Continuous Water Level Zoning For Hydrographic Surveys

Hydrographic survey data collected by the National Ocean Service (NOS) contain measurements of water depths in the coastal waters of the U.S. In order for these depths to be used in the making of the nautical charts, the measured depth value must be corrected to determine depth relative to Mean Lower Low Water (MLLW), which is the datum for NOS charts. The departure of the instantaneous water level from MLLW is called the tide correction and is due to variations in the phase of the astronomic tide, the difference between the long-term Mean Sea Level (MSL) and MLLW, and to other, non-tidal effects caused primarily by winds and river flows.

The schematic to the right shows the elevations and depths that are important in hydrographic surveying. The instantaneous Sea Surface differs from the MSL by the sum of the astronomic tide (A) and the residual or non-tidal water level (R). The MSL lies above the MLLW by the spatially-varying offset (O). S is the raw depth measurement, or sounding, and the tide correction is C. The depth displayed on a NOS chart is D. Recently, the Global Positioning System (GPS) has been used to make hydrographic surveying more accurate. Elevations measured using GPS are commonly referenced to one of several possible ellipsoids. The ellipsoidally-referenced sea surface is E, and the ellipsoidally-referenced MSL is M.

Today, discrete tide zoning is the method NOS uses to obtain the tide correction. For any location (x, y) and time (t) in a tide zone (k), the tide correction (C) is estimated by shifting the water level measured at one tide station (L_n) by a time increment unique to that zone (T_k) and multiplying the result by a factor unique to that zone (f_k). In equation form, this is:

However, discrete tide zoning has several known inaccuracies. Tide zoning rests on the simplifying assumption that the water level in each zone has a fixed relationship to the measured water level at a single, nearby gauge. In reality, the shape of the measured water level curve varies throughout coastal waters in a complex way. Also, zoning produces a discontinuity in the tide correction when a ship crosses from one zone to the next. Therefore, a new method of estimating tide corrections is being developed. This method does not depend on discrete tide zones, but instead generates a unique value of the correction for the exact time and location of the sounding; it can be thought of as continuous tide zoning. The new method separates the astronomic tide and the residual (or non-tidal) component and treats them differently. First, the method spatially interpolates each tidal constituent's amplitude and phase throughout the region and makes a tidal prediction. This predicted tide is then added to the residual component and the offset, which are computed by spatially interpolating the values at the shore. The new method is called Tidal Constituent And Residual Interpolation (TCARI).

The theory behind TCARI is as follows. In general, the tide correction (C) at any point in the hydrographic survey area can be expressed as the sum of the astronomic tide (A), the residual (or non-tidal) water level (R), and the offset of MSL from MLLW (O), or:

At a water level gauge, the value of each of the three components can be determined from measurements; A and O can be obtained by the analysis of historical water level data, and R must be determined from the observations at the time of the hydrographic survey. However, at any point located some distance away from any of the gauges, the value of each component is usually not known. Using TCARI, an estimate can be made by spatial interpolation. That is, by summing the weighted value of the component at each of the nearby water level gauges. For example, the residual water level is determined by summing the weighted value of the residual (R_n) at each gauge:

Here W^r is the weighting function and N_r is the number of stations where the residual water level is measured. Therefore, a set of weighting functions is needed, one weighting function for each water level station. And because there is often a different group of water level stations supplying data for each component of C, a different set of weighting functions is needed for each component. Therefore, a preliminary expression for C is:

A further refinement is to interpolate not the total astronomical tide, but the harmonic constants of each tidal constituent. Given an amplitude (a) and phase or epoch (f), the total astronomic tide can be expressed as:

where i denotes the number of the tidal constituent, Imax is the total number of constituents, and w the angular speed of the constituent (in degrees per hour). Therefore, the final resulting equation is:

The above equation for the tide correction can be applied to any location within the hydrographic survey region, provided there are adequate historical measurements of the water levels to determine the harmonic constants (a, f) and the offsets (O), and contemporary measurements to determine the residual (R) during the time of the survey.

The TCARI method is applied by creating a grid for the coastal region of interest and then generating the weighting functions for each set of components. As an example, a plot of the instantaneous astronomic tide in Galveston Bay, as determined by summing the spatially-interpolated harmonic constants, is shown in the figure to the right. The grid consists of squares, one-third of a nautical mile on a side. The astronomic tide is shown relative to MSL.

The TCARI method was tested for accuracy using post-processed kinematic GPS measurements of water levels collected by NOS in San Francisco Bay, California, and Galveston Bay, Texas. In each estuary, the water levels were measured by a small craft on several traverses of a fixed course and then referenced to the ellipsoid. The measurements themselves had root mean square (RMS) errors estimated to be 7 cm for San Francisco Bay and 9 cm for Galveston Bay. The results indicate that TCARI was more accurate tide zoning; RMS errors were lower than zoning by 1 cm in San Francisco Bay and 2 cm in Galveston Bay. However, both methods had RMS errors approximately equal to those of the measurements.

The weighting functions are computed as follows. Each function is assumed to obey the two-dimensional Laplace's Equation:

That is, W = 1 at gauge 'n' and W = 0 at all other gauges. There will be one function W(x, y) for each gauge used for each component. At land boundaries, the slope of W in the normal direction is proportional to the slope of W in the adjacent region.

There are other applications for TCARI. One is to estimate the ellipsoidally-referenced sea level (E) by substituting an ellipsoidally-referenced Mean Sea Level (M) for the offset O in the final equation for C. This sea level can be used for comparisons with Real-Time Kinematic GPS measurements of the sea surface. A second application is to use the weighting functions to generate spatial distributions of the differences between tidal datums (for example, the Mean High Water minus Mean Low Water, which equals the mean tide range). Such spatially-varying tidal datum planes are useful in re-referencing NOS chart depths to other datums such as the North American Vertical Datum of 1988 (NAVD88).

Please send comments or questions to:
   Dr. Kurt Hess
NOAA/NOS/Coast Survey Development Laboratory
Phone: (301) 713-2801 ext. 123
Fax: (301) 713-4501

 Last updated March 14, 2001
Constructed and maintained by Meredith Westington