NOAA Technical Memorandum NOS CS 6 

MODELING OF TIDAL DATUM FIELDS IN SUPPORT 

OF VDATUM FOR THE NORTH AND CENTRAL 

COASTS OF CALIFORNIA 

Silver Spring, Maryland 

September 2006 

n 0 a a National Oceanic and Atmospheric Administration 

U.S. DEPARTMENT OF COMMERCE 

National Ocean Service 

Coast Survey Development Laboratory 
Office of Coast Survey 

National Ocean Service 

National Oceanic and Atmospheric Administration 

U.S. Department of Commerce 

The Office of Coast Survey (OCS) is the Nation's only official chartmaker. As the oldest 

United States scientific organization, dating from 1807, this office has a long history. Today 

it promotes safe navigation by managing the National Oceanic and Atmospheric 

Administration's (NOAA) nautical chart and oceanographic data collection and information 

programs. 

There are four components of OCS: 

The Coast Survey Development Laboratory develops new and efficient techniques to 

accomplish Coast Survey missions and to produce new and improved products and 

services for the maritime community and other coastal users. 

The Marine Chart Division collects marine navigational data to construct and 

maintain nautical charts, Coast Pilots, and related marine products for the United 

States. 

The Hydrographic Surveys Division directs programs for ship and shore-based 

hydrographic survey units and conducts general hydrographic survey operations. 

The Navigation Services Division is the focal point for Coast Survey customer service 

activities, concentrating predominantly on charting issues, fast-response 

hydrographic surveys and Coast Pilot updates. 
NOAA Technical Memorandum NOS CS 6 

MODELING OF TIDAL DATUM FIELDS IN SUPPORT OF 

VDATUM FOR THE NORTH AND CENTRAL COASTS OF 

CALIFORNIA 

Edward P. Myers, Ill 

Kurt Hess 

September 2006 

n 0 a a National Oceanic and Atmospheric Administration 

U.S. DEPARTMENT National Oceanic and 

OF COMMERCE Atmospheric Administration 

Carlos Gutierrez, Secretary Conrad C. Lautenbacher, Jr., 

VADM USN (Ret.), Under Secretary 

Office of Coast Survey 

Captain Steven Barnum 

National Ocean Service 

John H, Dunnigan 

Assistant Administrator 

Coast Survey Development 

Laboratory 

Mary Erickson 
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. 
MODELING OF TIDAL DATUM FIELDS IN SUPPORT OF VDATUM FOR 

THE NORTH AND CENTRAL COASTS OF CALIFORNIA 

Edward P. Myers III and Kurt Hess 

Coast Survey Development Laboratory 

NOAA/NOS/OCS 
iii

TABLE OF CONTENTS

LIST OF FIGURES ....................................................................................................................... iii 

ABSTRACT.....................................................................................................................................v 

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

2. OFFSHORE TIDAL DATUMS ..................................................................................................3 

3. DATUMS INSIDE SAN FRANCISCO BAY.............................................................................7 

4. DATUMS INSIDE TOMALES BAY .......................................................................................11 

5. TIDAL MARINE GRID............................................................................................................12 

6. TOPOGRAPHY OF THE SEA SURFACE ..............................................................................14 

ACKNOWLEDGMENTS .............................................................................................................15 

REFERENCES ..............................................................................................................................15 
iv
v

LIST OF FIGURES 

Figure 1.1. North/Central California region of interest for calculation of tidal datums. . ...............2 

Figure 2.1. The unstructured grid used in the tidal inversion model. .............................................3 

Figure 2.2. HHW from the tidal inversion model. ..........................................................................5 

Figure 2.3. HW from the tidal inversion model. .............................................................................5 

Figure 2.4. LLW from the tidal inversion model. ...........................................................................5 

Figure 2.5. LW from the tidal inversion model. .............................................................................5 

Figure 2.6. MTL from the tidal inversion model. ...........................................................................5 

Figure 2.7. DTL from the tidal inversion model. ............................................................................5 

Figure 2.8. Tide gauges used for analysis of offshore datums. .......................................................6 

Figure 2.9. Datum comparisons between the tidal inversion model and the data for 

offshore region. ...............................................................................................................................6 

Figure 3.1. MHHW differences between data and model. .............................................................7 

Figure 3.2. Tide gauges used in the spatial interpolation of the datums. ........................................8 

Figure 3.3. MHHW from the tidal inversion model, corrected using the TCARI 

interpolation of MHHW differences between the data and the tidal inversion model.

.................................................................................................................................................9 

Figure 3.4. TCARI results for MHHW. ..........................................................................................9 

Figure 4.1. Tomales Bay tide gauges. ...........................................................................................10 

Figure 4.2. Interpolated MHHW in Tomales Bay. .......................................................................11 

Figure 4.3. Interpolated MHW in Tomales Bay. ..........................................................................11 

Figure 4.4. Interpolated MLLW in Tomales Bay. ........................................................................11 

Figure 4.5. Interpolated MLW in Tomales Bay. ...........................................................................11 

Figure 5.1. Algorithm used in generating the final datums for the VDatum grid. ........................12 
vi

Figure 5.2. VDatum grid. Brown represents points outside the region of interest, green 

are offshore and coastal points, light blue are points outside the tidal inversion grid 

but inside the coastline, dark blue are inside the tidal inversion grid and corrected 

with TCARI interpolation of data-model differences, and yellow points are inside 

Tomales Bay. . ......................................................................................................................13 

Figure 5.3. Enlargement of VDatum grid near San Francisco Bay. . ............................................13 

Figure 5.4. Bounding polygon, inside of which VDatum transformations are available. .............14 
vii

ABSTRACT 

In a project funded by NOAA’s National Marine Sanctuary Program, a VDatum 

application is developed for the northern and central coastal regions of California. The 

region of interest encompasses the Cordell Bank, Gulf of the Farallones, and Monterey 

Bay National Marine Sanctuaries. VDatum will enable bathymetric data in these 

sanctuaries to be seamlessly transformed between tidal, orthometric and ellipsoidal 

vertical datums. In this report, we highlight the creation of the tidal datum fields used as 

input to the VDatum software. A numerical model of tides in the Eastern North Pacific 

Ocean, the domain of which includes coastal California and its major bays and estuaries, 

is used to compute spatially varying fields of the tidal datums. The model results are used 

in combination with spatial interpolation techniques to provide a final set of tidal datum 

fields that match exactly at locations where observations are available. These are then 

combined with geoid and ellipsoidal models developed by NOAA’s National Geodetic 

Survey to complete the suite of transformations available in the VDatum software. 
viii
1

1. INTRODUCTION 

In a project supported by NOS’ National Marine Sanctuary Program, spatially varying 

tidal datums are computed offshore of central California and inside San Francisco Bay. 

The region of interest extends from Point Sal (34° 54.18’ N) in the south to Point Arena 

(38° 57.5’ N) in the north and is bounded in the west by a line running from (122°1.54’ 

W, 34°54.18’ N) to (125°00’ W, 38°57.5’ N). The region, shown in Figure 1.1, also 

includes Tomales Bay, Bodega Bay and San Francisco Bay up to the San JoaquinSacramento River delta. The tidal datums computed for this project include mean higher 

high water (MHHW), mean high water (MHW), mean lower low water (MLLW), mean 

low water (MLW), mean tide level (MTL) and diurnal tide level (DTL). Two numerical 

models are used to determine the spatial variations in these datums throughout the region 

of interest. The results, when combined with spatial variations of orthometric and 

ellipsoidal vertical datums provided by the National Geodetic Survey, will be 

incorporated into VDatum (Milbert and Hess, 2001), a software tool for transformation of 

height data. 

The numerical models used in this study include a tidal inversion model for the Eastern 

North Pacific Ocean (Myers and Baptista, 2001), and a tide interpolation model based on 

solution of Laplace’s equation (Hess, 2001a). Along the coast and in offshore regions, the 

tidal inversion model is used. Differences between this model and tide gauge data are 

made using a linear correction for each datum as a function of latitude. The tidal 

inversion model is also used in San Francisco Bay, though corrections cannot be made as 

a simple linear function. Instead, the tide interpolation model is used to simulate the 

differences between the tidal inversion model and the data. In those regions of San 

Francisco Bay that the inversion model does not cover, the tide interpolation model is 

used to compute the datums directly. The tidal inversion model does not extend into 

Tomales Bay, and datums there are likewise computed directly with the tide interpolation 

model. 

 

Section 2 describes how the vertical datums were calculated for the coastal and offshore 

regions. Sections 3 and 4 describe the procedures used to determine the datums inside 

San Francisco Bay and Tomales Bay, respectively. Section 5 summarizes how the final 

VDatum grid is generated using the results from these three approaches. 
2

Figure 1.1. North/Central California region of interest for calculation of tidal datums. 
3

2. OFFSHORE TIDAL DATUMS 

Outside of San Francisco Bay and Tomales Bay, tidal datums are computed using the 

tidal inversion model results. The unstructured grid used in the model is shown in Figure 

2.1. Boundary conditions for the grid were obtained using inversion techniques with the 

finite element model ADCIRC (Luettich et al., 1991). Using an initial set of boundary 

conditions from the TPX0.3 global tide model (Egbert et al., 1994), tide gauges within 

the grid were used to calibrate the inversion to the final set of boundary conditions. One 

of the primary strengths of the regional model over global tide models was that it 

provided enhanced representations of the tides along the continental shelf and coastal 

regions. 

Figure 2.1. The unstructured grid used in the tidal inversion model (Myers and Baptista, 

2001). 

Equilibrium tidal amplitudes and phases from the regional tide model are available for 

the O1, K1, Q1, P1, K2, N2, M2, S2, M4, MS4, M6, 2SM2, MN4, and μ2 tidal constituents. 

For each of the nodes in the grid, the amplitudes and phases are extracted and used to 

construct time series for a selected time period. For the purpose of this study, time series 
4

were generated for all of 1992 at 6-minute intervals using appropriate nodal factors and 

equilibrium arguments. The year 1992 was selected because the nodal factor for the 

strongest tidal constituent, M2, was approximately 1.0, its average value during the 19-

year tidal epoch. 

Once time series are generated at each of the nodes in the grid, they are analyzed using a 

program developed within CSDL to calculate tidal datums (Hess, 2001b). The program 

selects the high and low water marks over 25-hour time periods using a cubic 

interpolation method. These highs and lows from the time series are then used to 

determine each of the datums. These computed datums are displayed in Figures 2.2-2.7. 

There are 26 coastal tide gauges in this region (excluding San Francisco Bay) available 

for contrasting the regional tide model results with observations. However, many of these 

gauges are located within small embayments or near obstructions not represented in the 

model grid. Therefore, the nine tide gauges in Figure 2.8 were determined to be the best 

suitable (i.e. least obstructed) locations for comparison with the model results. 

The datum values at the tide gauges are computed by NOAA’s Center for Operational 

Oceanographic Products and Services (CO-OPS) and provided on their web pages as 

benchmark sheets. Differences between the data and the model are shown in Figure 2.9 

for the nine selected tide gauges. Also shown is a linear fit of these differences as a 

function of latitude. These linear fits to the differences can subsequently be added to the 

model results as a correction. Differences between the corrected results and the data, 

shown as the open circles in Figure 2.9, show that errors are now limited to less than 5 

cm and are generally between 0 and 3 cm. 
5

Figure 2.2. MHHW from the tidal inversion model. Figure 2.3. MHW from the tidal inversion model.

Figure 2.4. MLLW from the tidal inversion model. Figure 2.5. MLW from the tidal inversion model. 

Figure 2.6. MTL from the tidal inversion model. Figure 2.7. DTL from the tidal inversion model.
6

Figure 2.8. Tide gauges used for analysis of offshore datums. 

Figure 2.9. Datum comparisons between the tidal inversion model and the data for 

offshore region.
7

3. TIDAL DATUMS INSIDE SAN FRANCISCO BAY 

The tidal inversion model results inside San Francisco Bay indicate that the magnitudes 

of the tides are underrepresented when contrasted with data. As an example, differences 

between the data and model are shown for MHHW in Figure 3.1. The values in the data 

are generally larger in magnitude than those of the model. This is most likely associated 

with a need for more grid refinement in San Francisco Bay in order for the model to more 

accurately depict both the bathymetry and tides. 

Figure 3.1. MHHW differences between data and model.

In order to correct the model results in San Francisco Bay, the tide interpolation model 

TCARI (Hess, 2001a) was used to spatially interpolate the differences between the model 

and the data. These corrections could then be added to the model results to provide a 

more accurate representation of each datum. The gauges shown in Figure 3.2 are those 

that were not located in small tributaries/embayments, were not obstructed, and were 

inside or within one nautical mile of the tidal inversion grid. 

As will be discussed in Section 5, final calculations of the datums are provided on a 

VDatum grid of regularly spaced points. Within San Francisco Bay, the VDatum values 

are interpolated from the corrected model results. However, there may be some VDatum 

points that lie outside of the regional tide model grid but are still in the water area. For 

these areas, TCARI was also run to simulate the datums from available data (instead of 

simulating the differences between model and data). 
8

The corrected MHHW from the tidal inversion model is shown in Figure 3.3, where the 

correction was made using the TCARI interpolation of the data-model differences. The 

TCARI interpolation of the MHHW data is displayed in Figure 3.4. 

Figure 3.2. Tide gauges used in the spatial interpolation of the datums. 
9

Figure 3.3. MHHW from the tidal inversion model, corrected using the TCARI 

interpolation of MHHW differences between the data and the tidal inversion model. 

Figure 3.4. TCARI results for MHHW. 
10

4. TIDAL DATUMS INSIDE TOMALES BAY 

Since the regional tide model does not extend into Tomales Bay, TCARI is used to 

spatially interpolate the datums inside the bay based on available benchmark data. Tidal 

datums are calculated at the five tide gauges displayed in Figure 4.1. TCARI uses the 

data at these five locations and spatially interpolates them to a user-defined set of points 

dispersed throughout the bay, accounting for shoreline location. 

The results from the interpolation for MHHW, MHW, MLLW, and MLW are shown in 

Figures 4.2-4.5, respectively. The tidal range increases with distance into the bay, as seen 

by the increase in magitudes for each of the datums. The TCARI results are computed on 

a set of regularly spaced points (0.06 nautical miles spacing) that are subsequently 

triangulated into a mesh grid. 

Figure 4.1. Tomales Bay tide gauges. 
11

Figure 4.2. Interpolated MHHW in Tomales Bay. Figure 4.3. Interpolated MHW in Tomales Bay.

Figure 4.4. Interpolated MLLW in Tomales Bay. Figure 4.5. Interpolated MLW in Tomales Bay.
12

5. TIDAL MARINE GRID 

The final set of computed tidal datums are provided on a grid of regularly spaced points 

(0.01 degree spacing). All points that lie outside of the region of interest (Figure 1.1) are 

given a value of –88.8888. The general algorithm followed for determining the datum 

values on the VDatum grid is shown in Figure 5.1: 

Figure 5.1. Algorithm used in generating the final datums for the VDatum grid.

The final values in Figure 5.1 are color coded in their boxes. Figure 5.2 shows the 

VDatum points using the same color codes, i.e. which approaches were used in different 

parts of the region of interest. An enlargement of the VDatum grid near San Francisco 

Is the point inside 

the domain?

No -88.8888 

Yes 

Is the point inside 

Tomales Bay? 

Yes Is the point inside 

the TCARI grid? No 

Yes Interpolate from TCARI grid 

Use nearest TCARI value 

No 

Is the point inside 

San Francisco Bay? 

Yes Is point inside the 

tidal inversion grid? 

No Use the TCARI interpolation 

of the data 

Yes 

Interpolate from the regional tide model 

Correct the model by adding the TCARI interpolation of data-model differences 

(either interpolating from the TCARI grid or using the nearest TCARI point) 

Interpolate from the 

regional tide model or take 

nearest tide model value (if 

outside model domain)

Correct the model by adding the linear coastal correction 

No 
13

Bay is shown in Figure 5.3. A bounding polygon was created to delineate those areas 

where VDatum transformations are applicable for this region, as shown in Figure 5.4. 

Figure 5.2. VDatum grid. Brown represents points outside the region of interest, green 

are offshore and coastal points, light blue are points outside the tidal inversion grid but 

inside the coastline, dark blue are inside the tidal inversion grid and corrected with 

TCARI interpolation of data-model differences, and yellow points are inside Tomales 

Bay. 

Figure 5.3. Enlargement of VDatum grid near San Francisco Bay.
14

Figure 5.4. Bounding polygon (shown in red), inside of which VDatum transformations 

are available.

6. TOPOGRAPHY OF THE SEA SURFACE 

To reference the collective group of tidal datums to the orthometric class of datums, the 

topography of the sea surface (TSS) was used to relate MSL to the North American 

Vertical Datum of 1988 (NAVD88). Observed MSL-NAVD88 differences were spatially 

interpolated by NOS’ National Geodetic Survey using a linear Kriging technique on a 

grid of regularly structured points at 0.01 degree spacing. The TSS and tidal datum 

results were then input to the VDatum software for this region, along with 

transformations relating NAVD88 with the National Geodetic Vertical Datum of 1929 

(NGVD29), NAVD88 with the North American Datum of 1983 (NAD83), and NAD83 

with other ellipsoidal vertical reference systems. 
15

ACKNOWLEDGMENTS 

This work was made possible through support from the National Marine Sanctuary 

Program on a project coordinated by Christine Taylor. Development of this VDatum 

application for North/Central California fits into the vision that Dr. Bruce Parker had for 

building a national VDatum tool. Dr. Dennis Milbert and Jason Woolard of the National 

Geodetic Survey incorporated the tidal datums into the VDatum software which links 

them to the geoid and ellipsoidal models. 

REFERENCES 

Egbert, G.D., A.F. Bennett, and M.G.G. Foreman, 1994. TOPEX/POSEIDON Tides 

Estimated Using a Global Inverse Model. Journal of Geophysical Research, v. 99, 

No.C12, 24821-24852. 

Hess, K.W., 2001a. Spatial Interpolation of Tidal Data in Irregularly-Shaped Coastal 

Regions by Numerical Solution of Laplace’s Equation. Estuarine, Coastal and Shelf 

Science, v. 54, 175-192. 

Hess, K.W., 2001b: Generation of Tidal Datum Fields for Tampa Bay and the New York 

Bight. U.S. Department of Commerce, NOAA Technical Report NOS CS 11. 31 pp + 

appendices. 

Luettich, R.A., J.J. Westerink, and N.W. Scheffner, 1991. ADCIRC: An Advanced 

Three-Dimensional Circulation Model for Shelves, Coasts, and Estuaries. Department of 

the Army, US Army Corps of Engineers, Washington, DC. 

Milbert, D.G., and K.W. Hess, 2001. Combination of Topography and Bathymetry 

Through Application of Calibrated Vertical Datum Transformations in the Tampa Bay 

Region. In Proceedings of the 2nd Biennial Coastal GeoTools Conference. 

Myers, E.P., and A.M. Baptista, 2001. Inversion for Tides in the Eastern North Pacific 

Ocean. Advances in Water Resources, v. 24, 505-519.