Scott Polar Research Institute, University of Cambridge ...

The fractal properties of the underside of

Arctic sea ice

P. Wadhams & N.R. Davis

Scott Polar Research Institute, University of

Cambridge, Lensfield Road, Cambridge, CB2 1ER

UK

Abstract

Using data from an Arctic submarine cruise in spring 1987, the authors employautocorrelation and spectral analysis techniques to investigate the nature of theroughness of the ice underside and its geographical variation. It is found that thefractal roughness varies systematically over the Arctic and is negatively correlatedwith other commonly accepted parameters of ice roughness such as mean draft.The fractal dimension lies in the range 1.40 to 1.48 in the Arctic Basin, and 1.61to 1.77 in the Greenland Sea. Good consistency is demonstrated between twoways of computing the fractal dimension, using the autocorrelation at small lagsand the trend of the power spectrum in a wavelength range from 10 m to severalhundred metres.

1 Introduction

Most of our information on the thickness distribution of Arctic sea ice comes fromupward-looking sonar profiles generated by submarine transects of the ArcticOcean. This type of data also gives us the opportunity to study the roughnesscharacteristics of the ice underside. The statistical properties of ice undersideroughness are of importance to a number of problems which cannot be solved bya knowledge of the ice thickness distribution alone. These include the nature ofice-ocean frictional coupling; the way in which internal waves are stimulated bythe overlying sea ice cover, the interaction between ice and the seabed or offshorestructures; and the nature of acoustic scattering by the ice underside.

Studies of the characteristics of ice roughness began, for the case of theupper surface, with analyses of spectral peaks (Hibler[7]; Hibler and LeSchack[8]) in order to test for anisotropy of ridge orientation. Rothrock andThorndike[13] were the first to study the fractal properties of the ice underside inrelation to spectral characteristics. More recent work on fractal roughness hasbeen carried out by Bishop and Chellis [2, 3] on the small-scale properties of ice

Transactions on the Built Environment vol 5, © 1994 WIT Press, www.witpress.com, ISSN 1743-3509

354 Marine, Offshore and Ice Technology

keels; Connors et al. [6] on short sections of first- and multi-year ice; and Key andMcLaren [10] on a long trans- Arctic submarine profile obtained in 1970.

In May 1987 an extensive dataset of upward sonar profiles and otherimagery was obtained from the Arctic during a remote sensing experiment whichinvolved a submarine and two aircraft (Comiso et al.[5]; Wadhams[16, 17, 18];Wadhams et al.[21]; Wadhams and Martin [20]). The sonar profile extended fromthe North Pole to the ice edge in the southern Greenland Sea, covering severaldistinct ice regimes such as the central Arctic Basin; the heavily ridged zone northof Greenland; Fram Strait; and the Greenland Sea. In this paper we analyse thesedata in order to gain insight into the geometrical properties of the ice undersideand their geographical variation.

2 Autocorrelation function, power spectrum and fractal dimension

Let the draft of the ice underside be represented by y(x) where x is horizontaldistance. Within a sample space k the autocorrelation function Ryy (T) is given by

Ryy(T)=E[yk(x)yk(x + T)] (1)

where E indicates expected value and I is a lag. We need consider only positive

values of T since it can be shown that (Bendat and Piersol[l]) for a stationaryrandom process

Ryy(T)=Ryy(-T) (2)

We shall consider only the normalised autocorrelation function

Ryy(T) = Ryy(T)/Ryy(0) 0)

In the case of a dataset of finite length X,X

(4)

which becomes a sum rather than an integral if the dataset consists of drafts atfinite spacings rather than being continuous.

The general shape of Ryy(x) tells us the length scale over which the iceunderside shows significant correlation, i.e. the typical scales over whichindividual physical features of the ice bottom (floes, ridges, leads) extend. Aninformative simple measure of this is the lag at which the autocorrelation functionfirst falls to zero. Significant periodicity in the autocorrelation function wouldalso be evidence of similar periodicity in ice bottom features, e.g. a preferredspacing for ridges or leads.

A third, and very important, property of the autocorrelation function is thatat small lags it can yield insight into the roughness of the ice surface, defined interms of its fractal dimension. Consider a surface where for very small increments

% in x, the draft y varies as %#, i.e.

|y(x + T) - y(x)l =T#, T->0 (5)


Marine, Offshore and Ice Technology 355

Such a surface is said to obey the Lipschitz-Holder condition and a is the

Lipschitz exponent (Rothrock and Thorndike, [13]). Only if a = 1 is y

differentiable, while when a = 0 the derivative of y is infinite. Thus the range

0<cc< 1 defines a range of continuous but undifferentiable functions, with a beinga measure of the "roughness" of the surface. For such a surface, it can be shownusing (1) that

R(T)« R(0) - c i2 a (6)

for small lags. We can therefore test whether sea ice obeys the Lipschitz-Holdercondition and, if so, what is its Lipschitz exponent, by examining the variation ofR(i) near zero lag. Normalising (6) we find that

+ c (7)

so a log-log plot of (1 - R(x)) against T gives us a measure of a .

If the ice underside is indeed a rough - i.e. continuous but undifferentiable -function, the amount of detail displayed by any representation of it will increaseas the interval between successive horizontal increments decreases (assuming aperfect measurement technique, i.e. zero beamwidth). Such a profile is not asimple line, but rather a fractal (Mandelbrot [11]). Mandelbrot showed that a setof points m a plane can be assigned a dimension, the so-called Hausdorff orfractal dimension D, depending on the extent to which the set of points resemblesan area (dimension 2) or a smooth line (dimension 1). A rough ice profile willhave a dimension between 1 and 2, with the actual value being a measure of its'roughness". Mandelbrot showed that the Hausdorff dimension is related to theLipschitz exponent by

D = 2_a (g)

The power spectral density function is defined as the Fourier transform ofthe autocorrelation function, i.e.

00S(k) = J Ryy(T) exp(-i 2 n k T) di (9)

where k = 1A is the wave number. Mandelbrot showed that if a function obeysthe Lipschitz-Holder condition, then in most cases the power spectral densityvaries as kP, where

p = -2a-l (iQ)

Thus if p<-3 the profile is smooth, i.e. differentiable, while if -l<p<-3 the profileis rough. If p = -1 the power spectrum is not integrable and the surface has infinitevariance, i.e. can be described as "fully rough". In the special case of p = -3 thereis no natural length scale to the surface, which would therefore give the sameappearance when viewed under any degree of magnification. A number of naturalr̂faces have spectra which lie close to this state (Jaeger and Schuring[9],Nye[12]).



3 Dataset and Processing Technique

The dataset comprises corrected ice draft data from a 1987 cruise by a Britishsubmarine, extending from the North Pole via Fram Strait into the Greenland Sea.The data were collected during May 1987 using an upward-looking sonaroperating at 48 kHz and with a quoted beam width of less than 5°. The data wererecorded digitally and on paper chart.

Extensive data reduction was necessary to generate a clean and correcteddataset for this analysis. The digital data points were recorded at uniform timeintervals (of about 0.25 s) and these were converted into spatial intervals usingboat speed values calculated from navigational fixes on the SINS (ship's inertialnavigational system). In doing this account had to be taken of rates of accelerationand deceleration during speed alterations. The spatial data points were thencorrected to a uniform interval of 1 m using a square-law interpolation routine.Next the data, hitherto expressed as a range to the reflecting underside, wereconverted to draft data making use of a manual analysis of the chart records,insertion of sea level values wherever observed, generation of a smoothed curvethrough these values, and subtraction.

The corrected dataset was then divided into sections based on 50 km ofsubmarine track length. The analysis for autocorrelation function and powerspectral density was carried out over successive 2 km subsections within eachsection, with ensemble averaging. Each 2 km subsection had to be complete, withno gaps, for the spatial analysis to be valid. It was decided that sections containingfewer than 5 such complete subsections would not give reliable statistics andthese were therefore ignored.

The normalised autocorrelation function was computed using a GENSTATroutine for lags of up to 1000 m. The routine uses the indirect approach ofcomputing the spectral density using FFT (Fast Fourier Transform) techniques,then computing the inverse Fourier transform of the power spectrum. The powerspectral density was computed using conventional FFT techniques for wavenumbers of up to 0.5 m (the Nyquist frequency). Results for each subsection wereensemble averaged to yield results for a whole section, and the random errorcomputed as l/n*#, where n is the number of subsections in a section (Bendat andPiersol[l]). No attempt was made to estimate bias error.

4 Results

Figure 1 shows a typical result from a 50 km section, including theautocorrelation function with 90% confidence intervals and the power spectraldensity in log-log and log-lin forms, with standard error shown.

4.1 Autocorrelation functionThe general appearance of the autocorrelation functions is straightforward. Thereis some evidence of oscillations on a few of the sections, but the only sections forwhich significant secondary peaks are observed come from the ice edge region inFram Strait and include a great deal of open water. It is possible that a systematicpreferred spacing of leads is occurring (e.g. due to the presence of ice edge bands,which frequently have regular spacings (Wadhams [15]).

The lag at which the autocorrelation function first crosses zero is shown infigure 2. The results are plotted against latitude, and we have divided the trackinto four regions where ice conditions may be expected to vary significantly. The



submarine's general track was shown in figures 6 and 1 of Wadhams[18]. Fromthe North Pole the submarine sailed down towards northeast Ellesmere Island(region 1). At 85°N it turned southeastward and traversed north of Greenlandacross the heavily ridged Greenland Offshore Zone before beginning a transit ofFram Strait (region 2). Region 3 comprises Fram Strait and the northernGreenland Sea from 81°N to 78°N, while region 4 comprises the centralGreenland Sea from 78°N southward to 75°N. The plots show the latitude of thecentroid of each 50 km section used in the analysis.

It can be seen from figure 2 that there is little apparent trend in this zerocrossing lag, but when we average the results for each region (table 1) we find thatregions 1 and 2, in the Arctic Basin, have similar average values of 267 m for thisquantity, while regions 3 and 4 in the Greenland Sea have higher values of 358 mand 334 m, implying an order and coherence in the ice cover which extends over agreater distance. This may be related to the fact that the ice in the Greenland Seais broken up into large floes (smaller near the ice edge), and that the break-uppattern which creates these floes, together with the wide open water spacesbetween floes, give the underside profile an autocorrelation which extends overgreater lags than in the central Arctic, where the autocorrelation is dominated byridge systems.

TABLE 1. Mean values of roughness parameters for different regions of theArctic. i=calculated from short wavelength values, l=calculated from longwavelength values. "

Region First zero a DfromR p Dfromp Meancrossing lag draft(m) (m)

1 267 0.518 1.482 -2.146 1.427 5002 267 0.555 1.445 -2.096s 1.452 4.51

-2.2081 1.3963 358 0.391 1.609 -1.705 1.648 1.904 334 0.261 1.739 -1.467s 1.767 097

-1.5461 1.727

Figure 3 shows the Hausdorff dimension D as a function of latitudecalculated from the autocorrelation function as follows. Preliminary tests of the fitof data to eqn. (7) showed that a typical plot of In (1 - R) against In T has aslowly increasing slope for lags of up to 8 m, then a relatively constant slope forlags up to about 20 m, then a decreasing slope. The relationship is meant to beused at small lags to compute a, but it is likely that at the smallest lags theautocovariance is artificially increased by the fact that the sonar had a beamwidthof about 5° (giving a surface beam diameter of about 8 m) and so gives asmoothed profile. We therefore used the range 8-20 m lag to obtain a best value ofa for each section by linear regression. D is then calculated from a using (8).

It can be seen from figure 3, and also from the regional averages shown intable 1, that the Greenland Sea ice has a significantly higher fractal dimensionthan the ice in the Arctic Basin, and that the dimension increases as the mean draft



decreases. Within each group there are wide variations, however. It is interestingto compare the results with those obtained from a cruise by another Britishsubmarine to the Fram Strait and Arctic Basin in the summer of 1985 (Calvert[3]).Lipschitz exponents lay in the range 0.5 to 0.8, suggesting a lower fractalroughness in summer which may be due to smoothing of the ice surface bymelting.

4.2 Power spectrumAs can be seen from the example of figure 1, the power spectral densities whenplotted on a log-log scale show three distinct slope domains:i) At low wave numbers, i.e. very long wavelengths of 300 m or more, the decayof the spectrum with increasing wave number is slow, i.e. p in eqn. (10) is of theorder of -1. This corresponds to wavelengths where the autocorrelation functionshows little or no coherence.ii) At moderate wave numbers, corresponding to wavelengths of between about300 m and 10 m, the spectrum falls off as a power law, with p-values which areinvestigated further below.iii) At high wave numbers, corresponding to wavelengths of less than 10 m, thereis a "knee" in the spectrum followed by a more rapid fall-off in energy densitywith increasing k. We have already seen that the autocorrelation function hasanomalous properties at lags of 8 m or less, and that 8 m is approximately thesurface beam diameter of the sonar. It is likely, therefore, that the smoothing ofthe under-ice profile caused by sonar beamwidth is producing an anomalouslyrapid fall-off in energy at wave numbers corresponding to wavelengths of lessthan the surface beam diameter. We therefore regard this part of the spectrum asan artefact of the sonar system, and further investigation of its properties liesbeyond the scope of this paper. However, it is important to note that the locationof the "knee" provides a useful measure of the true surface beam diameter of thesonar. This is valuable in the interpretation of datasets where the sonar beamwidthis not known or is unspecified, since beamwidth must be known in order to makea valid correction to mean ice draft in the analysis of ice thickness distribution(Wadhams[15]) and to indicate the correct threshold parameter to use in therepresentation of ridge spacings by a three-parameter lognormal (Wadhams andDavy[19]).

Region (ii) was studied further by measuring the slope of the linear part ofthe log-log spectrum and defining the range of wave numbers over which thislinearity occurred. Results are given in figure 4. A few of the sections have twogradients shown, since the spectrum changed slope within the range of moderatewave numbers. This is reflected in the average values shown in table 1, where forregions 2 and 4 there are two values given, in which the shorter and longer of thetwo wavelength ranges are used For most of the sections we noticed a remarkableconsistency; the log-log plots show a straight line from about 10 m wavelength,corresponding to the beginning of beamwidth-induced smoothing, to a wavenumber corresponding to 200-300 m wavelength. In the Greenland Sea the rangeof constant slope sometimes extended right out to the maximum wavelength limit(2 km). The exponents show some very low values, especially in the Fram Straitsections, but mostly these too lie within a limited range.

Table 1, and the trends of figure 4, show that in regions 1 and 2, coveringthe Arctic Basin, the mean slope of the power spectrum is remarkably consistent,at about -2.1 to -2.2. In Fram Strait, however, it drops significantly to -1.7 and thedecline continues in the lower-latitude Greenland Sea to -1.5.



Using (8) and (10) we can obtain an independent measure of the Hausdorffdimension from p. These are shown in figure 3 plotted against D as calculatedfrom the autocorrelation function. The average values for each region shown intable 1 demonstrate a remarkable consistency between D as calculated from theautocorrelation function at small lags (8-20 m), and D as calculated from thepower spectrum using wavelengths of up to several hundred metres. This showsthat we can use either technique with confidence as our preferred way of derivingthe fractal dimension.

5 Conclusions

Our general conclusions arer-i) The Arctic ice underside is a rough surface which obeys the Lipschitzcondition, with exponents in the range 0.5 to 0.8.ii) The autocorrelation function at small lags (8 m to 20 m) has a uniform slopefrom which the Lipschitz exponent and Hausdorff dimension may be calculated.At very small lags the autocorrelation function is affected by the finite beamwidthof the echo sounder.iii) The power spectral density of the ice surface S(k) varies as kP, where p takesaverage values between -1.9 and -2.2 at wavelengths between 10 m and 200-300m. Using Mandelbrot's relation (10) the Lipschitz exponent can be derived fromthe p-value and agrees well with exponents estimated from the shape of theautocorrelation function.iv) At wavelengths below 10 m the power spectrum falls off more rapidly, whichis again a product of the smoothing effect of sonar beamwidth upon the iceprofile. This can be used as a way of determining the effective surface beamdiameter of the sonar, which is important for making corrections to otherstatistical parameters derived from sonar data.v) The fractal "roughness" is greatest in the Greenland Sea (1.61 to 1.77 onaverage, depending on the method of calculation), and is consistent, andsignificantly lower, over the rest of the Arctic Basin region sampled (1.40 to 1.48on average). This is despite the fact that the physical roughness (expressed interms of mean and r.m.s. ice draft) is lowest in the Fram Strait region andGreenland Sea.

It is interesting to compare these results with earlier studies. Those ofBishop and Chellis[2,3] and Connors et al.[6] were based on very short lengths ofdata; they obtained fractal dimensions of 1.2 to L7 for ice keels [3], 1.4 for asmall stretch of first-year ice and 1.6 for a small stretch of multi-year ice [6], Theanalysis of Key and McLaren[10], covering a long stretch of sonar profile, wasdone in a different way from the present study, by finding the variance of draftincrements at different lags. In their fig. 2 they show a (there called H) for twodifferent ranges of lag, 3-15 m and 15-75m, obtaining ranges of 0.55 to 0.78 forshort lags and 0.15 to 0.45 for longer lags. These correspond to fractal dimensionsof 1.45 to 1.22, and 1.85 to 1.55 respectively. These results are consistent with ourown, but since their dataset did not include the Greenland Sea it was not possiblefor them to test whether D changes significantly in an environment where the iceis melting and is affected by the open ocean.

References

1. Bendat, J.S. and A.G.Piersol (1986). Random Data Analysis and MeasurementProcedures. 2nd ed., Wiley, New York.

2. Bishop, G.C. and S.E. Chellis (1989). Fractal dimension: a descriptor of icekeel surface roughness. Geophys. Res. Letters, 7(5(9), 1007-1010.



3. Bishop, G.C and S.E.Chellis (1990). A fractal description of ice keel small-scale surface roughness. In Sea Ice Properties and Processes (ed. S.F.Ackley,W.F.Weeks), 141-145, CRREL Monograph 90-1, US Army Cold Regions Res.& Engng Lab., Hanover, N.H.

4. Calvert, LA. (1988). Low lag autocorrelations of the under side of sea ice. Rept.for Diploma in Mathematical Statistics, Univ. Cambridge.

5.Comiso, J.C., P. Wadhams, W.B.Krabill, R.N.Swift, J.P.Crawford andW.B.Tucker IQ (1991). Top/bottom multisensor remote sensing of Arctic seaice. /. Geophys. Res., 9<5(C2), 2693-2709.

6. Connors, D.N., E.R. Levine and R.R. Shell (1990). A small-scale under-icemorphology study in the high arctic. In Sea Ice Properties and Processes (ed.S.F.Ackley, W.F.Weeks), 145-151, CRREL Monograph 90-1, US Army ColdRegions Res. & Engng. Lab., Hanover N.H.

7. Hibler, W.D. En (1972). Two dimensional statistical analysis of Arctic sea iceridges. In Sea Ice (ed. T. Karlsson), 261-275, Natl. Res. Counc. of Iceland,Reykjavik.

8. Hibler, W.D. ffl and L.A.LeSchack (1972). Power spectrum analysis ofundersea and surface sea-ice profiles. /. GlacioL, 11, 345-356.

9. Jaeger, R.M. and D.J.Schuring (1966). Spectrum analysis of terrain of MareCognitum. /. Geophys. Res.,ll(3Q\ 5954-5970.

10. Key, J. and A.S. Mclaren (1991). Fractal nature of the sea ice draft profile.Geophys. Res. Letters, 7S(8), 1437-1440.

11. Mandelbrot, B.B. (1977). Fractals. Form, Chance and Dimension. Freeman,San Francisco.

12. Nye, J.F. (1973). A note on the power spectra of sea-ice profiles. AIDJEXBull., 21, 20-21.

13. Rothrock, D.A. and A.S.Thorndike (1980). Geometric properties of theunderside of sea ice. /. Geophys. Res., 85(C7), 3955-3963.

14. Wadhams, P. (1981). Sea-ice topography of the Arctic Ocean in the region70°W to 25°E. Phil. Trans. Roy. Soc., Lond., A302, 45-85.

15. Wadhams, P. (1983). A mechanism for the formation of ice edge bands. /.Geophys. Res., 88(C5), 2813-2818.

16. Wadhams, P. (1988). The underside of Arctic sea ice imaged by sidescansonar. Nature, Lond., 333, 161-164.

17. Wadhams, P. (1990). Evidence for thinning of the Arctic ice cover north ofGreenland Nature, Lond., 345,795-797.

18. Wadhams, P. (1992). Sea ice thickness distribution in the Greenland Sea andEurasian Basin, May 1987. /. Geophys. Res., 97, 5331-5348.

19. Wadhams, P. and T. Davy (1986). On the spacing and draft distributions forpressure ridge keels. /. Geophys. Res., 91 (C9), 10697-10708.

20. Wadhams, P. and S. Martin (1990). Processes determining the bottomtopography of multiyear Arctic sea ice. In Sea Ice Properties and Processes(ed. S.RAckley, W.F.Weeks), 136-141, CRREL Monograph 90-1, US ArmyCold Regions Res. & Engng. Lab., Hanover N.H.

21. Wadhams, P., N.R. Davis, J.C. Comiso, J. Crawford, G. Jackson, W. Krabill,R. Kutz, C.B. Sear, R. Swift and W.B. Tucker IE (1990). Concurrent remotesensing of Arctic sea ice from submarine and aircraft. Intl. J. Remote Sensing,72(9), 1829-1840.



.00_Autocovariance

Lipschitz exponent

0.520

Housclorrp dimension

1 .480

LOG 1 0 (energy density)

7

G

5

4

3

2

1

0

-1-5 -4 -3 -2 -1 0

LOG 10 (wave number)

I OG10 (energy density)8 .

G

5

3

2

1

0

- 1 I u .0.00 0.10 0.20

GOO 800Lag m

Spoc.Lral slope

1200

FIG.l. Autocorrelation function(top) and wave number spectrumfor a 50 km length of track inArctic Basin (about 87°N 55°W).

wave number m~l

0.30 0.40 0.50



JzCD

GOO

5OO

4-00

(3 3OO

§r\i

2OO

TOOReg 4 j Reg 3 [ Reg 2 P Reg 1

75 8O

Latitude N

FIGURE 2. Lag at which autocorrelation function first crosses zero, plottedagainst latitude. The four regions are explained in the text.

2.0

Q

18

1.6

1.2

1.0 Req 4 ! Reg 3 Reg 2 j Reg 1

Autocorrelation

Power Spectrum

75 80 85 90

Latitude N

FIGURE 3. Hausdorff (fractal) dimension D plotted as function of latitude,calculated from autocorrelation function at small lags (continuous line) andpower spectrum slope (dotted line).



Slope _-(of

PowerSpectrum -2

-3

1000Range

of 100Validity

10

180 85

Latitude (N)90

FIGURE 4. The exponent of the power law which best fits the wave numberspectrum, with the range of wavelengths over which this slope is valid. Theisolated spot values refer to sections in which two slopes occurred, for short andlong wavelength ranges; the spot value refers to the long wavelength range,while the short wavelength value forms part of the continuous plot


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