computer software such

as spreadsheet programs, and have an internet browser and should have some basic scientific

knowledge of tides and water levels, and some knowledge of the legal and practical significance of

tidal datums (e.g, NOS, 2000). 
2
3

2. BACKGROUND

2.1 Characteristics of the Tides

The purpose of this section is to provide a brief overview of tidal terms, tidal variations, and

tidal characteristics. This overview provides context for the detailed description of the tidal datum

computation methodology that follows.

The word “tides” is a generic term used to define the alternating rise and fall of the oceans with

respect to the land, produced by the gravitational attraction of the moon and sun. To a much smaller

extent, tides also occur in large lakes, in the atmosphere, and within the solid crust of the earth, acted

upon by these same gravitational forces of the moon and sun. Additional non-astronomical factors

including configuration of the coastline, local depth of the water, ocean-floor topography, and other

hydrographic and meteorological influences may play an important role in altering the range of tide,

the time interval between high and low waters, and times of arrival of the tides. 

There are three basic types of tides: semidiurnal (twice-daily), mixed (also twice-daily), and

diurnal (daily) (Figure 1). The first type, semidiurnal, has two high waters (high tides) and two low

waters (low tides) each tidal day. A tidal day is the time of rotation of the Earth with respect to the

Moon, and its mean value is approximately equal to 24.84 hours. Qualitatively, the two high waters

for each tidal day must be almost equal in height. The two low waters of each tidal day also must

be approximately equal in height. The second type, mixed, is similar to the semidiurnal except that

the two high waters and the two low waters of each tidal day typically have marked differences in

their heights. When there are differences in the heights of the two high waters, they are designated

as higher high water and lower high water; when there are differences in the heights of the two lows,

they are designated as higher low water and lower low water. The third type, diurnal, has one high

water and one low water each tidal day. 

The most important modulations of the tides are those associated with the phases of the moon

relative to the sun. Spring tides are tides occurring during the new and full moon phases. These are

the tides of the greatest amplitude, thus the highest and lowest waters are recorded at these times

during each lunar month. Neap tides are tides occurring approximately midway between the time

of new and full moon. The neap tidal range is usually 10 to 30 percent less than the mean tidal range.

In addition to spring and neap tides, there are lesser, but significant monthly modulations due to the

elliptical orbit of the moon about the earth (perigee and apogee) and yearly modulations due to the

elliptical orbit of the earth about the sun (perihelion and aphelion). Modulations in mixed and

diurnal tides are especially sensitive to the monthly north and south declinations of the moon relative

to the earth’s equator (tropic tides and equatorial tides), and to the yearly north and south

declinations of the sun (equinoxes and solstices). Although the astronomical influences of the moon

and sun upon the earth would seem to imply a uniformity in the tide, the type of tide can vary both

with time at a single location and geographically along the coast. The transition from one type to

another is usually gradual either temporally or spatially, resulting in hybrid or transition tides. In

Figure 2, semidiurnal tides are illustrated by the one-month plot for New York, mixed-type tides are

illustrated by the plot for San Francisco, and the plot for Pensacola illustrates a diurnal tide. 
4

Figure 1. A depiction of the three primary kinds of tides. From the top panel

downward they are semidiurnal, mixed, and diurnal. Standard tidal terminology is

used to describe the various aspects of the tides. The zero on these graphs is

illustrative of the relationship of the tides to Mean Sea Level (MSL). 

Figure 3 shows the gradual spatial transitions from mixed to diurnal to mixed and back to diurnal

in the Gulf of Mexico. It is important to know the location of these transition zones because they

limit how far the datum computation procedures described in this document can be applied

successfully.
5

COMPARISON OF TIDE TYPES OVER ONE CALENDAR MONTH

-0.500

0.500

1.500

2.500

3.500

4.500

5.500

6.500

7.500

8.500

01-01-02 01-06-02 01-11-02 01-16-02 01-21-02 01-26-02 01-31-02 

Time (hours)

Elevation (meters)

Semidiurnal - Fort Pulaski, GA

Mixed - San Francisco, CA

Diurnal - Pensacola, FL

Figure 2. Comparison of types of tide over one-calendar month.

Figure 3. An illustration of the spatial variability of the type of tide in the Gulf

of Mexico.
6

Figure 4. An illustration of the effect of the regression of the moon’s nodes on the water levels at

Puget Sound, WA. The heavy black curve is the annual mean range, or the difference in height

between mean high water and mean low water. The time elapsed between trough to trough, or

peak to peak, is the period of oscillation of the regression, and is about 18.6 years. The more

rapidly varying curve is the monthly mean range. Changes in the monthly mean range are due to

rapidly changing meteorological and oceanographic conditions.

There is a variation in the path of the moon about the earth that has a period of about 18.6 years, and

is called the regression of the moon’s nodes. The regression of the nodes introduces an important

variation into the amplitude of the monthly and annual mean range of the tide, as may be seen in

Figure 4. It is the regression of the moon’s nodes which forms the basis of the definition of the

National Tidal Datum Epoch (NTDE). Because the variability of the monthly mean range is larger

than the regression of the nodes, the National Tidal Datum Epoch is defined as an even 19 year

period so as not to bias the estimate of the tidal datum. 
7

Greenwich interval local interval = + 0 069 . L

2.2 Lunitidal Intervals

The high and low water phase of the tide at a particular place follows the passage of the moon

over the local meridian by nearly a constant interval (Marmer, 1951). The moon, in its apparent

rotation about the earth, crosses a given meridian 50 minutes later each day. Thus, the high or low

water at given place seems to occur about 50 minutes later each day. The moon’s upper meridian

passage refers to the instant when the moon is directly above the given meridian. The moon’s lower

passage refers to the moon being 180o

 distant in longitude. 

A lunitidal interval is the interval in time between the moon’s passage over the local or

Greenwich meridian and the following high or low water. The average of all high water intervals

for all phases of the moon is called the mean high water lunitidal interval. It is abbreviated HWI.

Similarly, the average of all low water intervals for all phases of the moon is called the mean low

water lunitidal interval, and is abbreviated LWI. 

In this handbook, HWI and LWI are calculated as Greenwich intervals. Greenwich intervals

refer to the time interval between passage of the moon over the meridian of Greenwich, and the local

high or low water of interest. This calculation is found in the section on “Comparison of

Simultaneous High and Low Waters”. Equivalent 19-year HWI and LWI values are computed at

the subordinate station, by using the accepted values of Greenwich HWI and LWI at the control

station. The relation in hours between Greenwich and local intervals may be given by,

where L is the west longitude of the local meridian in degrees. If L is east longitude, L is negative

(Hicks, 1989). Assistance in determining when the moon passes the Greenwich meridian may be

obtained from the U.S. Naval Observatory website, http://aa.usno.navy.mil/AA. Assistance in

determining accurate local and GMT (UTC) times may be obtained from the website of the National

Institute of Standards and Technology (NIST), http://www.nist.gov.

2.3 Other Signals in Water Level Measurements

Tides are not the only factor causing the sea surface height to change. Additional factors include

waves and wave setup; ocean and river currents; ocean eddies; temperature and salinity of the ocean

water; wind; barometric pressure; seiches; and relative sea level change. All of these factors are

location dependent, and contribute various amounts to the height of the sea surface. Examples are:

wind setup - up to about 1 meter (~3.2 feet); ocean eddies - up to about 25 centimeters (~0.8 foot);

upper ocean water temperature - up to about 35 centimeters (~1.1 foot); ocean currents or ocean

circulation - about 1 meter; and global sea level rise (about 0.3 meter (1 foot) per century).

It is NOS procedure to not separate out other sea level effects from the tides for computation

tidal datums. There are no mathematical or statistical filters applied to the data before processing.

Tidal tabulations will include, for instance, the effects of storm surge. NOS does use a combination

mechanical/numerical filter to remove the unwanted effects of high frequency wind waves and

currents. The filter is part of the physical design of the sensor and the data collection algorithm in

the data collection platform (Scherer, 1986).
8

2.4 Tide Station Networks

The NOS National Water Level Program (NWLP) provides unique water level and ancillary data

sets and information to users in support of a wide variety of critical activities. A priority of the

NWLP is to provide the basic data for the vertical, tidal datum control for the nation. For instance,

these data specifically support NOAA’s Nautical Charting and safe navigation programs. The

instrumentation of the NWLP consists of water level stations in the National Water Level

Observation Network (NWLON), and any short-term stations operating for special projects such as

hydrographic surveys, photogrammetry, and United States Army Corps of Engineers (USACE)

dredging activities. The NWLON (Figure 5) is composed (as of year 2000) of approximately 175

long term stations distributed around the country and the world. The applications supported by the

NWLON include: nautical charting, hydrography, remote sensing for shoreline, boundary

determination, navigation, channel dredging and harbor improvements, tsunami and storm surge

warnings, tide predictions, environmental monitoring and habitat restoration, climate and global

change, international lake level regulation, international treaty compliance, and international

boundary determination. 

Except for water level stations in the Great Lakes, most of the stations in the NWLON are in

coastal areas that come under the influence of the tide to a significant degree and are referred to as

control tide stations. These stations have accepted tidal datums computed over the NDTE or at least

over a several year period from which equivalent NTDE accepted values have been computed. This

network provides direct datum control for a nearby areas and control for short-term stations for a

larger geographic area. The extent of datum control depends upon the complexity of the coastal

zone in terms of changes in tidal characteristics, localized effects of river runoff and wind, and

differences in long-term sea level trends. 

The NWLON is designed to provide a nationwide fundamental tidal datum control network.

Most applications require tidal datum information at a higher resolution than that provided by the

NWLON network spacing. Depending on the application, networks of shorter-term stations are

established. 

Control tide stations are generally those which have been operated for 19 or more years, are

expected to continuously operate in the future, and are used to obtain a continuous record of the

water levels in a locality. Control tide stations are sited to provide datum control for national

applications, and located in as many places as needed for datum control. As the records from such

a station constitute basic water level data for present and future use, during the installation and

maintenance of the station, the aim is to obtain the highest degree of reliability and precision that

is practical. The essential equipment of a control tide station includes an automatic water level

sensor, protective well, shelter, back-up water level sensor, a system of bench marks, and possibly

ancillary geophysical instruments. 

Secondary water level stations are those which are operated for less than 19 years but more than

1 year, and have a planned finite lifetime. Secondary stations provide control in bays and estuaries

where localized tidal effects are not realized at the nearest control station. Observations at a

secondary station are not usually sufficient for a precise independent determination of tidal datums,
9

 Figure 5. Locations of U.S. NWLON water level stations.

but when reduced by comparison with simultaneous observations at a suitable control tide station

very satisfactory results may be obtained. Secondary tide stations may also provide data for the

reduction of soundings in connection with hydrographic surveys. 

Tertiary water level stations are those which are operated for more than a month but less than

1 year. Short-term water level measurement stations (secondary and tertiary) may have their data

reduced to equivalent 19-year tidal datums through mathematical simultaneous comparison with a

nearby control station. Short-term data, often at several locations, are collected routinely to support

hydrographic surveying. In the Great Lakes, seasonal data are compared to simultaneous

observations from adjacent stations for datum determination in harbors.

The locations of tide stations are organized into a hierarchy (Figure 6). Control (or primary)

tide stations, secondary stations and tertiary stations are located at strategic locations for network

coverage. The site selection criteria include spatial coverage of significant changes in tidal

characteristics such as: changes in tide type, changes in range of tide, changes in time of tide,

changes in daily mean sea level and changes in long term mean sea level trends. Other criteria
10

include coverage of critical navigation areas and transitional zones, historical sites, proximity to the

geodetic network, and the availability of existing structures, such as piers suitable for the location

of the scientific equipment. 

Site reconnaissance is performed prior to the installation of a new station. Field site visits are

made to aid in the design, make measurements, and render technical drawings; to recover bench

marks, or plan for new bench marks; and to obtain permission, permits, agreements, etc. The field

parties take into consideration the requirements for the installation and protection of the instruments.

The most important considerations are the presence of a suitable structure, the necessary benchmark

locations, adequate water depth, special materials that might be needed to prevent marine fouling

and corrosion, availability of telephone and electrical service, site security, and lightning protection.

There are numerous types of tide gauges and sensors that can be used for tidal datum

computation purposes. NOS specifications for tide station installations and data processing and

reduction for NOS hydrographic surveys are found in NOS Hydrographic Survey Specifications and

Deliverables (NOS,2003). International references include the Manual on Sea level Measurement

and Interpretation (IOC, 2000). The latest update on NOS water level measurement systems and

capabilities are found in Mero, 1998. For tidal datum applications, it is important for gauge sensors

to be carefully calibrated with either frequent calibration checks or cycled swaps of calibrated

sensors for long-term installations. The sensor “zero” must be precisely related to either a tide staff

and/or the bench marks through staff/gauge comparisons or direct leveling between the sensor and

the bench marks. Vertical stability of the sensor “zero”, both physically and internally, must be

monitored and any movement taken into account in the data reduction and datum computation.

2.5 Bench marks and Differential Leveling

A network of bench marks is an integral part of every water level measurement station. A bench

mark is a fixed physical object or mark (sometimes referred to as a monument) used as a reference

for a vertical datum. For example, a tidal bench mark is a mark near a tide station to which the tidal

datums are referenced. Since gauge measurements are referenced to the bench marks, it follows that

the overall quality of the datums is partly dependent on both the quality of the bench mark

installation and the quality of the leveling between the bench marks and the gauge.

Bench marks have site selection considerations much like the tide stations they support. The

first consideration is longevity; bench marks are sited to minimize susceptibility to damage or

destruction. Bench marks are sited to ease future recovery (locating and leveling to the mark) and

to ensure accessibility (open, overhead clearance). Bench marks must also be placed in the most

stable structure for the locality. Preference should be given to disks set in bedrock, in large man

made structures with deep foundations, or installation of stainless steel rods driven to substantial

resistance. Since bench marks are vulnerable to natural disturbances, such as geologic and soil

activity, in addition to damage inflicted by man, more bench marks are installed around stations with

longer term data series. At primary control stations, where 19 years of observations have been

conducted or are planned, a network of at least ten bench marks is installed in the vicinity of the
11

Figure 6. Illustration of tide station network hierarchy.
12

station. Five bench marks are installed at secondary and tertiary stations. At least three bench

marks are installed at short term (less than 30 days) stations.

Bench marks are leveled whenever a new tide station is established, or when data collection is

discontinued at a tide station. Bench marks are also leveled before and after maintenance is

performed at a station, and at least annually to perform stability checks. In addition, whenever new

bench marks are installed, the existing bench marks are re-leveled. Differential levels (Figure 7) are

used to check the elevation differences between bench marks, to extend vertical control, and to

monitor the stability of the water level measurement gauge. The quality of leveling is a function of

the procedures used, the sensitivity of the leveling instruments, the precision and accuracy of the

rod, the attention given by surveyors, and the refinement of the computations. Bench marks are

leveled by either compensator leveling instruments, or by an electronic digital barcode system.

Compensator-type leveling instruments require double running. However, under certain

circumstances, electronic digital/barcode systems allow for single running. The User’s Guide for

the Installation of Bench Marks and Leveling Requirements for Water Level Stations (Hicks et al.,

1987) and NOAA Manual NOS N.S. 1 (Floyd, 1978) provide detailed guidelines for bench mark

installations and leveling. The Standards and Specifications for Geodetic Control Surveys includes

interim Federal Geodetic Control Subcommittee specifications and procedures to incorporate

electronic digital/barcode levels. The National Tidal Benchmark System (NTBMS) provides datum

information for previously and currently occupied tidal measurement locations. The number of

stations in the NTBMS is approximately 6000. Bench marks may become invalid due to crustal

movement, and may also be invalidated by changes in local tidal characteristics due to dredging,

erosion, and accretion. In many cases, bench marks in the NTBMS have not been releveled in many

years, resulting in some uncertainty in their validity. At present, about 2000 stations have bench

marks with valid published elevations.
13

Figure 7. A schematic diagram of extending vertical control inland from the tidal

datum by the method of differential leveling.

2.6 Data Processing Procedures

Data collected from the field must be processed and tabulated before tidal datums are computed.

This section reviews the procedures used to perform this processing. Raw unedited water level data

collected from tide gauges should undergo preliminary quality control such as checking for outliers,

gaps, distorted tide curves, and undocumented shifts in datum or scale factor. NOS collects data at

6-minute intervals, but other intervals can be used successfully. Six-minute intervals were chosen

to easily allow times of the high and low tides to be estimated to the nearest tenth of an hour. The

procedure of preliminary review consists of manually or automated checks of the data, making

relevant comparisons with backup sensors, tide staffs, nearby stations, or predicted tides, and

scanning any automated instrument and data acquisition reports. Small gaps in data should be filled

to get a time series that is as continuous as possible. NOS fills small gaps up to 3 to 4 hours using

least-squares curve fits of the 6-minute data. Longer gaps in data up to 3-days are filled in a

hierarchical sense depending on location and availability of source data. If data from backup sensors

are not available, the gaps are inferred using data from nearby stations or predicted tides. Gaps are

left in the data if source data to fill them are insufficient. 

Once the gaps are filled, and depending upon the type of station, the tabulation process is carried

out that includes generation of hourly heights, generation of high and low tides, and selection of

higher high and lower low waters. NOS tabulates the times and heights and high and low waters

by using a least-squares polynomial curve fit to the 6-minute tide data. Hourly heights are tabulated

as every tenth 6-minute interval value (Figures 8a and b). After this the monthly means computed

for the various datums. The time-series data, hourly heights, high and low waters, higher highs and

lower lows, and monthly means are subsequently verified by a senior analyst (Figure 9). 
14

8454049 QUONSET POINT RI - Six Minute - Water Level : October 2002

6.000

6.500

7.000

7.500

8.000

8.500

9.000

10-01-02 00:00 10-06-02 00:00 10-11-02 00:00 10-16-02 00:00 10-21-02 00:00 10-26-02 00:00 10-31-02 00:00

Elevation Relative to Station Datum (meters)

Figure 8a and b. Observed 6-minute data for one month and results from the tabulation

of the tide.
15

 Jan 28 2003 08:24 HIGH/LOW WATER LEVEL DATA October, 2002 

 Station: 8454049 T.M.: 0 W 

 Name: QUONSET POINT, RI Units: Meters 

 Type: Mixed Datum: Station Datum 

 Note: > Higher-High/Lower-Low [] Inferred Tide Quality: Verified 

 

 High Low High Low 

 Day Time Height Time Height Day Time Height Time Height 

 --- ---- ------ ---- ------ --- ---- ------ ---- ------ 

 1 7.5 8.037 2.4 7.326 16 > 9.7 [ 8.292] 2.6 7.394 

 > 20.2 8.071 > 12.9 7.197 > 21.3 8.782 14.6 7.563 

 2 8.8 8.000 2.6 7.173 17 10.6 8.345 > 6.0 7.470 

 > 21.4 8.176 > 14.3 7.066 > 22.8 8.323 > 15.4 7.245 

 3 9.5 8.233 3.2 7.157 18 10.7 8.257 4.0 7.248 

 > 22.3 8.314 > 15.6 7.049 23.3 8.230 16.7 7.196 

 4 10.5 8.525 4.1 7.163 19 > 11.8 8.296 > 4.3 7.140 

 > 23.1 8.599 > 16.3 7.057 > 23.4 8.292 17.1 7.204 

 5 > 11.5 8.632 4.4 7.109 20 > 5.0 7.066 

 23.8 8.466 > 17.1 6.873 12.4 8.209 > 17.5 6.994 

 6 > 5.8 6.670 21 0.4 [ 8.128] 5.8 7.036 

 12.2 8.477 18.2 6.832 > 12.8 8.297 18.1 7.090 

 7 > 0.5 8.582 > 6.4 6.961 22 0.9 8.142 > 6.5 6.999 

 > 13.3 8.819 19.2 6.969 > 13.4 8.216 19.0 7.040 

 8 1.3 8.457 6.9 6.888 23 1.4 [ 8.075] > 6.9 7.013 

 > 14.0 8.644 > 20.1 6.877 > 13.7 [ 8.180] > 19.1 6.915 

 9 2.3 8.355 > 7.9 6.852 24 2.1 7.934 7.3 6.969 

 > 14.9 8.631 20.9 6.986 > 14.7 8.164 19.9 7.093 

 10 3.4 8.316 > 8.2 6.969 25 2.9 [ 7.993] > 8.0 7.047 

 > 15.8 8.497 21.2 7.086 > 15.4 8.156 > 20.3 7.136 

 11 4.3 8.240 > 9.4 7.129 26 3.8 [ 8.061] 8.3 7.204 

 > 16.7 8.455 22.1 7.305 > 16.2 8.607 23.5 7.389 

 12 5.2 8.295 > 10.3 7.380 27 4.6 7.974 > 9.1 7.090 

 > 17.7 8.462 > 17.1 8.216 21.9 7.348 

 13 5.9 8.266 0.5 7.481 28 5.4 7.860 > 10.5 7.064 

 > 18.7 8.344 11.8 7.461 > 17.9 8.008 

 14 6.8 8.077 > 2.2 7.401 29 6.2 7.949 1.5 7.243 

 > 20.1 8.161 > 12.7 7.190 > 18.6 8.042 > 11.6 7.109 

 15 8.3 8.156 2.0 7.349 30 7.3 [ 8.052] > 1.5 7.197 

 20.9 8.273 > 14.1 7.344 > 20.0 [ 8.154] 13.0 7.211 

 31 8.3 8.215 2.1 7.239 

 > 20.7 8.290 > 14.1 7.222 

 Highest Tide: 8.819 13.3 Hrs Oct 7 2002 

 Lowest Tide: 6.670 5.8 Hrs Oct 6 2002 

 

 Monthly Means: MHHW 8.357 

 MHW 8.272 DHQ 0.085 

 MTL 7.707 GT 1.266 HWI 0.42 Hrs 

 DTL 7.724 MN 1.131 LWI 6.13 Hrs 

 MSL 7.668 

 MLW 7.141 DLQ 0.050 

 MLLW 7.091 

 

Figure 9. Example of a Monthly Tabulation of the Tide
16

2.7 The National Tidal Datum Epoch

As mentioned in section 2.1, a specific nineteen year period designated as a National Tidal

datum Epoch (NTDE) is used to compute tidal datums because it is the closest full year to the 18.6-

year nodal cycle, the period required for the regression of the moon’s nodes to complete a circuit

of 360/ of longitude (Schureman, 1941). The NTDE is used as the fixed period of time for the

determination tidal datums because it includes all significant tidal periods, is long enough to average

out the local meteorological effects on sea level, and by specifying the NTDE, a uniform approach

is applied to the tidal datums for all stations.

The relative secular sea level change, as well as the variability of the change by geographic

region, is readily seen (Figure 10) where the yearly mean sea level is plotted against time. For

datum computation, the National Tidal Datum Epoch is used as the fixed period of time for the

determination of tidal datums because it includes all significant tidal periods, is long enough to

average out the local meteorological effects on sea level, and by specifying the NTDE, uniformity

is applied to all the tidal datums. However, because of relative sea level change, as the years pass,

tidal datums become out of date for navigational purposes and for other applications. Thus, a new

NTDE must be considered periodically (Hicks, 1981). The policy of NOS is to consider a new tidal

datum epoch every 25 years to appropriately update the tidal datums to account for the global sea

level change and long-term vertical adjustment of the local landmass (e.g., due to subsidence or

glacial rebound)(Gill et al, 1998). Figure 11 shows the effect of sea level rise on the elevation of

MTL over various NDTE periods. NOS will be updating from the 1960-78 NTDE to a 1983-2001

NTDE in 2003. Estimated relative sea level trends compiled from observations at U.S. tide stations

are found in Zervas (2000).
17

Figure 10. Relative sea level change at several locations in the U.S.
18

San Francisco, CA - Variations in MTL Across National 

Tidal Datum Epochs

2.200

2.300

2.400

2.500

2.600

2.700

2.800

2.900

3.000

3.100

3.200

1-001-041-081-121-161-201-241-281-321-361-401-44

1-481-52

1-561-601-641-681-72

1-761-80

1-84

1-88

1-921-96

time (months)

mean tide level (meters)

1941-59 MTL :2.700m 1960-78 MTL: 2.728m

Figure 11. Illustration of the long term changes in sea level causing the need to update

tidal datums such as mean tide level (MTL). The graph shows the annual values of mean

tide level with horizontal lines drawn to show the 19-year MTL values for the 1941-59 and

1960-78 NTDE. 
19

3. GENERAL TIDAL DATUM COMPUTATION PROCEDURES

3.1 Datum Computation Procedures Overview

A vertical datum is termed a tidal datum when it is defined by a certain phase of the tide. Tidal

datums are local datums and should not be extended into areas which have differing hydrographic

characteristics without substantiating measurements. In order that they may be recovered when

needed, such datums are referenced to fixed points known as bench marks.

1. Make Observations - Tidal datums are computed from continuous water level observations

over specified lengths of time. Observations are made at specific locations called tide stations. Each

tide station consists of a water level gauge or sensor(s), a data collection platform or data logger and

data transmission system, and a set of tidal bench marks established in the vicinity of the tide station.

NOS collects water level data at 6-minute intervals.

2. Tabulate the Tide - Once the 6-minute interval data are quality controlled and any small gaps

filled, the data are processed by tabulating the high and low tides and hourly heights for each day.

Tidal parameters from these daily tabulations of the tide are then reduced to mean values, typically

on a calendar month basis for longer period records or over a few days or weeks for shorter-term

records.

3. Compute Tidal Datums - First reduction tidal datums are determined directly by averaging

values of the tidal parameters over a 19-year NDTE. Equivalent NDTE tidal datums are computed

from tide stations operating for shorter time periods through comparison of simultaneous data

between the short-term station and a long term station.

4. Compute Bench Mark Elevations - Once the tidal datums are computed from the

tabulations, the elevations are referenced to the bench marks established on the land using the

elevation differences established by differential leveling between the tide gauge sensor “zero” and

the bench marks during the station operation. The bench mark elevations and descriptions are

disseminated by NOS through a station specific published bench mark sheet. Connections between

tidal datum elevations and geodetic elevations are obtained after leveling between tidal bench marks

and geodetic network bench marks. Traditionally, this has been accomplished using differential

leveling, however GPS surveying techniques can also be used (NGS, 1997).

A primary determination of any tidal datum is based directly on the average of observations over

a 19 year period. For example, a primary determination of Mean High Water is based directly on the

average of the high waters over a 19 year period. Tidal datums must be specified with regard to the

NTDE. Although many tidal datums are discussed in this report, the principal tidal datums include

Mean Higher High Water (MHHW), Mean High Water (MHW), Mean Sea Level (MSL), Mean Low

Water (MLW), and Mean Lower Low Water (MLLW) (Marmer, 1951 and NOS, 2000).

MHHW is defined as the arithmetic mean of the higher high water heights of the tide observed

over a specific 19-year Metonic cycle (the National Tidal Datum Epoch). Only the higher high

water of each pair of high waters of a tidal day is included in the mean (Figure 1). For stations with
20

shorter series, a comparison of simultaneous observations is made with a primary control tide station

in order to derive the equivalent of a 19-year value (Marmer, 1951).

MHW is defined as the arithmetic mean of all of the high water heights observed over a specific

19-year Metonic cycle (the NTDE). For stations with shorter series, a comparison of simultaneous

observations is made with a primary control tide station in order to derive the equivalent of a 19-year

value. 

MSL is defined as the arithmetic mean of hourly heights observed over a specific 19-year

Metonic cycle (the NTDE). Shorter series are specified in the name, like monthly mean sea level

or yearly mean sea level (e.g., Marmer, 1951; Hicks, 1985).

MLW is defined as the arithmetic mean of all of the low water heights observed over a specific

19-year Metonic cycle (the NTDE). For stations with shorter series, a comparison of simultaneous

observations is made with a primary control tide station in order to derive the equivalent of a 19-year

value.

MLLW is defined as the arithmetic mean of the lower low water heights of the tide observed

over a specific 19-year Metonic cycle (the NTDE). Only the lower low water of each pair of low

waters of a tidal day is included in the mean. For stations with shorter series, a comparison of

simultaneous observations is made with a primary control tide station in order to derive the

equivalent of a 19-year value.

In addition, the Mean Tide Level (MTL), Diurnal Tide Level (DTL), Mean Range (Mn), Diurnal

High Water Inequality (DHQ), Diurnal Low Water Inequality (DLQ), and Great Diurnal Range (Gt)

are defined as follows: 

MTL is a tidal datum equivalent to the average of MHW and MLW.

 

DTL is a tidal datum equivalent to the average of MHHW and MLLW.

Mn is the difference in elevation between MHW and the MLW. 

DHQ is the difference in elevation between MHHW and MHW. 

DLQ is the difference in elevation between MLW and MLLW. 

Gt is the difference in elevation between MHHW and MLLW. 

3.2 Other Vertical Datums and Their Relationship to Tidal Datums

In addition to tidal datums, other vertical datums are determined and employed for various

applications. Examples are fixed datums of the National Geodetic Reference System, or the

National Geodetic Vertical Datum (NGVD 1929) (previously referred to as the Sea Level Datum

of 1929), or the North American Vertical Datum of 1988 (NAVD 88). NGVD 1929 is a fixed datum
21

adopted as a standard geodetic reference for heights and was derived from a general adjustment of

the first order leveling nets of the US and Canada, in which MSL was held fixed as observed at 26

stations in the US and Canada. Numerous adjustments have been made to these leveling networks

since originally established in 1929. The North American Vertical Datum of 1988 (NAVD 88)

involved a simultaneous least-squares, minimum constraint adjustment of the Canadian-Mexican-US

leveling observations. Local MSL was held fixed at Father Point/Rimouski, Quebec, Canada, as the

single constraint. The North American Vertical Datum of 1988 (NAVD 88) and International Great

Lakes Datum of 1985 (IGLD 85) are both based upon this simultaneous, least-squares, minimum

constraint adjustment of Canada, Mexico, and U.S. leveling observations. These fixed geodetic

datums (e.g., NGVD 1929 and NAVD 88) do not reflect the changes in sea level and because they

represent a “best” fit over a broad area, their relationship to local mean sea level differs from one

location to another. MSL is a tidal datum often confused with NGVD 1929 and they are not

equivalent. NGVD 1929 was replaced by NAVD 88 and the National Geodetic Survey no longer

supports the NGVD 1929 system. 

Figure 12 shows the datums related to Station Datum (STND) at San Francisco Bay, CA. The

elevation of the primary bench mark (PBM 180 1936), is 5.794 m above STND. The Highest Water

Level (HWL) recorded at San Francisco is 4.462 m above station datum, and the Lowest Water

Level (LWL) is 0.945 m . HWL and LWL are not tidal datums, but are the extreme values of the

maximum and minimum water levels recorded at the station. For San Francisco Bay, the value of

NGVD 29 is below Mean Sea Level (MSL), and NAVD88 is lower still. MSL pertains to local

mean sea level and should not be confused with NAVD 88, the ellipsoid or the superseded NGVD

29. Figures 13a and 13b show why the direct transfer of tidal datum relationships through NAVD

88, NGVD 29 or the ellipsoidal differences, even within the same bay, estuary or river, may not be

accurate. The graph illustrates that tidal datums are local datums relative to the land and great care

must be taken to extrapolate tidal datum differences and relationships to geodetic datums. In some

instances, linear interpolation can be used to estimate datum relationships between two known points

along a stretch of shoreline that is not very complicated in a topographic and bathymetric sense. 

When in doubt of the relationship of a tidal datum to a geodetic datum, establishment of a tide

station and connection to geodetic datum using differential levels or GPS is recommended for most

applications. NOS establishes geodetic connections at the NWLON stations through differential

levels between tidal bench marks and geodetic bench marks. Use of GPS survey equipment to

occupy tidal bench marks is the emerging state-of-the-art method for making the connections. See

the NOS Web-sites at www.tidesandcurrents.noaa.gov and www.ngs.noaa.gov for further

information on geodetic and tidal datum elevations on bench marks.
22

Figure 12. A tidal datum stick diagram for San Francisco, CA showing the

relationships of the various tidal and geodetic datums.
23

 

Figure 13a. Locations of stations plotted in Figure 13b.
24

Tida l Da tum s a nd NGVD Re la tive to NAVD88

a t Sa int Johns Rive r, Florida

-1.50

-0.50

0.50 Jacksonville Beach

872-0291

Little Talbot Island

872-0194

Mayport

872-0220

Fulton

872-0221

Dame Point

872-0219

Main St. Bridge

872-0226

Longbranch

872-0242

Ortega River Ent.

872-0296

Orange Park

872-0374

Green Cove Springs

872-0496

Palatka

872-0774

Station

Elevation in Meters

NA V D88

MHW relative to NA V D88

MSL relative to NA V D88

MLLW relative to NA V D88

NGV D relative to NA V D88

Figure 13b. The relationship of the geodetic datums to tidal datums in the St. John’s River, FL. 

The perspective is going upstream from left to right. The graph was constructed by assigning

NAVD88 to a relative zero value at each site, and adjusting the other datums accordingly.
25

3.3 Steps Required to Compute Tidal Datums at Short-Term Stations

Due to time and resource constraints, primary determinations of tidal datums ( i.e. using 19 years

of data) are not practical at every location along the entire coast where tidal datums are required. At

intermediate locations, a secondary determination of tidal datums can usually be made using

observations covering much shorter periods than 19 years. Results are corrected to an equivalent

mean value by comparison with a suitable control tide station (Marmer, 1951).

Conceptually, the following steps need to be completed in order to compute equivalent NTDE

tidal datums listed in section 3.1 at short term stations using the method of comparison of

simultaneous observations:

1) Select the time period over which the simultaneous comparison will be made.

2) Select the appropriate control tide station for the subordinate station of interest based on

location, tidal characteristics, and availability of data.

. 3) Obtain the simultaneous data from subordinate and control stations and obtain or tabulate

the tides and compute monthly means, as appropriate.

4) Obtain the accepted NTDE values of the tidal datums at the control station from NOS via

the CO-OPS Website (www.tidesandcurrents.noaa.gov)

5) Compute the mean differences and/or ratios (as appropriate) in the tidal parameters between

the subordinate and control station over the period of comparison.

6) Apply the mean differences and ratios computed in step 5, above, to the accepted values at

the control station to obtain equivalent or corrected NTDE values for the subordinate

station. The computations use slightly different formulas depending on the type of tide.

These differences are explained in section 3.4 and in Chapter 4.

3.4 Datum Computation Methods

There are some key datum computation methods used by NOS (in step 6, above) that differ

slightly depending upon the tidal characteristics and the type of tide.

Standard Method. This method is generally used for the West Coast and Pacific Island stations

and is also called the Range Ratio Method. First, equivalent NTDE values for MTL, Mn, DHQ and

DLQ are determined by comparison with an appropriate control. From these, the following are then

computed:

MLW = MTL - (0.5 x Mn)

MHW = MLW + Mn

MLLW= MLW - DLQ

MHHW = MHW + DHQ
26

Modified-Range Ratio Method. This method is generally used for the East Coast, Gulf Coast

and Caribbean Island stations. First, equivalent NTDE values for MTL, DTL, Mn and Gt as

determined by comparison with an appropriate control. The difference from the Standard Method

is that ratios of the DHQ and DLQ values are not used to compute MHHW and MLLW because

numerically the values are very small for semidiurnal tide areas. A Gt ratio about DTL is used

instead. From these, the following are computed:

MLW = MTL - (0.5 x Mn)

MHW = MLW + Mn

MLLW= DTL - (0.5 x Gt)

MHHW = MLLW + Gt

Direct Method. This method is usually used only when a full range of tidal values are not

available. For example, direct MHW can be computed for situations when low waters are not

recorded, such as in the upper reaches of a marsh. Since MTL, DTL, and Mn and Gt cannot be

determined if low waters are cut-off, equivalent NTDE values for MHW and MHHW datums are

determined directly by comparison of high tides with an appropriate control using the available part

of the tidal cycle. 

3.5 Accuracy

Generalized accuracies for datums computed at secondary or tertiary stations in terms of the

standard deviation error for the length of the record are summarized in Table 1 (see Swanson, 1974).

These values were calculated using accepted datums for control station pairs in the NWLON. The

values in Table 1 are the confidence intervals for the tidal datums based on the standard deviation.

 

Table 1. Generalized accuracy of tidal datums for East, Gulf, and West Coasts when determined

from short series of record and based on the standard deviation( one-sigma). From Swanson (1974).

Series Length

(months)

East Coast

(cm) (ft.)

Gulf Coast

(cm) (ft.)

West Coast

(cm) (ft.)

1 3.96 0.13 5.48 0.18 3.96 0.13

3 3.05 0.10 4.57 0.15 3.35 0.11

6 2.13 0.07 3.65 0.12 2.43 0.08

12 1.52 0.05 2.74 0.09 1.82 0.06

It is helpful to view the data in Table 1 graphically (Figure 14). The Swanson error curves are

similarly shaped for each coast. Ranked by coast, the errors are smallest for the East and West

coasts, with the largest errors on the Gulf coast. The largest errors coincide with the least data, and

decrease asymptotically to a finite value with increasing data. This handbook contains examples

which apply datum computation techniques to water level data with series lengths of about 1 week,

and also for twelve months. Examples containing a week of data or less should be interpreted as

having generalized errors greater than or equal to those for a 1-month data series shown on the first
27

Estimated Error in Tidal Datums vs. Length of Series

0.000

0.010

0.020

0.030

0.040

0.050

0.060

0.070

02468

101214

TIME (months)

One-Standard Deviation 

(meters)

East Coast

Gulf Coast

West Coast

Figure 14. Estimated Error in Tidal Datums from Swanson (1974)

row of Table 1. Examples containing a year of data have generalized errors that correspond to the

fourth row.

Th

e uncertainty in the value of the tidal datum translates into a horizontal uncertainty of the location

of a marine boundary when the tidal datum line is surveyed to the land (Demarcating and Mapping

Tidal Boundaries, 1970). Table 2 expresses the uncertainty in the marine boundary as a function

of the slope of the land. A slope of 1% means that the land rises 1 meter for every 100 meters of

horizontal distance. This is illustrated in Figure 15. 
28

Figure 15. Let x be the horizontal distance inland, and y be the

vertical rise of the land. By definition tan(") = y/x. Likewise,

the cotangent of ", denoted cot("), is given by cot(") = x/y .

y

y

x

∆ ∆ x ~

∆∆ ∆ x y

x

y

~ cot( = y α ) 

In Figure 15, denote the horizontal distance inland as x, and denote the vertical elevation change

as y. Denote the error (i.e., vertical error from Table 1) in the tidal datum as )y. The upper bound

on )y is 0.03 m from Table 1, based upon 12 months of data from the Gulf coast. This vertical error,

)y, translates into a horizontal error in determining the location of the tidal datum on the ground.

Denote this horizontal error as )x. Given the errors of Table 1, if one knows the slope of terrain,

given by y/x, (also presented in the first column of Table 2 in percent form), one can estimate the

uncertainty of the horizontal location )x, by the following formula,

Rearranging, we find,

Returning to Table 2, the first column of Table 2 is the percent of the slope of the terrain, or

(y/x)*100%. The reciprocal of the slope, (x/y), represents cot("). The second column of Table 2

merely represents the slope in degrees. The first row represents very gentle slopes, and the last row

represents steep slopes. The horizontal uncertainty is suggested in column three. The horizontal

uncertainty )x, is defined as ()y)x[cotangent(")], using from Table 1, a )y = 0.03 m, which

represents an upper bound for the vertical uncertainty in a tidal datum based on comparison of

observations for a year’s period of time at a subordinate station. The greatest errors in the

determination of the marine boundary obviously occur for relatively flat terrain, which is

characteristic of broad sections of the Atlantic and Gulf Coasts, and wetlands.

Thus, an engineer or surveyor who needs to determine the location of private property or state

owned tide lands in a coastal zone (Figure 15) with a slope of 0.1% could potentially experience an

error of over 30 meters (at the 1-sigma level, twice this at the 2-sigma or 95% confidence level)

given a year of observations at a subordinate station installed in the local area. Fewer observations

(e.g., one week of data) would make the error larger as illustrated by Tables 1 and 2. 
29

Table 2. Error in position of marine boundary as a function of the slope of the land (12 month series).

% of Slope Degree of Slope (degrees) Error (meters)

0.1 0.05 32.3

0.2 0.1 14.9

0.5 0.3 6.1

1 0.6 3

2 1 1.5

5 3 0.61

10 6 0.3

15 9 0.18

20 11 0.15

30 17 0.09

50 27 0.06

100 45 0.03
30
31

4. WORKED EXAMPLES OF TIDAL DATUMS

4.1 Procedural Steps

The procedural steps required to compute tidal datums at short-term stations are illustrated in

this manual using examples. The first three examples illustrate the method of comparison of

simultaneous observations using monthly means from a control and subordinate station in a region

of similar tidal characteristics to produce equivalent datums at the subordinate station. Even though

these examples use 12 months of data, data sets with more or less months use the same procedure.

The second three examples cover the three types of tide using the tide by tide comparison (TBYT)

procedure. These are computed by comparison of individual simultaneous high and low waters

between the tertiary station and a primary control, or an acceptable secondary control station, in a

region of similar tidal characteristics. The last example is one that shows the use of the direct

comparison of mean values for high water datum determination.

The examples found in this Chapter provide details of the formulas and procedures required to

complete steps 4 through 6 found in Section 3.3. Note that the Standard Method is used on the

West Coast, and the Modified Range Ratio method is used for a Gulf Coast example (see section

3.4). The Modified Range Ratio method was adopted procedurally for both diurnal and semidiurnal

tide types because the ratios formed from numerically small ranges of tide and small inequalities

using the Standard Method become inconsistent from month to month. So in these cases MHHW

and MLLW are computed using DTL and Gt. 

4.2 Comparison of Monthly Means

4.2.1 Modified Range Ratio Method - Semidiurnal Tides

The first example describes computing tidal datums for secondary stations with a semidiurnal

tide. The procedure involves selecting a suitable control station with known 19-year values of the

datums, and reducing the subordinate station data to equivalent 19-year mean values. Along the East

Coast of the United States the tides are predominantly semidiurnal over large distances of the coast,

and many candidate example cases exist. A simple case is offered by the comparison of Fort

Pulaski, GA to Charleston, SC. Their locations are shown in Figure 16. For this example, Fort

Pulaski is being treated as the subordinate station for which datums need to be computed and

Charleston is being treated as the control station.
32

Figure 17a. Time series plots of monthly mean tide level (MTL) for Fort Pulaski, GA and

Charleston, SC.

 Figure 16. NWLON station locations along the South Atlantic Bight.

From the CO-OPS website, http://www.tidesandcurrents.noaa.gov/, verified, historical, and

monthly mean water level data, and accepted datums, may be downloaded. A year of monthly mean

data are shown in Tables 3 and 4. Figures 17a and 17b show plots of a subset of these monthly

means. The data were downloaded from the hyperlink U.S. and Global Coastal Stations, under

Verified/Historical Water Level Data.
33

Figure 17b. Time series plot of the monthly mean range of tide (Mn) for Ft. Pulaski, GA and

Charleston, SC.

Table 3. NOS monthly mean water levels for Fort Pulaski, GA. These data are verified, monthly

mean values. STND refers to station datum, an arbitrary, vertical reference point at a given station.

Data are in Meters above STND and Times are on UTC (GMT)

8670870 FORT PULASKI, SAVANNAH RIVER , GA from 199703 to 199802

Year Mo MHHW MHW DTL MTL MSL MLW MLLW GT MN DHQ DLQ HWI LWI

1997 3 3.326 3.254 2.159 2.156 2.201 1.059 0.991 2.335 2.195 0.072 0.068 0.54 6.94

1997 4 3.422 3.344 2.259 2.248 2.296 1.151 1.095 2.327 2.193 0.078 0.056 0.55 6.97

1997 5 3.412 3.306 2.208 2.188 2.238 1.071 1.004 2.408 2.235 0.106 0.067 0.54 6.94

1997 6 3.508 3.389 2.320 2.291 2.342 1.193 1.131 2.377 2.196 0.119 0.062 0.62 7.01

1997 7 3.467 3.340 2.253 2.225 2.278 1.110 1.040 2.427 2.230 0.127 0.070 0.55 6.98

1997 8 3.460 3.371 2.309 2.295 2.350 1.218 1.159 2.301 2.153 0.089 0.059 0.55 6.98

1997 9 3.495 3.415 2.369 2.353 2.408 1.292 1.243 2.252 2.123 0.080 0.049 0.61 7.08

1997 10 3.426 3.353 2.321 2.305 2.359 1.257 1.216 2.210 2.096 0.073 0.041 0.62 7.06

1997 11 3.337 3.233 2.182 2.158 2.206 1.083 1.026 2.311 2.150 0.104 0.057 0.58 7.02

1997 12 3.336 3.221 2.139 2.118 2.169 1.014 0.942 2.394 2.207 0.115 0.072 0.61 7.03

1998 1 3.365 3.257 2.152 2.135 2.182 1.014 0.939 2.426 2.243 0.108 0.075 0.55 6.95

1998 2 3.506 3.411 2.304 2.300 2.349 1.189 1.101 2.405 2.222 0.095 0.088 0.58 7
34

MTL

MHW MLW = +

2 . (1)

Table 4. NOS monthly mean water levels for Charleston, SC. These data are verified, monthly

mean values. The data fields have the same meaning as in Table3. 

Data are in Meters above STND and Times are on UTC (GMT)

8665530 CHARLESTON, COOPER RIVER ENTRANCE , SC from 199703 to 199802 

Year Mo MHHW MHW DTL MTL MSL MLW MLLW GT MN DHQ DLQ HWI LWI

1997 3 2.555 2.476 1.672 1.662 1.700 0.847 0.789 1.766 1.629 0.079 0.058 0.41 6.65

1997 4 2.648 2.574 1.763 1.751 1.797 0.928 0.877 1.771 1.646 0.074 0.051 0.41 6.66

1997 5 2.609 2.506 1.699 1.673 1.717 0.840 0.789 1.820 1.666 0.103 0.051 0.41 6.67

1997 6 2.725 2.614 1.818 1.792 1.836 0.970 0.910 1.815 1.644 0.111 0.060 0.47 6.77

1997 7 2.673 2.557 1.758 1.730 1.774 0.904 0.843 1.830 1.653 0.116 0.061 0.46 6.70

1997 8 2.683 2.595 1.807 1.789 1.832 0.984 0.932 1.751 1.611 0.088 0.052 0.45 6.66

1997 9 2.732 2.654 1.870 1.851 1.896 1.048 1.008 1.724 1.606 0.078 0.040 0.51 6.72

1997 10 2.672 2.600 1.824 1.807 1.851 1.014 0.975 1.697 1.586 0.072 0.039 0.48 6.71

1997 11 2.579 2.478 1.692 1.669 1.709 0.860 0.805 1.774 1.618 0.101 0.055 0.46 6.68

1997 12 2.572 2.459 1.661 1.635 1.673 0.811 0.749 1.823 1.648 0.113 0.062 0.45 6.71

1998 1 2.588 2.471 1.665 1.634 1.678 0.797 0.742 1.846 1.674 0.117 0.055 0.39 6.63

1998 2 2.724 2.637 1.821 1.811 1.854 0.984 0.917 1.807 1.653 0.087 0.067 0.39 6.7

Table 5. NOS accepted tidal datums for Charleston. Data are accepted. These are the 19-year

values of the tidal datums at Charleston, SC, and have been computed by first reduction.

Data are in Meters above STND, time intervals are on UTC (GMT).

8665530 Charleston, Cooper River Entrance, SC, USA 

Station MHHW MHW DTL MTL MSL MLW MLLW GT MN 

8665530 2.527 2.423 1.643 1.622 1.658 0.817 0.759 1.768 1.606 

 DHQ DLQ HWI LWI

0.104 0.058 0.35 6.57

Definition, MTL. Mean Tide Level, MTL, the average of MHW and MLW, is defined by

the equation

 

This value is already calculated and is presented in the above Tables 3-5. MTL is the starting

point for the calculation of the equivalent 19 year datums at Fort Pulaski, based on one year of

simultaneous observations at Fort Pulaski and Charleston. The subordinate station, Fort Pulaski, is

designated as station A, and the control station, Charleston is designated station B. MTL at Fort

Pulaski must be reduced to a 19-year equivalent value by using the information from Charleston. In

Table 6, column A contains the MTL values from Table 3, and the values in column B are from Table

4. The column designated A-B to the right of column B, is the difference between the two columns.

The entry, SUMS, is the sum of the differences A-B. The SUMS are divided by the TOTAL
35

MTL MTL

N (MTL (I) MTL (I)) ( )

I

N

CORRECTED FOR A ACCEPTED FOR B A B = +





 



 −

=

∑ 1

2

1

MONTHS to produce the MEANS, or the mean difference. The entry, ACCEPTED FOR B, is the

19-year accepted mean value of MTL for Charleston from Table 5. The corrected value of MTL at

Fort Pulaski, CORRECTED FOR A, is the sum of MEANS + ACCEPTED FOR B. CORRECTED

FOR A is the result of this procedure: MTL at Fort Pulaski has been corrected by comparison with

an appropriate control. The form of the correction is

where I goes from 1 to N, and I represents an individual month and N is the total number of months.

In this case, N = 12. If the number of months change, N changes as appropriate. The entry MEANS

in Table 6 is equivalent to the second term on the right hand side of Equation 2, ACCEPTED FOR

B is the first term of the right hand side, and CORRECTED FOR A is the final result on the left hand

side.

Table 6. Worksheet for calculating the corrected MTL for the subordinate station. The purpose

of this worksheet is to solve Equation (2). 

(A) SUBORDINATE STATION 8670870 FORT PULASKI, SAVANNAH RIVER

(B) STANDARD STATION 8665530 CHARLESTON, COOPER RIVER ENTRANCE

Mon Year M T L

 A B A - B

 METER METER METER 

Mar 1997 2.156 1.662 0.494

Apr 1997 2.248 1.751 0.497 

May 1997 2.188 1.673 0.515 

Jun 1997 2.291 1.792 0.499

Jul 1997 2.225 1.730 0.495

Aug 1997 2.295 1.789 0.506

Sep 1997 2.353 1.851 0.502

Oct 1997 2.305 1.807 0.498

Nov 1997 2.158 1.669 0.489

Dec 1997 2.118 1.635 0.483

Jan 1998 2.135 1.634 0.501

Feb 1998 2.300 1.811 0.489

SUMS 5.968

TOTAL MONTHS 12.000

MEANS 0.497 

ACCEPTED FOR B 1.622

CORRECTED FOR A 2.119
36

DTL DTL

N (DTL (I) DTL (I))

I

N

 ( ). CORRECTED FOR A ACCEPTED FOR B A B = +





 



 −

=

∑ 1

1

4

DTL

MHHW MLLW = +

2 . (3)

Definition, DTL. Diurnal Tide Level, DTL, is defined as 

DTL is also provided in Tables 3-5 using the above Equation 3 from the MHHW and MLLW

data in those tables, and is entered into Table 7 below. In Table 7, Column A is DTL from the

values in Table 3. Column B is DTL from values in Table 4. The third column in Table 7 is the

difference field A-B. The difference field is summed and stored in SUMS, the mean is computed

and stored in MEANS. The ACCEPTED FOR B entry for DTL is obtained from Table 5. The

corrected DTL value at Fort Pulaski is the sum of MEANS + ACCEPTED FOR B, and is stored in

CORRECTED FOR A entry of Table 7. The mathematical form of the correction is given by

Equation 4, 

Table 7. Worksheet for calculating the corrected DTL for a subordinate station. This

worksheet solves Equation 4.

Mon Year DTL

 A B A - B 

 METER METER METER 

Mar 1997 2.159 1.672 0.487

Apr 1997 2.259 1.763 0.496

May 1997 2.208 1.699 0.509 

Jun 1997 2.320 1.818 0.502

Jul 1997 2.253 1.758 0.495 

Aug 1997 2.309 1.807 0.502

Sep 1997 2.369 1.870 0.499 

Oct 1997 2.321 1.824 0.497 

Nov 1997 2.182 1.692 0.490

Dec 1997 2.139 1.661 0.478

Jan 1998 2.152 1.665 0.487

Feb 1998 2.304 1.821 0.483

SUMS 5.925 

TOTAL MONTHS 12.000

MEANS 0.494

ACCEPTED FOR B 1.643

CORRECTED FOR A 2.137
37

Mn Mn

N

Mn I

Mn I CORRECTED FOR A ACCEPTED FOR B

A

I B

N

 = × (6). 



 



 =

∑ 1

1

( )

( )

Definition, MN. The Mean Range, Mn, is defined by 

Mn = MHW − MLW (5).

Mn is given in Tables 3-5 and was calculated by applying the above equation. The results are

organized in Table 8 below. Column A is the Mn from Table 3 and Column B is the Mn acquired

from Table 4. Column A/B is the ratio of the values in Column A divided by those in Column B.

The sum of the ratios is stored in SUMS. The mean ratio is stored in MEANS. The accepted value

of Mn from Charleston is obtained from Table 5. The corrected value for Mn at Fort Pulaski is

determined by multiplying ACCEPTED FOR B by MEANS, and is shown in CORRECTED FOR

A in Table 8. The mathematical form of the correction is represented by Equation 6,

Table 8. Worksheet for correcting the Mn at a subordinate station. This worksheet solves

Equation 6. 

Mon Year Mn

 A B A / B

 METER METER RATIO

Mar 1997 2.195 1.629 1.347

Apr 1997 2.193 1.646 1.332

May 1997 2.235 1.666 1.342

Jun 1997 2.196 1.644 1.336

Jul 1997 2.230 1.653 1.349

Aug 1997 2.153 1.611 1.336

Sep 1997 2.123 1.606 1.322

Oct 1997 2.096 1.586 1.322

Nov 1997 2.150 1.618 1.329

Dec 1997 2.207 1.648 1.339

Jan 1998 2.243 1.674 1.340

Feb 1998 2.222 1.653 1.344

SUMS 16.038

TOTAL MONTHS 12.000

MEANS 1.337

ACCEPTED FOR B 1.606

CORRECTED FOR A 2.146

Definition, Gt. The Great Diurnal Range (Gt) is defined as 

Gt = MHHW − MLLW (7)
38

Gt Gt

N

Gt I

Gt I CORRECTED FOR A ACCEPTED FOR B

A

I B

N

 = × (8). 



 



 =

∑ 1

1

( )

( )

In Table 9 below the values of Gt are obtained from Table 3 and Table 4. The Column A/B is

the ratio of the values in Column A divided by the values in Column B. The A/B column is summed

and the mean ratio determined. The accepted Gt at Charleston is obtained from Table 5. The

CORRECTED for A is the product of MEANS times ACCEPTED for B. The mathematical form

of the corrected Gt is given by Equation 8,

Table 9. Worksheet for calculating the corrected Gt at a subordinate station. The purpose of

this worksheet is to solve Equation 8.

Mon Year G t 

 A B A / B 

 METER METER RATIO

Mar 1997 2.335 1.766 1.322

Apr 1997 2.327 1.771 1.314

May 1997 2.408 1.820 1.323 

Jun 1997 2.377 1.815 1.310 

Jul 1997 2.427 1.830 1.326 

Aug 1997 2.301 1.751 1.314

Sep 1997 2.252 1.724 1.306 

Oct 1997 2.210 1.697 1.302 

Nov 1997 2.311 1.774 1.303 

Dec 1997 2.394 1.823 1.313 

Jan 1998 2.426 1.846 1.314 

Feb 1998 2.405 1.807 1.331 

SUMS 15.778

TOTAL MONTHS 12.000 

MEANS 1.315 

ACCEPTED FOR B 1.768 

CORRECTED FOR A 2.325

Once the corrected values of MTLCORRECTED FOR A, DTLCORRECTED FOR A, Mn CORRECTED FOR A, and

Gt CORRECTED FOR A, are determined by the above process the following Equations 9-12 may be used

to determine the tidal datums at the subordinate station. 

MLWcorrected for A = MTLcorrected for A - 1⁄2 Mncorrected for A (9)

MHWcorrected for A = MLWA + Mncorrected for A (10)

MLLWcorrected for A = DTLcorrected for A - 1⁄2 Gtcorrected for A (11)

MHHWcorrected for A = MLLWA + Gtcorrected for A (12)
39

MLW MTL Mn

MHW MLW Mn

MLLW DTL Gt

MHHW MLLW Gt

= − × = − =

= + =+ =

= −×= − =

= + = + =

05 2119 05 2146 1046

1046 2146 3192

05 2137 05 2 325 0 974

0 974 2 325 3299

. . .( . ) .

.. .

. . .( . ) .

. . ..

 

 

In each of the four equations above, the numerical values of MTLCORRECTED FOR A,

DTLCORRECTED FOR A, MnCORRECTED FOR A, and GTCORRECTED FOR A, are entered into Equations 9-12 from

the appropriate CORRECTED FOR A entry from Tables 6-9 above. Applying Equations 9-12 for

Fort Pulaski, the tidal datums are

Appendix 1. Part A, contains a composite example spreadsheet of the calculations described

above. For completeness, values for the diurnal inequalities may also be computed from the values

obtained in equations 9-12. 

Definition, DHQ. Diurnal High Water Inequality, is defined as

DHQ = MHHW - MHW. (13)

Definition, DLQ. Diurnal Low Water Inequality, is defined as

DLQ = MLW - MLLW. (14)

Applying these equations to the values computed above, the diurnal inequalities at Ft. Pulaski

are

DHQ = MHHW - MHW = 3.299 - 3.192 = 0.107

DLQ = MLW - MLLW = 1.046 - 0.974 = 0.072.
40

 Figure 18. A map showing tide stations in the Gulf of Mexico.

4.2.2 Modified Range Ratio Method - Diurnal Tides

 Along much the Gulf Coast of the United States the tides are predominantly diurnal. However,

over short distances of the coast, the tides may in one location be diurnal, and in another location

nearby, semidiurnal or mixed. Few easy candidate example cases exist. However, a simple case is

offered by the comparison of Panama City Beach, FL to Pensacola, FL, as shown in Figure 18. 

From the CO-OPS website, http://www.tidesandcurrents.noaa.gov/, verified historical monthly

mean water level data, and accepted datums, may be downloaded. A year’s worth of monthly values

for Panama City Beach and Pensacola are shown in Tables 10 and 11. A subset of these monthly

mean values are plotted in the graphs found in Figures 19a and 19b. These data were downloaded

from the hyperlink U.S. and Global Coastal Stations, under Verified/Historical Water Level Data.

The equations used to compute the tidal datums at the subordinate station, Panama City Beach, have

been completely developed in the previous section on semidiurnal tides, and are applied in a

straightforward manner to diurnal tides.
41

Figure 19a. Time series plots of monthly mean diurnal tide level (DTL) for Pensacola, FL and

Panama City, FL.

Figure 19b. Time series plots of monthly mean diurnal range of tide (Gt) for Panama City

beach, FL and Pensacola, FL.
42

Table 10. NOS monthly mean water levels for Panama City Beach, FL.

Data are in Meters above STND and Times are on UTC (GMT)

8729210 PANAMA CITY BEACH , FL from 199603 to 1997020

Year Mo MHHW MHW DTL MTL MSL MLW MLLW GT MN DHQ DLQ 

1996 3 8.502 8.479 8.321 8.327 8.298 8.175 8.140 0.362 0.304 0.023 0.035

1996 4 8.536 8.492 8.363 8.357 8.354 8.222 8.190 0.346 0.270 0.044 0.032

1996 5 8.603 8.571 8.409 8.399 8.392 8.227 8.215 0.388 0.344 0.032 0.012

1996 6 8.621 8.610 8.411 8.416 8.399 8.223 8.201 0.420 0.387 0.011 0.022

1996 7 8.679 8.660 8.476 8.478 8.462 8.296 8.273 0.406 0.364 0.019 0.023

1996 8 8.708 8.669 8.540 8.529 8.526 8.388 8.372 0.336 0.281 0.039 0.016

1996 9 8.721 8.699 8.563 8.573 8.558 8.448 8.404 0.317 0.251 0.022 0.044

1996 10 8.780 8.741 8.608 8.600 8.605 8.459 8.436 0.344 0.282 0.039 0.023

1996 11 8.706 8.704 8.523 8.531 8.514 8.357 8.340 0.366 0.347 0.002 0.017

1996 12 8.607 8.606 8.402 8.412 8.388 8.219 8.197 0.410 0.387 0.001 0.022

1997 1 8.540 8.521 8.351 8.342 8.347 8.163 8.162 0.378 0.358 0.019 0.001

1997 2 8.531 8.480 8.350 8.329 8.345 8.178 8.169 0.362 0.302 0.051 0.009

Table 11. NOS monthly mean water levels for Pensacola, FL.

Data are in Meters above STND and Times are on UTC (GMT)

8729840 PENSACOLA, PENSACOLA BAY , FL from 199603 to 199702 

Year Mo MHHW MHW DTL MTL MSL MLW MLLW GT MN DHQ DLQ 

1996 3 2.826 2.826 2.662 2.666 2.655 2.506 2.498 0.328 0.320 0.000 0.008

1996 4 2.868 2.855 2.728 2.722 2.724 2.589 2.587 0.281 0.266 0.013 0.002

1996 5 2.908 2.908 2.730 2.730 2.731 2.552 2.552 0.356 0.356 0.000 0.000

1996 6 2.914 2.911 2.721 2.724 2.718 2.536 2.527 0.387 0.375 0.003 0.009

1996 7 2.944 2.940 2.757 2.759 2.750 2.578 2.571 0.373 0.362 0.004 0.007

1996 8 2.989 2.973 2.833 2.833 2.825 2.694 2.678 0.311 0.279 0.016 0.016

1996 9 3.029 3.015 2.875 2.883 2.872 2.751 2.721 0.308 0.264 0.014 0.030

1996 10 3.100 3.068 2.934 2.923 2.934 2.779 2.767 0.333 0.289 0.032 0.012

1996 11 3.022 3.022 2.843 2.843 2.840 2.663 2.663 0.359 0.359 0.000 0.000

1996 12 2.909 2.903 2.715 2.712 2.704 2.520 2.520 0.389 0.383 0.006 0.000

1997 1 2.865 2.865 2.686 2.686 2.684 2.507 2.507 0.358 0.358 0.000 0.000

1997 2 2.858 2.839 2.696 2.691 2.696 2.542 2.533 0.325 0.297 0.019 0.01
43

Table 12. NOS accepted tidal datums for Pensacola. These are the 19-year, accepted, official

values of the tidal datums at Pensacola, FL, and have been computed by first reduction. Note that

HWI -- Greenwich Mean High Water Interval in Hours, and LWI -- Greenwich Mean Low Water

Interval in Hours, are not calculated for this station because the type of tide at Pensacola is diurnal.

The data are in meters above station datum.

Data are in Meters above STND

8729840 Pensacola, Pensacola Bay, FL, USA

Station MHHW MHW DTL MTL MSL MLW MLLW GT MN 

8729840 2.911 2.902 2.719 2.719 2.713 2.536 2.524 0.387 0.366

 DHQ DLQ HWI LWI

 0.009 0.012 N/A N/A

Corrected MTL for Panama City Beach. Mean Tide Level, MTL, the average of MHW and

MLW, is defined by Equation 1, and is presented in the above Tables 10-12. MTL is the starting

point for the calculation of the equivalent 19 year datums at Panama City Beach, based on one year

of monthly means between there and Pensacola. The subordinate station, Panama City Beach, is

designated as station A, and the control station, Pensacola is designated station B. MTL at Panama

City Beach must be reduced to a 19-year equivalent value by using the information from Pensacola.

In Table 13 below the values in column A are the MTL values from Table 10, those values in

column B are from Table 11. The column to the right of B, is the difference, A-B. The procedure

is identical to the semidiurnal case and shall be reiterated here. The entry, SUMS, is the sum of the

differences A-B. The SUMS are divided by the TOTAL MONTHS to produce the MEANS, or the

mean differences. The entry, ACCEPTED FOR B, is the accepted value of MTL for Pensacola

from Table 12. The corrected value of MTL at Panama City Beach, CORRECTED FOR A, is the

sum of MEANS + ACCEPTED FOR B. As a result of this procedure MTL observed at Panama

City Beach for the period from March 1996 to February 1997 has been corrected to an equivalent

19-year mean value of MTL by comparison with an appropriate control. 

Table 13. Worksheet for calculating the corrected MTL for the subordinate station. The

purpose of this worksheet is to solve Equation (2). 

(A) SUBORDINATE STATION 8729210 PANAMA CITY BEACH

(B) STANDARD STATION 8729840 PENSACOLA, PENSACOLA BAY

Mon Year M T L

 A B A - B

 METER METER METER 

Mar 1996 8.327 2.666 5.661

Apr 1996 8.357 2.722 5.635 

May 1996 8.399 2.730 5.669 

Jun 1996 8.416 2.724 5.692

Jul 1996 8.478 2.759 5.719

Aug 1996 8.529 2.833 5.696

Sep 1996 8.573 2.883 5.690

Oct 1996 8.600 2.923 5.677

Nov 1996 8.531 2.843 5.688

Dec 1996 8.412 2.712 5.700

Jan 1997 8.342 2.686 5.656

Feb 1997 8.329 2.691 5.638
44

SUMS 68.121

TOTAL MONTHS 12.000

MEANS 5.677 

ACCEPTED FOR B 2.719

CORRECTED FOR A 8.396

Corrected DTL for Panama City Beach. DTL is obtained from Tables 10-12 and entered into

Table 14 below. Column A is DTL from Table 10, Column B is DTL from Table 11, and the third

column is the difference field A-B. The differences are summed and stored in SUMS, the mean is

computed and stored in MEANS. The ACCEPTED FOR B is calculated by applying the above

equation to the MHHW and MLLW entries in Table 12. The corrected DTL value at Panama City

Beach is the sum of MEANS + ACCEPTED FOR B, and is stored in CORRECTED FOR A. 

Table 14. Worksheet for calculating the corrected DTL for Panama City Beach. The purpose

of this Table is to solve Equation 4 for Panama City Beach.

Mon Year DTL

 A B A - B 

 METER METER METER 

Mar 1996 8.321 2.662 5.659

Apr 1996 8.363 2.728 5.635

May 1996 8.409 2.730 5.679 

Jun 1996 8.411 2.721 5.690

Jul 1996 8.476 2.757 5.719 

Aug 1996 8.540 2.833 5.707

Sep 1996 8.563 2.875 5.688 

Oct 1996 8.608 2.934 5.674 

Nov 1996 8.523 2.843 5.680

Dec 1996 8.402 2.715 5.687

Jan 1997 8.351 2.686 5.665

Feb 1997 8.350 2.696 5.654

SUMS 68.137 

TOTAL MONTHS 12.000

MEANS 5.678

ACCEPTED FOR B 2.719

CORRECTED FOR A 8.397

Corrected MN for Panama City Beach. The Mean Range, Mn, is defined by Equation 5 and

is obtained from Tables 10-12. The results are organized in Table 15 below.

Table 15. Worksheet for correcting the Mn at a subordinate station. 

Mon Year M N

 A B A / B

 METER METER RATIO

Mar 1996 0.304 0.320 0.950

Apr 1996 0.270 0.266 1.015

May 1996 0.344 0.356 0.966

Jun 1996 0.387 0.375 1.032

 Jul 1996 0.364 0.362 1.006

Aug 1996 0.281 0.279 1.007

Sep 1996 0.251 0.264 0.951
45

Oct 1996 0.282 0.289 0.976

Nov 1996 0.347 0.359 0.967

Dec 1996 0.387 0.383 1.010

Jan 1997 0.358 0.358 1.000

Feb 1997 0.302 0.297 1.017

SUMS 11.897

TOTAL MONTHS 12.000

MEANS 0.991

ACCEPTED FOR B 0.366

CORRECTED FOR A 0.363

Column A is the difference, MHW - MLW, from Table 10. Column B expresses the same

information but is acquired from Table 11. Column A/B is the ratio of Column A divided by

Column B. The sum of the ratios is stored in SUMS. The mean ratio is stored in MEANS. The

accepted value of MN from Pensacola is obtained by applying Equation 5 to Table 12. The

corrected value for MN at Panama City Beach is determined by multiplying ACCEPTED FOR B

by MEANS, and is stored in CORRECTED FOR A. The mathematical form of the correction is

represented by Equation 6.

Corrected Gt at Panama City Beach. The Great Diurnal Range (GT) is defined by Eq. 7. In

Table 16 below the values of Gt are obtained from the data in Tables 10-12 and entering GtA into

Column A for the subordinate and GtB into Column B for the control. The Column A/B is the ratio

of the values in column A divided by the values in column B. The A/B column is summed and the

mean ratio determined. The accepted GT at Pensacola is obtained by applying Equation 7 to the

accepted data at Pensacola from Table 12. The CORRECTED at A is the product of MEANS times

ACCEPTED FOR B. The mathematical form of the corrected GT is given by Equation 8.

Table 16. Worksheet for calculating the corrected Gt at a subordinate station.

Mon Year G T 

 A B A / B 

 METER METER RATIO

Mar 1996 0.362 0.328 1.104

Apr 1996 0.346 0.281 1.231

May 1996 0.388 0.356 1.090 

Jun 1996 0.420 0.387 1.085 

Jul 1996 0.406 0.373 1.088 

Aug 1996 0.336 0.311 1.080

Sep 1996 0.317 0.308 1.029 

Oct 1996 0.344 0.333 1.033 

Nov 1996 0.366 0.359 1.019 

Dec 1996 0.410 0.389 1.054 

Jan 1997 0.378 0.358 1.056 

Feb 1997 0.362 0.325 1.114 

SUMS 12.983

TOTAL MONTHS 12.000 

MEANS 1.082

ACCEPTED FOR B 0.387

CORRECTED FOR A 0.419

Once the corrected values of MTLCORRECTED FOR A, DTLCORRECTED FOR A, MnCORRECTED FOR A, and
46

MLW MTL Mn

MHW MLW Mn

MLLW DTL Gt

MHHW MLLW Gt

= − × = − =

= + =+ =

= −×= − =

= + = + =

05 8 396 05 0 363 8 214

8 214 0 363 8577

05 8 397 05 0 419 8188

8188 0 419 8 607

. . .(. ) .

.. .

. . .(. ) .

. . ..

 

 

GTCORRECTED FOR A, are determined by the above process the Equations 9-12 may be used to determine

the other tidal datums at the subordinate station. For Panama City Beach, the tidal datums are

4.2.3 Standard Method - Mixed Tides

 The mixed tides case requires methodology called the standard method (section 3.4). Like the

previous cases, the problem involves selecting a suitable control station with known 19-year mean

values of the datums, and reducing the subordinate station data to their equivalent 19-year mean.

Along the West Coast of the United States the tides are predominantly mixed over large distances

of the coast, and many candidate example cases exist. A simple case is offered by the comparison

of Alameda, CA (subordinate station) to San Francisco, CA (control station), both located in central

California within San Francisco Bay (see Figure 20.) From the CO-OPS website,

http://www.tidesandcurrents.noaa.gov/, verified and historical monthly mean water level data, and

accepted datums, may be downloaded. A year’s worth of monthly values for Alameda and San

Francisco are shown in Tables 17 - 19 and monthly mean values for MTL and Mn are plotted in

Figures 21a and 21b. This data is downloaded from the hyperlink U.S. and Global Coastal Stations,

under Verified/Historical Water Level Data. A full example computational spreadsheet for the full

comparison is found in Appendix 1.
47

Figure 20. Map of California showing the locations of NWLON stations. 
48

Simultaneous Comparison of Mean Tide Level (MTL) for San 

Francisco and Alameda, CA

1.000

1.500

2.000

2.500

3.000

3.500

3-97 4-97 5-97 6-97 7-97 8-97 9-97 10-97 11-97 12-97 1-98 2-98

time (months)

mean tide level (meters)

MTL - San Francisco

MTL - Alameda

1960-78 MTL - San Francisco

Figure 21a. Time series plots of monthly mean tide level (MTL) for Alameda,

CA and San Francisco, CA.

Simultaneous Comparison of Mean Range of Tide (Mn) for 

San Francisco and Alameda, CA

0.800

0.900

1.000

1.100

1.200

1.300

1.400

1.500

1.600

1.700

1.800

3-97 4-97 5-97 6-97 7-97 8-97 9-97 10-97 11-97 12-97 1-98 2-98

time (months)

mena range of tide (meters)

Mn - Alameda

Mn- San francisco

1960-78 Mn - San Francisco

Figure 21b. Time series plots of monthly mean range of tide (Mn) for Alameda,

CA and San Francisco, CA.
49

Table 17. NOS monthly mean water levels for Alameda, CA. The data are verified.

Data are in Meters above STND and Times are on UTC (GMT)

9414750 ALAMEDA, SAN FRANCISCO BAY , CA from 199703 to 199802

Year Mo MHHW MHW DTL MTL MSL MLW MLLW GT MN DHQ DLQ HWI LWI

1997 3 2.941 2.809 2.000 2.058 2.029 1.306 1.059 1.882 1.503 0.132 0.247 7.95 1.42

1997 4 2.865 2.749 1.889 1.978 1.951 1.206 0.913 1.952 1.543 0.116 0.293 8.01 1.50

1997 5 2.998 2.832 1.989 2.066 2.039 1.299 0.979 2.019 1.533 0.166 0.320 8.05 1.53

1997 6 3.118 2.900 2.067 2.131 2.105 1.363 1.016 2.102 1.537 0.218 0.347 8.02 1.50

1997 7 3.136 2.911 2.090 2.147 2.123 1.384 1.044 2.092 1.527 0.225 0.340 8.05 1.54

1997 8 3.137 2.953 2.151 2.196 2.175 1.439 1.164 1.973 1.514 0.184 0.275 8.05 1.54

1997 9 3.061 2.945 2.155 2.199 2.177 1.453 1.250 1.811 1.492 0.116 0.203 8.03 1.52

1997 10 3.072 2.967 2.159 2.216 2.191 1.464 1.247 1.825 1.503 0.105 0.217 8.01 1.54

1997 11 3.226 3.060 2.221 2.298 2.265 1.535 1.216 2.010 1.525 0.166 0.319 8.02 1.53

1997 12 3.263 3.039 2.196 2.267 2.231 1.495 1.129 2.134 1.544 0.224 0.366 8.01 1.54

1998 1 3.285 3.070 2.247 2.303 2.265 1.537 1.209 2.076 1.533 0.215 0.328 7.93 1.46

1998 2 3.269 3.082 2.391 2.414 2.375 1.746 1.512 1.757 1.336 0.187 0.234 7.98 1.55

Table 18. NOS monthly mean water levels for San Francisco, CA. The data are verified. 

Data are in Meters above STND and Times are on UTC (GMT)

9414290 SAN FRANCISCO, SAN FRANCISCO BAY , CA from 199703 to 199802

Year Mo MHHW MHW DTL MTL MSL MLW MLLW GT MN DHQ DLQ HWI LWI

1997 3 3.481 3.359 2.670 2.731 2.715 2.102 1.860 1.621 1.257 0.122 0.242 7.52 0.70

1997 4 3.411 3.294 2.561 2.648 2.634 2.002 1.710 1.701 1.292 0.117 0.292 7.53 0.84

1997 5 3.550 3.388 2.661 2.741 2.727 2.094 1.772 1.778 1.294 0.162 0.322 7.53 0.89

1997 6 3.675 3.462 2.740 2.809 2.795 2.156 1.804 1.871 1.306 0.213 0.352 7.50 0.87

1997 7 3.699 3.480 2.768 2.830 2.816 2.180 1.836 1.863 1.300 0.219 0.344 7.53 0.89

1997 8 3.708 3.532 2.833 2.882 2.872 2.231 1.957 1.751 1.301 0.176 0.274 7.50 0.89

1997 9 3.627 3.514 2.833 2.878 2.870 2.242 2.039 1.588 1.272 0.113 0.203 7.50 0.86

1997 10 3.645 3.544 2.845 2.905 2.894 2.265 2.046 1.599 1.279 0.101 0.219 7.48 0.89

1997 11 3.805 3.643 2.914 2.995 2.978 2.347 2.023 1.782 1.296 0.162 0.324 7.53 0.90

1997 12 3.842 3.625 2.895 2.970 2.948 2.314 1.948 1.894 1.311 0.217 0.366 7.53 0.88

1998 1 3.857 3.644 2.939 3.001 2.981 2.357 2.020 1.837 1.287 0.213 0.337 7.51 0.79

1998 2 3.836 3.653 3.063 3.103 3.082 2.552 2.291 1.545 1.101 0.183 0.261 7.58 0.82
50

Table 19. NOS tidal datums for San Francisco. Data are accepted. These are the 19-year values

of the tidal datums at San Francisco, CA, and have been computed by first reduction. The data are

in meters above station datum.

Data are in meters above STND, time intervals in hours (UTC)

9414290 San Francisco, San Francisco Bay, CA, USA 

Station MHHW MHW DTL MTL MSL MLW MLLW GT MN 

9414290 3.536 3.353 2.646 2.728 2.713 2.103 1.759 1.777 1.250

 DHQ DLQ HWI LWI

0.183 0.344 7.56 0.83

Corrected MTL at Alameda. MTL, Equation 1, is the starting point for the calculation of the

equivalent 19 year datums at Alameda, based on one year of observations. The subordinate station,

Alameda, is designated as station A, and the control station, San Francisco is designated station B.

MTL at Alameda must be reduced to a 19-year equivalent value by using the information from San

Francisco. In Table 20 below the values in column A are the MTL values from Table 17, those in

column B are from Table 18. The column to the right of B, is the difference, A-B. The entry,

SUMS, is the sum of the differences A-B. The SUMS are divided by the TOTAL MONTHS to

produce the MEANS, or the mean difference. The entry, ACCEPTED FOR B, is the accepted value

of MTL for San Francisco from Table 19. From Equation 2, the corrected value of MTL at

Alameda, CORRECTED FOR A, is the sum of MEANS + ACCEPTED FOR B. 

Table 20. Worksheet for calculating the corrected MTL for Alameda. The purpose of this

worksheet is to solve Equation (2). 

(A) SUBORDINATE STATION 9414750 ALAMEDA, SAN FRANCISCO BAY 

(B) STANDARD STATION 9414290 SAN FRANCISCO, SAN FRANCISCO BAY

Mon Year M T L

 A B A - B

 METER METER METER 

Mar 1997 2.058 2.731 -0.673

Apr 1997 1.978 2.648 -0.670

May 1997 2.066 2.741 -0.675 

Jun 1997 2.131 2.809 -0.678

Jul 1997 2.147 2.830 -0.683

Aug 1997 2.196 2.882 -0.686

Sep 1997 2.199 2.878 -0.679

Oct 1997 2.216 2.905 -0.689

Nov 1997 2.298 2.995 -0.697

Dec 1997 2.267 2.970 -0.703

Jan 1998 2.303 3.001 -0.698

Feb 1998 2.414 3.103 -0.689

SUMS -8.220 

TOTAL MONTHS 12.000

MEANS -0.685 

ACCEPTED FOR B 2.728 

CORRECTED FOR A 2.043
51

Corrected MN at Alameda. The Mean Range, Mn, is defined by Equation 5 and calculated

values are found in Tables 17-19. The results are organized in Table 21 below. 

Table 21. Worksheet for correcting the Mn at a subordinate station. The purpose of this

worksheet is to solve Equation 6.

Mon Year M N

 A B A / B

 METER METER RATIO

Mar 1997 1.503 1.257 1.196

Apr 1997 1.543 1.292 1.194

May 1997 1.533 1.294 1.185

Jun 1997 1.537 1.306 1.177

Jul 1997 1.527 1.300 1.175

Aug 1997 1.514 1.301 1.164

Sep 1997 1.492 1.272 1.173

Oct 1997 1.503 1.279 1.175

Nov 1997 1.525 1.296 1.177

Dec 1997 1.544 1.311 1.178

Jan 1998 1.533 1.287 1.191

Feb 1998 1.336 1.101 1.213

SUMS 14.198

TOTAL MONTHS 12.000

MEANS 1.183

ACCEPTED FOR B 1.250

CORRECTED FOR A 1.479

Column A is the difference, MHW - MLW, from Table 17. Column B expresses the same

information but acquired from Table 18. Column A/B is the ratio of Column A divided by Column

B. The sum of the ratios is stored in SUMS. The mean ratio is stored in MEANS. The accepted

value of Mn from San Francisco is obtained by applying Equation 5 to Table 19. The corrected

value for Mn at Alameda is determined by multiplying ACCEPTED FOR B by MEANS, and is

stored in CORRECTED FOR A. 

Definition, DHQ. The Diurnal High Water Inequality (DHQ) was previously defined by

Equation 13 as 

DHQ MHHW MHW = − 

In Table 22 below, the monthly values of DHQ were previously calculated by NOS by applying

Equation 13 to the values obtained from Tables 17 and 18. DHQA was then entered into Column

A for the subordinate station and DHQB into Column B for the control station. Column A/B is the

ratio of A/B. The A/B column is summed and the mean ratio determined. The accepted DHQ at San

Francisco is obtained by applying Eq. 13 to the accepted data at San Francisco from Table 19. The

CORRECTED FOR A is the product of MEANS times ACCEPTED FOR B. 
52

DLQ MLW MLLW = −

DHQ DHQ N

DHQ I

DHQ I CORRECTED FOR A FOR B

A

I B

N

 ACCEPTED = × (15). 



 



 =

∑ 1

1

( )

( )

The mathematical form of the corrected DHQ is given by Eq. 15,

Table 22. Worksheet for calculating the corrected DHQ at a subordinate station. This

worksheet solves Eq. 15.

Mon Year DHQ 

 A B A / B 

 METER METER RATIO

Mar 1997 0.132 0.122 1.082

Apr 1997 0.116 0.117 0.991

May 1997 0.166 0.162 1.025 

Jun 1997 0.218 0.213 1.023 

Jul 1997 0.225 0.219 1.027 

Aug 1997 0.184 0.176 1.045

Sep 1997 0.116 0.113 1.027 

Oct 1997 0.105 0.101 1.040 

Nov 1997 0.166 0.162 1.025 

Dec 1997 0.224 0.217 1.032 

Jan 1998 0.215 0.213 1.009 

Feb 1998 0.187 0.183 1.022 

SUMS 12.348

TOTAL MONTHS 12.000 

MEANS 1.029 

ACCEPTED FOR B 0.183 

CORRECTED FOR A 0.188 

Definition, DLQ. The Diurnal Low Water Inequality (DLQ) was previously defined by

Equation 14 as 

 

 

In Table 23 below, DLQ values were previously calculated by NOS by applying Equation 13 to

values from Tables 17 and 18. DLQA and DLQB values were entered into Column A for the

subordinate and into Column B for the control. Column A/B is the ratio of A/B. The A/B column

is summed and the mean ratio determined. The accepted DLQ at San Francisco is obtained by

applying Eq. 15 to the accepted data at San Francisco from Table 19. The CORRECTED at A is the

product of the MEANS times the ACCEPTED FOR B. 
53

MLW MTL Mn

MHW MLW Mn

MLLW MLW DLQ

MHHW MHW DHQ

A CORRECTED FOR A CORRECTED FOR A

A A CORRECTED FOR A

A A CORRECTED FOR A

A A CORRECTED FOR A

= −

= +

= −

= +

 

 

 

 

 (17)

 (18)

 (19)

 

1 2

20

/

. ()

DLQ DLQ N

DLQ I

DLQ I CORRECTED FOR A FOR B

A

I B

N

 ACCEPTED = × (16). 



 



 =

∑ 1

1

( )

( )

The mathematical form of the corrected DLQ is given by Equation 16,

Table 23. Worksheet for calculating the corrected DLQ at a subordinate station. This

worksheet solves Equation 16.

Mon Year DLQ 

 A B A / B 

 METER METER RATIO

Mar 1997 0.247 0.242 1.021

Apr 1997 0.293 0.292 1.003

May 1997 0.320 0.322 0.994

Jun 1997 0.347 0.352 0.986 

Jul 1997 0.340 0.344 0.988 

Aug 1997 0.275 0.274 1.004

Sep 1997 0.203 0.203 1.000

Oct 1997 0.217 0.219 0.991 

Nov 1997 0.319 0.324 0.985 

Dec 1997 0.366 0.366 1.000

Jan 1998 0.328 0.337 0.973 

Feb 1998 0.234 0.261 0.897 

SUMS 11.842

TOTAL MONTHS 12.000 

MEANS 0.987

ACCEPTED FOR B 0.344

CORRECTED FOR A 0.339

Once the corrected values of MTLCORRECTED FOR A, MnCORRECTED FOR A, DHQCORRECTED FOR A, and

DLQCORRECTED FOR A, are determined by the above process. The following Eqs. 17-20 may be used

to determine the tidal datums at the subordinate station.
54

MLW MTL Mn

MHW MLW Mn

MLLW MLW DLQ

MHHW MHW DHQ

A CORRECTED FOR A CORRECTED FOR A

A A CORRECTED FOR A

A A CORRECTED FOR A

A A CORRECTED FOR A

= − = − =

= + =+ =

= − =− =

= + =+ =

 

 

 

 

 

 

 

1 2 2 043 0 5 1479 1304

1304 1479 2 783

1304 0 339 0 965

2 783 0188 2 971

/ . .(. ) .

.. .

.. .

. . ..

In each of the four equations above, the values of MTLCORRECTED FOR A, MnCORRECTED FOR A,

DHQCORRECTED FOR A, and DLQCORRECTED FOR A, are entered from the appropriate CORRECTED FOR

A entry from Tables 20-23 above. MLWA, as derived from Eq. 17, is entered into Eqs. 18 and 19.

MHWA , as derived from Eq. 18, is entered into Eq. 20. For Alameda, the tidal datums are

Appendix 1. Part B, contains a composite example spreadsheet of the calculations described above.

4.3 Comparison of Simultaneous High and Low Waters (Tide by Tide Analysis)

The method of Comparison of Simultaneous High and Low Waters or, Tide by Tide Analysis

(TBYT), is used to compute datums form short-term stations. For instance, if the simultaneous

water level measurements at the subordinate and control stations exist for less than a month or if the

data series starts and ends in the middle of two consecutive months. The next three examples

illustrate this comparison method for three different types of tide. For each type, equivalent 19-year

datums are computed, in addition to lunitidal intervals. The example data may be downloaded from

the CO-OPS website. All data are in meters, referenced to station datum, and are on the Greenwich

time meridian. Referencing data series to Greenwich meridian is now the standard NOS practice.

When calculating time differences in the phase of tide between two stations, it is necessary to factor

in any differences in the time zone that the observations were recorded on. The present NOS

practice of operating all tide stations on GMT. Five days of simultaneous observations are used in

this example for illustration only. The estimated vertical error in the resultant datums for this series

length, extrapolating the curve from Figure 14, is expected to be greater than 0.04 m. At least one

month of data ensures closure on the significant monthly variations of the tide and provides a proper

number of values to get meaningful statistics from the comparisons. Using less than one month of

data is proper as long as the user realizes the higher uncertainties in the resulting datums. 

The examples follow the procedural steps outlined in section 4.1.2. First, the data from the

control and subordinate stations are obtained and put into tabular form for comparisons.

Summations of the tides and their simultaneous differences are computed on the same tabular form.

Then, based on the summations and differences, a specific sequence of equations is used to form the

differences and ratios needed to compute the equivalent NDTE datums. Appendix 2 contains a

complete example TYBT spreadsheet for Alameda and San Francisco.

4.3.1 Modified Range Ratio - Semidiurnal Tides

The following exercise illustrates the Modified Range Ratio Method to compute equivalent 19-

year datums at Fort Pulaski, GA, using Charleston, SC as a control station, for a short period of

observations. 
55

Figure 22. The menu driven interface to the CO-OPS database through the internet. 

The interface is completed for the Fort Pulaski data of the tide by tide comparison

between Charleston and Fort Pulaski.

Data for this example may be downloaded from http://www.tidesandcurrents.noaa.gov/. Click

on the hyperlink for “Water Level Observations”, under the heading “Verified / Historical Water

Level Data” , click on the hyperlink for U.S. and Global Coastal Stations . The next page is a

database menu form, illustrated in Figure 22, filled out as appropriate for Fort Pulaski, GA.
56

Note that in the scroll box under “Time Interval”, “W3 - High/Lows”, is selected. By clicking

on the button captioned, “View Data/View Report”, the database returns the information presented

in Table 24, and Figure 23. The additional hourly water level data presented in Figure. 28, is

obtained by selecting “W2- Hourly Heights”, instead of “W3- High/Lows”. The data for Charleston,

SC are shown in Table 25, and Figure 24. The composite graph of Fort Pulaski and Charleston is

shown in Figure 25.

Table 24. NOS High and Low Waters for 8670870 Fort Pulaski, Savannah River, GA,

USA from 19960304 to 19960308. Data are verified. Data are in meters above Station

Datum (STND). Times are on UTC (GMT). The fields in the table have the following

meanings. Station--Unique seven character identifier for the station. Date Time--Date and

time the data were collected by the DCP. WL_Value--Water level height. Type –

Designation of Water level height. LL = lower low water, L = low water, H = High water,

HH Higher High water. Infer--A flag when that when set to 1 indicates that the water level

value has been inferred. Limit--A flag that when set to 1 indicates that either the maximum

or minimum expected water level height limit was exceeded.

Station Date Time WL_Value Type Infer Limit

8670870 1996/03/04 00:00 3.066 H 0 0

8670870 1996/03/04 06:06 0.864 LL 0 0

8670870 1996/03/04 12:18 3.178 H 0 0

8670870 1996/03/04 18:36 0.956 L 0 0

8670870 1996/03/05 00:42 3.280 HH 0 0

8670870 1996/03/05 06:48 0.908 LL 0 0

8670870 1996/03/05 12:42 3.227 H 0 0

8670870 1996/03/05 19:18 0.973 L 0 0

8670870 1996/03/06 01:30 3.244 HH 0 0

8670870 1996/03/06 07:36 0.658 LL 0 0

8670870 1996/03/06 13:36 3.153 HH 0 0

8670870 1996/03/06 19:48 0.678 L 0 0

8670870 1996/03/07 01:54 3.061 H 0 0

8670870 1996/03/07 08:18 0.641 L 0 0

8670870 1996/03/07 14:06 3.077 HH 0 0

8670870 1996/03/07 20:24 0.610 LL 0 0

8670870 1996/03/08 02:36 3.027 H 0 0

8670870 1996/03/08 09:06 0.165 LL 0 0

8670870 1996/03/08 14:54 2.681 H 0 0

8670870 1996/03/08 21:06 0.429 L 0 0
57

Table 25. NOS High and Low Waters for 8665530 Charleston, Cooper River Entrance,

SC, USA from 19960304 to 19960308. Data are verified. Data are in meters above Station

Datum (STND). Times are on UTC (GMT). The fields in the table have the same meanings

as Table 24.

Station Date Time WL_Value Type Infer Limit

8665530 1996/03/04 06:00 0.633 LL 0 0

8665530 1996/03/04 12:12 2.411 H 0 0

8665530 1996/03/04 18:30 0.792 L 0 0

8665530 1996/03/05 00:36 2.493 HH 0 0

8665530 1996/03/05 06:36 0.730 LL 0 0

8665530 1996/03/05 12:54 2.401 H 0 0

8665530 1996/03/05 19:06 0.768 L 0 0

8665530 1996/03/06 01:06 2.407 HH 0 0

8665530 1996/03/06 07:06 0.531 L 0 0

8665530 1996/03/06 13:18 2.321 HH 0 0

8665530 1996/03/06 19:36 0.502 LL 0 0

8665530 1996/03/07 01:48 2.227 H 0 0

8665530 1996/03/07 08:00 0.423 LL 0 0

8665530 1996/03/07 14:00 2.241 H 0 0

8665530 1996/03/07 20:12 0.484 L 0 0

8665530 1996/03/08 02:24 2.252 HH 0 0

8665530 1996/03/08 08:54 0.114 LL 0 0

8665530 1996/03/08 14:54 1.848 H 0 0

8665530 1996/03/08 20:42 0.291 L 0 0

Figure 23. The data returned from the CO-OPS database for Fort Pulaski, GA. Referring

to Figure 22, the selection of “W3-Highs/Lows”, returns the data denoted by the diamond

symbols. Their values are also labeled. The rest of the curve is obtained by selecting

“W2-Hourly Heights”, and resubmitting the data request.
58

Figure 24. The data returned from the CO-OPS database for Charleston, SC. 

Type or Designation Conversion at Subordinate Station

In using Tide by Tide comparisons, it is important that the each of high and low water tide

designations are matched between subordinate and control stations (e.g. higher high waters at the

subordinate station should be matched with corresponding higher high waters at the control station).

 Individual tides are designated during tabulation at each station independent of control/subordinate

relationship. If the control station is well chosen for the subordinate based upon geographic

proximity and type of tide, the corresponding tide designations will agree more often than not.

However, the user must be prepared for occurrences in which tide designations differ between the

subordinate and control, and must develop a consistent rule that can be applied to resolve

differences. This occurs especially in areas with small inequalities and significant effects of

meteorology on water levels where a higher high water at the subordinate may actually be

designated a lower high water at the control through the independent pairing process at each station.

In these cases, the fundamental rule is that the tide designation at the subordinate station is changed

to agree with the corresponding tide designation at the control station, if necessary. If the tide

designation at the subordinate is already the same as that at the control, a conversion is not done.

It is important to note that the numerical value of the water level at the subordinate station is not

altered; changes are reflected only in how this numerical value is categorized. 
59

Figure 25. A graph of Charleston (upper curve) and Fort Pulaski (lower curve) data. 

The high water and low water values required for the tidal datum calculations are printed

to illustrate the tide by tide comparison. Refer to this graph when considering the

organization of Table 26 below.

For example, in Figure 25, the first pair of water level values used in the analysis consist of

3.178 m at the subordinate and 2.411 m at the control. From Table 25, the value of 2.411 m is

categorized as an “H”. From Table 24 at the subordinate, 3.178 m is also categorized as an “H”.

These values match. In Table 26, the control station is labeled “B”, and the subordinate is labeled

“A”. The categories of {HH, H, L, LL} in Tables 24 and 25, are designated as HHW, LHW, HLW,

and LLW in Table 26; translating into words as “higher high water, lower high water, higher low

water, and lower low water”, respectively. In Table 26, the value of 2.411 m is entered as the first

value under LHWB , and 3.178 m is the first value under LHWA . They are placed under the LHW

columns because of the “H” designation at the control. Their difference, , 3178 2 411 0 767 ... − =

is the first entry under the column )LHW. The next tide pair are low waters with heights of 0.792

m at the control and 0.956 m at the subordinate. From Table 25, the 0.792 m at the control is an

“L”, which corresponds to the HLWB column in Table 26. The value of 0.956 m at the subordinate

is also an “L”, placing it as the first entry under the HLWA column. Their difference,

0 956 0 792 0164 .. . − = , is the first entry under )HLW. The next pair, 2.493 and 3.280, are

entered under HHW because of the HH at the control. Their difference, 0.787 m, is the first entry

under )HHW. The next pair, 0.730 and 0.980, are the first entries under the LLW columns,

because of the “LL” designation at the control. Their difference, , is the 0 908 0 730 0178 ... − =

first entry under )LLW. This completes the first row of Table 26, and serves as a model for filling
60

HHW m A = = 12 704

4 3176 21 . . () 

in the rest of Table 26. So far, type conversion at the subordinate has not occurred. However,

continuing through this table example type conversion at the subordinate occurs a total of six times

as explained below.

The first instance of type conversion occurs in a HLW pairing. The pair 0.531 and 0.658 in

Figure 25, requires type conversion at the subordinate. In Table 25 for the control, 0.531 is

designated as an “L”. However, in Table 24 at the subordinate, the 0.658 value is designated as an

“LL”. To rectify this mismatch at the subordinate, the “LL” is changed to an “L”. Thus, these two

values remain paired in Table 26, by entering the 0.658 under the HLWA column. Type conversion

at the subordinate occurs for the next three low water pairings. In the 0.502 and 0.678 pair, the

0.678 is changed from an “L” to an “LL”. In the next low water pair, the value of 0.641 is changed

from an “L” to an “LL”. In the next pair, the value of 0.610, is changed from an “LL” to an “L”.

These changes are all reflected in Table 26. Among the high water conversions, in the pair

consisting of 2.241 and 3.077, the value 3.077 is changed from an “HH” to an “H”. In the next pair,

2.252 and 3.027, the 3.027 is changed from an “H” to an “HH”. 

Tidal Datums

Table 26 is the fundamental starting point for computing equivalent 19-year tidal datums at the

subordinate from individual high and low waters from a short series of data. As in the prior chapter,

the sum of the values in a particular column is stored in the “SUMS” row. The number of entries

is in the “NUMBER” row, and the average is stored in the “MEANS” position. For example, the

sum of the higher high waters at A, is 12.704 m. The average is denoted by 

Table 26. Comparison of simultaneous observations for 96-3-4 TO 96-3-8. 

Datums computed for the 1960-1978 tidal epoch. (A) designates the subordinate station,

8670870, FORT PULASKI, SAVANNAH RIVER, GA. (B) designates the control

8665530, CHARLESTON, COOPER RIVER ENTRANCE, SC.

 (A) STATION (B) STATION (A) - (B)

HEIGHT OF HEIGHT OF HEIGHT DIFFERENCE

HHWA LHWA LLWA HLWA HHWB LHWB LLWB HLWB )HHW )LHW )LLW )HLW 

3.280 3.178 0.908 0.956 2.493 2.411 0.730 0.792 0.787 0.767 0.178 0.164

3.244 3.227 0.678 0.973 2.407 2.401 0.502 0.768 0.837 0.826 0.176 0.205

3.153 3.061 0.641 0.658 2.321 2.227 0.423 0.531 0.832 0.834 0.218 0.127

3.027 3.077 0.165 0.610 2.252 2.241 0.114 0.484 0.775 0.836 0.051 0.126

 2.681 0.429 1.848 0.291 0.833 0.138

SUMS: 12.704 15.224 2.392 3.626 9.473 11.128 1.769 2.866 3.231 4.096 0.623 0.760

NUMBER: 4 5 4 5 4 5 4 5 4 5 4 5

 ———— ———— ——— ———— ———— ———— ———— ———— ————— ————— ————— ————— 

MEANS: HHWA LHWA LLWA HLWA HHWB LHWB LLWB HLWB )HHW )LHW )LLW )HLW

3.176 3.045 0.598 0.725 2.368 2.226 0.442 0.573 0.808 0.819 0.156 0.152
61

GT HHW LLW A AA = − =− = 3176 0 598 2 578 . . . (26)

( ) ( )

( ) ( )

DHQ HHW LHW

DLQ HLW LLW

A AA

A AA

=× − =× − =

=× − =× − =

05 05 3176 3045 0 066

05 05 0 725 0598 0

. .. . .

. .. . .

 (22)

064 (23)

Mn HW LW A AA = −=− = 3110 0 662 2 448 . . . (27)

HWA = + = 3176 3045

2

3110 . . . (24)

LWA = + = 0 725 0598

2 0 662 . . . (25)

Similar averages are computed as shown in Table 26 for , , for the LHWA LLWA HLWA

subordinate station and for , , , fo HHWB LHWB LLWB HLWB r the control station. In addition,

sums and averages of the four height differences are calculated as shown for

∆ ∆∆∆ HHW LHW HLW LLW ,,, . 

The first set of calculations establishes the diurnal inequalities, the ranges of tide and the mean

high water and mean low water elevations at the subordinate station for the comparison period. 

Directly from Table 26, the Diurnal High Water Inequality at A (DHQA), and the Diurnal Low Water

Inequality at A (DLQA) are given by Equations 22 and 23,

The mean high water at A, denoted by, , is HWA computed from the mean of and HHWA

LHWA , or

Likewise, the mean low water at A, denoted by, LW , is computed from the mean of A

HLWA and ,or LLWA

 

 

The Great Diurnal range at A, GtA, is given by

The Mean range at A, MnA, is given by
62

DTL HHW LLW ( ) ( ) A AA =× + =× + = 05 05 3176 0598 1887 . . . . . (28)

MTL HW LW ( ) ( ) A AA =× + =× + = 05 05 3110 0 662 1886 . . . . . (29)

( ) ∆

∆ ∆

HW

HHW LHW . . = . ()+ = + = 2

0808 0819

2

0814 32

( ) ∆

∆ ∆

LW

HLW LLW = + = + = 2

0152 0156

2 0154 . . . (33)

The Diurnal Tide Level at A, DTLA, is given by

The Mean Tide Level at A, MTLA, is given by

The second set of calculations establishes the differences and ratios (as appropriate depending upon

method used) in the various tabulated parameters between the subordinate and control station over

the comparison time period. The average height differences for HHW, LHW, HLW and LLW are

read from Table 26 as 0.808, 0.819, 0.152 and 0.156m, respectively.

The diurnal high water inequality difference is then computed by

∆ ∆∆ DHQ HHW LHW =× − × 05 30 . = 0.5 (0.808 - 0.819) = - 0.006 ( ) ( )

and the diurnal low water inequality difference is computed by

∆ ∆∆ DLQ HLW LLW =× − × 05 31 . = 0.5 (0.152 - 0.156) = - 0.002 ( ) ( )

The mean High Water difference, , is computed from the high water differences by ∆HW

The mean Low Water difference, , is given by, ∆ LW
63

∆ ∆∆ DTL HHW LLW =× + =× + = 0 5 0 5 0 808 0156 0 482 . . . . . (35) ( ) ( )

∆ ∆∆ MTL HW LW =× + =× + = 05 05 0814 0154 0 484 . . . . . (36) ( ) ( )

Mn Mn

Mn Mn

DHQ DHQ

DHQ DHQ

DLQ DLQ

DLQ DLQ

RATIO

A

A

RATIO

A

A

RATIO

A

A

= − = =

= − = − − =

= − = − − =

∆

∆

∆

2.449

2.449 - 0.660 (38)

 (39)

 (40)

1369

0 066

0 066 0 006

0 917

0 064

0 064 0 002

0 970

.

.

. (. ) .

.

. (. ) .

Gt

Gt

Gt Gt RATIO

A

A

= − = − = ∆

2 578

2 578 0 652 1339 .. . . (41)

∆∆ ∆ Mn HW LW = −=−= 0814 0154 0 660 . . . (34)

∆∆ ∆ Gt HHW LLW = − =−= 0808 0156 0 652 . . . (37)

The Mean Range difference, )Mn, is given by,

The Diurnal Tide Level difference, )DTL, is given by,

The Mean Tide Level difference, )MTL, is given by,

The Great Diurnal range difference, )Gt, is given by,

The Mean Range Ratio, MnRATIO , the Diurnal High Water Inequality Ratio, DHQRATIO, and the

Diurnal Low Water Inequality Ratio, DLQRATIO, are given by 

The Great Diurnal Ratio, GtRATIO , is given by

Table 5 presents the accepted 19-year tidal datums for Charleston, the control station for this

example. For this example, the accepted values is for MTL, DTL, MN and Gt are required to derive

the accepted datums at the control station using the modified range ratio method (from table 5).
64

Mn Mn Mn CORRECTED ACCEPTED RATIO FOR A FOR B

 (44)

= ×

=×= 1606 1369 2198 .. .

MTL MTL MTL CORRECTED ACCEPTED FOR A FOR B

 (42)

= +

=+ =

∆

1622 0 484 2106 .. .

DTL DTL DTL CORRECTED ACCEPTED FOR A FOR B

 (43)

= +

=+ =

∆

1643 0 482 2125 .. .

Gt Gt Gt CORRECTED ACCEPTED RATIO FOR A FOR B

 (45)

= ×

=×= 1768 1339 2 367 .. .

MLLW DTL Gt

m

A CORRECTED CORRECTED = − ×

= −×

=

 FOR A FOR A

 (46)

05

2125 05 2 367

0 942

.

. ..

.

MHHW MLLW Gt

m

A CORRECTED = +

= +

=

A FOR A

 (47)

0 942 2 367

3309

. .

.

The MTLCORRECTED FOR A is given by,

The DTLCORRECTED FOR A is given by,

The Mn CORRECTED FOR A is given by,

The GtCORRECTED FOR A is given by,

From the above results, the equivalent 19-year datums at Fort Pulaski, GA based on a short term

period of simultaneous observations with a control tide station can be computed. Then equations

following those found in equations 9 through 12 are used.

The Mean Lower Low Water at A, MLLWA, is given by,

The Mean Higher High Water at A, MHHWA, is given by,
65

MLW MTL Mn

m

A CORRECTED CORRECTED = − ×

= −×

=

 FOR A FOR A

 (48)

05

2106 05 2198

1007

.

. ..

.

MHW MLW Mn

m

A CORRECTED = +

= +

=

A FOR A

 (49)

1007 2198

3205

. .

.

The Mean Low Water at A, MLWA, is given by,

The Mean High Water at A, MHWA, is given by,

Lunitidal Intervals

Lunitidal Intervals can provide a measure of the average difference in time of tide between two

locations. When sufficient data are available and monthly mean values are computed for a station,

the monthly Greenwich Mean High Water Interval (HWI) and Greenwich Mean Low Water Interval

(LWI) are among the means routinely computed by NOS for stations with semidiurnal and mixed tide

types. These stations usually have a one-to-one correspondence between the tides (high and low

waters) and the transits of the moon relative to the Greenwich meridian. As stated earlier, the

Greenwich Intervals represent the mean difference in time between the upper and lower passage of

the moon over the Greenwich Meridian and the following high and low tides observed at a station.

The monthly mean HWI and LWI can be used to compute the average time difference in the phase

of tide between different locations by subtracting one from the other. When only a short series of

data is available, as in the present example, the time differences between individual corresponding

tides are computed, and then the means of these differences are computed to provide this information.

In this section we compute the equivalent 19-year Greenwich High Water Interval (HWI) and

Low Water Interval (LWI) at the subordinate station. To compute the tidal datums above, the notion

of time was handled implicitly, simply by organizing the pairs of high or low values sequentially in

Table 26. Here, the time aspect is treated explicitly. Table 27, like Table 26, is based upon the

information presented in Tables 24 and 25. Table 27 includes the water level heights for stations A

and B to show correct matching, but the elevations are not used to compute the HWI and LWI. The

first pair of corresponding high waters have heights of 3.178 m and 2.411 m(Figure 34). These are

the first entries under the HW column of stations A and B, respectively. The time associated with

3.178 is 1996/03/04 12:18 GMT (Table 27), and the time associated with 2.411 is 1996/03/04 12:12

GMT. These times have been converted from hours: minutes format to decimal hours in Table 27.

 The difference, 12.3-12.2 = 0.1, is the first entry under the HW Hours Time Difference column. The

next pair of values are low waters occurring at 18.6 hours and 18.5 hours on March 4, and are entered

under the LW columns of Station A and B, respectively. The difference of 0.1 hours is the first entry

under the LW Time difference column. This is the method by which Table 27 is properly

constructed. Note that Table 27 has a simpler layout than Table 26, because the computation of HWI

and LWI does not distinguish between higher high water and lower high water, or, lower low water

and higher low water.
66

HWI HWI HWI

hours

CORRECTED ACCEPTED FOR A FOR B

 (50)

= +

=+= 0 35 012 0 47 .. .

LWI LWI LWI

hours

CORRECTED ACCEPTED FOR A FOR B

 (51)

= +

=+= 657 0 26 683 ...

The mean high water time difference is 0.12 hours, and the mean low water time difference is

0.26 hours (Table 27). From Table 5, the accepted HWI at Station B is 0.35 hours, and the accepted

LWI interval is 6.57 hours. 

 

Likewise,
67

Table 27. Calculation of the mean time differences between the subordinate and control stations. Themeans computed below are entered into Equations 48 and 49 to estimate the Greenwich lunitidal intervals at thesubordinate station. The times entered below are in GMT.  (A) SUBORDINATE STATION 8670870 FORT PULASKI, SAVANNAH RIVER  (B) STANDARD STATION 8665530 CHARLESTON, COOPER RIVER ENTRANCE  (A) STATION (B) STATION (A) - (B) (A) STATION (B) STATION  TIME OF TIME OF TIME DIFFERENCE HEIGHT OF HEIGHT OF HW LW HW LW HW LW HW LW HW LW  DATE HOURS HOURS DATE HOURS HOURS HOURS HOURS METERS METERS METERS METERS  MAR 4 12.3 4 6.1 MAR 4 12.2 0.1 3.178 2.411  18.6 18.5 0.1 0.956 0.792  5 0.7 5 6.8 5 0.6 6.6 0.1 0.2 3.280 0.908 2.493 0.730  12.7 19.3 12.9 19.1 -0.2 0.2 3.227 0.973 2.401 0.768  6 1.5 6 7.6 6 1.1 7.1 0.4 0.5 3.244 0.658 2.407 0.531  13.6 19.8 13.3 19.6 0.3 0.2 3.153 0.678 2.321 0.502  7 1.9 7 8.3 7 1.8 8.0 0.1 0.3 3.061 0.641 2.227 0.423  14.1 20.4 14.0 20.2 0.1 0.2 3.077 0.610 2.241 0.484  8 2.6 8 9.1 8 2.4 8.9 0.2 0.2 3.027 0.165 2.252 0.114  14.9 21.1 14.9 20.7 0.0 0.4 2.681 0.429 1.848 0.291  SUMS 1.1 2.3  NUMBER 9 9  MEANS 0.12 0.26 
68

4.3.2 Modified Range Ratio - Diurnal Tides

A comparison of Simultaneous High and Low Waters, also referred to as a Tide by Tide

Comparison is used to compute short term datums if the simultaneous water level measurements at

the subordinate and control exist for less than a month. Consider Figure 26, which illustrates the

period of time from March 10 through March 16, 1996 at Panama City Beach and Pensacola, FL. 

The moon was in the tropics during this time period, producing a clean example of simultaneous

diurnal tides at the subordinate and control. The data downloaded from the CO-OPS website are

presented in Tables 28 and 29.

In Figure 26, the values of the high and low waters are plotted to facilitate understanding. The

first step is to track the high and low waters correctly. Generally, this can be done by allowing a day

to elapse to make sure that a peak and trough cycle have been resolved by each time series. Hence,

the analysis begins with the trough of March 11, 1996, and records each high water and low water

pair thereafter. 

In Table 30, Panama City Beach, the subordinate station, is designated A, and Pensacola, the

control station, is B. The left pair of columns represents the height of the high and low waters at

Panama City Beach. Units are meters. The middle pair represent the height of the high and low

waters at Pensacola. The right pair of columns represent the difference between the high waters at

A and B, and the difference between the low waters at A and B. For example the first entry in the

right pair is in the LW column, and is simply the low water at A minus the low water at B, or 7.843 -

2.271 = 5.572. For the HW field, the difference is the high water at A minus the high water at B, or

8.167 - 2.578 = 5.589. Focusing now on the leftmost pair, the sum of the high waters at A is 42.253

from 5 observations, hence the mean HHW ( )is HHW A 8.451. Likewise, the mean lower low water,

( ) is 8.014. The higher high water at B, ( ) LLW A HHWB , is 2.806. The lower low water at B, ( )is LLWB

2.383. The mean difference of the high waters between A and B,( ) is 5.645. The mean ∆HHW

difference of the low waters ( ) is 5.632. ∆LLW

 
69

Table 28. NOS high and low waters for 8729210 Panama City Beach, FL, USA from 

1996-03-10 to 1996-03-16. Data are verified. Data are in meters above STND. Times are

on UTC (GMT).

Station Date Time WL_Value Type Infer Limit

8729210 1996/03/10 04:24 7.869 LL 0 0

8729210 1996/03/10 18:48 8.222 HH 0 0

8729210 1996/03/11 03:06 7.843 LL 0 0

8729210 1996/03/11 17:54 8.168 HH 0 0

8729210 1996/03/12 04:06 7.814 LL 0 0

8729210 1996/03/12 18:12 8.405 HH 0 0

8729210 1996/03/13 05:48 7.960 LL 0 0

8729210 1996/03/13 20:06 8.522 HH 0 0

8729210 1996/03/14 09:30 8.093 LL 0 0

8729210 1996/03/14 21:00 8.544 HH 0 0

8729210 1996/03/15 09:00 8.108 LL 0 0

Table 29. NOS high and low waters for 8729840 Pensacola, Pensacola Bay, FL, USA 

from 1996-03-10 to 1996-03-16. Data are verified. Data are in meters above STND. 

Times are on UTC (GMT). 

Station Date Time WL_Value Type Infer Limit

8729840 1996/03/10 05:24 2.234 LL 0 0

8729840 1996/03/10 20:36 2.637 HH 0 0

8729840 1996/03/11 06:48 2.271 LL 0 0

8729840 1996/03/11 20:54 2.579 HH 0 0

8729840 1996/03/12 06:54 2.228 LL 0 0

8729840 1996/03/12 21:36 2.740 HH 0 0

8729840 1996/03/13 09:00 2.320 LL 0 0

8729840 1996/03/13 23:48 2.862 HH 0 0

8729840 1996/03/14 11:18 2.444 LL 0 0

8729840 1996/03/15 00:54 2.886 HH 0 0

8729840 1996/03/15 11:24 2.454 LL 0 0

8729840 1996/03/16 02:06 2.963 HH 0 0

8729840 1996/03/16 12:06 2.579 LL 0 0

8729840 1996/03/17 03:36 2.978 HH 0 0
70

Figure 26. A comparison of corresponding highs and lows for diurnal tides. Panama City is

the upper curve and Pensacola is the lower curve.

Table 30. Worksheet for computing a tide by tide comparison of high and low waters, diurnal

example. Refer to Figure 26 to see how the high and low waters are matched.

(A) SUBORDINATE STATION 8729210 PANAMA CITY BEACH 

(B) STANDARD STATION 8729840 PENSACOLA, PENSACOLA BAY 

(A) STATION (B) STATION (A) - (B)

HEIGHT OF HEIGHT OF HEIGHT DIFFERENCE

HHW LLW HHW LLW HHW LLW

METERS METERS METERS METERS METERS METERS

 7.843 2.271 5.572

8.167 2.578 5.589

8.405 7.814 2.740 2.228 5.665 5.586

8.522 7.960 2.862 2.320 5.660 5.640 

8.544 8.093 2.886 2.444 5.658 5.649

8.615 8.108 2.963 2.454 5.652 5.654 

8.267 2.977 2.578 5.689 

 ___ ___ ____ HHWA HHWB )HHW 

SUMS 42.253 14.029 28.224

NUMBER 5 5 5

MEANS 8.451 2.806 5.645

 ____ ____ ____

 LLWA LLWB )LLW

SUMS 48.085 14.295 33.790

NUMBER 6 6 6 

MEANS 8.014 2.383 5.632
71

( )( )

Gt MHHW MLLW

DTL MHHW MLLW

ACCEPTEDFORB

ACCEPTEDFORB

= − = − =

=× + =× + =

2 911 2 524 0 387

05 05 2 911 2 524 2 719

.. .

. .. . .

 (57)

 (58)

Gt HHW LLW A AA = − =−= 8 451 8 014 0 437 . . . (52)

DTL HHW LLW ( ) ( ) A AA =× + =× + = 05 05 8 451 8 014 8 232 . . . . . (53)

∆∆ ∆ Gt HHW LLW = − = 5.645- 5.632 = 0.013 (54)

Gt

Gt

Gt Gt RATIO

A

A

= − = − = ∆

0 437

0 437 0 013

1031 .

. . . (55)

( )

( )

∆ ∆∆ DTL HHW LLW A A =× +

× +

05

05 5645 5632

.

 = = 5.638 .. . (56)

Tidal Datums

Recalling the definition of Gt is MHHW - MLLW, the Gt at A for this time period, GtA, is formed

by Equation 52,

Similarly, the DTL at A may be defined as Equation 53,

The average Gt difference, is given by Equation 54, 

The Gt ratio, is defined by Eq. 55,

The DTL difference, is defined by Eq. 56,

Recalling from Table 12, the accepted 19 year values of MHHW and MLLW for Pensacola, are 2.911

and 2.524, respectively, the accepted Gt and DTL values may be determined as,
72

Gt Gt Gt CORRECTED ACCEPTED RATIO FOR A FOR B

 (59)

= ×

= ×= 0 387 1031 0 399 .. .

DTL DTL DTL CORRECTEDFORA ACCEPTEDFORB DIFFERENCE = +

 (60) = += 2 719 5638 8 357 ...

MHHW MLLW Gt A A CORRECTEDFORA = +

 =+ = 8156 0 399 8555 .. . (62)

MLLW DTL Gt A CORRECTEDFORA CORRECTEDFORA = −×

= −× =

05

8 357 05 0 399 8156

.

 . . (. ) . (61)

The Gt, corrected for A, is given by Equation 59,

DTL, corrected for A is given by Equation 60,

Recalling Equations 11 and 12 , MLLW and MHHW may now be determined at Panama City Beach

as,

These are the tidal datums for Panama City Beach, FL based on seven days of observations and

adjusted to equivalent 19-year mean values through comparison of simultaneous observations at

Pensacola, FL, a control station with accepted datums.

Lunitidal Intervals

Values for accepted and monthly HWI and LWI are not computed from tabulations for stations

classified as diurnal, such as the case for Pensacola and panama City, FL. Instead, the values for

Tropic Higher High Water Interval (TcHHWI), and the Tropic Lower Low Water Interval (TcLLWI)

produced by harmonic analyses are used. Thus, this example concludes with the computation of the

equivalent 19-year TcHHWI and TcLLWI at Panama City Beach. The values of TcHHWI and

TcLLWI may be obtained from CO-OPS by request.

In Table 31 below, to use the TcHHWI and TcLLWI at Pensacola, one should technically be

using the mean differences in time between the higher high waters and lower low waters. The tides

at Panama City Beach and Pensacola are dominated by the diurnal tide and during this period when

there are no secondary tides, in effect all high waters and low waters are actually higher high and

lower low waters. Because the characteristics of the tide at the two stations are similar, using

TcHHWI and TcLLWI should serve to supply time information at Panama City Beach. 
73

TcHHWI TcHHWI HWI

hours

CORRECTED ACCEPTED FOR A FOR B

 (63)

≈ +

≈+= 395 300 6 95 .. .

TcLLWI TcLLWI LWI

hours

CORRECTED ACCEPTED FOR A FOR B

 (64)

≈ +

≈ += 1527 300 18 27 .. .

Likewise,

These tropic intervals are used to estimate average time differences in the tide between diurnal

stations, however they are most accurate during tropic tides (maximum north and south declination

of the moon each month ) and typically are inaccurate during times of equatorial tides (when the

moon is on the equator each month).
74

Table 31. Computation of the mean time differences between Panama City Beach and Pensacola, FL. (A) SUBORDINATE STATION 8729210 PANAMA CITY BEACH  (B) STANDARD STATION 8729840 PENSACOLA, PENSACOLA BAY (A) STATION (B) STATION (A) - (B) (A) STATION (B) STATION  TIME OF TIME OF TIME DIFFERENCE HEIGHT OF HEIGHT OF  HW LW HW LW HW LW HW LW HW LW  DATE HOURS HOURS DATE HOURS HOURS HOURS HOURS METERS METERS METERS METERS  MAR 10 18.8 11 3.1 MAR 11 0.8 2.3 7.843 2.271 11 17.9 14.9 3.0 8.167 2.578  12 18.2 12 4.1 12 15.6 0.9 2.6 3.2 8.405 7.814 2.740 2.228  13 20.1 13 5.8 13 17.8 3.0 2.3 2.8 8.522 7.960 2.862 2.320  14 21.0 14 9.5 14 18.9 5.3 2.1 4.2 8.544 8.093 2.886 2.444  16 1.1 15 9.0 15 20.1 5.4 5.0 3.6 8.615 8.108 2.963 2.454  16 8.0 16 21.6 6.1 1.9 8.267 2.977 2.578  SUMS: 15.0 18.0  NUMBER: 5 6  MEANS: 3.00 3.00
75

Table 32. NOS high and low waters for 9414750 Alameda, San Francisco Bay, CA,

USA from 1997-03-01 to 1997-03-07. Data are verified. Data are in meters above

STND. Times are on UTC (GMT). 

Station Date Time WL_Value Type Infer Limit

9414750 1997/03/01 00:12 2.332 H 0 0

9414750 1997/03/01 05:36 1.655 L 0 0

9414750 1997/03/01 12:06 2.748 HH 0 0

9414750 1997/03/01 19:06 1.306 LL 0 0

9414750 1997/03/02 01:42 2.250 H 0 0

9414750 1997/03/02 06:24 1.744 L 0 0

9414750 1997/03/02 13:06 2.815 HH 0 0

9414750 1997/03/02 20:12 1.214 LL 0 0

9414750 1997/03/03 03:06 2.304 H 0 0

9414750 1997/03/03 07:48 1.784 L 0 0

9414750 1997/03/03 14:06 2.768 HH 0 0

9414750 1997/03/03 21:24 1.039 LL 0 0

9414750 1997/03/04 04:12 2.352 H 0 0

9414750 1997/03/04 09:06 1.725 L 0 0

9414750 1997/03/04 15:18 2.814 HH 0 0

9414750 1997/03/04 22:18 0.848 LL 0 0

9414750 1997/03/05 05:12 2.468 H 0 0

9414750 1997/03/05 10:06 1.642 L 0 0

9414750 1997/03/05 16:30 2.940 HH 0 0

9414750 1997/03/05 23:06 0.729 LL 0 0

9414750 1997/03/06 06:06 2.641 H 0 0

9414750 1997/03/06 11:12 1.527 L 0 0

9414750 1997/03/06 17:30 3.077 HH 0 0

9414750 1997/03/07 00:00 0.705 LL 0 0

9414750 1997/03/07 06:48 2.784 H 0 0

9414750 1997/03/07 12:06 1.361 L 0 0

9414750 1997/03/07 18:24 3.139 HH 0 0

4.3.3 Standard Method - Mixed Tides

This section presents a mixed tides tide by tide case between Alameda and San Francisco, CA

(Tables 32 and 33). Figures 27-29 show a short series of observations from Alameda, San Francisco,

and a composite graph of the two. The analysis is different from the purely diurnal case because it

involves the tabulation of the full set of observations of Higher High Water (HHW), Lower High

Water (LHW), Lower Low Water (LLW), and Higher Low Water (HLW) at both stations.

Tabulating the means from Figure 38 requires a more complex table structure (Table 34). A full

example spreadsheet showing all aspects of this comparison is found in Appendix 2.
76

Table 33. NOS high and low waters for 9414290 San Francisco, San Francisco Bay,

CA, USA from 1997-03-01 to 1997-03-07. Data are verified. Data are in meters above

STND. Times are on UTC (GMT).

Station Date Time WL_Value Type Infer Limit

9414290 1997/03/01 04:54 2.458 L 0 0

9414290 1997/03/01 11:48 3.340 HH 0 0

9414290 1997/03/01 18:30 2.111 LL 0 0

9414290 1997/03/02 01:24 2.853 H 0 0

9414290 1997/03/02 05:36 2.532 L 0 0

9414290 1997/03/02 12:36 3.417 HH 0 0

9414290 1997/03/02 19:36 2.008 LL 0 0

9414290 1997/03/03 03:00 2.901 H 0 0

9414290 1997/03/03 07:06 2.568 L 0 0

9414290 1997/03/03 13:42 3.361 HH 0 0

9414290 1997/03/03 21:00 1.857 LL 0 0

9414290 1997/03/04 04:06 2.933 H 0 0

9414290 1997/03/04 08:24 2.511 L 0 0

9414290 1997/03/04 15:00 3.391 HH 0 0

9414290 1997/03/04 21:30 1.669 LL 0 0

9414290 1997/03/05 05:00 3.021 H 0 0

9414290 1997/03/05 09:18 2.438 L 0 0

9414290 1997/03/05 15:54 3.488 HH 0 0

9414290 1997/03/05 22:30 1.535 LL 0 0

9414290 1997/03/06 05:42 3.165 H 0 0

9414290 1997/03/06 10:24 2.335 L 0 0

9414290 1997/03/06 17:00 3.603 HH 0 0

9414290 1997/03/06 23:12 1.498 LL 0 0

9414290 1997/03/07 06:30 3.292 H 0 0

9414290 1997/03/07 11:12 2.172 L 0 0

9414290 1997/03/07 17:48 3.648 HH 0 0
77

Table 34. Worksheet for calculating a tide by tide analysis for a mixed tide case.

COMPARISON OF SIMULTANEOUS OBSERVATIONS FOR 97 3 1 TO 97 3 7 

(A) SUBORDINATE STATION 9414750 ALAMEDA, SAN FRANCISCO BAY 

(B) STANDARD STATION 9414290 SAN FRANCISCO, SAN FRANCISCO BAY

 (A) STATION (B) STATION (A) - (B)

 HEIGHT OF HEIGHT OF HEIGHT DIFFERENCE

HHWA LHWA LLWA HLWA HHWB LHWB LLWB HLWB )HHW )LHW )LLW )HLW 

2.748 2.250 1.306 1.744 3.340 2.853 2.111 2.532 -0.592 -0.603 -0.805 -0.788

2.815 2.304 1.214 1.784 3.417 2.901 2.008 2.568 -0.602 -0.597 -0.794 -0.784

2.768 2.352 1.039 1.725 3.361 2.933 1.857 2.511 -0.593 -0.581 -0.818 -0.786

2.814 2.468 0.848 1.642 3.391 3.021 1.669 2.438 -0.577 -0.553 -0.821 -0.796 

2.940 2.641 0.729 1.527 3.488 3.165 1.535 2.335 -0.548 -0.524 -0.806 -0.808

3.077 2.784 0.705 1.361 3.603 3.292 1.498 2.172 -0.526 -0.508 -0.793 -0.811

3.139 3.648 -0.509 

SUMS: 20.301 14.799 5.841 9.783 24.248 18.165 10.678 14.556 -3.947 -3.366 -4.837 -4.773

NUMBER: 7 6 6 6 7 6 6 6 7 6 6 6

 SSSS SSSS SSSS SSSS SSSS SSSS SSSS SSSS SSSS SSSS SSSS SSSS

MEANS: HHWA LHWA LLWA HLWA HHWB LHWB LLWB HLWB )HHW )LHW )LLW )HLW

2.900 2.467 0.974 1.631 3.464 3.028 1.780 2.426 -0.564 -0.561 -0.806 -0.796

Tidal Datums

Table 34 is structured as follows. There are three major groups (A), (B), and (A)-(B). Each of

these are further subdivided into four columns. Under (A) and (B), HHW, LHW, LLW, HLW, both

appear; the differences appear in (A)-(B) as )HHW ,)LHW, )LLW, and )HLW. In addition,

Table 34 is structured in chronological order with respect to Figure 29 (Figures 27 and 28 support

Figure 29 by treating the time series separately), the HHWA being paired with the HHWB. Thus, the

first entry under )HHW, -0.592, is equal to the difference of the HHWA -HHWB = 2.748 - 3.340.

The chief feature of Table 34 is that the pairs (HHWA ,HHWB ) are matched higher high water to

higher high water, lower low water to lower low water. Thus, the difference fields represent

differences between pairs of simultaneous observations, as a visual check with Figure 29

demonstrates. The means at A and B are denoted by the overbar, the mean difference is denoted with

an overbar and is preceded by the ) operator. The means and the mean differences are stored in the

MEANS row. 
78

Figure 27. The water level time series for Alameda with the highs and lows plotted. The data

are referenced to station datum.

Figure 28. The water level time series for San Francisco with the highs and lows plotted. The

data are referenced to station datum.
79

9414290 SAN FRANCISCO, SAN FRANCISCO BAY CA

9414750 ALAMEDA, SAN FRANCISCO BAY CA

2.853 2.901 2.933

3.021

3.165

3.292 3.340 3.417 3.361 3.391

3.488

3.603 3.648

2.458 2.532 2.568 2.511

2.438

2.335

2.172 2.111

2.008

1.857

1.669

1.535 1.498

2.332

2.250 2.304 2.352

2.468

2.641

2.784 2.748 2.815 2.768 2.814

2.940

3.077 3.139

1.655

1.744 1.784 1.725

1.642

1.527

1.361 1.306

1.214

1.039

0.848

0.700 0.729 0.705

1.200

1.700

2.200

2.700

3.200

3.700

3-1-97

0:00

3-2-97

0:00

3-3-97

0:00

3-4-97

0:00

3-5-97

0:00

3-6-97

0:00

3-7-97

0:00

Meters

Figure 29. A composite graph of Alameda (lower curve) and San Francisco (upper

curve). The time series at Alameda is represented by the dashed curve. The data are

referenced to station datum at each site.
80

( ) ( )

( ) ( )

∆ ∆∆

∆ ∆∆

DHQ HHW LHW

DLQ HLW LLW

= × − = × − − − =−

= × − = ×− −− =

05 05 0564 0561 0 002

05 05 0 796 0806 0 005

. . . (. ) .

. . . (. ) .

 (71)

 (72)

( ) ( )

( ) ( )

DHQ HHW LHW

DLQ HLW LLW

A AA

A AA

=× − =× − =

=× − =× − =

05 05 2 900 2 467 0 216

05 05 1631 0 974 0 328

. .. . .

. .. . .

 (65)

 (66)

( ) ( )

( ) ( )

HW HHW LHW

LW HLW LLW

A AA

A AA

=× + =× + =

=× + =× + =

05 05 2 900 2 467 2 684

05 05 1631 0 974 1302

. .. . .

. .. . .

 (67)

 (68)

( ) ( )

Mn HW LW

MTL HW LW

A AA

A AA

= −= −=

=× + =× + =

2 684 1302 1382

05 05 2 684 1302 1993

.. .

. .. . .

 (69) 

 (70)

( ) ( )

∆∆ ∆

∆ ∆∆

Mn HW LW

MTL HW LW

= − =− − − =

= × + = × − + − =−

0562 0801 0 239

05 05 0562 0801 0 682

. (. ) .

. . . (. ) .

 (75)

 (76)

( ) ( )

( ) ( )

∆ ∆∆

∆ ∆∆

HW HHW LHW

LW HLW LLW

= × + = × − + − =−

= × + = × − + − =−

05 05 0564 0561 0562

05 05 0 796 0806 0801

. . . (. ) .

. . . (. ) .

 (73)

 (74)

Directly from Table 34, the Diurnal High Water Inequality at A (DHQA), and the Diurnal Low

Water Inequality at A (DLQA) is given by Equations 65 and 66,

The Mean High Water Height at A, , and the Mean Low Water Height At A, , is HWA LWA

calculated in Equations 67 and 68, 

From Equations 26 and 28, the Mean Range at A, (MnA), and Mean Tide Level at A (MTLA) are

shown in Equations 69 and 70,

Returning to Table 34 to utilize the mean differences, the difference in DHQ and DLQ, denoted

by ) DHQ and ) DLQ, respectively, are calculated as,

The Mean High Water Difference, , and M ∆HW ean Low Water Difference, , are given ∆ LW

by

The Mean Range Difference, , and the Mean Tide Level Difference, , are given by ∆ Mn ∆ MTL
81

Mn

Mn

Mn Mn

DHQ DHQ

DHQ DHQ

DLQ DLQ

DLQ DLQ

RATIO

A

A

RATIO

A

A

RATIO

A

A

= − = − =

= − = − − =

= − = − =

∆

∆

∆

1382

1382 0 239 1209

0 216

0 216 0 002 0 991

0 328

0 328 0 005

1015

.

. . .

.

. (. ) .

.

. . .

 (77)

 (78)

 (79)

MTL MTL MTL

Mn Mn Mn

DHQ DHQ DHQ

DLQ DLQ DLQ

CORRECTEDFORA ACCEPTEDFORB

CORRECTEDFORA ACCEPTEDFORB RATIO

CORRECTEDFORA ACCEPTEDFORB RATIO

CORRECTEDFORA ACCEPTEDFORB RATIO

= + = +− =

= × =×=

= × =×=

= × = ×=

∆ (80)

 (81)

 (82)

 (83)

2 728 0 682 2 046

1250 1209 1511

0183 0 991 0181

0 344 1015 0 349

. (. ) .

.. .

.. .

.. .

MLW MTL Mn

MHW MLW Mn

MLLW MLW DLQ

MHHW MHW DHQ

A CORRECTEDFORA CORRECTEDFORA

A A CORRECTEDFORA

A A CORRECTEDFORA

A A CORRECTEDFORA

= − =− =

= + =+=

= − =− =

= + =+=

05 2 046 05 1511 1290

1290 1511 2 801

1290 0 349 0 941

2 801 0181 2 982

. . .(. ) .

.. .

.. .

...

 (84)

 (85)

 (86)

 (87)

HWI HWI HWI

hours

CORRECTED ACCEPTED FOR A FOR B

 (88)

= +

=+= 7 56 0 35 7 91 .. .

LWI LWI LWI

hours

CORRECTED ACCEPTED FOR A FOR B

 (89)

= +

=+= 083 0 71 154 ...

Three quantities designated as MnRATIO, DHQRATIO, and DLQRATIO, may now be calculated as

Four quantities designated as MTLCORRECTED FOR A, MnCORRECTED FOR A, DHQCORRECTED FOR A, and

DLQCORRECTED FOR A, may now be calculated as

Employing the Standard Method the tidal datums at Alameda are,

These are the tidal datums for Alameda, CA based on seven days of observations and adjusted

to equivalent 19-year mean values through comparison of simultaneous observations at San

Francisco, CA, a control station with known accepted datums. 

Lunitidal Intervals 

The accepted 19-year values at San Francisco (Table 19) includes HWI and LWI. Using the

mean differences from Table 35, the equivalent HWI and LWI at Alameda are determined to be 7.91

and 1.54 hours, respectively. 

Likewise,

 
82

Table 35. Computation of the mean time differences between Alameda and San Francisco, CA.(A) SUBORDINATE STATION 9414750 ALAMEDA, SAN FRANCISCO BAY(B) STANDARD STATION 9414290 SAN FRANCISCO, SAN FRANCISCO BAY (A) STATION (B) STATION (A) - (B) (A) STATION (B) STATION  TIME OF TIME OF TIME DIFFERENCE HEIGHT OF HEIGHT OF  HW LW HW LW HW LW HW LW HW LW  DATE HOURS HOURS DATE HOURS HOURS HOURS HOURS METERS METERS METERS METERS MAR 1 12.1 1 5.6 MAR 1 11.8 0.3 2.748 1.655 3.340  19.1 18.5 0.6 1.306 2.111  2 1.7 2 6.4 2 1.4 5.6 0.3 0.8 2.250 1.744 2.853 2.532  13.1 20.2 12.6 19.6 0.5 0.6 2.815 1.214 3.417 2.008  3 3.1 3 7.8 3 3.0 7.1 0.1 0.7 2.304 1.784 2.901 2.568  14.1 21.4 13.7 21.0 0.4 0.4 2.768 1.039 3.361 1.857  4 4.2 4 9.1 4 4.1 8.4 0.1 0.7 2.352 1.725 2.933 2.511  15.3 22.3 15.0 21.5 0.3 0.8 2.814 0.848 3.391 1.669  5 5.2 5 10.1 5 5.0 9.3 0.2 0.8 2.468 1.642 3.021 2.438  16.5 23.1 15.9 22.5 0.6 0.6 2.940 0.729 3.488 1.535  6 6.1 6 11.2 6 5.7 10.4 0.4 0.8 2.641 1.527 3.165 2.335  17.5 7 0.0 17.0 23.2 0.5 0.8 3.077 0.705 3.603 1.498  7 6.8 12.1 7 6.5 11.2 0.3 0.9 2.784 1.361 3.292 2.172  18.4 17.8 0.6 3.139 3.648  SUMS: 4.6 8.5  NUMBER: 13 12  MEANS: 0.35 0.71
83

MHW MHW

N (MHW (I) MHW (I)) ( )

I

N

CORRECTED FOR A ACCEPTED FOR B A B = +





 



 −

=

∑ 1

90

1

MHHW MHHW

N (MHHW (I) MHHW (I)) ( )

I

N

CORRECTED FOR A ACCEPTED FOR B A B = +





 



 −

=

∑ 1

91

1

Comparison of Low-water Cut-off Station with Control Sation Requiring Use 

of Direct Method

-1.600

-1.400

-1.200

-1.000

-0.800

-0.600

-0.400

-0.200

0.000

0.200

03-25-00 00:00 03-26-00 00:00 03-27-00 00:00 03-28-00 00:00 03-29-00 00:00 03-30-00 00:00 03-31-00 00:00 04-01-00 00:00

Time (UTC)

Elevation Relative to MHW (meters)

Richmond Hamilton AFB

Figure 30. Comparison of data for a typical 7-day period showing low waters cutoff at the subordinate station.

4.4 Direct Method

The Direct Method is usually used only when a full range of tidal values are not available. For

example, direct MHW can be computed for situations when low waters are not recorded, such as in

the upper reaches of a marsh. Since MTL, DTL, and Mn and Gt cannot be determined if low waters

are cut-off, equivalent NTDE values for MHW and MHHW datums are determined directly by

comparison of high tides with an appropriate control using the available part of the tidal cycle. The

equation for computation of MHW from monthly means or using a TBYT would be:

and the equation for MHHW would be:

where N would represent either monthly means or the number of individual tides depending on the

type of comparison. Figure 30 shows the nature of the tide curve of a station requiring the Direct

Method for datum determination.

The following two tables show the direct comparisons for 3-month time period used to obtain MHW
89

and MHHW datums for the station pair shown in Figure 30.

Table 36. Worksheet for computing MHW at subordinate station using the Direct Method.

(A) SUBORDINATE STATION 9415126 HAMILTON AFB, SAN FRANCISCO BAY 

(B) STANDARD STATION 9414863 RICHMOND, SAN FRANCISCO BAY

Mon Year M H W

 A B A - B

 METER METER METER 

Mar 2000 0.893 5.121 -4.228

Apr 2000 0.873 5.113 -4.240

May 2000 0.889 5.127 -4.238 

SUMS -12.706 

TOTAL MONTHS 3.000

MEANS -4.235 

ACCEPTED FOR B 5.128 

CORRECTED FOR A 0.893

Table 37. Worksheet for computing MHHW at subordinate station using the Direct

Method

(A) SUBORDINATE STATION 9415126 HAMILTON AFB, SAN FRANCISCO BAY 

(B) STANDARD STATION 9414863 RICHMOND, SAN FRANCISCO BAY

Mon Year M H H W

 A B A - B

 METER METER METER 

Mar 2000 1.013 5.242 -4.229

Apr 2000 0.964 5.208 -4.244

 May 2000 1.040 5.291 -4.251

SUMS -12.724 

TOTAL MONTHS 3.000

MEANS -4.241 

ACCEPTED FOR B 5.310 

CORRECTED FOR A 1.069
90

5.0 SUMMARY

The techniques used in this handbook are the NOS methodology for the computation of tidal

datums. These techniques must be used to perform work according to NOS standards and

specifications. To perform tidal datum computations, the tides at the subordinate and control must

have similar tidal characteristics and similar ranges. The selection of the proper control station is

more difficult for diurnal tides than for semidiurnal or mixed, because diurnal tides frequently change

their character both temporally at a site, and spatially over short distances. Lastly, all measurements

contain some error or uncertainty in their values. Vertical errors translate into horizontal errors when

surveyed to land. Horizontal errors are exacerbated by low slopes. In general, the accuracy of tidal

datums increases as the amount of data used in the computations increases.

5.1 SUMMARY OF TIDAL DATUMS

Regardless of type of tide or datum computation methodology, the following tidal datums and

tidal parameters are typically desired:

Datums:

Mean Higher High Water (MHHW) 

Mean High Water (MHW) 

Mean Tide Level (MTL) 

Diurnal Tide Level (DTL)

Mean Sea Level (MSL)

Mean Low Water (MLW)

Mean Lower Low Water (MLLW)

Ranges of Tide and Inequalities

Mean Range of Tide (Mn)

Great Diurnal Range of Tide (Gt)

Diurnal High Water Inequality (DHQ)

Diurnal Low Water Inequality (DLQ)

Lunitidal Intervals (not computed for Diurnal Tides)

High Water Interval (HWI)

Low Water Interval (LWI)

5.2 SUMMARY OF PROCEDURES FOR COMPUTATION OF TIDAL DATUMS

A vertical datum is called a tidal datum when it is defined by a certain phase of the tide. Tidal

datums are local datums and should not be extended into areas which have differing hydrographic

characteristics without substantiating measurements. In order that they may be recovered when

needed, such datums are referenced to fixed points known as bench marks.
91

Basic Procedures:

1. Make Observations - Tidal datums are computed from continuous observations of the water

level over specified lengths of time. Observations are made at specific locations called tide stations.

Each tide station consists of a water level gauge or sensor(s), a data collection platform or data logger

and data transmission system, and a set of tidal bench marks established in the vicinity of the tide

station. NOS collects water level data at 6-minute intervals.

2. Tabulate the Tide - Once water level observations are quality controlled and any small gaps

filled, the data are processed by tabulating the high and low tides and hourly heights for each day.

Tidal parameters from these daily tabulations of the tide are then reduced to mean values, typically

on a calendar month basis for longer period records or over a few days or weeks for shorter-term

records.

3. Compute Tidal Datums - First reduction tidal datums are determined directly by meaning

values of the tidal parameters over a 19-year NDTE. Equivalent NDTE tidal datums are computed

from tide stations operating for shorter time periods through comparison of simultaneous data

between the short-term station and a long term station.

4. Compute Bench Mark Elevations - Once the tidal datums are computed from the tabulations,

the elevations are transferred to the bench marks established on the land through the elevation

differences established by differential leveling between the tide gauge sensor “zero” and the bench

marks during the station operation. The bench mark elevations and descriptions are disseminated by

NOS through a published bench mark sheet for each station. Connections between tidal datum

elevations and geodetic elevations are obtained after leveling between tidal bench marks and geodetic

network bench marks. Traditionally, this has been accomplished using differential leveling, however

GPS surveying techniques can also be used (NGS, 1997).

The locations of tide stations are organized into a hierarchy:

Control tide stations are generally those which have been operated for 19 or more years, are

expected to continuously operate in the future, and are used to obtain a continuous record of the water

levels in a locality. Control tide stations are sited to provide datum control for national applications,

and located in as many places as needed for datum control. 

Secondary water level stations are those which are operated for less than 19 years but more than

1 year, and have a planned finite lifetime. Secondary stations provide control in bays and estuaries

where localized tidal effects are not realized at the nearest control station. Observations at a

secondary station are not usually sufficient for a precise independent determination of tidal datums,

but when reduced by comparison with simultaneous observations at a suitable control tide station

very satisfactory results may be obtained. 

Tertiary water level stations are those which are operated for more than a month but less than 1

year. Short-term water level measurement stations (secondary and tertiary) may have their data

reduced to equivalent 19-year tidal datums through mathematical simultaneous comparison with a

nearby control station.

Control (or primary) tide stations, secondary stations and tertiary stations are located at strategic
92

locations for network coverage. The site selection criteria include spatial coverage of significant

changes in tidal characteristics such as: changes in tide type, changes in range of tide, changes in time

of tide, changes in daily mean sea level and changes in long term mean sea level trends. Other

criteria include coverage of critical navigation areas and transitional zones, historical sites, proximity

to the geodetic network, and the availability of existing structures, such as piers suitable for the

location of the scientific equipment. 

Procedure for Simultaneous Comparison:

Conceptually, the following steps need to be completed in order to compute equivalent NTDE

tidal datums at short term stations using the method of comparison of simultaneous observations:

1) Select the time period over which the simultaneous comparison will be made.

2) Select the appropriate control tide station for the subordinate station of interest.

. 3) Obtain the simultaneous data from subordinate and control stations and obtain or tabulate the

tides and compute monthly means, as appropriate.

4) Obtain the accepted NTDE values of the tidal datums at the control station from NOS via the

CO-OPS Website (www.tidesandcurrents.noaa.gov)

5) Compute the mean differences and/or ratios (as appropriate) in the tidal parameters between

the subordinate and control station over the period of simultaneous comparison.

6) Apply the mean differences and ratios computed in step 5, above, to the accepted values at

the control station to obtain equivalent or corrected NTDE values for the subordinate station.

The computations use slightly different formulas depending on type of tide. These

differences are explained in section 3.4 and in Chapter 4.

Datum Computation Methods:

There are some key datum computation methods used by NOS (in step 6, above) that differ

slightly depending upon the tidal characteristics and the type of tide.

Standard Method. This method is generally used for the West Coast and Pacific Island stations

and is also called the Range Ratio Method. First, equivialent NTDE values for MTL, Mn, DHQ and

DLQ are determined by comparison with an appropriate control. From these, the following are then

computed:

MLW = MTL - (0.5 x Mn)

MHW = MLW + Mn

MLLW= MLW - DLQ

MHHW = MHW + DHQ
95

Modified-Range Ratio Method. This method is generally used for the East Coast, Gulf Coast

and Caribbean Island stations. First, equivalent NTDE values for MTL, DTL, Mn and Gt as

determined by comparison with an appropriate control. The difference from the Standard Method

is that ratios of the DHQ and DLQ values are not used to compute MHHW and MLLW because

numerically the values are very small for semidiurnal tide areas. A Gt ratio about DTL is used

instead. From these, the following are computed:

MLW = MTL - (0.5 x Mn)

MHW = MLW + Mn

MLLW= DTL - (0.5 x Gt)

MHHW = MLLW + Gt

Direct Method. The Direct Method is usually used only when a full range of tidal values are not

available. For example, direct MHW can be computed for situations when low waters are not

recorded, such as in the upper reaches of a marsh. Since MTL, DTL, and Mn and Gt cannot be

determined if low waters are cut-off, equivalent NTDE values for MHW and MHHW datums are

determined directly by comparison of high tides with an appropriate control using the available part

of the tidal cycle. 

ACKNOWLEDGMENTS

The first draft of this document was prepared by John Schultz and Briah Connor under contract

to Neptune Sciences, Inc. Subsequent revisions were the result reviews of several oceanographers

in the Center for Operational Products and Services (CO-OPS) and the results of practical use by

external users of the first draft.
96
97

6. REFERENCES

Bowditch, N., The American Practical Navigator, An Epitome of Navigation, Defense Mapping

Agency Hydrographic/Topographic Center, Bethesda, MD, pp. 873, 1995. 

Floyd, R.P., Geodetic Bench Marks, NOAA Manual NOS NGS 1, U.S. Department of Commerce,

NOAA, National Ocean Survey, Rockville, MD, pp.51, September 1978.

Gill, S.K., J.R. Hubbard, and W.D. Scherer, Updating the National Tidal Datum Epoch for the United

States, in Proceedings Ocean Community Conference, Marine Technology Society Annual

Conference, November 16-19, 1998, 1040-1043, 1998.

Hicks, S.D., Tidal Datums and Their Uses - A Summary, Shore and Beach, 27-33, 1985.

Hicks, S.D., P.C. Morris, H.A. Lippincott, and M.C. O’Hargan, Users’s Guide for the Installation of

Bench Marks and Leveling Requirements for Water Levels, NOAA National Ocean Service,

Silver Spring, MD, pp. 73, 1987.

International Oceanographic Commission (IOC), Manual on Sea level Measurement and

Interpretation, Volume 3 - Reappraisals and recommendations as of the Year 2000, Manuals and

Guides 14, UNESCO 2000.

 Marmer, H.A., Tidal Datum Planes, NOAA National Ocean Service, Special Publication No. 135,

U.S. Coast and Geodetic Survey, U.S. Govt. Printing Office, revised ed.,1951.

Mero, T.N., NOAA/National Ocean Service Application of Real-Time Water Levels, in Proceedings,

Volume 2, Ocean Community Conference, The Marine Technology Society Annual Conference,

November 16-19, 1998, 1036-1039,1998.

National Geodetic Survey, Guidelines for Establishing GPS-Derived Ellipsoid Heights (Standards:

2CM and 5CM) Version 4.3, NOAA Technical Memorandum NOS NGS-58, November 1997.

National Ocean Service, Tide and Current Glossary, NOAA National Ocean Service, Silver Spring,

MD, 2000.

National Ocean Service, Tidal Datums and Their Applications, NOAA Technical Report NOS COOPS 1, Center For Operational Oceanographic Products and Services, Silver Spring, MD, 2000.

National Ocean Service, NOS Hydrographic Surveys Specifications and Deliverables, NOAA

National Ocean Service, Silver Spring, MD, pp. 91, 2003.

Scherer, W.D., National Ocean Service’s Next Generation Water level Measurement System, Vol.

4, International Congress of Surveying, Toronto, Ontario, Canada, pp 232-243, 1986.
98

Schureman, P., Manual of Harmonic Analysis and Prediction of Tides, Special Publication No. 98

(1940), U.S. Coast and Geodetic Survey, U.S. Govt. Printing Office, revised ed.,1941. 

Swanson, R.L., Variability of Tidal Datums and Accuracy in Determining Datums from Short Series

of Observations, NOAA Tech. Rep. NOS 64, Silver Spring, MD, pp. 41, 1974.

Zervas, C., Sea Level Variations of the United States 1854 - 1999, NOAA Technical Report NOS

CO-OPS 36, Silver Spring, MD, pp186, 2001.
A-1

Appendix 1:

EXAMPLES OF TIDAL DATUM COMPUTATION

SPREADSHEETS

A. SEMIDIURNAL TIDE TYPE: MODIFIED RANGE RATIO

METHOD

B. MIXED TIDE TYPE: STANDARD METHOD
A-2

SUBORDINATE STATION A: FORT PULASKI, GA ELEVATIONS IN METERS, TIME IN HOURS

CONTROL STATION B: CHARLESTON, SC

MTL DTL HWI

Station A Station B A - B Station A Station B A - B Station A Station B A - B

1997 03 2.156 1.662 0.494 2.159 1.672 0.487 0.54 0.41 0.13

1997 04 2.248 1.751 0.497 2.259 1.763 0.496 0.55 0.41 0.14

1997 05 2.188 1.673 0.515 2.208 1.699 0.509 0.54 0.41 0.13

1997 06 2.291 1.792 0.499 2.320 1.818 0.502 0.62 0.47 0.15

1997 07 2.225 1.730 0.495 2.253 1.758 0.495 0.55 0.46 0.09

1997 08 2.295 1.789 0.506 2.309 1.807 0.502 0.55 0.45 0.10

1997 09 2.353 1.851 0.502 2.369 1.870 0.499 0.61 0.51 0.10

1997 10 2.305 1.807 0.498 2.321 1.824 0.497 0.62 0.48 0.14

1997 11 2.158 1.669 0.489 2.182 1.692 0.490 0.58 0.46 0.12

1997 12 2.118 1.635 0.483 2.139 1.661 0.478 0.61 0.45 0.16

1998 01 2.135 1.634 0.501 2.152 1.665 0.487 0.55 0.39 0.16

1998 02 2.300 1.811 0.489 2.304 1.821 0.483 0.58 0.30 0.28

SUMS 5.968 5.925 1.70

# MONTHS 12.000 12.000 12.00

MEANS 0.497 0.494 0.14

ACCEPTED VALUE FOR B: 1.622 1.643 0.35

CORRECTED AT A: 2.119 2.137 0.49

MN GT LWI

Station A Station B A /B Station A Station B A / B Station A Station B A - B

1997 03 2.195 1.629 1.347 2.335 1.766 1.322 6.94 6.65 0.29

1997 04 2.193 1.646 1.332 2.327 1.771 1.314 6.97 6.66 0.31

1997 05 2.235 1.666 1.342 2.408 1.820 1.323 6.94 6.67 0.27

1997 06 2.196 1.644 1.336 2.377 1.815 1.310 7.01 6.77 0.24

1997 07 2.230 1.653 1.349 2.427 1.830 1.326 6.98 6.70 0.28

1997 08 2.153 1.611 1.336 2.301 1.751 1.314 6.98 6.66 0.32

1997 09 2.123 1.606 1.322 2.252 1.724 1.306 7.08 6.72 0.36

1997 10 2.096 1.586 1.322 2.210 1.697 1.302 7.06 6.71 0.35

1997 11 2.150 1.618 1.329 2.311 1.774 1.303 7.02 6.68 0.34

1997 12 2.207 1.648 1.339 2.394 1.823 1.313 7.03 6.71 0.32

1998 01 2.243 1.674 1.340 2.426 1.846 1.314 6.95 6.63 0.32

1998 02 2.222 1.653 1.344 2.405 1.807 1.331 7.00 6.73 0.27

SUMS 16.038 15.779 3.67

# MONTHS 12.000 12.000 12.00

MEANS 1.337 1.315 0.31

ACCEPTED VALUE FOR B: 1.606 1.768 6.57

CORRECTED AT A: 2.146 2.325 6.88

METERS

MLW(A) = MTL(A) - 1/2 MN(A) 1.046

MHW(A) = MLW (A) + MN(A) 3.193

MLLW(A) = DTL(A) - 1/2 GT(A) 0.974

MHHW(A) = MLLW(A) +GT(A) 3.299

HOURS

GHWI: 0.49 GLWI: 6.88
A-3

SUBORDINATE STATION A: ALAMEDA, CA ELEVATIONS IN METERS;TIMES IN HOURS

CONTROL STATION B: SAN FRANCISCO, CA

MTL MN HWI

Station A Station B A - B Station A Station B A /B Station A Station B A - B

1997 03 2.058 2.731 -0.673 1.503 1.257 1.196 0.54 0.41 0.13

1997 04 1.978 2.648 -0.670 1.543 1.292 1.194 0.55 0.41 0.14

1997 05 2.066 2.741 -0.675 1.533 1.294 1.185 0.54 0.41 0.13

1997 06 2.131 2.809 -0.678 1.537 1.306 1.177 0.62 0.47 0.15

1997 07 2.147 2.830 -0.683 1.527 1.300 1.175 0.55 0.46 0.09

1997 08 2.196 2.882 -0.686 1.514 1.301 1.164 0.55 0.45 0.10

1997 09 2.199 2.878 -0.679 1.492 1.272 1.173 0.61 0.51 0.10

1997 10 2.216 2.905 -0.689 1.503 1.279 1.175 0.62 0.48 0.14

1997 11 2.298 2.995 -0.697 1.525 1.296 1.177 0.58 0.46 0.12

1997 12 2.267 2.970 -0.703 1.544 1.311 1.178 0.61 0.45 0.16

1998 01 2.303 3.001 -0.698 1.533 1.287 1.191 0.55 0.39 0.16

1998 02 2.414 3.103 -0.689 1.336 1.101 1.213 0.58 0.30 0.28

SUMS -8.220 14.197 1.70

# MONTHS 12.000 12.000 12.00

MEANS -0.685 1.183 0.14

ACCEPTED VALUE FOR B: 2.728 1.250 0.35

CORRECTED AT A: 2.043 1.479 0.49

DHQ DLQ LWI

Station A Station B A /B Station A Station B A /B Station A Station B A - B

1997 03 0.132 0.122 1.082 0.247 0.242 1.021 6.94 6.65 0.29

1997 04 0.116 0.117 0.991 0.293 0.292 1.003 6.97 6.66 0.31

1997 05 0.166 0.162 1.025 0.320 0.322 0.994 6.94 6.67 0.27

1997 06 0.218 0.213 1.023 0.347 0.352 0.986 7.01 6.77 0.24

1997 07 0.225 0.219 1.027 0.340 0.344 0.988 6.98 6.70 0.28

1997 08 0.184 0.176 1.045 0.275 0.274 1.004 6.98 6.66 0.32

1997 09 0.116 0.113 1.027 0.203 0.203 1.000 7.08 6.72 0.36

1997 10 0.105 0.101 1.040 0.217 0.219 0.991 7.06 6.71 0.35

1997 11 0.166 0.162 1.025 0.319 0.324 0.985 7.02 6.68 0.34

1997 12 0.224 0.217 1.032 0.366 0.366 1.000 7.03 6.71 0.32

1998 01 0.215 0.213 1.009 0.328 0.337 0.973 6.95 6.63 0.32

1998 02 0.187 0.183 1.022 0.234 0.261 0.897 7.00 6.73 0.27

SUMS 12.349 11.841 3.67

# MONTHS 12.000 12.000 12.00

MEANS 1.029 0.987 0.31

ACCEPTED VALUE FOR B: 0.183 0.344 6.57

CORRECTED AT A: 0.188 0.339 6.88

METERS

MLW(A) = MTL(A) - 1/2 MN(A) : 1.304

MHW(A) = MLW (A) + MN(A): 2.782

MLLW(A) = MLW(A) - DLQ(A): 0.964

MHHW(A) = MHW(A) +DHQ(A): 2.971

HOURS

GHWI: 0.49 GLWI: 6.88
A-4
B-1

Appendix 2:

EXAMPLE OF TIDAL DATUM COMPUTATION

SPREADSHEET FOR TIDE BY TIDE COMPARISON
B-2

TIDES: Comparison of Simultaneous Observations (Tide by 

A) Subordinate Station: 9414750 Alameda, SanFrancisco BayB) Standard Station: 9414290 San Francisco, San Francisco Bay

Lat: 37 46.3 nLat: 37 48.4 n Time Meridian:

 units: meters

Long: 122 17.9 wLong: 122 27.9 w(A) 0° (UTC) (B) 0° (UTC)

Date (A) Station = (B) Station = (A) Station = (B) Station = (A) - (B) 

TIME TIME HEIGHT (STATION DATUM) HEIGHT (MLLW) TIME DIFFERENCE 

HHW LHW HLW LLW HHW LHW HLW LLW HHW LHW HLW LLW HHW LHW HLW LLW HHW LHW HLW 

03/01/97 12.1 19.1 11.8 18.5 2.748 1.306 3.340 2.111 0.3 03/02/97 1.7 6.4 1.4 5.6 2.250 1.744 2.853 2.532 0.3 

13.1 20.2 12.6 19.6 2.815 1.214 3.417 2.008 0.5 

03/03/97 3.1 7.8 3.0 7.1 2.304 1.784 2.901 2.568 0.1 

14.1 21.4 13.7 21.0 2.768 1.039 3.361 1.857 0.4 

03/04/97 4.2 9.1 4.1 8.4 2.352 1.725 2.933 2.511 0.1 

15.3 22.3 15.0 21.5 2.814 0.848 3.391 1.669 0.3 

03/05/97 5.2 10.1 5.0 9.3 2.468 1.642 3.021 2.438 0.2 

16.5 23.1 15.9 22.5 2.940 0.729 3.488 1.535 0.6 

03/06/97 6.1 11.2 5.7 10.4 2.641 1.527 3.165 2.335 0.4 

17.5 24.0 17.0 23.2 3.077 0.705 3.603 1.498 0.5 

03/07/97 6.8 12.1 6.5 11.2 2.784 1.361 3.292 2.172 0.3 

18.4 17.8 3.139 3.648 0.6 

HHW (A) LHW HLW (A) LLW (A) HHW (B) LHW (B) HLW (B) LLW (B) 

HHW = Higher High Water Sums : 20.301 14.799 9.783 5.841 24.248 18.165 14.556 10.678 LHW = Lower High Water Number : 7 6 6 6 7 6 6 6 HLW = Higher Low Water Means : 2.900 2.467 1.631 0.974 3.464 3.028 2.426 1.780 LLW = Lower Low Water Average HW time difference : 0.35 minutes

 Average LW time difference : 0.71 minutesAverage HW/LW time difference : 0.53 minutes
B-3

TIDES: Comparison of Simultaneous Observations (Tide by T

Eq. #

0.217 = DHQ (A) = 0.5*(HHW (A) - LHW (A)) 22 Standard Method: Datum Planes (relative to stat0.329 = DLQ (A) = 0.5*(HLW (A) - LLW (A)) 23 2.984 = MHHW (A) = MHW (A) + DHQ co2.683 = HW (A) = 0.5*(HHW (A) + LHW (A)) 24 2.802 = MHW (A) = MLW (A) + MN corre1.302 = LW (A) = 0.5*(HLW (A) + LLW (A)) 25 1.291 = MLW (A) = MTL corrected for A - 1.927 = Gt (A) = HHW (A) - LLW (A) 26 0.941 = MLLW (A) = MLW (A) - DLQ corr1.381 = Mn (A) = HW (A) - LW (A) 271.937 = DTL (A) = 0.5*(HHW (A) + LLW (A)) 281.993 = MTL (A) = 0.5*(HW (A) + LW (A)) 29

-0.001 = D DHQ =0.5*( D HHW - D LHW) 30

0.005 = D DLQ= 0.5*(D HLW - D LLW) 31 Modified Range Ratio Method: Datum Planes (relative to station datum)

-0.562 = D HW = 0.5*(D HHW + D LHW) 32 2.977 = MHHW (A) = MLLW (A) + Gt corr-0.801 = D LW = 0.5*(D HLW +D LLW) 33 2.802 = MHW (A) = MLW (A) + MN correc

0.238 = D MN = D HW - D LW 34 1.291 = MLW (A) = MTL corrected for A - 

-0.685 = D DTL = 0.5*(D HHW + D LLW) 35 0.945 = MLLW (A) = DTL corrected for A -0.682 = D MTL = 0.5*(D HW + D LW) 36

0.242 = D GT = D HHW - D LLW 371.209 = MN RATIO = MN (A) / (MN (A) - D MN) 380.993 = DHQ RATIO = DHQ (A) / (DHQ (A) - D DHQ) 391.017 = DLQ RATIO = DLQ (A) / (DLQ (A) - DDLQ) 401.144 = Gt RATIO = Gt (A) / (Gt (A) - D Gt) 41

Note 1: Accepted values are from NOAA, Accepted Datums Epoch:1960-78

Internet address:http://www.co-ops.nos.noaa.gov/

Station MHW MLW MSL Mn MTL MHHW MLLW DHQ DLQ Gt DTL HWI LWI

94142 3.353 2:28 2.713 1.250 2.728 3.536 1.759 0.183 0.344 1.777 2.646 7.560 0.830

Eq#

2.046 = MTL corrected for A = MTL accepted for B + ∆ MTL 42

1.961 = DTL corrected for A = DTL accepted for B + ∆ DTL 43

1.511 = MN corrected for A = MN accepted forB * MN ratio 442.033 = GT corrected for A = GT accepted for B * GT ratio 450.182 = DHQ corrected for A = DHQ accepted for B * DHQ ratio 150.350 = DLQ corrected for A = DLQ accepted for B * DLQ ratio 16

 Compiled by: Computed by: Verified by:

(Date) (Date) (Date)
B-4