Vella and SesyCeoinformation Science Journal, Vol. 3, No. l,pp. 24-28

Gridding Digitized Bathymetry in theStraits of Malacca

Muhammad Nur Jason P. Vella and Shahrum Ses

Depariment of Geomatics EngineeringFaculty of Ceoinformaiion Scienceand Engineering

Universi ti Technologi MalaysiaSkudai, 81310

Johor Malaysiashalzrulll@Jksg.utm.my

Abstract

Maritime charts in the form of bathymetry from the Straits of Malacca were digitised to form the data setthat was operated on by the Kriging algorithm to provide gridded digital data sets. There were two forms ofalgorithms tested, the Kriging and the Minimum Curvature, of these two Kriging was found to be the mostsuitable. Along with these two algorithms different grid sizes were also experimented with, these were 1minute and 5 minutes (I' & 5'). The I' grid size and the Kriging were found to be the best combination forthe gridding of the digitised bathymetry.

The bathymetry was digitised 'on screen' using Golden Software's Didger3. The maps were of mixed scales1:200000 and 1:300000, scanned at a resolution of 150 dpi, this was done in order to decrease the size of thedigital file, namely the TIF format. Affine polynomial calibrations were performed on all charts and werewithin acceptable limits. A total of 18357 points were digitised induding soundings and coastlines, the mini­mum height being -358.86 m the maximum being 0 m. All soundings were transformed to Mean Sea Level(MSL) using the tidal information published by the Department of Surveying and Mapping Malaysia (JU­PEM) Tide Tables for the year 1998.

A comparison of the gridded (Kriging) digitised points is accomplished through comparing an independentset of digitised points (approximately 10%) with the gridded digitised data. Also a comparison with ETOP05is made, the results are as follows: a mean of the differences of -0.332 m and the standard deviation of themean of ±7.784m, for the digitised data set and a mean of the differences of -0.281 m and the standard devia­tion of the mean of ±15.812m for ETOP05. This tends to show that ETOPOS is not as representative of thebathymetry of the Straits of Malacca and that the gridding process followed by the authors is suitable enoughfor the data to be used in other applications, such as tidal studies or current analysis for offshore and coastalengineering processes.

Key words: Digitising, Kriging, Minimum Curvature.

1. Introduction

Most branches of science and engineering requiresome form of way to derive an ordered data setfrom randomly distributed data. In this instanceit is the need to acquire an ordered data set fromrandomly distributed soundings on maritime charts

of the Straits of Malacca. The subject area rangesfrom Latitude south IOta Latitude north 6.5° andLongitude east 98° to Longitude east 103.5°. Onscreen digitisingis used as a way of capturing thesoundings, this is more efficient and convenient than

25Geoinformation Science Journal, Vol. 3, No.1

the usual form of tablet digitising, but nevertheless of 18357 points were digitised including soundingsdoes have its drawbacks. and coastlines, the minimum height being -358.86 m

the maximum being 0 m.Before any type of gridding is carried out or for thatmatter digitising, it is essential to fully understandthe use of the data set once completed. In this caseit is to be used to derive accurate tide models ofthe Straits of Malacca. This procedure requires atleast a I' evenly gridded data set, globally there isa 5' data set ETOP05. This data set ETOP05 couldeasily have been used and saved a lot of time andeffort, however it has been shown that ETOP05 isnot necessarily that reliable in the subject area. Afterdeciding on a suitable grid size for the results, itis necessary to find suitable data sources that canprovide such results. The maritime charts werechosen, as they are the most easily accessible datasource for soundings. It was not possible to attain theraw soundings from the Navy as they are regardedas sensitive data not meant for public domain. In all,five charts of the Straits of Malacca were purchasedand from these charts it was possible to determinethat the soundings were all approximately spacedat least T to 2' apart. Therefore this data source issuitable for on screen digitising.

The data were all digitised on screen, after scanningthe charts. Once the scanning had been accomplished,the charts were then calibrated and the digitisingcarried out. This process was lengthy but muchquicker than had conventional table digitisingmethods been used. The data were gridded at severaldifferent combinations in order to determine whichcombination was the most suitable. Comparisonswere made to ETOP05 and an independent data setnot included in the gridding process.

2. Digitizing

The five charts used for digitising were, starting fromthe most southern MAL 521, MAL 532, MAL 540,MAL 553 and MAL 565, all these charts overlappedeach other by at least 5% in the south eastern corner.Positions are based on the Revised Kertau Datum andthe soundings are based on the Lowest AstronomicalTide (LAT). The datum for the soundings to be usedin the tidal modelling process is Mean Sea Level(MSL), the soundings were converted to MSL byapplying a constant offset to each different chart,and this offset was attained from the nearest tidegauge benchmark to the particular chart in question.The positions after being digitised were convertedto WGS84, these positions are used in the griddingprocedures. Figure 1, shows the distribution of thedigitised soundings and the respective charts. A total

Figure 1: The digitised data used in the griddingprocess.

Before on screen digitising occurs the charts mustbe converted to digital form, this is done using theScan Plus III 813SC digital scanner. For this purposethe charts were scanned in black and white onlyand at a low resolution so as to save memory, thishelps when using the on screen digitising software.Once this is completed the charts are ready to becalibrated by the software, via transformation. Themost common transformation is Affine Polynomial.Affine transformations are a subset of bilineartransformations. Bilinear transformations accountfor rotation, shift, and differential scaling in X andY. The X and Y axis orthogonality may change, butparallel lines remain parallel. Affine polynomialtransformations of a plane change squares intoparallelograms and change circles into ellipses of thesame shape and orientation. For a discussion of thisand other techniques refer to Wolberg (1990). Theaffine polynomial transformation results may not beas good as when using a higher order polynomial.

3. Gridding

Once the data have been digitised it is possible toproceed with the gridding. This is done in orderto attain homogenous evenly spaced data (X,Y,Z)triplets. Gridding methods produce a regularlyspaced, rectangular array of Z values from irregularlyspaced XYZ data. The term "irregularly spaced"means that the points follow no particular patternover the extent of the map, so there are many "holes"

26 Geoinformation Science Journal, Vol. 3, No.1

where data are missing. Gridding fills in these holesby interpolating Z values at those locations whereno data exists.

A grid is a rectangular region comprised of evenlyspaced rows and columns. The intersection of a rowand column is called a grid node. Rows contain gridnodes with the same Y coordinate, and columnscontain grid nodes with the same X coordinate.Gridding generates a Z value at each grid node byinterpolating or extrapolating the data values.

Grid file columns and rows are sometimes referredto as X grid lines and Y grid lines, respectively. Thereare many gridding techniques to choose from, as towhich one is used can depend on the type of data andspacing of the irregular data (if indeed it is irregular).All gridding methods require at least three non­collinear data points. Some methods require moredata points. For example, a higher-order polynomialfit needs more than three data points: there must beat least as many data as there are degrees of freedom.However, the number of data points in this case isirrelevant as there are thousands of data points.

The set of digitised bathymetry data is combinedwith the GTOP030 base for the region. TheGTOP030 Z value is set to 0 m, this is done in orderto control the gridding in close to the coast and toprevent the gridding algorithm from gridding landareas as ocean areas. This is masking and although itis not a very sophisticated way, it is quick and easyto implement. The draw back is that the griddingalgorithm must now process an exceptional amountof extra data.

Two gridding methods are used; these are Krigingand Minimum Curvature.

Kriging: is a geostatistical gridding method thathas proven useful and popular in many fields. Thismethod produces visually appealing maps fromirregularly spaced data. Kriging attempts to expresstrends suggested in your data, so that, for example,high points might be connected along a ridge ratherthan isolated by bull/s-eve type contours. Krigingis a very flexible gridding method. You can acceptthe Kriging defaults to produce an accurate grid ofyour data, or Kriging can be custom-fit to a data setby specifying the appropriate variogram model. Thevariogram is a measure of how quickly things changeon the average. The underlying principle is that, onthe average, two observations closer together aremore similar than two observations farther apart.Because the underlying processes of the data oftenhave preferred orientations, values may change

more quickly in one direction than another. As such,the variogram is a function of direction.

The variogram is a three-dimensional function. Thereare two independent variables (the direction 8, theseparation distance h) and one dependent variable(the variogram value y(8,h)). The variogram modelmathematically specifies the spatial variability of thedata set and the resulting grid file. The interpolationweights, which are applied to data points duringthe grid node calculations, are direct functions ofthe variogram model. The variogram (XY plot) is aradial slice (like a piece of pie) from the variogramgrid, which can be thought of as a "funnel shaped"surface. This is necessary because it is difficult todraw the three-dimensional surface, let alone tryto fit a three dimensional function (model) to it. Bytaking slices, it is possible to draw and work withthe directional experimental variogram in a familiarform (- an XY plot).

Minimum Curvature: is widely used in the earthsciences. The interpolated surface generated byMinimum Curvature is analogous to a thin, linearlyelastic plate passing through each of the data valueswith a minimum amount of bending. MinimumCurvature generates the smoothest possible surfacewhile attempting to honour your data as closelyas possible. Minimum Curvature is not an exactinterpolator, however. This means that your data arenot always honoured exactly. Minimum Curvatureproduces a grid by repeatedly applying an equationover the grid in an attempt to smooth the grid. Eachpass over the grid is counted as one iteration. Thegrid node values are recalculated until successivechanges in the values are less than the MaximumResiduals value, or the maximum number ofiterations is reached (Maximum Iteration field). Thecode used by Surfer 7, fully implements the conceptsof tension as described and detailed in Smith andWessel (1990). This routine first fits a simple planarmodel using least squares regression:

AX+BY+C=Z(X,Y)

IGTOP030 is a 30' 'global digital elevationmodel produced by the USGS.

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The Minimum Curvatu re algorithm generates thesurfac e tha t interpolates th e avai lable data andsolves the modified biha rmonic diff erential equationwith ten sion :

Unlike Smi th and Wessel (1990), the fixed nodesare defined as the average of th e neighbour ingobserved va lues. Tha t is, consi der a rectangle thesize and shape of a gri d cell. The neighbou rhood ofa grid node is defined by this rectan gle cen tred onthe grid node. If there are any observed da ta withinthe neighbourhood of a grid node, the valu e of thatgrid no de is fixed eq ua l to the arithmetic average ofcontained da ta .

Ceoinforrna tion Science Journal, Vol. 3, No.1 27

Thus, there are four step s to gene rate the fina l grid The sur face usin g Krig ing is show n in Figure 2, thisusing the Minimum Curvature method . Firs t, the surface is grid ded at T intervals in X and Y withleast squa res regression model is fit to the dat a. con tour intervals at 10 m. Figure 2 shows a muchSecond, the valu es of the planar regression mod el mor e smo othe r surface than compared to that of theat the data locations are sub trac ted from the data surface prod uced by Minimum Curvature in Figurevalues; this yields a set of residual data values . 3. Both figures also show the 10')\) of dat a re taine dThird, th e Minim um Curvature algo rithm is used to fro m the or igina l se t to do the com parisons with.interp olate the res id ua ls at the grid nodes. Fourth,the values of the planar regression mo de l at thegrid nod es are adde d. to the int er polat ed residuals,yieldi ng a fina l inter pola ted surface.

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Figure 2 : Ver ifica tion surface using Krigin g

There are th ree sets of ass ociated boundaryconditions:

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Figure 3 : Verification sur face using Min imumCurvature

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28 Geoinformation Science Journal, Vol. 3, No.1

Figure 3 represents the digitised bathymetry griddedat Tusing Minimum Curvature and shows a slightlyless smooth result than Kriging. This is to be expectedas kriging when computing the interpolationweights considers the inherent trustworthiness ofthe data. If the data measurements are exceedinglyprecise and accurate, the interpolated surface goesthrough each and every observed value. If the datameasurements are suspect, the interpolated surfacemay not go through an observed value, especially ifa particular value is in stark disagreement with theneighbouring observed values. This is an issue ofdata repeatability.

4. Comparisons

In order to gauge some kind of quality measurementfor the final gridded surface and of which griddingmethod is the most suitable, two tests are carriedout. These are: 1. Comparison with ETOP05 grid,and 2. Comparison with a retained data set, whichexcludes 10% of the digitised data from the griddingprocess.

The later is the more important test as it gives adirect check on how well the gridding algorithmshandle the data, whereas the former test is toprovide a comparison with the global bathymetrymodel, ETOP05. Table 1 represents the comparisonsundertaken as mentioned previously.

KRIG Me ETOPO 5

I' • •

x 0.332 -1.244 nil

cri- 7.784 10.674 nil

Num 824 824 nil

5' • • •

x -0.503 -1.757 -0.281

cri- 9.720 9.403 15.812

Num 824 824 824

Table 1 : Results of comparisons (all necessary unitsin metres)

It is seen how gridding the data using Kriging and agridding interval of 1'gives the better results, othergrid sizes were not tried so it is inconclusive as tothe most suitable gridding interval for this data set,what is shown is the most suitable according to theauthors needs.

The Kriging method although slower in computing

the surface, better fits the control data than that ofthe surface computed using Minimum Curvature.ETOP05 is a 5'global model therefore it is notexpected to compare as well with the control dataas the digitised sets. Also it is to be kept in mind thatthe control data itself comes from the same digitisedset, so what these comparisons really are checking isthe digitising process and the algorithm's ability tomap the surface appropriately. In order to performa thorough and conclusive check, it would bebetter to compare with in-situ reduced bathymetrysoundings. However due to the obvious financialconstraints of doing this, the control data as is, isconsidered acceptable.

5. Results

Of the digitised data there are some 8000 pointsexcluding the coastlines that are digitised and of this10% are excluded for comparisons. Kriging was foundto be the better algorithm to use for gridding with theother choice of Minimum Curvature. Tfor this studywas shown to be the most suitable gridding interval,however this is not shown conclusively, as othergrid sizes apart from 5'were not used. Comparisonsare made to ETOP05 and were found to be not asgood when compared to the other methods.

ETOP05 is better suited to the open oceans, where inthis study it has been constrained to coastal shallowwaters. It is recommended that ETOP05 not be usedin any further studies where bathymetry is neededin a shallow water context, until further studies within-situ data can be used for the comparisons.

It is recommended that for this study a griddinginterval of T'be used and that Kriging be chosen asthe method of gridding, as the comparisons showthis is the better of the two methods tested.

References

Wolberg, George., (1990). Digital Image Warping,IEEE Computer Society Press Monograph, 340 pp.

Smith, W. H. F., and Wessel, P. (1990).Griddingwith Continuous Curvature Splines in Tension,Geophysics, v. 55, n. 3, p. 293-305.