2.2.1 Sample Preparation

It is important in textural analysis to achieve as high a quality of input image as possible. Emphasis has therefore been placed on careful sample preparation. Thin sections are cut so that they can be microtomed and polished from both sides in order to ensure a smooth surface devoid of artificial grooves (cf. Reference EickenEicken and Lange, 1991). Final section thickness amounts to approximately 0.5 mm.

2.2.2 Recording and Digitization of Images

When a thin section of ice is placed between crossed polarizers, individual grains appear in different colours and shades. This is due to the fact that, when entering an ice sample, a wave of linearly polarized light breaks up into two components travelling at different speeds (birefringence). When leaving the birefringent medium, the component rays display a characteristic phase lag. Upon recombination in the upper polarizer above the sample, some wavelengths are filtered out of the original beam of white light by destructive interference — the crystals appear coloured. The exhibited colour depends on the two factors that determine the phase lag between the waves. In this case these are the section thickness and the tilt angle of the optic axis which for ice coincides with the crystallographic caxis. Crystals exhibit no birefringence in propagation directions parallel to their optic axes. With increasing tilt angle and decreasing section thickness, so-called higher-order interference colours give way to orange and grey of increasingly darker shade (for a more precise and comprehensive account consult, for example, Reference Gribble and HallGribbley and Hall (1985)). Crystals with vertically oriented taxes appear completely dark.

Apart from displaying characteristic interference colours, anisotropic crystals appear in different shades as the orientation of the crystal changes with respect to the planes of polarization. If the horizontal projection of its caxis, i.e. the azimuth, corresponds to either plane of polarization, a crystal appears completely dark (maximum extinction). At angles of 45 ° between the polarization plane and the c-axis azimuth, minimum extinction corresponding to a grey-value maximum occurs. Consequently, the grey-value distribution of an image consisting of an assortment of crystals varies as the orientation of the thin section with respect to the polarizers. Thus, samples such as the one featured in Figures 2 and 3 appear quite dissimilar when viewed under different polarizer orientations and have to be recorded in a normalized fashion for consistent, comparable results. This problem can be solved by recording a thin section twice, such that one image exhibits the minimum and the other the maximum median grey value G _md(min) and G _md(max) displayed by the sample (Figs 2 and 3). These two different orientations of the polarizer/ analyser pair are easily determined interactively when recording a sample and relocated at a later stage because they are singular and invariant with regard to different recording conditions. The difference

contains information on the degree of crystal-optical alignment of the sample (not considering possible artifacts due to equivocal mapping of colours into grey tones). The larger D, the larger the degree of caxis alignment (Fig. 3). For crystals with randomly distributed caxes, Dapproaches 0. The highest degree of alignment corresponds to a value of Dachieved for a single crystal extending over the whole field of view. The angle between the two orientations of polarizer/analyser during recording ideally (i.e. in the case of preferred alignment) measures 45 °. Analysis of grain shape is eased by ensuring that the rows and columns of an image are aligned in parallel with the vectors of polarization. If this is the case, the same is true on average for the elongation axes of crystals, since caxes of ice crystals are oriented at right angles to the basal plane which coincides with the tabular morphology of grains.

Fig. 3. Grey-value histograms of the sample depicted in Figure 2, recorded at the two orientations of polarizer/analyser for which the median grey value G_mdis at its minimum and maximum, respectively.

In addition to pictures taken in linearly polarized light, the sample is also recorded in circularly polarized light with the aid of two λ/4 plates placed between sample and polarizers. In circularly polarized light, any shading of crystals due to their orientation with respect to the plane of polarization is removed because the electro-magnetic field vector rotates about the axis of light propagation. Hence, only the tilt of crystals’ caxes and section thickness are expressed in the grey-value distribution (cf. section 2.3).

The recordings in plain light supply the basis for studies of sample porosity. As a ray of ordinary light passes through a medium, it is refracted or reflected by internal phase boundaries. As a result of internal scattering and absorption, pores appear dark in ice thin sections with a bright background (Fig. 4a). This effect is more pronounced for gas than for liquid inclusions because the difference in refractive indices with respect to ice, which determines the limiting angle of internal reflection, is larger for the former. Illuminating a sample merely from the sides over a dark background causes pores to appear as bright speckles in a matrix of dark ice due to the same phenomena (Fig. 4b). Before the recording and digitization of these two images, artificial grey-value gradients resulting from uneven lighting, etc. are removed through the accentuation with the aid of false-colour look-up tables (LUT) and subsequent local compensation of CRT voltage.

Fig. 4. An example of a sea-ice thin section, introduced in Figure 2, recorded in plain transmitted light with pores appearing dark (left, Fig. 4a) and in incident light with pores appearing bright (right, Fig. 4b).

2.2.3 Filtering Images in Preparation for Analysis

Pores may appear as dark speckles due to internal scattering and absorption in images recorded between crossed polarizers. This is particularly true for grains with internal inclusions recorded at orientations of minimum extinction, as is evident in the example shown in Figure 2b. Here, pores might erroneously be interpreted as small crystals by the grain-boundary identification algorithm. To minimize these errors, images recorded between crossed polarizers are subjected to non-linear filtering routines prior to analysis.

Since non-linear filters are also used for the determination of size distributions as described below, an illustration of this technique and other local transforms employed in the analysis is appropriate. Essentially, the method consists of translating a mask (B) of a given size pixel by pixel across the features (A) of an image (Fig. 5). For each position of the kernel B, the pixel that is currently at the origin (in this case the centre) is set to either the maximum or minimum value of the set of pixels covered by the whole of B. The former convolution is referred to as dilation (expressed through the symbol ⊕), as it expands high grey-value objects in the image (Fig. 5). The resulting image represents the union (denoted by U) of features (A) with the translated elements (b) of the kernel B (Reference SerraSerra, 1982):

The latter convolution is termed erosion (expressed through the symbol ⊖) resulting in the shrinking of high grey-value features. It corresponds to the intersection (denoted by ∩) of A with the translated elements b of Β (Reference SerraSerra, 1982);

The process of sequentially dilating and eroding an image is referred to as a morphological opening (Reference SerraSerra, 1982):

where B^Tis the reflection of Β about the origin; here, B^T= B. By morphologically opening an image with a mask of size 2b + 1, all features smaller or equal in size to 2b and exhibiting lower grey values than their surroundings will be removed.

Fig. 5. Schematic representation of the morphological opening of a set of two dark objects (denoted A) with a quadratic mask (denoted B) by performing an erosion-dilation sequence.

Opening the images of sea-ice samples recorded between crossed polarizers with masks 5×5 pixels in size proved to remove the majority of signals, i.e. more than 90%, resulting from pores within grains. This is illustrated in Figure 6 which represents the image of Figure 2b after the morphological opening. Errors introduced into automated grain-boundary recognition through misinterpretation of inclusions can therefore be kept to a minimum. For samples with larger pores, e.g. glacial ice, larger masks have to be employed for the morphological opening. It has to be taken into account during textural analysis that the morphological filters employed might efface crystal features up to 2b in size. This is noticeable in Figure 6 as a slight blurring of outlines. For specimens with inclusions as large as the crystals, the method obviously fails altogether.

Fig. 6. Sample depicted in Figure 2 after morphological opening with a mask 5 by 5 pixels in size. Note the disappearance of intraeranular bores.

The second step in image preparation consists of low-pass filtering all frames recorded to remove high-frequency noise generated by the camera or the analogue/digital(A/D)-converter, thereby improving the efficiency of grain-boundary recognition algorithms. The image S(x,y)is convolved with a 3 × 3 kernel h(u,v)such that the resulting picture Sprime(x,y)is given by:

with m = 3, k = (m− 1)/2, (x, y) and (u, v)being the coordinates of picture and kernel elements, respectively (Reference HaberäckerHaberäcker, 1985). For low-pass filtering, the kernel is specified by the matrix

As a result, pixels are set to a value that corresponds to the average of all nine elements covered by the mask.

2.4.1 Grain Boundaries

Image segmentation is aimed at identifying grain boundaries in images recorded between crossed polarizers or identifying pores in images recorded in plain light. Since identification of grains or grain boundaries is a prerequisite to determination of the majority of textural parameters, segmentation is of considerable importance in automated textural analysis. If we assume that a grain in the broadest sense can be perceived as a domain of homogeneous crystal-lattice orientation (possibly including other phases), then the process of segmentation consists of the identification of (1) foreign component phases, which have to be analysed separately, and (2) domain boundaries. Item (1) has been taken care of by image-preparation algorithms described in the previous sections. With regard to (2), image analysis of thin sections can at best yield information on crystal-optical rather than crystallographic orientation of domains. In the majority of cases, however, this information suffices completely.

Grey-tone recordings access only a fraction of the full information inherent in a thin section exhibiting interference colours higher than the first order. Difficulties resulting from this shortcoming can be resolved by cutting sections to a final thickness corresponding to low-order white interference colours. Alternatively, one may assume that no significant information is lost when taking grey-tone recordings of coloured sections. The significance of this information loss is covered in section 3.1, for now the latter assumption shall suffice.

Two approaches were taken to devise an algorithm that identifies grain boundaries by evaluating the grey-value distributions of images taken between crossed polarizers. Whereas one is fairly elaborate and yields information on the crystal-optical orientation of phases by analysing sequences of images recorded under varying orientations of polarizers (cf. Reference EickenEicken, 1991), the other, presented below, is simple and has been proven to yield satisfactory results even for samples of complex texture (Reference EickenEicken and Lange, 1991). The latter algorithm recognizes grain boundaries as locally confined zones of high contrast, achieved by Sobel-filtering of both images taken between crossed polarizers as a first step (i.e. with minimum and maximum median grey value, see Figures 2 and 3). This convolution computes grey-value gradients in x- and y-direction with kernels h_x and h_y defined as (Reference HaberäckerHaberäcker, 1985):

Thus, high grey values are assigned to regions of high contrast and vice versa. By adding both convoluted images (and setting all values >255 to 255), information about image contrast is combined for the two singular polarizer orientations. The resulting image is binarized about a threshold Gthr with each pixel Gbeing assigned a new value of G′ of either 0 or 255, so that

An example of this segmentation procedure is given in Figure 7 for the sea-ice sample introduced above. Dynamic determination of the threshold, such that it corresponds to a given areal fraction of grain boundaries in the image, as suggested by Reference Russ and RussRuss and Russ (1987) and other authors, is of little value in this case because the grain-boundary density varies considerably between samples. Consequently, the threshold has been fixed at a static level that corresponds to the minimum grey-level fluctuation attributed to a grain boundary. It is assumed that all values below the threshold are associated with intragranular phenomena (undulose extinction, sub-grains, etc.) which are not considered in the segmentation process (an estimate of the errors resulting from this approach will be discussed in section 3.1). Based on analysis of both glacial and sea-ice samples, G _thrhas been set to 128 for our studies.

Fig. 7. Sample depicted in Figure 2 after the segmentation process (black: grains; white: grain boundaries).

2.4.2 Pores

The distinction between pores and grains is best made by evaluating the images recorded in plain light. As discussed above, pores appear dark in plain transmitted light due to internal absorption (Fig. 4a). The same physical cause has the adverse effect when light is shone through a sample in narrow beams from the sides only; pores appear as bright speckles in a dark matrix (Fig. 4b). In principle, two approaches can be taken to segment images. Either spatial contrast along the perimeter of inclusions is used to determine pore boundaries, or pores and ice are distinguished by their differences in grey value. For the samples analysed in this study, the first approach does not lead to convincing results. Grey-level contrast within inclusions can be so high as to render correct identification of pore boundaries impossible (such as in groups of pores in the lower left corner of Figure 4a and Figure 4b). Further difficulties arise as the changes in grey value between crystals and pores often appear as gradual transitions owing to scattering in the boundary zone.

Segmentation through binarization of images, based on a grey-value threshold to distinguish between crystals and pores, holds more promise. The grey-value histogram of the example of a sea-ice thin section recorded in transmitted plain light is shown in Figure 8. The pronounced peak at high grey values corresponds to the ice, whereas the drawn-out tail represents pores. Where is one to fix the threshold G _thrthat separates crystal and pore fraction in the histogram? The lack of any self-evident division is due to the gradual grey-value transition that characterizes pore boundaries. Here, not even interactive manual setting of thresholds is warranted to yield reproducible results. Determination of G_thr is computed from the magnitude of the median grey value G_medof the entire image’s grey-value distribution. Here, a quadratic equation with empirically determined coefficients has been employed in estimating G_thr:

Comparison with manually segmented images shows that this approach yields satisfactory results, while ensuring independent, objective analysis of samples. Results are improved significantly by combining the two images recorded in plain light before binarization about G_thr-The binarized image, based on Figure 4a and Figure 4b, is shown in Figure 9. The entire segmentation procedure comprises these steps: (1) inversion of the image recorded in plain light such that each grey value G₀is replaced by G_n= 255 − G₀, (2) addition of the two images with all grey values > 255 set to 255, and (3) binarization about a threshold G_thrsuch that all grey values < G_thrare set to 0 (ice) and all others to 255 (pores), G_thrbeing determined according to Equation (7).

Fig. 8. Grey-value histogram of the plain-light image shown in Figure 4a. The peak at high grey values corresponds to the ice phase; pores are represented by the lower end of the spectrum.

Fig. 9. Sample depicted in Figure 4 after the segmentation process (black: pores; white: ice; porosity A_pA= 24%, mean chord size c_m= 0.4 mm J.

2.5.1 Determination of Porosity and the Effect of Finite Section Thickness

Stereologically, one of the simplest parameters to extract from an image segmented into ice and pores is the fraction of the total area covered by pores A_PA From Reference DelesseDelesse (1847; cited in Reference WeibelWeibel, 1979) we know that porosity V_PV is given by the simple formula:

provided there is a uniform distribution of pores in the direction normal to Α_PA . Unfortunately, a thin section of 0.5 mm thickness with pores down to a few tenths of a millimetre in size bears little resemblance to the ideal concept of a planar, truly two-dimensional sample. Thus, in practice, we are not determining A_PA but rather the projection of the porosity V_PV into the plane of vision (Fig. 10).

In order to account for this artificial increase in V_PV by sampling from a volume of thickness t, one can correct determinations of Α_PA with a compensating factor K_t (Reference WeibelWeibel, 1980):

K_t is a function both of tand the size of inclusions (which in this case are assumed to be uniform in size). For spherical pores of diameter d, one obtains

The porosity determined on a section of, for example, 0.5 mm thickness with pores of diameter 1 mm would then be too high by a factor of 1.75. The pore geometry of sea-ice samples, in particular, is of such complexity as to necessitate a more thorough treatment of issues relating to section-thickness effects (see also the instructive monographs by Reference WeibelWeibel (1979, Reference Weibel1980)). As the grain-size of glaciological samples typically ranges one order of magnitude above the thickness of thin sections, correction of the thickness effect for grain-size determinations was deemed unnecessary.

Fig. 10. Effects of finite section thickness, with pores from the entire volume projected on to the video image with a resultant overestimation of porosity. Note that pore size will be underestimated for spherical pores because of truncation. Inclined boundaries between crystals may induce interference fringes or feign abnormally broad grain boundaries.

2.5.2 Linear Analysis: Textural Parameters Obtained from Test Lines Intersecting with Grains or Pores

Principally, linear analysis consists of extracting the lengths of lines intercepting features within a sample (Reference WeibelWeibel, 1979). Here, we consider intercepts of horizontal and vertical lines, i.e. rows and columns of the digital image, with grains or pores. Ideally, these test lines should be randomly distributed. However, such random sampling has to rely on elaborate interpolation schemes when working on orthogonal digital grids. The concept for textural analysis of glaciological samples presented here does not justify these procedures, even more so, because the digitization procedure ensures that the mean crystal-optical orientation of grains is parallel to the frame’s principal axes (see section 2.2.2). On average, this procedure precludes that horizontally elongated grains or pores (e.g. columnar crystals or brine layers) are aligned at 45 ° with respect to the image’s reference frame (see, for instance, Figs 2 and 4).

A simple, albeit powerful stereological descriptor, the grain-boundary or perimeter density Β_A follows immediately from the number of chords Νextracted from a binarized image of area Aand resolution b(expressed by a pixel’s lateral dimension) according to

Thus, B_A describes the length of grain perimeters or boundaries per unit area. Samples composed of small or highly tortuous grains would consequently exhibit relatively high values of Β_A . Though computation of this parameter is straightforward, the implications of resolution and orthogonal geometry of the digital image entering into the equation are much less so. From a crystallographic viewpoint, and depending on the characteristics of a given sample (e.g. distribution of other phases, direct impingement of grains), Β_A may have to be modified by an appropriate “internal-surface” factor. The even simpler point-count measure P_A of the boundary density, as given by the number of pixels per unit area belonging to grain boundaries, is not considered here. Since it does not strictly depend on the grain’s surface area but also on the width of the boundary zone, it corresponds essentially to porosity Α_PA which can also be viewed as a pixel measure of “boundary space” per unit area.

The statistical descriptors of the intercept or chord-length distribution f(c_i)(illustrated for the sea-ice example in Figure 11), i.e. its mean (C_m), standard deviation (σ)and skewness (f), represent another set of useful textural parameters. They are defined as (Reference LloydLloyd, 1984)

Whereas C_m describes the mean grain-size of a sample, fis a measure of the asymmetry inherent in an intercept distribution. For f< 0, the curve is skewed towards lower values than the normal distribution; for f> 0, it tails towards higher values. As has been shown, fcan be used in automated discrimination between different sea-ice textural classes (Reference EickenEicken and Lange, 1991). The analysis of chord sizes does not take into account the location of intercepts within an image nor does it identify the set of chords that constitute an individual grain. This circumstance may be viewed both as a great advantage and a serious shortcoming of linear analysis in general. It can be quite disadvantageous in some studies not to consider individual grains, and not to record their sizes and shapes one by one. Yet, facing complex textures (as in the case of sea ice), one of the strengths of linear analysis is that it does not require the identification of individual grains, which, for stereological reasons, can be associated with a large error, no matter whether performed automatically or manually. Even so, grain-sizes can be computed from the chord-size distribution. As shown by Reference CroftonCrofton (1868) and Reference DaviesDavies (1962), the mean area Aand mean perimeter Βof features with convex outlines are described by the first and third moment C_m1 and C_m3 of the chord-size distribution according to

with the kth moment defined as

Typical values computed from an example of a chord-size distribution are indicated in the caption to Figure 11.

Fig. 11. Chord-size distribution of the example of sea ice shown in Figures 2, 6 and 7 as obtained through linear analysis (mean chord size C_m= 1.8 mm, σ = 1.5 mm, f = 3.4, computed mean grain area A = 18 mm², computed mean grain perimeter Β = 32 mm; see section 2.5.2 for further explanation of symbols and parameters). Also shown is the feature-size distribution determined through morphological opening.

The parameter I, which has been defined as the ratio between mean chord sizes determined along each of the image’s two principal axes, describes the mean elongation of grains. Isometric grains correspond to I− 1, whereas for elongated grains I< 1. More elaborate measures of grain geometry, such as boundary curvature, can be extracted from the chord-size distribution based on the relations given by Reference DeHoff and EliasDeHoff (1967).

2.5.3 Morphological Filters as Digital Sieves

In order to overcome the restrictions of linear analysis outlined above and to come closer to describing the “true” grain-size distribution, morphological filters were relied on to extract information on the size distribution of cross-sectional areas displayed in a thin section (see Figure 11 for the corresponding example). As outlined in section 2.2.3 and Figure 5, the morphological opening can be quite effective in removing low grey-value features from an image. By applying a sequence of openings with increasing mask size to the binarized image with delineated grain boundaries, features of increasing sizes are extinguished. For each step, the areal fraction (Α_A)of features removed during individual openings is recorded (Fig. 5; cf. Reference SerraSerra, 1982).

By referring to areal rather than numerical density, the peaks at low intercept lengths typical for chord-size frequency diagrams are not evident in morphological analysis, in which more weight is given to larger features (Fig. 11). As it has been described here, the digital sieve does not distinguish between individual particles or entities as does its hardware prototype, but considers grains and grain protrusions as equivalent (Fig. 5). Analysis of grain-sizes requires identification and enumeration of grains as discussed below. The similarity between the distribution of chords and morphological sizes typical for granular ice samples is mostly due to stereological factors (cf. Fig. 11).

2.5.4 Automated Enumeration of Features

Similar to point-density methods discussed in section 2.5.2, one can determine the density of grains N_A per unit area in a thin section. This has been achieved here by using an algorithm that automatically identifies all sets of pixels bounded by closed curves, assuming these are individual grains. Ν_A is easily determined by simply counting grains manually or by using a semi-automated system after manual tracing of grain boundaries. Errors can result if different cross-sections of a single, not fully convex grain are interpreted as individual crystals (Fig. 12). This stereological problem of two-dimensional sampling of three-dimensional structures has been discussed by Reference RigsbyRigsby (1968) and Reference HookeHooke (1969) for glacial ice. Experience shows that the problem may be of equal or higher importance in the study of sea ice (see, for instance, the concave grains in Figure 2). However, automated techniques are prone to substantial errors of a different kind. In cases where contrast between adjacent grains is low (either because the lattice mismatch between crystals is very small or because of video-recording conditions), the segmentation algorithm might not completely identify the boundary, leaving the grain bounded by an open curve. This is schematically shown in Figure 12, but it is also evident in our example with highly aligned grains and sub-grains visible (Fig. 2b, bottom left). In the corresponding segmented image (Fig. 7), grain boundaries cannot be fully traced in this area. As a consequence, the counting algorithm will consider adjacent grains connected through a perforated boundary as one crystal. Whereas this does not represent a problem in linear analysis, for example, where the chord-size distribution is not particularly affected, these small perturbations prove to be a source of larger errors in the automated enumeration of grains. Thus, an image consisting of tens of grains with boundaries properly identified except for one pixel within each boundary would be counted as one crystal (Fig. 12), although parameters such as the grain-boundary density would deviate by a fraction of a per cent.

Fig. 12. Distribution of grain cross-sections in a horizontal thin section and for a vertical cut (upper left and right), and identification of grains through automated analysis (crystal 1 regarded as two grains, crystals 2 and 3 as one grain due to incomplete segmentation).

2.5.5 Edge Effects: Correcting for Truncation Errors

Features located near the margins of an image are likely to be truncated by the boundary. This circumstance has to be taken into account for size determinations, either by disregarding clipped grains in the course of the analysis or through later correction of the resulting data. The first option holds most promise for operations such as enumeration of grains or pores by counting only half of all features cut by the image margins. In chord-size analysis, a similar approach was taken by placing a frame within the image and considering only chords whose centre points fall within this frame. The width of this boundary zone is determined by the largest feature present within the sample. Restrictions are placed on this method by elongated grains cut at an oblique angle. Alternatively, intercept data have also been corrected after determination according to a method suggested by Reference Hougardy and StienenHougardy and Stienen (1978). Here, the true length of truncated intercepts is computed from the chord-size distribution of individual rows and columns.

In general, the problem of edge effects is intricately linked to the problem of sampling errors. The more features that are recorded within an image, the less important is the artificial reduction in, for example, mean intercept length due to truncation. Truncation significantly alters results only as the grains or pores recorded become very large compared to the image size. In cases where constrictions with regard to core diameter, etc. do not allow for larger samples, it also becomes increasingly difficult to quantify truncation errors reliably. Ultimately, the problem may be resolved either through a sufficient sampling density or possibly by relying on textural parameters, such as grain-boundary densities, that are not subject to truncation errors.

2.5.6 Computing “True” Pore- or Grain-Sizes from Chord-Size Distributions

When sampling along test lines arranged on a sampling plane, the parameters obtained through linear analysis, such as mean chord size, will underestimate the true feature size, assuming that the objects are of spherical or near-spherical shape. Planar cuts through grains or pores will mostly generate smaller-than-maximum cross-sections, whereas the test lines will mostly miss the centre points of an object. Of the various correction procedures that have been devised (cf. Reference WeibelWeibel (1980) for a detailed description), the method of Reference Cahn and FullmanCahn and Fullman (1956) is particularly useful in chord-size analysis. They determined the number of chords n(c)dc with lengths between cand c-dc to be

where n(D)dDis the number of spherical particles between Dand D-dDin diameter. For discrete intercept lengths as obtained from digital analysis, Reference BockstiegelBockstiegel (1966) derived a solution to the equation yielding the number of spherical particles between and in diameter:

According to Bockstiegel’s method, three-dimensionally valid grain-size distributions have been computed for c. 50 representative samples of granular ice composed of convex, isometric grains (both glacial and sea ice). In many cases, nonsensical size distributions with negative or abnormally high frequencies resulted; almost all computed distributions did not correspond to the true picture. The most obvious explanation for this inexactitude, which has also been reported by Reference AlleyAlley (1987) for semi-automated analysis of firn samples, would be differences in the assumed and actual sample geometry. The method is bound to fail when applied to textures comprising indented, non-spherical grains, which are frequently observed in sea-ice studies (cf. Fig. 12). Yet, even for particles almost spherical in shape, part of the explanation is associated with the very principles of automated digital analysis. The approximation of comparatively small features within an image by orthogonal picture elements is rather unrealistic and results in abnormal distributions of small chord sizes. It could be argued, however, that the information gained from three-dimensionally valid size distributions is generally marginal, especially when considering the effects of resolution and analysis procedure on the outcome. Thus, it appears most appropriate to restrict computation of derived parameters to cases where they are specifically needed for analysis and where all significant errors involved in the computation can be quantified.