Interpolation of Hydrographic Survey Data

Janet Burroughes, Dr Ken George and Dr Vic Abbott
Institute of Marine Studies, University of Plymouth, UK.

Abstract

This paper considers the interpolation of historic data, collected using a single beam echo sounder, onto a regular grid.  The use of conventional software provides a number of interpolation methods, including the application of Triangular Irregular Networks (TINs), Inverse Distance Weighting (IDW) and Kriging.  These methods are, however, found to produce artificial, interpolation artefacts when applied to sounding data, concentrated along lines which cross narrow, deep channels.  The paper develops an interpolation method to overcome this problem, based on the inverse distance weighting (IDW) technique.  The results obtained during testing of the method on historic, single beam, echo sounder data collected in the Truro River, south west Cornwall, UK are presented.  These results demonstrate significant success in reducing artificial artefacts of interpolation.

Introduction

This paper considers the interpolation of single beam, echo sounder data collected in the Truro River, south west Cornwall.  It forms part of a study into long term changes in the river and is based on surveys carried out over the past forty years, using a number of different surveying techniques. 

Bathymetric data are usually available in one of two forms:

1.   Published navigational charts, providing both spot soundings and depth contours, known as isobaths.  The isobaths may be reduced to a series of depth values by specifying the position and depth at frequent intervals along their length.

2.   Unpublished soundings charts or working drawings, comprising a larger number of spot soundings.  Frequently these soundings are concentrated along survey lines running approximately perpendicular to the direction of the depth contours, with relatively large areas devoid of depth information existing between each line.

For several applications, depth information needs to be specified on a grid (often a rectangular grid).  These include:

1.   The comparison of bathymetric data sets from diverse sources.

2.   Analysis of temporal trends.

3.   Numerical modelling using finite differences.

This process involves the application of a method of spatial interpolation between the soundings.  In order to achieve this interpolation at a specified point, one of a variety of methods must be applied.  The details of this process vary from method to method; three commonly used methods are as follows:

1.   Inverse Distance Weighting (IDW), in which a radius of interpolation is defined around each specified point.  All data values, linearly weighted, within this radius are then used to contribute to the value assigned at the point.  The weighting applied to each value is inversely proportional to the distance of the sounding from the point.  Since the output values form a weighted average they cannot be greater than the highest or less than the lowest input value. Hence, IDW cannot create ridges or valleys (Watson & Philip, 1985).  Furthermore, as the influence of each input point on the interpolated values is distance related, IDW will not preserve ridges or valleys, (Philip & Watson, 1982).

2.   Triangular Irregular Network (TIN), where the domain is divided into triangles, with the criterion that each triangle should be as close to equilateral as possible.  This method maintains existing sounding values, with interpolation between these data points being performed along the edges of the defined triangles.

3.   Kriging is a statistical interpolation method, based on regionalised variable theory.  This method assumes that the spatial variation in depth values exhibits the same pattern of variation over all parts of the surface.   The mathematical function to be applied during kriging is chosen by consideration of the spatial variation of depth values within a particular data set.  This is achieved by comparing graphs of semi-variance of the actual data with those of data values predicted by each mathematical function, plotted against the distance between pairs of data points.  These graphs are known as semi-variograms.  For more details of the kriging interpolation method see Oliver (1990).

Application to the Truro River

The Truro River forms one of the major tributaries flowing into the Fal estuary, in southwest Cornwall, UK.  The area of the Truro River of particular interest to commercial navigation stretches northwards from its confluence with the Fal estuary in the south, to the tidal limit of the river, situated in the centre of Truro.  The relative locations of the Fal Estuary and its tributaries are shown in Figure 1.

The upper section of this stretch of river varies in width from less than 40m to more than 200m.  It practically dries out at low water, revealing extensive mud flats on either side of a steep-sided channel some 30m wide.  This channel contains a stream of river water, which continues to flow even at extreme low tide.  The bathymetric data collected in the Truro River, and supplied by Truro Harbour Master, takes the form of a large number survey lines forming cross sections across the river channel.  The data was supplied at a scale of 1:2500, giving an average along line sounding spacing of 11m and between line intervals of about 40m.  Additional soundings have been input, manually along the centre line of the main channel.

It is evident that in order to obtain reasonable results from lines of soundings spaced 40m apart, a radius of 50m is appropriate for use in a method of spatial interpolation.  Since the channel is only approximately 30m wide, the use of a circular zone of interpolation around each specified point means that averaging will take place across the channel.  The extreme case is a point in the bed of the channel, whose depth will be determined by averaging depths on the slopes on either side, as well as soundings adjacent to it within the channel.  Consequently, the averaged depth assigned to points in the vicinity of the channel will be unrealistically shoal.  The overall effect of this limitation with each interpolation method is to produce a series of artificial ridges across the channel, the centre of which correspond with the centre of the 40 metre ‘gap’ between lines of soundings (Figure 2).

Overcoming the Interpolation Problem

Initial attempts to eliminate this interpolation problem involved locating the line of maximum depths along the channel.  Additional data points were added to the sounding data along this line, by means of manual interpolation between adjacent deepest depths.  The IDW interpolation model was then reapplied to this manually improved point data file.  This method exhibited a moderate degree of success in reducing interpolation artefacts, but some artificial ridges of smaller horizontal extent and lesser depth differential remained after the interpolation process.  The results are displayed in Figure 3.

At University of Plymouth, a program is being developed to completely overcome this interpolation problem.  Within this program each sounding in the data set is assigned to one of three categories.  Bearing in mind the general trend of the river channel is north south, the categories are assigned as follows:

1.   W = west of the channel bed

2.   C = in the channel bed

3.   E = east of the channel bed

Each point in the rectangular grid on to which points are to be interpolated is similarly assigned to one of the three zones W, C or E.  The allocation of zones to the sounding data is illustrated in Figure 4.

To date, these zones have been assigned by manually editing soundings and grid cells within the data files.  Clearly for large and/or multiple data sets it would be desirable to devise a method of automating this process. 

Interpolation was performed by the method of inverse distance weighting, with an interpolation radius of 50m, but taking the zones into account, thus:

Zone of Specified Point                                Zone of Sounding

West                                                             West or Channel

Channel                                                         Channel

East                                                              Channel or East

The success of zoning the channel and mud bank areas in the interpolation process is clearly shown in Figure 5.   It can be seen that allowing for the channel by zone allocation removes artificially interpolated ridges across the channel otherwise produced by interpolation (previously illustrated in Figure 2).

Discussion and Conclusions

The results of interpolation of lines of soundings onto a regular, rectangular grid were found to be far more realistic, particularly when the existence of the narrow, relatively deep channel was specifically taken into account during the interpolation process. The characteristics of fine, tide washed sediments, such as those found in the Truro River, would be inconsistent with the formation of the series of holes and ridges placed in the channel by interpolation without specific treatment of the existence of this channel.  The zoning method proposed is intended to eliminate this problem successfully.

Currently the program developed at the University of Plymouth requires manual assignment of the zones (west, channel or east), in order to perform this refined method of data interpolation.  To allow the program to be easily applied to large and/or multiple data sets further development would be desirable to automate the zone assignment process. Trend analysis using a number of historic data sets, from the Truro River, illustrates an application which would benefit from the automation of zone assignment within the program.

References

Oliver, M.A.  ‘Kriging: A Method of Interpolation for Geographical Information Systems’.  International Journal of Geographic Information Systems 4: no. 4, pp 313-332, 1990.

Philip, G.M. and Watson, D.F., (1982). ‘A Precise Method for determining Contoured Surfaces’, Australian Petroleum Exploration Association Journal 22: pp 205-212.

Watson, D.F. and Philip, G.M., (1985). ‘A Refinement of Inverse Distance Weighted Interpolation’, Geo-Processing, 2: pp 315-327.

Fig. 1: Relative locations of the Fal estuary and Truro River

Fig. 2: Interpolation of bathymetric data without

Fig. 3: Interpolation of bathymetric data with manual interpolation of soundings along the centre line of the channel

Fig. 4: Allocation of the three zones to bathymetric data in part of the upper Truro River

Fig. 5: Interpolation of bathymetric data using Zoned Inverse Distance Weighting to allow for the channel