NOAA Technical Memorandum NOS CS 4

 

VDATUM FOR PUGET SOUND: GENERATION OF THE GRID 

AND POPULATION WITH TIDAL DATUMS AND SEA SURFACE 

TOPOGRAPHY

 

 

 

Silver Spring, Maryland 

January 2004

noaa 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 (CS) 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 CS: 

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 4

VDATUM FOR PUGET SOUND: GENERATION OF THE GRID 

AND POPULATION WITH TIDAL DATUMS AND SEA SURFACE 

TOPOGRAPHY

Kurt W. Hess 

 Office of Coast Survey 

 

Stephen A. White 

 National Geodetic Survey 

 

 

 

 

January 2004 

noaa National Oceanic and Atmospheric Administration 

U.S. DEPARTMENT National Oceanic and National Ocean Service 

OF COMMERCE Atmospheric Administration Richard W. Spinrad, Ph.D. 

Donald Evans, Secretary Conrad C . Lautenbacher, Jr., Assistant Administrator 

 VADM USN (Ret.), Under Secretary 

 

Office of Coast Survey Coast Survey Development 

Captain Roger L. Parsons Laboratory 

 Bruce B. Parker 
 

 

 

 

 

 

 

 

 

 

 

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.
TABLE OF CONTENTS

LIST OF FIGURES ....................................................................................................................... iv 

LIST OF TABLES......................................................................................................................... iv 

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

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

2. TIDAL DATUM FIELDS .......................................................................................................... 1 

3. DIGITIZED SHORELINE ......................................................................................................... 1 

4. THE BOUNDING POLYGON .................................................................................................. 5 

5. VDATUM GRID POINTS ......................................................................................................... 6

6. POPULATION OF THE VDATUM GRID WITH TIDAL DATUMS..................................... 8 

7. COMPARISON OF VDATUM GRIDDED VALUES WITH OBSERVED DATUMS .......... 9 

8. SUMMARY OF THE VDATUM GRID GENERATION PROCESS....................................... 9 

9. GENERATION OF THE GRIDDED TOPOGRAPHY OF THE SEA SURFACE ................ 10 

ACKNOWLEDGEMENTS.......................................................................................................... 11 

REFERENCES ............................................................................................................................. 12 

APPENDIX A. COPY OF AMS PAPER..................................................................................... 13 

APPENDIX B. TIDE STATIONS AND TIDAL DATUMS ....................................................... 19 

APPENDIX C. NOS TIDE STATION NUMBERS AND NAMES........................................... 21 

APPENDIX D. SUMMARY OF COMPARISONS WITH CO-OPS STATION DATA............ 23 

APPENDIX E. SUMMARY OF COMPARISONS WITH NGS BENCHMARK DATA.......... 25 

 iii
LIST OF FIGURES 

Figure 1. Sample coastline in Puget Sound showing the NGDC shoreline (thin line) and the 

newer, EVS shoreline (thick line). In this figure, the maximum difference in position is 

approximately 0.1 nmi. ........................................................................................................... 2 

Figure 2. Initial Puget Sound EVS shoreline for MHW as extracted from NOS Chart No. 18440. 

Note the presence of numerous small lakes; the next figure shores the revised EVS shoreline 

with the lakes removed. .......................................................................................................... 2 

Figure 3. Digitized shoreline for Puget Sound from NOS Chart No. 18440, revised to exclude 

lakes. ....................................................................................................................................... 3 

Figure 4. The mainland polygon as formed by a closed boundary segment................................... 4 

Figure 5. Line segment with numbered points that defines a river intersecting the chart border... 4 

Figure 6. Bounding polygon (solid line) and mainland polygon (gray area) for Puget Sound....... 5 

Figure 7. Schematic showing the VDatum grid points (+ symbols), an 'A cell' and a 'B cell'. The 

A cell is centered on the VDatum grid point, while the corners of the B cell are the VDatum 

points. The symbol * denotes a corner of an A cell, i is the longitude index, and j is the

latitude index........................................................................................................................... 7 

Figure 8. Portion of the VDatum grid for Puget Sound. The small dark squares show locations of 

non-null values and the '+' symbols show locations of null values. The curved line shows 

the MHW coastline, white areas show water (i.e., locations where a non-null tidal datum 

value will be found), and gray areas are land. ........................................................................ 7 

Figure 9. VDatum grid points for Puget Sound. The areas with null values are shown in gray, and 

the areas covered by non-null tidal datum values are shown as white within the gray area. . 8 

Figure 10. The TSS field (m) for Puget Sound and the locations of the tide stations (denoted by

filled triangles) used to generate the field of values. ............................................................ 11 

 

LIST OF TABLES 

Table 1. Summary of Steps in Production of the VDatum Grid ........................................9 

Table 2. Location and elevation information for NOAA tide stations.................................10 

 iv
ABSTRACT 

The generation and population of the VDatum tidal grid for Puget Sound, Washington, is 

discussed. The VDatum grid has a spacing of approximately 0.1 nautical miles. Points in the grid 

are first determined to be either land or water, depending on their location relative to a digitized

coastline. Land points are given null values. For water points, the tidal datum fields are searched

to provide a value. The tidal datum fields were generated by spatial interpolation, taking landwater boundaries into account, of datum values at 69 historical stations. The sea surface 

topography, or difference between local mean sea level and the NAVD 88 geopotential surface, 

is generated by a minimum curvature algorithm using data at 11 tide stations. 

Key Words: tides, tidal datums, Puget Sound, North American Vertical Datum of 1988, mean 

sea level, spatial interpolation, coast line. 

 v
 vi
1. INTRODUCTION 

 

The National Ocean Service (NOS) is developing a vertical datum transformation tool called 

VDatum (Milbert, 2002; Parker, 2002). For previous VDatum grids (Hess, 2001), a tide model 

was used to determine which VDatum points represented water and which represented land. If a 

Vdatum point was within a specified distance from a tidal data point, the point was considered to 

be water, and the datum values for that point were taken from the tide model. However, points in 

water areas where there were no tidal values were left unfilled. A new approach for the 

generation and population of the VDatum grid is described herein and applied to Puget Sound. In 

the new approach, the VDatum grid points are first determined to be land or water, based on their 

relationship to the digitized coastline, and then the tidal data are searched for a value. 

 

2. TIDAL DATUM FIELDS 

Tidal datum fields for Puget Sound, Washington, were generated on a 0.125 nautical miles (nmi) 

tidal grid using TCARI (Hess, 2002; 2003). The grid was generated using NOAA’s National 

Geophysical Data Center’s (NGDC’s) digitized, high-resolution shoreline. The method for 

generation of the datum fields is discussed in Hess and Gill (2003) (see Appendix A), but the 

final tidal datum fields described here were re-generated using tidal values updated by NOS’ 

Center for Operational Oceanographic Products and Services (CO-OPS) to the most recent 

National Tidal Datum Epoch, 1983-2001 (see Appendix B). 

3. DIGITIZED SHORELINE 

Initially, the NGDC shoreline was used (as it was for the tidal datum fields) to create the 

VDatum grid. However, a more recent digitized shoreline is available from CSDL’s Extracted 

Vector Shoreline (EVS) project. The new shoreline differs from the older shoreline (Figure 1) 

due to physical changes in the coast and because newer charts have been updated using 

geographical positions revised using GPS. Both sets of data use the NAD 83 horizontal datum. 

The EVS shoreline for Mean High Water (MHW) from NOS Chart 18440 (‘Puget Sound’), valid 

for 2002, was reformatted (using rd_evs.f), and subject to the concatenation process (using 

concat.f) to connect short line segments. This shoreline is shown in Figure 2. In the EVS data 

file, geographic positions are in decimal degrees, with six places to the right of the decimal point, 

and line segments are separated by records containing two zeros. Notice that chart borders were 

included as ‘coastline’ in the EVS for this chart, as were some bridges. A digitized Mean Lower 

Low Water (MLLW) shoreline is also available. 

Since the original shoreline data contained numerous small lakes, which were not needed for the 

project, they were manually removed to create a new digitized shoreline. That shoreline is shown 

in Figure 3. 
 

Figure 1. Sample coastline in Puget Sound showing the NGDC shoreline (thin line) and the newer, EVS 

shoreline (thick line). In this figure, the maximum difference in position is approximately 0.1 nmi. 

Figure 2. Initial Puget Sound EVS shoreline for MHW as extracted from NOS Chart No. 18440. Note the 

presence of numerous small lakes; the next figure shores the revised EVS shoreline with the lakes removed. 

 2
Figure 3. Digitized shoreline for Puget Sound from NOS Chart No. 18440, revised to exclude lakes.

The next step was to create the ‘mainland’ polygon, a single, continuous, closed (i.e., with 

matching end points) line segment that encloses all the land area in the chart except for islands. 

The mainland polygon (Figure 4) is used as a reference landmass when determining whether

other closed line segments are islands or lakes. The polygon was created in several steps. The 

first step was to manually remove the bridges in the EVS that created a discontinuous body of 

water for the Sound. The bridges were in the Hood Canal at Port Gamble, in the lower Sound at 

Tacoma, and at Bremerton. The removal was accomplished by inserting new segment end points, 

removing the points representing the bridges, and re-concatenating the resulting segments. 

Several other bridges were allowed to remain since they connected small islands with the

mainland and did not cut off embayments from the Sound proper. 

Following the removal of bridges, the next step was to move the southern chart boundary line 

further south and the eastern chart boundary line further east. The main problem with these 

boundaries was that some rivers, which themselves are part of the shoreline, intersected the 

boundaries (Figure 5), thereby creating a discontinuity along a border. The segments of these 

rivers closest to the boundaries were removed, and then the borders were moved down and to the 

right. The resulting mainland polygon is shown in Figure 4. 

 3
 

Figure 4. The mainland polygon as formed by a closed boundary segment.

 

Figure 5. Line segment with numbered points that defines a river intersecting the chart border. 

 4
The final step was to search the coastline file to identify islands and any remaining lakes using

the mainland polygon. The points defining the mainland polygon were renumbered so that the 

points appear in counterclockwise (CCW) order (using program landolakes.f). Each remaining

closed line segment was analyzed as follows. For the latitude of a point in the segment, the

closest point at the same latitude in the mainland polygon was found. Since the mainland 

polygon’s points are numbered CCW, the orientation of the mainland polygon indicates whether 

the segment is inside or outside the mainland polygon. If the line segment is inside the mainland 

polygon, it must enclose water (i.e., be a lake); if outside, the line segment must enclose land 

(i.e., be an island). 

4. THE BOUNDING POLYGON 

Points in the VDatum grid were further constrained by allowing water, or non-null, tidal values 

only within a bounding polygon. The bounding polygon can be used, for example, to exclude 

tidal areas that may be within the coastline rectangle but for which no tidal data are available.

The bounding polygon created for the Puget Sound region is shown in Figure 6. 

Figure 6. Bounding polygon (solid line) and mainland polygon (gray area) for Puget Sound.

 5
5. VDATUM GRID POINTS 

The VDatum grid for the Puget Sound area was generated by a new program, vgridder.f, which 

is a modification of the TCARI grid program pa.f. Here, the grid origin is at latitude0 and 

longitude0, and extends to latitude1 and longitude1. The longitudinal spacing between points is 

delx, and the latitudinal spacing is dely. The VDatum grid consists of points as defined by

longitudei = longitude0 + (i - 1)*delx 

latitudej = latitude0 + (j - 1)*dely 

 

where the index i denotes longitude and index j denotes latitude. The range of i is 1 to imax and 

the range of j is 1 to jmax, where 

imax = ( longitude1 - longitude0 )/delx 

jmax = ( latitude1 - latitude0 )/dely 

For Puget Sound, the grid origin is at latitude0 = 47.0167 (47° 1' N), longitude0 = -123.1833 

(123° 11' W). The upper right corner is at latitude1 = 48.1833 (48° 11' N), longitude1 = -122.1667 

(122° 10' W). The spacing between points is delx = 0.0025 deg, dely = 0.0018 deg. This results 

in a distance between points of approximately 200 m (0.1 nmi). 

Each point in the VDatum grid is designated as being either a null point (i.e., having no valid 

tidal datum transfer values) or a non-null point. Several steps were required to make that 

determination. Note that for the following discussion, an ‘A cell’ is defined as the rectangular 

area, centered on a VDatum grid point, whose corners are the centers of the four surrounding ‘B

cells’ (Figure 7). A B cell is defined as the rectangular area whose corners are VDatum points. 

In the first step in identifying null and non-null points, each point is checked to determine 

whether it lies within the bounding polygon. Then, if it does, the A cell in the VDatum grid is 

checked for the possibility of containing shoreline. Specifically, if any point along the border of

the A cell lies within the water area, the VDatum point is designated as water, or non-null. This 

means that a VDatum point can lie a distance as much as 1⁄2delx or 1⁄2dely on the land side of the 

shoreline and still be considered to in the water. 

Next, all B cells are checked to determine whether at least one corner is an non-null point; if so, 

then the B cell is considered to be water. Then, if a VDatum point is surrounded by four water B

cells, that point is made non-null. Finally, one or more layers of water cells can be added to the B

cells, and non-null VDatum points subsequently added if they are surrounded by water B cells. 

A sample of the VDatum grid with no added layers is shown in Figure 8, and the final grid for

the entire Puget Sound area is shown in Figure 9. 

 6
j+1 + + + + + 

 * - - - - - - - - - * 

 : : 

j + : + : + - - - - - - - - -+ + 

 : A cell : : : 

 * - - - - - - - - - * : B cell : 

 : : 

j-1 + + + - - - - - - - - -+ + 

 i-1 i i+1 i+2 i+3 

Figure 7. Schematic showing the VDatum grid points (+ symbols), an 'A cell' and a 'B cell'. The A cell is 

centered on the VDatum grid point, while the corners of the B cell are the VDatum points. The symbol * 

denotes a corner of an A cell, i is the longitude index, and j is the latitude index. 

Figure 8. Portion of the VDatum grid for Puget Sound. The small dark squares show locations of non-null 

values and the '+' symbols show locations of null values. The curved line shows the MHW coastline, white 

areas show water (i.e., locations where a non-null tidal datum value will be found), and gray areas are land. 

 7
Figure 9. VDatum grid points for Puget Sound. The areas with null values are shown in gray, and the areas 

covered by non-null tidal datum values are shown as white within the gray area. 

6. POPULATION OF THE VDATUM GRID WITH TIDAL DATUMS 

The VDatum grid was populated (using the program vpop.f) with tidal datums using the fields 

previously generated using TCARI. For each non-null point in the VDatum grid, the tidal datum 

values within a circle of user-specified radius (here 0.02 deg) are selected and a weighted mean 

(using the inverse distance squared) is computed. After the non-null points are filled in the above 

manner, a search is made for unfilled points. For these locations, the datums from an adjacent, 

filled non-null point are used. 

The input tidal datums are referenced to mean sea level (MSL). The output files, which are also 

referenced to MSL, are for the datums of mean higher high water (MHHW), MHW, mean low 

water (MLW), MLLW, mean tide level (MTL) and diurnal tide level (DTL).

 8
7. COMPARISON OF VDATUM GRIDDED VALUES WITH OBSERVED DATUMS

Comparisons were made (using the program cp.f) of the tidal datum values for MHHW, MHW, 

MLW, and MLLW obtained from the VDatum gridded files and the datum values at both the 

water level stations and at the benchmarks. Both a root mean squared error (RMSE) and a 

standard deviation (SD) were computed for the four values at each location. 

For the water level stations, the datums from the TideSheet052 file (Hess and Wilson, in prep.) 

that were available for generating the tidal datum fields were used (Appendix B). The 

comparison at 68 stations show a mean RMSE of 0.2 cm and a mean SD of 0.2 cm (Appendix

C). The maximum RMSE, 2.9 cm, and maximum SD, 2.7 cm, occurred at the same location: 

station 9445293 at Pleasant Harbor on Hood Canal. Pleasant Harbor was too small a feature to be 

included in the tidal datum grid, so tidal datum values there were taken from adjacent VDatum

points. 

For the datums at the benchmarks, data from NGS were used. Note that the new method of 

generating the VDatum grid locations allows for non-null VDatum points to occur on land that is 

within a small distance (about equal to the grid point spacing) of the shoreline. The comparison 

at 67 locations show a mean RMSE of 0.6 cm and a mean SD of 0.6 cm (Appendix D). The 

maximum RMSE, 6.6 cm, and maximum SD, 2.6 cm, occurred at the same location: station 

9444900 (or PID of TR0559) at Port Townsend. These errors apparently are due to the presence 

of older tidal data in the NGS database. 

8. SUMMARY OF THE VDATUM GRID GENERATION PROCESS 

The following table summarized the process of generating the grid and populating the grid with 

tidal datum values. 

Table 1. Summary of steps in production of the VDatum grid. 

 Computer

Step Action Program

1 Obtain digitized shoreline (preferably EVS) rd_evs.f 

2 Concatenate shoreline concat.f 

3 Clean up segments clean.f 

4 Remove bridges -

5 Make closed, long segment -

6 Make CCW, Remove Lakes landolakes.f 

7 Create bounding polygon -

8 Create grid points vgridder.f 

9 Fill points with tidal datum values vpop.f 

10 Compare with CO-OPS and NGS data cp.f 

 9
9. GENERATION OF THE GRIDDED TOPOGRAPHY OF THE SEA SURFACE 

VDatum uses a gridded Topography of the Sea Surface (TSS), which is the elevation of the 

North American Vertical Datum 1988 (NAVD 88) relative to local mean sea level (LMSL). The 

TSS required orthometric height relationships for the NOAA tide stations where elevation 

information has been compiled. The tide stations and associated elevation information used in

the computation of the TSS are presented in Table 2. The average NAVD 88-to-MSL offset was 

-1.291 m, and the maximum difference in the offset values 0.17 m. All data are based on the 

most recent National Tidal Datum Epoch (1983-2001). A positive value means that the NAVD 

88 reference value is further from the center of the Earth than the local mean sea-level surface. 

The values in the table show that the southern Sound (Tacoma, 9446484), MSL sits above 

NAVD 88 by 1.356 m, and in the northern Sound (Port Townsend, 9444900), MSL sits above 

NAVD 88 by 1.185 m. This difference probably reflects the fact that the southern Sound 

contains fresher, and therefore, less dense water, and so must have a higher sea surface to

maintain a horizontal pressure balance. 

Table 2. Location and elevation information for NOAA tide stations.

Station 

Number 

Latitude Longitude LMSL NAVD 88 NAVD 88 

-LMSL

9446969 47.0600 -122.9033 2.546 1.227 -1.319

9445958 47.5617 -122.6233 2.080 0.769 -1.311

9447110 47.5850 -122.3617 2.028 0.727 -1.301

9447130 47.6050 -122.3383 2.023 0.715 -1.308

9445133 47.7483 -122.7267 1.978 0.728 -1.250

9447427 47.8133 -122.3833 1.959 0.637 -1.322

9444900 48.1117 -122.7583 1.522 0.337 -1.185

9446484 47.2667 -122.4133 2.094 0.758 -1.336

9447659 47.9800 -122.2233 1.976 0.620 -1.356

9449424 48.8633 -122.7583 1.610 0.293 -1.317

9444122 48.1400 -123.4133 1.282 0.097 -1.185

A continuous surface (Figure 10) was produced by the minimum curvature interpolation method 

(Smith and Wessel, 1990), using data at the 11 tide stations in Table 2. This method creates a 

surface that honors the data as closely as possible. The grid point locations are the same as those 

used for the tidal datums. The maximum residuals value used was 0.00005 meters, and to control 

the amount of bowing on the interior and at the edges of the grid, internal and boundary tensions 

of 0.3 were utilized. The mean difference between the derived TSS grid and values at the tidal 

benchmarks where known relationships exist was 0.001 meters. It should be noted that tide 

stations leveled in NAVD 88 were not available in Hood Canal. Therefore, caution should be 

taken in using the TSS field in this area.

 10
Figure 10. The TSS field (m) for Puget Sound and the locations of the tide stations (denoted by filled 

triangles) used to generate the field of values.

 11
ACKNOWLEDGEMENTS

Digital coastline was provided by CSDL’s Annie Raymond, working in the Extracted Vector 

Shoreline project begun by CSDL’s Roger Johnson. Dennis Milbert of NOS’ National Geodetic 

Survey (NGS) provided insights into generating, populating, and testing the VDatum grid, and 

provided NGS benchmark data. 

REFERENCES

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

U.S. Department of Commerce, National Oceanic and Atmospheric Administration, Silver 

Spring, Maryland, NOAA Technical Report NOS CS 11, 43 pp. 

_____, 2002: Spatial interpolation of tidal data in irregularly-shaped coastal regions by

numerical solution of Laplace’s equation. Estuarine, Coastal and Shelf Science, 54(2), 175-

192. 

_____, 2003: Water level simulation in bays by spatial interpolation of tidal constituents, 

residual water levels, and datums. Continental Shelf Research, 23(5), 395-414. 

_____, and S. K. Gill, 2003: Puget Sound Tidal Datums by Spatial Interpolation. Proceedings, 

Fifth Conference on Coastal Atmospheric and Oceanic Prediction and Processes. Am. 

Meteorological Soc., Seattle, August 6-8, 2003. Paper 6.1, 108 - 112 (see Appendix A). 

_____, and R. Wilson (in preparation): TideSheet: An Astronomical Tide Database. 48 pp. 

Milbert, D.G., 2002: Documentation for VDatum (and VDatum Tutorial); Vertical Datum 

Transformation Software. Ver. 1.06 (http://nauticalcharts.noaa.gov/bathytopo/vdatum.htm). 

Parker, B. P., 2002: The integration of bathymetry, topography, and shoreline, and the vertical 

datum transformations behind it. International Hydrographic Review (3) 3 (November 2002). 

Smith, W.H.F., and P. Wessel, 1990. Gridding with Continuous Curvature Splines in Tension. 

Geophysics, 55(3), 293 - 305. 

 12
APPENDIX A. COPY OF AMS PAPER

6.1 PUGET SOUND TIDAL DATUMS BY SPATIAL INTERPOLATION

Kurt W. Hess* and Stephen K. Gill, National Ocean Service, NOAA 

1. INTRODUCTION

The National Ocean Service (NOS) of the 

National Oceanic and Atmospheric Administration 

(NOAA) requires tidal datum information such as

mean high water (MHW) and mean lower low

water (MLLW) to support nautical charting, 

navigational safety, shoreline photogrammetry,

and marine boundary determination. In addition, 

tidal datum information is needed for referencing 

NOS’ bathymetric data (which is referenced to 

MLLW) to any one of the other vertical elevation 

reference systems. A software tool under

development at NOS called VDatum (Milbert, 

2002) is designed to transform among 

approximately 30 vertical reference datums. To be 

applicable over coastal waters, VDatum requires

tidal datum fields, where the field describes the 

two dimensional, horizontal variability of the datum

elevation. Tidal datum fields for VDatum have 

been produced by NOS for Tampa Bay and 

coastal southern Louisiana, the New York Bight, 

central coastal California, and Delaware Bay

(Hess, 2001). Once VDatum has been established 

for a region, data can be incorporated into 

integrated bathymetric-topographic Digital

Elevation Models for use in coastal GIS 

applications (Parker et al., 2001; Gesch and 

Wilson, 2002). VDatum will also be needed for

carrying out the kinematic-GPS hydrographic

surveying that NOS is planning to implement. 

NOS routinely collects water level 

observations at shore-based stations along U.S. 

coasts and analyzes them to produce tidal datums.

As described above, there is an important need to 

obtain two-dimensional tidal datum fields that 

cover the coastal waters between the water level 

stations. This paper discusses a method for

obtaining tidal datum fields in Puget Sound, 

Washington, by the method of spatial interpolation 

of tidal data. 

_____________________________________

 

*Corresponding author address: Kurt W. Hess,

Coast Survey Development Laboratory, Rm. 7826, 

1315 East-West Highway, Silver Spring, MD 

20910; e-mail: kurt.hess@noaa.gov.

2. TIDES AND DATUMS

Tidal datums at water level stations are 

elevation values that are determined from a time

series of observations. For stations located along 

the coasts of the U.S. (except for the Great 

Lakes), the analysis starts with the identification of

all the tidal extrema (highs and lows) in the record, 

and continues with the selection (within a 25-hour

time period) of the higher of the two highs and the 

lower of the two lows. If only one high water is

present in the time period, it is categorized as a 

higher high. Thus, for high water (for example), 

each day has either a high and a higher high, or a 

single higher high. The average of all the highs

and the higher highs is called the Mean High 

Water (MHW), and the average of just the higher

highs is called the Mean Higher High Water

(MHHW). The process for producing Mean Low

Water (MLW) and MLLW from the low waters is

similar. The average of the MHW and the MLW is

called the Mean Tide Level (MTL) and the average 

of the MHHW and the MLLW is called the Diurnal 

Tidal Level (DTL). Mean Sea Level (MSL) is the 

average of the hourly water levels. Where MSL is

not computed, the MTL or DTL can be used as

approximations. For further information on tidal 

datums, see Gill and Schultz (2001). 

Observations made in a limited time period are 

adjusted to represent equivalent values for a 19-

year National Tidal Datum Epoch (NTDE). The 

present NTDE of 1983-2001 was just implemented 

in April 2003 and replaced the previous 1960-1978 

NTDE period. This recent epoch will give more

accurate datums for locations where apparent sea 

levels are changing rapidly due to local land 

subsidence caused by mineral and ground water

extraction, isostatic rebound following the last ice 

age, or tectonic motion. 

Tidal datum values at NOS water level stations

are routinely computed and are available to the 

public in the form of the station benchmark sheets.

Within the Puget Sound, Washington, study area 

(between latitudes 47° 3' N and 48° 11' N, and 

longitudes 123° 11' W and 122° 10' W) there are 

69 stations with historical tidal datum values. 

Those used in this study are shown in Figure 1. In 

 13
this area, the value of the elevation of MHHW

above MSL varies from 0.8 m at the northern end 

to 2.0 m at the southern end, and the MLLW varies

from -1.6 m at the northern end to -2.5 m at the 

southern end. These changes are generally

correlated with changes in the range of tide. 

Figure 11. The Puget Sound study area with 

historical tide stations (squares), water cells in the 

computational grid (white area), the main axis of

the Sound (solid line, with several straight 

segments, in white area), and the main axis of

Hood Canal (dashed line). 

3. SPATIAL INTERPOLATION 

Spatial interpolation is used to generate the 

tidal datum fields. The interpolation method 

requires the datum field, f, to satisfy Laplace’s

Equation (LE), 

  (1) 2 f 0

and the solution is found numerically on a grid. No 

water depth data are used. The solution field 

matches the input data at the water level stations

and takes into account land forms by the use of a 

specialized land-water boundary condition: 

 







f

n

a

f

n  (2) 

where n is the normal direction, α is a constant, 

and the overbar signifies a local average. The 

interpolation method was shown to give useful 

digital representations of amplitude and phase 

distributions as produced by numerical models of

tidally dominated bays, as well as datum planes, 

such as the ellipsoidally-referenced MSL or the 

MSL-to-MLLW difference (Hess, 2003; Hess,

2002). 

The first step in the application of the 

interpolation method to Puget Sound, Washington, 

was to create a regular grid of square cells. A cell 

size of 231 m (0.125 nautical miles) was chosen to 

resolve many of the narrow passages in the 

Sound. This yields an undifferentiated grid with 

329 cells in the eastward direction and 560 cells in 

the northward direction. A digitized coastline, 

obtained from the web site of NOAA’s National 

Geophysical Data Center, was used to define the 

land-water boundary. A small number of natural 

passages were too narrow to be resolved 

automatically, and so were added by manual 

editing of 70 cells. The resulting grid (Fig. 1) has

37,280 water cells. 

The next step was to apply the LE 

interpolation to the required datum levels. Tidal 

datums for stations were used for boundary values

if the station was adjacent to water; of those 

available (69), 60 were used (Fig.1). The solution 

method for the LE, successive over-relaxation, 

required approximately 1,700 iterations to reach 

convergence for each of the datum fields (MHHW, 

MHW, MLW, and MLLW). Convergence was

defined as when the maximum change between 

iterations in the numerical solution at any cell was

less than 2.5 x 10-5 times the difference between 

the maximum and minimum input datum values. 

The spatially-interpolated tidal datums for MHHW, 

MHW, MLW, and MLLW for Puget Sound are 

shown in Figures 2 to 5, respectively.

 14
Figure 2. Contours of the interpolated MHHW tidal 

datum field (m).

4. DISCUSSION

The accuracy of the computations is, in 

general, difficult to assess since there are few

published datum fields based solely on 

observations. However, the accuracy was

estimated in the following manner. For the MHHW 

datum, 60 additional approximations of the datum

field were generated. To generate each new field, 

one of the 60 stations was removed as input, a 

different station for each new field. Then the value 

from each new field at the cell where the datum

input was removed was compared to the value at 

the same cell in the original field, where the datum

input was used. The root mean square (RMS),

average, and maximum differences (original value 

minus new value) are 3.4 cm, -0.3 cm, and 16.4 

cm, respectively. The maximum difference, at 

Bush Point on the west side of Whidbey Island (at 

48° 2.0' N and 122° 36.2' W), occurs in a region 

Figure 3. Contours of the interpolated MHW tidal 

datum field (m).

with a relatively large horizontal datum gradient 

(appx. 3 cm per km).

Another useful, but non-rigorous, assessment 

of accuracy was made by plotting the interpolated 

tidal datums along the axes of the main channels

(shown in Figure 1) and the datums at the nearby

tide stations (Figure 6). The results show that the 

interpolated solution varies little from the input 

data values. 

A third approach to assessing accuracy is to 

compare interpolated fields with the datums from

tidal hydrodynamic models. For example, a tidal 

model for Puget Sound has been applied to the 

task of computing tidal datum fields (Mofjeld et al., 

2002). In a comparison of the spatially interpolated 

solution and the modeled datum at 471 locations, 

the RMS difference for all datums was 5.2 cm. 

Statistics on the differences between the two 

models are shown in Table 1. 

 

 15
Figure 4. Contours of the interpolated MLW tidal 

datum field (m).

 TABLE 1 

Differences in tidal datum fields (interpolated 

minus hydrodynamic model-based values).

Tidal 

Datum

Avg. Diff. 

(cm)

RMS Diff. 

(cm)

Max. Abs.

Diff. (cm)

MHHW -5.8 6.4 11.8

MHW -2.6 3.3 7.5

MLW -1.3 2.9 10.9

MLLW -5.9 7.0 18.4

Figure 5. Contours of the interpolated MLLW tidal 

datum field (m).

Since hydrodynamic models incorporate the 

physics of water flow, they can produce datum

fields that more accurately represent bathymetric

influences. However, these models typically

require long periods of time (months to years) to 

calibrate and, because of the complexity of the 

flow field, usually do not exactly match the data at 

the tide stations. By contrast, spatial interpolation 

matches the data at the tide stations but involves

no tidal physics. NOS is exploring the approach of

taking the results from hydrodynamic models,

where they exist, and then spatially-interpolating 

the errors to produce final, corrected datum fields. 

Finally, since datum values at locations other

than water level stations are difficult to obtain, 

especially offshore, NOS has plans to develop 

rapidly-deployable buoys with GPS positioning 

capability, but these would not be ready for a few

years. 

 16
Figure 6. For four tidal datums, the interpolated 

elevations are shown along the main channel 

(solid line) and data from nearby tide stations

(squares), and along Hood Canal (dashed line)

and nearby stations (triangles). The main axis and 

Hood Canal axis are shown in Fig.1. 

5. SUMMARY AND CONCLUSIONS

The spatial interpolation of tidal datums is a 

fast and relatively accurate method of producing 

tidal datum fields. For Puget Sound, the MHHW

field ranges from 0.8 m at the northern end to 2.0 

m at the southern end, and the MLLW ranges from

-1.6 m at the northern end to -2.5 m at the 

southern end. The relative accuracy of the MHHW

field was estimated to be 3.4 cm. In coastal 

regions where a tidal hydrodynamic model has

been calibrated, the modeled datum field can be 

corrected by interpolating the errors at the tide 

stations to produce a more accurate field. These 

datum fields will facilitate the development of the 

VDatum tool for the Puget Sound region. 

REFERENCES

Gesch, D., and R. Wilson, 2002: Development of a 

seamless multisource topographic/ bathymetric

elevation model for Tampa Bay. Marine 

Technology Society Journal, 35(4), 58–64. 

Gill, S. K., and J. R. Schultz, 2001: Tidal Datums

and Their Applications. Silver Spring, Maryland. 

NOAA Special Publication NOS CO-OPS 1, 111p. 

+ appendix. [Available from Center for Operational 

Oceanographic Products and Services, Rm. 6633, 

1325 East-West Highway, Silver Spring, Maryland 

20910] 

Hess, K.W., 2001: Generation of Tidal Datum

Fields for Tampa Bay and the New York Bight. 

NOAA Technical Report NOS CS 11, 43p. 

[Available from Coast Survey Development 

Laboratory, Rm. 7806, 1315 East-West Highway, 

Silver Spring, MD 20910] 

_____, 2002: Spatial interpolation of tidal data in 

irregularly-shaped coastal regions by numerical 

solution of Laplace’s equation. Estuarine, Coastal 

and Shelf Science, 54(2), 175-192. 

_____, 2003: Water level simulation in bays by

spatial interpolation of tidal constituents, residual 

water levels, and datums. Continental Shelf 

Research, 23(5), 395-414. 

Milbert, D.G., 2002: Documentation for VDatum

(and VDatum Tutorial); Vertical Datum

Transformation Software. Ver. 1.06 (http:// 

nauticalcharts.noaa.gov/bathytopo/vdatum.htm).

Mofjeld, H. O., A. J. Venturato, V. V. Titov, F. I. 

González, and J. C. Newman, 2002. Tidal Datum

Distributions in Puget Sound, Washington, Based 

on a Tidal Model. NOAA Technical Memorandum

OAR PMEL-122, 35pp. [Available from the Pacific

Marine Environmental Laboratory, 7600 Sand 

Point Way, Seattle, WA 98115] 

Parker, B.B., D. Milbert, R. Wilson, J. Bailey, and 

D. Gesch, 2001: Blending bathymetry and 

topography: the Tampa Bay demonstration project. 

Proceedings, U.S. Hydrographic Conference 2001, 

Norfolk, VA, The Hydrographic Society of America, 

11 pp. 

 17
 18
APPENDIX B. TIDE STATIONS AND TIDAL DATUMS

Table B.1. Tide stations and datums relative to MSL. Number is the 7-digit NOS station number. In the 

column with heading NB, the letter ‘a’ denotes stations not used in generating the tidal datum fields. 

N Number Latitude Longitude MHHW MHW MLW MLLW MTL DTL NB 

 1 9444705 48.0583 -122.9167 0.945 0.732 -0.731 -1.463 -9.999 0.000 a 

 2 9444900 48.1117 -122.7500 1.074 0.867 -0.762 -1.522 0.000 0.053 a 

 3 9445016 47.9267 -122.6167 1.253 0.997 -0.997 -1.844 -9.999 0.000 

 4 9445017 47.9250 -122.6800 1.220 0.976 -0.975 -1.798 -9.999 0.000 

 5 9445059 47.8583 -122.5800 1.280 1.006 -1.036 -1.859 -9.999 0.000 

 6 9445088 47.8150 -122.6583 1.332 1.061 -1.060 -1.932 -9.999 0.000 

 7 9445133 47.7483 -122.7267 1.396 1.125 -1.101 -1.978 0.000 0.012 

 8 9445246 47.7617 -122.8500 1.447 1.155 -1.159 -2.073 -9.999 0.000 

 9 9445269 47.7117 -122.8217 1.463 1.158 -1.158 -2.042 -9.999 0.000 

10 9445272 47.8000 -122.8583 1.414 1.156 -1.158 -2.054 0.000 -0.001 

11 9445293 47.6650 -122.9117 1.463 1.158 -1.189 -2.073 -9.999 0.000 a 

12 9445296 47.6417 -122.8267 1.444 1.155 -1.156 -2.070 -9.999 0.000 

13 9445326 47.6033 -122.9700 1.430 1.161 -1.158 -2.039 -9.999 0.000 

14 9445388 47.5083 -123.0517 1.415 1.125 -1.124 -2.051 -9.999 0.000 

15 9445441 47.4183 -122.9000 1.533 1.225 -1.225 -2.158 -9.999 0.000 

16 9445478 47.3583 -123.0983 1.497 1.198 -1.198 -2.112 -9.999 0.000 

17 9445526 47.9183 -122.5450 1.295 1.039 -1.043 -1.887 -9.999 0.000 

18 9445639 47.7967 -122.4933 1.375 1.116 -1.115 -1.975 -9.999 0.000 

19 9445683 47.7467 -122.4767 1.359 1.097 -1.098 -1.942 -9.999 0.000 

20 9445717 47.7250 -122.6383 1.487 1.219 -1.219 -2.091 -9.999 0.000 

21 9445719 47.7250 -122.6383 1.487 1.219 -1.219 -2.091 -9.999 0.000 

22 9445753 47.7050 -122.5250 1.463 1.188 -1.159 -2.012 -9.999 0.000 

23 9445832 47.6433 -122.6150 1.491 1.226 -1.225 -2.112 -9.999 0.000 a 

24 9445882 47.6200 -122.5150 1.402 1.158 -1.158 -2.042 -9.999 0.000 

25 9445901 47.6100 -122.6600 1.555 1.280 -1.280 -2.103 -9.999 0.000 

26 9445913 47.5967 -122.5100 1.463 1.189 -1.189 -2.042 -9.999 0.000 

27 9445938 47.5733 -122.5433 1.448 1.186 -1.185 -2.045 -9.999 0.000 

28 9445958 47.5617 -122.6233 1.498 1.230 -1.212 -2.080 0.000 0.009 

29 9445993 47.5233 -122.5167 1.494 1.219 -1.189 -2.042 -9.999 0.000 

30 9446025 47.5117 -122.4633 1.454 1.189 -1.188 -2.060 -9.999 0.000 

31 9446248 47.4000 -122.3283 1.475 1.204 -1.207 -2.079 -9.999 0.000 

32 9446273 47.3867 -122.4650 1.542 1.264 -1.253 -2.131 0.000 0.006 

33 9446281 47.3833 -122.8233 1.838 1.555 -1.554 -2.478 -9.999 0.000 

34 9446291 47.3783 -122.6400 1.707 1.433 -1.433 -2.286 -9.999 0.000 

35 9446366 47.3417 -122.7750 1.829 1.554 -1.555 -2.469 -9.999 0.000 a 

36 9446369 47.3400 -122.5883 1.494 1.250 -1.250 -2.103 -9.999 0.000 

37 9446375 47.3333 -122.5067 1.512 1.244 -1.240 -2.112 -9.999 0.000 

38 9446451 47.3017 -122.6817 1.744 1.460 -1.460 -2.365 -9.999 0.000 

39 9446484 47.2667 -122.4133 1.511 1.246 -1.224 -2.093 0.000 0.011 a 

40 9446486 47.2717 -122.5517 1.612 1.338 -1.341 -2.225 -9.999 0.000 

41 9446489 47.2817 -122.9233 1.838 1.554 -1.555 -2.475 -9.999 0.000 

42 9446491 47.2800 -122.6517 1.719 1.442 -1.442 -2.338 -9.999 0.000 

43 9446545 47.2550 -122.4317 1.503 1.235 -1.234 -2.106 -9.999 0.000 a 

44 9446583 47.2467 -122.8617 1.814 1.524 -1.524 -2.441 -9.999 0.000 

45 9446628 47.2150 -123.0833 1.996 1.691 -1.540 -2.332 0.000 0.076 

46 9446638 47.2100 -122.7533 1.768 1.493 -1.463 -2.347 -9.999 0.000 

47 9446666 47.1967 -122.9383 1.890 1.585 -1.585 -2.499 -9.999 0.000 

 19
Table B.1. (Continued). 

N Number Latitude Longitude MHHW MHW MLW MLLW MTL DTL NB 

48 9446671 47.1667 -122.7633 1.856 1.521 -1.521 -2.466 -9.999 0.000 

49 9446705 47.1800 -122.6750 1.757 1.475 -1.454 -2.351 0.000 0.011 

50 9446714 47.1733 -122.6033 1.741 1.463 -1.460 -2.368 -9.999 0.000 

51 9446742 47.1583 -123.0083 1.899 1.603 -1.607 -2.527 0.000 -0.002 

52 9446800 47.1417 -122.9033 1.859 1.576 -1.579 -2.521 -9.999 0.000 

53 9446807 47.1000 -122.8967 1.879 1.591 -1.598 -2.535 0.000 -0.004 

54 9446828 47.1183 -122.6650 1.750 1.466 -1.469 -2.368 -9.999 0.000 

55 9446969 47.0600 -122.9033 1.908 1.597 -1.597 -2.530 -9.999 0.000 

56 9447029 47.5350 -122.3217 1.417 1.143 -1.143 -1.966 -9.999 0.000 a 

57 9447110 47.5850 -122.3600 1.442 1.179 -1.161 -2.027 0.000 0.009 

58 9447111 47.5850 -122.3600 1.442 1.179 -1.161 -2.027 0.000 0.009 a 

59 9447130 47.6050 -122.3383 1.439 1.175 -1.159 -2.023 0.000 0.008 

60 9447131 47.6033 -122.3383 1.438 1.176 -1.158 -2.021 0.000 0.009 a 

61 9447265 47.6883 -122.4033 1.405 1.143 -1.147 -2.003 -9.999 0.000 

62 9447427 47.8133 -122.3833 1.365 1.106 -1.107 -1.960 -9.999 0.000 

63 9447659 47.9800 -122.2233 1.404 1.138 -1.122 -1.976 0.000 0.008 

64 9447725 48.0450 -122.2100 1.396 1.137 -1.140 -1.905 -9.999 0.000 

65 9447773 48.0650 -122.2883 1.432 1.158 -1.128 -1.951 -9.999 0.000 

66 9447814 47.9400 -122.3567 1.378 1.125 -1.125 -1.981 -9.999 0.000 

67 9447827 47.9783 -122.5500 1.280 0.951 -0.948 -1.792 -9.999 0.000 

68 9447854 48.0333 -122.6033 1.067 0.854 -0.853 -1.615 -9.999 0.000 

69 9447855 48.0267 -122.5433 1.454 1.183 -1.183 -2.033 -9.999 0.000 

70 9447856 48.0333 -122.3767 1.417 1.152 -1.152 -2.012 -9.999 0.000 

71 9447883 48.1050 -122.5700 1.432 1.158 -1.159 -2.012 -9.999 0.000 

72 9448094 48.1367 -122.3667 1.411 1.143 -1.162 -2.015 0.000 -0.009 

 20
APPENDIX C. NOS TIDE STATION NUMBERS AND NAMES.

Table C.1. NOS station numbers and names in the Puget Sound region. 

 N NOS Name 

 Station 

 Number 

 1 9444705 GARDINER LNDG PORST DISCOVERY WA 

 2 9444900 PORT TOWNSEND ADMIRALTY INLET WA 

 3 9445016 FOULWEATHER BLUFF TWIN SPITS WA 

 4 9445017 PORT LUDLOW ADMIRALY INLET WA 

 5 9445059 PORT GAMBLE HOOD CANAL WA 

 6 9445088 LOFALL WA 

 7 9445133 BANGOR WA 

 8 9445246 WHITNEY POINT WA 

 9 9445269 ZELATCHED POINT DABOB BAY WA 

 10 9445272 QUILCENE DABOB BAY HOOD CANAL WA 

 11 9445293 PLEASANT HARBOR HOOD CANAL WA 

 12 9445296 SEABECK HOOD CANAL WA 

 13 9445326 TRITON HEAD WA 

 14 9445388 AYOCK POINT WA 

 15 9445441 LYNCH COVE DOCK WA 

 16 9445478 UNION HOOD CANAL WA 

 17 9445526 HANSVILLE WA 

 18 9445639 KINGSTON APPLE TREE COVE WA 

 19 9445683 POINT JEFFERSON PUGET SOUND WA 

 20 9445717 POULSBO WA 

 21 9445719 POULSBO WA 

 22 9445753 PORT MADISON BAINBRIDGE ISLAND WA 

 23 9445832 BROWNSVILLE WA 

 24 9445882 EAGLE HARBOR BAINBRIDGE ISLAND WA 

 25 9445901 TRACYTON DYES INLET PUGET SND WA 

 26 9445913 PORT BLAKELY PUGET SOUND WA 

 27 9445938 CLAM BAY WA 

 28 9445958 BREMERTON WA 

 29 9445993 SOUTH COLBY-HARPER YUKON HBR WA 

 30 9446025 POINT VASHON WA 

 31 9446248 DES MOINES WA 

 32 9446273 BURTON QUARTERMASTER HBR VASHON WA 

 33 9446281 ALLYN WA 

 34 9446291 WAUNA CARR INLET PUGET SOUND WA 

 35 9446366 VAUGHN CASE INELT PUGET SOUND WA 

 36 9446369 GIG HARBOR PUGET SOUND WA 

 37 9446375 NEILL POINT WA 

 38 9446451 GREEN POINT WA 

 39 9446484 TACOMA COMMENCEMENT BAY WA 

 40 9446486 TACOMA NARROWS BRIDGE WA 

 41 9446489 WALKERS LANDING WA 

 42 9446491 ARLET WA 

 43 9446545 TACOMA WA 

 44 9446583 BALLOW WA 

45 9446628 SHELTON OAKLAND BAY WA 

 

 21
Table C.1. (Continued) 

 N NOS Name 

 Station 

 Number 

 46 9446638 LONGBRACH FILUCE B PUGET SND WA 

 47 9446666 ARCADIA TOTTEN INLET WA 

 48 9446671 DEVIL'S HEAD WA 

 49 9446705 YOMAN POINT ANDERSON ISLAND WA 

 50 9446714 STEILACOOM WA 

 51 9446742 BARRON POINT LITTLE SKOOKUM INLET ENT WA 

 52 9446800 DOFFLEMEYER POINT WA 

 53 9446807 BUDD INLET SOUTH OF GULL HARBOR WA 

 54 9446828 DUPONT NISQUALLY REACH WA 

 55 9446969 OLYMPIA WA 

 56 9447029 DUWAMISH RIVER (8TH AVE SOUTH) WA 

 57 9447110 LOCKHEED SHIPYARD TEST CONTROL WA 

 58 9447111 LOCKHEED SHIPYARD TEST SITE WA 

 59 9447130 SEATTLE PUGET SOUND WA 

 60 9447131 SEATTLE PUGET SOUND (BACKUP) WA 

 61 9447265 MEADOW POINT WA 

 62 9447427 EDMONDS WA 

 63 9447659 EVERETT WA 

 64 9447725 EBEY SLOUGH POSSESSION SOUND WA 

 65 9447773 TULALIP TULALIP BAY WA 

 66 9447814 GLENDALE POSSESSION SOUND WA 

 67 9447827 DOUBLE BLUFF ADMIRALTY INLET WA 

 68 9447854 BUSH POINT WHIDBEY ISLAND WA 

 69 9447855 HOLLY HARBOR FARMS WA 

 70 9447856 SANDY POINT SARATOGA PASSAGE WA 

 71 9447883 GREENBANK WHIDBEY ISLAND WA 

72 9448094 KAYAK PT PORT SUSAN WA 

 22
APPENDIX D. SUMMARY OF COMPARISONS WITH CO-OPS STATION DATA. 

Table D.1. For each station, the RMS error (RMSE) and standard deviation (SD) between the observed 

values of MHHW, MHW, MLW, and MLLW and the corresponding values obtained from the VDatum

gridded files. Under ‘Status’, an entry of ‘null’ means that no VDatum values were found. 

 n Number Latitude Longitude RMSE(m) SD(m) Status 

 1 9444705 48.05833 -122.91666 0.00000 0.00000 null 

 2 9444900 48.11167 -122.75000 0.00361 0.00330 

 3 9445016 47.92667 -122.61667 0.00218 0.00206 

 4 9445017 47.92500 -122.68000 0.00179 0.00167 

 5 9445059 47.85833 -122.58000 0.00062 0.00059 

 6 9445088 47.81500 -122.65833 0.00086 0.00083 

 7 9445133 47.74833 -122.72667 0.00113 0.00095 

 8 9445246 47.76167 -122.85000 0.00079 0.00070 

 9 9445269 47.71167 -122.82166 0.00147 0.00110 

 10 9445272 47.80000 -122.85833 0.00057 0.00056 

 11 9445293 47.66500 -122.91167 0.02884 0.02680 

 12 9445296 47.64167 -122.82667 0.00085 0.00077 

 13 9445326 47.60333 -122.97000 0.00128 0.00124 

 14 9445388 47.50834 -123.05167 0.00157 0.00155 

 15 9445441 47.41833 -122.90000 0.00049 0.00049 

 16 9445478 47.35833 -123.09834 0.00062 0.00062 

 17 9445526 47.91833 -122.54500 0.00227 0.00215 

 18 9445639 47.79667 -122.49333 0.00086 0.00085 

 19 9445683 47.74667 -122.47667 0.00532 0.00531 

 20 9445717 47.72500 -122.63834 0.00015 0.00013 

 21 9445719 47.72500 -122.63834 0.00015 0.00013 

 22 9445753 47.70500 -122.52500 0.00308 0.00297 

 23 9445832 47.64333 -122.61500 0.00000 0.00000 null 

 24 9445882 47.62000 -122.51500 0.00105 0.00085 

 25 9445901 47.61000 -122.66000 0.00033 0.00032 

 26 9445913 47.59667 -122.51000 0.00037 0.00036 

 27 9445938 47.57333 -122.54333 0.00140 0.00126 

 28 9445958 47.56167 -122.62334 0.00031 0.00030 

 29 9445993 47.52333 -122.51667 0.00293 0.00189 

 30 9446025 47.51167 -122.46333 0.00158 0.00101 

 31 9446248 47.40000 -122.32833 0.00033 0.00032 

 32 9446273 47.38667 -122.46500 0.00065 0.00064 

 33 9446281 47.38334 -122.82333 0.00013 0.00013 

 34 9446291 47.37833 -122.64000 0.00063 0.00061 

 35 9446366 47.34167 -122.77500 0.00695 0.00678 

 36 9446369 47.34000 -122.58833 0.00017 0.00017 

 37 9446375 47.33333 -122.50667 0.00133 0.00133 

 38 9446451 47.30167 -122.68166 0.00125 0.00123 

 39 9446484 47.26667 -122.41333 0.01680 0.01327 

 40 9446486 47.27167 -122.55167 0.00069 0.00069 

 41 9446489 47.28167 -122.92333 0.00000 0.00000 

 42 9446491 47.28000 -122.65166 0.00026 0.00025 

 43 9446545 47.25500 -122.43166 0.01237 0.01224 

 44 9446583 47.24667 -122.86166 0.00057 0.00057 

 

 23
Table D.1. (Continued). 

 n Number Latitude Longitude RMSE SD Status 

 45 9446628 47.21500 -123.08334 0.00113 0.00041 

 46 9446638 47.21000 -122.75333 0.00052 0.00040 

 47 9446666 47.19667 -122.93833 0.00146 0.00136 

 48 9446671 47.16667 -122.76334 0.00656 0.00656 

 49 9446705 47.18000 -122.67500 0.00111 0.00029 

 50 9446714 47.17333 -122.60333 0.00089 0.00086 

 51 9446742 47.15833 -123.00833 0.00044 0.00043 

 52 9446800 47.14167 -122.90334 0.00107 0.00098 

 53 9446807 47.10000 -122.89667 0.00070 0.00058 

 54 9446828 47.11833 -122.66500 0.00145 0.00144 

 55 9446969 47.06000 -122.90334 0.00007 0.00006 

 56 9447029 47.53500 -122.32166 0.00000 0.00000 null 

 57 9447110 47.58500 -122.36000 0.00020 0.00017 

 58 9447111 47.58500 -122.36000 0.00020 0.00017 

 59 9447130 47.60500 -122.33833 0.00022 0.00021 

 60 9447131 47.60333 -122.33833 0.00170 0.00141 

 61 9447265 47.68833 -122.40334 0.00052 0.00044 

 62 9447427 47.81333 -122.38333 0.00034 0.00033 

 63 9447659 47.98000 -122.22334 0.00034 0.00034 

 64 9447725 48.04500 -122.21000 0.00030 0.00029 

 65 9447773 48.06500 -122.28833 0.00147 0.00056 

 66 9447814 47.94000 -122.35667 0.00108 0.00095 

 67 9447827 47.97833 -122.55000 0.00173 0.00155 

 68 9447854 48.03333 -122.60333 0.00452 0.00448 

 69 9447855 48.02667 -122.54333 0.00000 0.00000 null 

 70 9447856 48.03333 -122.37666 0.00056 0.00049 

 71 9447883 48.10500 -122.57000 0.00025 0.00025 

 72 9448094 48.13667 -122.36667 0.00073 0.00038 

 

 error summary 

 number = 68 

 avg. rmse = 0.00203 

 avg. std = 0.00183 

 max. rmse = 0.02884 at i= 11 

 max. std = 0.02680 at i= 11 

 24
APPENDIX E. SUMMARY OF COMPARISONS WITH NGS BENCHMARK DATA. 

Table E.1. For each station, the RMS error (RMSE) and standard deviation (SD) between the observed 

values of MHHW, MHW, MLW, and MLLW and the corresponding values obtained from the VDatum

gridded files. The PID is the NGS’s identification number. Under ‘Status’, an entry of ‘null’ means that 

no VDatum values were found.

 n PID Number Latitude Longitude RMSE(m) SD(m) Status 

 1 SY0225 9447814 47.93916 -122.35666 0.00081 0.00062 

 2 SY0227 9447814 47.93916 -122.35666 0.00081 0.00062 

 3 TR0186 9447725 48.05138 -122.17916 0.00000 0.00000 null 

 4 TR0184 9447725 48.05361 -122.17555 0.00000 0.00000 null 

 5 TR0185 9447725 48.05277 -122.17722 0.00000 0.00000 null 

 6 SY0003 9447659 47.97944 -122.21555 0.00000 0.00000 null 

 7 SY0004 9447659 47.97944 -122.21583 0.00000 0.00000 null 

 8 SY0004 9447659 47.97944 -122.21583 0.00000 0.00000 null 

 9 SY0002 9447659 47.98055 -122.21527 0.00000 0.00000 null 

 10 SY0128 9447427 47.81138 -122.38250 0.00056 0.00021 

 11 SY0126 9447427 47.81305 -122.38027 0.00060 0.00025 

 12 SY0282 9447130 47.60194 -122.33250 0.00000 0.00000 null 

 13 SY0328 9447130 47.59944 -122.32861 0.00000 0.00000 null 

 14 SY0288 9447130 47.60416 -122.33361 0.00928 0.00138 

 15 SY0283 9447130 47.60194 -122.33388 0.00930 0.00137 

 16 SY0284 9447130 47.60166 -122.33500 0.00931 0.00137 

 17 SY0286 9447130 47.60305 -122.33444 0.00929 0.00137 

 18 SY0290 9447130 47.60388 -122.33750 0.00926 0.00141 

 19 SY0289 9447130 47.60444 -122.33500 0.00928 0.00138 

 20 SY0287 9447130 47.60305 -122.33583 0.00928 0.00138 

 21 SY0865 9446969 47.05166 -122.90250 0.00068 0.00059 

 22 SY0866 9446969 47.05194 -122.90388 0.00071 0.00062 

 23 SY0869 9446969 47.04583 -122.90055 0.00000 0.00000 null 

 24 SY0868 9446969 47.05000 -122.90138 0.00073 0.00063 

 25 SY0867 9446969 47.04972 -122.90083 0.00000 0.00000 null 

 26 SY0743 9446828 47.11888 -122.66305 0.00205 0.00164 

 27 SY0741 9446828 47.11888 -122.66305 0.00205 0.00164 

 28 SY0738 9446828 47.11694 -122.66500 0.00261 0.00234 

 29 SY0740 9446828 47.11694 -122.66444 0.00267 0.00240 

 30 SY0739 9446828 47.11750 -122.66500 0.00239 0.00207 

 31 SY0757 9446714 47.17250 -122.60166 0.00220 0.00048 

 32 SY0756 9446714 47.17111 -122.60111 0.00236 0.00061 

 33 SY0755 9446714 47.17388 -122.59694 0.00526 0.00441 

 34 SY0754 9446714 47.17250 -122.60111 0.00216 0.00046 

 35 SY0536 9446545 47.26416 -122.41194 0.01254 0.01236 

 36 SY0535 9446545 47.25888 -122.41916 0.00000 0.00000 null 

 37 SY0534 9446545 47.25416 -122.42916 0.00000 0.00000 null 

 38 SY0532 9446545 47.25500 -122.43361 0.01270 0.01253 

 39 SY0533 9446545 47.24916 -122.43444 0.00000 0.00000 null 

 40 SY0541 9446545 47.24583 -122.43444 0.00000 0.00000 null 

 41 SY2987 9446491 47.28277 -122.65805 0.00186 0.00180 

 42 SY0536 9446484 47.26416 -122.41194 0.01436 0.01420 

 43 SY0535 9446484 47.25888 -122.41916 0.00000 0.00000 null 

 44 SY0534 9446484 47.25416 -122.42916 0.00000 0.00000 null 

 45 SY0840 9446375 47.33222 -122.50555 0.00245 0.00179 

 46 SY0808 9446281 47.38972 -122.82361 0.00057 0.00049 

 

 25
Table E.1 (Continued). 

 n PID Number Latitude Longitude RMSE(m) SD(m) Status 

 47 SY0807 9446281 47.38972 -122.82361 0.00057 0.00049 

 48 SY0805 9446281 47.38361 -122.82527 0.00052 0.00044 

 49 SY0806 9446281 47.38500 -122.82666 0.00054 0.00045 

 50 SY0804 9446281 47.38444 -122.82750 0.00053 0.00043 

 51 SY0645 9446254 47.38722 -122.46166 0.00624 0.00076 

 52 SY0646 9446254 47.38638 -122.45722 0.00618 0.00163 

 53 SY0647 9446254 47.38638 -122.45611 0.00620 0.00179 

 54 SY2364 9446248 47.40277 -122.32833 0.00206 0.00062 

 55 SY0917 9445958 47.56361 -122.62916 0.00000 0.00000 null 

 56 SY0922 9445958 47.56305 -122.62416 0.01042 0.00164 

 57 SY0920 9445958 47.56305 -122.62527 0.01043 0.00165 

 58 SY0919 9445958 47.56361 -122.62694 0.01044 0.00165 

 59 SY0915 9445958 47.56277 -122.63055 0.00000 0.00000 null 

 60 SY0902 9445938 47.57111 -122.55111 0.00288 0.00262 

 61 SY0902 9445938 47.57111 -122.55111 0.00288 0.00262 

 62 SY0900 9445938 47.57138 -122.55166 0.00343 0.00316 

 63 SY0901 9445938 47.57583 -122.54833 0.00486 0.00464 

 64 SY4915 9445683 47.75583 -122.47000 0.01276 0.01269 

 65 SY1272 9445478 47.35694 -123.10305 0.00159 0.00159 

 66 SY1275 9445478 47.35722 -123.09944 0.00066 0.00063 

 67 SY1274 9445478 47.35694 -123.10083 0.00081 0.00079 

 68 SY1271 9445478 47.35722 -123.10305 0.00159 0.00159 

 69 SY1236 9445388 47.50750 -123.05750 0.00135 0.00113 

 70 SY1238 9445388 47.49972 -123.06055 0.00877 0.00865 

 71 SY5514 9445326 47.60305 -122.98222 0.00390 0.00379 

 72 SY1163 9445246 47.76138 -122.85000 0.00237 0.00072 

 73 SY1162 9445246 47.76194 -122.85000 0.00237 0.00082 

 74 SY1164 9445246 47.76138 -122.85000 0.00237 0.00072 

 75 SY0959 9445133 47.74666 -122.72638 0.01622 0.00789 

 76 TR2726 9444900 48.11138 -122.76166 0.06637 0.02576 

 77 TR0559 9444900 48.11583 -122.75194 0.06569 0.02565 

 78 TR2693 0 48.07916 -123.04416 0.00000 0.00000 null 

 79 TR0852 0 48.08027 -123.04527 0.00000 0.00000 null 

 80 TR0851 0 48.08027 -123.04527 0.00000 0.00000 null 

 81 TR0850 0 48.08000 -123.04527 0.00000 0.00000 null 

 82 TR0186 0 48.05138 -122.17916 0.00000 0.00000 null 

 83 TR0184 0 48.05361 -122.17555 0.00000 0.00000 null 

 84 SY0783 0 47.28250 -122.92250 0.00000 0.02565 

 85 SY0782 0 47.28333 -122.92305 0.00000 0.02565 

 86 SY0274 0 47.58416 -122.36138 0.00000 0.02565 

 87 SY0273 0 47.58416 -122.36250 0.00000 0.02565 

 88 SY0272 0 47.58194 -122.36277 0.00000 0.02565 

 89 SY0271 0 47.57833 -122.36277 0.00000 0.02565 

 90 SY0270 0 47.57444 -122.36277 0.00000 0.02565 

 91 SY0269 0 47.57916 -122.35388 0.00000 0.00000 null 

 92 SY0268 0 47.57500 -122.35416 0.00000 0.02565 

 

 26
Table E.1 (Continued). 

 

 93 SY0267 0 47.57888 -122.35444 0.00000 0.00000 null 

 94 SY0266 0 47.57694 -122.35500 0.00000 0.00000 null 

 95 SY0265 0 47.57111 -122.34472 0.00000 0.02565 

error summary 

 number = 67 

 avg. rmse = 0.00587 

 avg. std = 0.00627 

 max. rmse = 0.06637 at i= 76 

 max. std = 0.02576 at i= 76 

 27