ASSESSING TEXTURAL VARIATION IN LAMINATED SANDS USING DIGITAL IMAGE ANALYSIS OF THIN SECTIONS ELMER H. VAN DEN BERG, VICTOR F. BENSE, AND WOLFGANG SCHLAGER Faculty of Earth and Life Sciences, Vrije Universiteit, De Boelelaan 1085, 1081 HV, Amsterdam, the Netherlands e-mail:
[email protected]
ABSTRACT: Although between-laminae parameter variations in sediments have been studied in the past, measured parameters were generally limited to grain-size distribution and total porosity. The magnitude and directional characteristics of physical properties of laminated sediments also depend on the shape and orientation of particles, packing, pore-size characteristics, and properties of the pore network. Leaving out a complete characterization of granular sediment in models of physical properties could result in a systematic mismatch between modeled and measured values. However, tools for high-resolution quantification of these spatial variable parameters are not readily available. In order to provide such a tool, an image-analysis procedure is presented based on digital images of thin sections. The procedure defines a sequence of rectangular measurement windows, with their long sides oriented parallel to stratification, and measures the sediment characteristics for each window. Logs of parameters values, oriented perpendicular to stratification, are thus created. After verification of the image-analysis procedure using comparative laboratory measurements, the procedure is applied to two thin-section images of unconsolidated fine-grained, laminated sands. The quantitative variations observed in the logs of grain and pore characteristics correlate well with bedding observed on the original images. Overall, the image-analysis procedure proved to be a useful and versatile tool for quantifying variations of textural characteristics on sub-lamina scale.
INTRODUCTION
Naturally deposited sediments commonly show small-scale variation in textural characteristics, most notably grain size, shape, orientation, and packing owing to variations in sedimentation processes. These textural variations find expression in the form of either fine stratification or lamination, which may have a preferential orientation, e.g. cross-bedding (Reineck and Singh 1980). The textural variations translate into variations in physical properties (e.g., permeability) on the same scale, which grouped together at larger scale, may have significant bearing on magnitude and directional characteristics of effective physical properties (Meyer and Krause 2001; Pickup et al. 1994; Potter and Pettijohn 1977). Until recently, most studies dealing with determination of textural characteristics on lamina and sublamina scale apply either experimental laboratory techniques (Cheel and Middleton 1986; Emery 1978) or point counting of thin sections (Hartkamp-Bakker 1993). These techniques, however, are notoriously labor intensive, may not be suitable for unconsolidated sediments, and do not quantify all sediment characteristics relevant to the geophysical properties studied. Past developments in digital image-analysis of thin sections allow efficient measurement of a large number of textural properties, including pore space, but up to now they have been used to characterize bulk properties, assuming sample homogeneity (Anselmetti et al. 1998; Ehrlich et al. 1991; Garboczi et al. 1999; Scha¨fer and Teyssen 1987). Only recently have digital image-analysis techniques been presented to study textural variations in stratified sediment, focusing mainly on granular characteristics (Cooper 1997; Francus 1998; Francus and Karabanov 2000; Williams et al. 1998). The aim of the present study is to develop and test a new image-analysis algorithm that enables automatic determination at high resolution of both solid and void phase characteristics in profile over thin sections. Technically this is done by automatically moving a rectangular sampling window JOURNAL OF SEDIMENTARY RESEARCH, VOL. 73, NO. 1, JANUARY, 2003, P. 133–143 Copyright q 2003, SEPM (Society for Sedimentary Geology) 1527-1404/03/073-133/$03.00
in regularly spaced steps over digital images of thin sections and measuring the textural characteristics at each location. To capture maximum sedimentological variation, sampling profiles were oriented perpendicular to bedding, thereby assuming that variation parallel to bedding is limited and can be neglected. In order to generate a consistent sampling strategy to efficiently detect heterogeneity in laminated sands, general rules for optimum sample size and sampling interval are devised. Performance of the imageanalysis procedure is verified by laboratory measurements of grain-size distribution characteristics and total porosity of artificially packed uniformly sized glass spheres and natural sands. The procedure is demonstrated with two digital images of unconsolidated, fine-grained fluvial and eolian deposits having low to no clay and silt content. In order to determine a limited number of key parameters, which explain most of the sedimentological variation, the data set was processed afterwards with standard factor analysis. Additionally, by applying factor analysis insight is gained on the interrelationship between parameters that are traditionally studied using experimental laboratory techniques. The last two steps must be seen as examples of how the data set generated with image-analysis can be utilized. TEST SAMPLES
The image-analysis procedure was verified by comparing grain-size distribution characteristics and total porosity resulting from image-analysis with laboratory measurements. In order to avoid possible bias between both measurement techniques caused by irregular particle shape, uniformly sized glass spheres 350 mm in diameter (Fig. 1A) were also used. Two geological samples were used to test whether the image-analysis procedure could reproduce the laminae successions that were visually observed in the samples. A first box-core sample (I21) was taken from a Holocene eolian deposit in the coastal dunes of the Netherlands. The material consists of fine- to medium-grained, well-sorted sand. The box-core sample (100 3 80 3 40 mm) was taken over a noticeable, convex-upward dipping foreset lamina (Fig. 1B). Its thickness is approximately 40 mm and is positioned at the top of a cross-bedded unit that is 150 to 200 mm thick. The shape of the foreset and the absence of a sharp crest suggests the sediments were deposited by a relatively rapid eolian filling of a depression (Zagwijn 1986). A second box-core sample (B2) was taken from a parallel-laminated fluvial deposit from the southern part of the Netherlands. The sediment, consisting of fine to medium sand with laminae thickness ranging from 2 to 5 mm, was probably deposited in either a levee or crevasse-splay environment (Fig. 1C). Iron oxide was precipitated in several thin bands, ranging in thickness between 2 and 10 mm. The (mineral) grain-size distributions of the three samples were measured with a laser particle diffraction spectrophotometer (Fritsch Laser Particle Sizer A22, C-version). Prior to measurement each sample was treated with 30% HCl and 30% H2O2 to remove carbonates and organic matter, respectively. Because the sediment predominantly contains small quantities of organic mater and carbonates (shell fragments), the difference between mineral grain-size and true particle-size distribution is small. Iron oxides may go into solution as colloids, and be measured as such in the clay size range (Konert and Vandenberghe 1997). Grain sizes are expressed as (projected) area diameter of spherical particles and frequencies as weight ratio. The total porosity of the same material, sampled separately in the field with core plugs, was determined by measuring the weight loss after oven-drying water-saturated samples for 24 hours at 60 degrees Celsius.
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FIG. 1.—Thin section of impregnated sample containing glass beads (A) with gray-scale image and binary (black and white) image after classification. Placement of box cores for eolian (I21, part B) and fluvial (B2, part C) field samples with gray-scale images and binary images after classification.
The box-core samples were dried for a period of 6 weeks in the laboratory and were vacuum-impregnated with a blue-stained epoxy resin of low viscosity. The amount of staining used for the reference sample and box-core B2 was 6 times more intense than the staining used in box-core I21 (10 gram resin and 0.1 gram Sudan Blue Oil). The blocks were then cut along the long axes and thin sections of standard thickness (30 mm) were prepared. IMAGE ANALYSIS
Details of the steps from the acquisition of digital images to image measurements of grain and pore characteristics are presented in publications by Ehrlich et al. (1991), Ehrlich et al. (1984), McCreesh et al. (1991), Russ (1998), and Ruzyla (1992) (Fig. 2). In this study no new image-analysis techniques are introduced. The novel aspect lies in applying existing imagemeasurement techniques to a limited rectangular area or so-called ‘‘measurement window,’’ which is part of a sequence of measurement windows, covering a mosaic of digital images (Fig. 2). In each of the measurement windows a number of solid and void phase characteristics are determined, in this way generating high-resolution logs across sediment laminae. Consequently, we present only a brief description of the image-analysis procedures, and focus primarily on the construction of these transects. Image Acquisition and Preprocessing Digital images were obtained by illuminating the thin sections with a flat-bed light source and acquiring digital photographs with a standard dig-
ital scanner (LEAF Lumina). The RGB images were saved in uncompressed TIFF format. The pixel resolution (Res) decreases from 3.98 mm for box core I21, 5.38 mm for images of the glass-spheres sample, to 12.5 mm for box core B2. In order to generate continuous transects of sediment characteristics over thin sections, a number of slightly overlapping (2%) images in series were acquired. The direction of the transects is always perpendicular to stratification, and in order to provide ample sample space for the digital image-analysis, the long sides of images (length/height ratio: 1.22) were oriented parallel to stratification. A photograph of an empty background without thin section was made in order to compensate for possible non-uniform illumination by the light source (Russ 1998). Spectral noise was reduced by applying a median filter. Both preprocessing steps were performed on a PC using the public-domain SCION IMAGE software (Scion Image, http://www.scioncorp.com). In order to ensure consequent pore and grain identification during image segmentation, the histograms of the red, green, and blue images were converted to resemble the histograms of a reference image using so-called ‘‘histogram matching’’ (Gupta 1991). The matched images were subsequently joined geometrically into mosaics (Figs. 1A, B, C). Both operations were performed using the commercially available remote-sensing software ERMAPPER (Dunne et al. 2001). Segmentation between Solid and Pore Phase The image mosaics were segmented into regions containing solid phase (grain sections) and regions containing void phase (pore space), using the maximum likelihood classification technique (Lillesand and Kiefer 1987).
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FIG. 2.—Schematic representation of the image-analysis procedures, software used for carrying out these steps, and operational characteristics of the automated image measurements with example output.
For each feature to be identified in the image (opaque grain sections, translucent grain sections, or pore space) the histogram characteristics are determined, by defining and analyzing regions (training areas) that contain only pixels belonging to one of the features. By assuming that histogram characteristics approach Gaussian probability distributions, an undefined pixel is assigned to the feature in which it most likely would occur. Definition of training areas is the only subjective user-related step in the segmentation procedure and thus liable to bias. However, supervised classification best suited the type of images that were used in this study, because it enabled a more direct definition of different features through use of training areas. Finally, classified images were converted to binary blackand-white images by grouping translucent and opaque grain sections, leaving only two features: grain sections (black pixels) and pore space (white pixels). Both operations were carried out using ERMAPPER software. Postprocessing Owing mainly to viewing the thin section with transmitted light, overlapping projections of grains in the thin section cause grain sections to touch in digital images. The best way to avoid this overlap is to acquire digital images with backscattered electron microscopy, which scans only the surface of a thin section (Fens 2000). Despite this advantage, we used conventional transmitted-light microscopy. Several automatic algorithms for separating touching grain sections are available in the literature, e.g., watershed segmentation and cycles of erosion and dilations operations (Russ 1998). In this study we applied the separation algorithm presented in Van den Berg et al. (2002). The algorithm detects the characteristic sharp contact wedges in the outline of touching grain sections and creates an intersection after checking if the angle of the contact wedge is smaller than a user-defined threshold value. Despite its straightforward operation, only
60% of all separations (sum of correct, omitted, and erroneous) were correct, which is slightly better than the watershed segmentation method (51%). Because manually separating grain sections in the mosaics was considered impractical, only the automatic separation algorithm was applied in this study, inevitably introducing bias into the measurement results. The processed binary images are presented in Figure 1. IMAGE MEASUREMENT
With image measurement, quantitative data were generated on size, shape, outline, and spatial arrangement of grains and pore space (Table 1), which were subsequently converted into 35 sedimentological parameters (Table 2). All the measurements concern sections exposed in a two-dimensional image of essentially three-dimensional features and thus must be considered apparent characteristics (Russ 1998). Because this study is mainly concerned with quantifying parameter variations, only apparent paTABLE 1.—Basic parameters measured with image-analysis, which are used as input in the calculation of sedimentological characteristics (Table 2). Parameter Atot Ag Lg, Lg,sum a,b-axis a r n Apb, Apth Ap, Apb,sum, Apth,sum lp
Description area of measurement window area of grain section perimeter of grain section, sum of all Lg within the measurement window apparent long and short axis of an ellipse fitted to the particle outline angle between the apparent long axis relative to horizontal distance of the center of a grain section to the center of its nearest neighbor number of grain sections in the measurement window area of a pore body, area of a pore throat area of all pores, area of all pore bodies, and area of all pore throats within the measurement window length of skeletonized pore branch
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E.H. VAN DEN BERG ET AL. TABLE 2.—Overview of sedimentological characteristics. Variable
Description
Remarks
Statistical variables
Grain size, dg
dg 5 Ï4Ag /p
Equivalent area diameter Mean dg , Median dg , Std of grain sections. dg , Skew dg , Kurt dg (graphical method, Pettijohn et al. 1987)
Grain shape, q
q 5 b/a
Aspect ratio, ranging be- Mean q, Median q, Std q, Skew q, Kurt q. tween 0 to 1, ellipsoidal to spherical, respectively.
Specific surface area, Ssp
Ssp 5
Grain orientation, ug and R g
C5 n i ·cos(2a), ug average grain orien- ug1 and ug2 , first and tation, R g orientation second most frequent S5 n i ·sin(2a) occurring angle interstrength, n i number of ug 5 [(arctan(S/C) 1 k·908)] val. observations within with k 5 0, 2 or 4 angle interval, i. depending on quadrant R g 5 (S 2 1 C 2 ) 0.5 r 0.5 ra 5 , re 5 Nearest neighborhood n Ïn/A tot distance ratio, R, rangR 5 ra /re es from ,1 (clustered), 51 (random) to .1 (ordered).
Packing, R
4L g,sum p (A tot 2 Ap )
O O
Specific surface expressed as surface area per unit volume of solid material.
O
Pore body size, dpb Diameter of largest inscribed Diameter of defined pore Mean dpb , Median dpb , bodies. circle based on Ultimate Std d pb , Skew d pb , Eroded Points of pore network Kurt dpb . Pore throat size, dpth 5 Ï4Apth /p dpth
Equivalent area diameter Mean dpth , Median dpth , of defined pore throats. Std d pt j , Skew d pt j , Kurt dpth .
Orientation of pore As grain orientation, ug , R g network up , R p
As grain orientation.
up1 and up2, first and second most frequent occurring angle interval.
Length of pore net- Length based on the number of All length ,5 pixels Mean lp , Median lp , Std work, lp pixels in a pore network were excluded. lp , Skew lp , Kurt lp . branch number of pore throats Apparent connec- z* 5 Pore network is increasnumber of pore bodies tivity, z* ingly connected with higher values of z*. Total porosity, w t
w t 5 Ap /A tot wpb 5 Apb,sum /A tot
The part of the total porosity, which is taken up by pore bodies.
Porosity of pore wpth 5 Apth,sum /A tot throats, wpth
The part of the total porosity, which is taken up by pore throats.
Porosity of pore bodies, wpb
rameter values are studied, assuming that each is representative of its threedimensional equivalent. The measurement procedures were fully automated in the image-analysis software program SCION IMAGE. Characterization of Granular Material For each grain that lies completely within a measurement window the basic parameters presented in Table 1 were determined. Example results of characterizing the reference sample are illustrated in Figure 3. The relative cumulative weight frequency of the grain-size distribution (Fig. 3B) was calculated by summing the ratios between the individual grain area and the total area of the grains in the window (Kellerhals et al. 1975). Because of noise in the binary images, all small pixel clusters with an → FIG. 3.—Example image of the reference glass beads sample and output of textural characteristics. A) The original binary image, B) the resulting cumulative grain-size distribution, C) the histogram of aspect ratio, and D) the orientation histogram of particles together with derived (statistical) parameters.
TEXTURE OF LAMINATED SANDS USING IMAGE ANALYSIS equivalent diameter smaller than 30 mm (ø 45 pixels in the reference sample; 25 pixels in I21; 5 pixels in B2) were excluded from size measurement. The aspect ratio, q, is descriptive only of the general outline of the grain sections and is equivalent to grain sphericity (Fig. 3C). Grain sections with aspect ratios smaller than a user-defined threshold value of ⅔ were used in the fabric analysis (Schwan 1989). The mean grain orientation ug and the mean resultant length Rg were computed (Davis 1986) and first and second most frequently occurring orientations were extracted (Fig. 3D). The perimeter of the aggregate of grain sections contained within the measurement window is the sum of perimeters of individual grains. The total perimeter of sectioned spherical particles in a thin section is the twodimensional equivalent of the three-dimensional surface area of the same material (Rink and Schopper 1978). By assuming spherical particles of the sediment, the total sum of perimeters determined with image-analysis was converted to specific surface area per unit volume of solid material (Ssp). Packing of grains is defined as the spatial arrangement of grains in an aggregate (Allen 1985; Dullien 1991). A method to characterize particle packing is provided by packing parameter, R, which is the ratio of the observed mean of the nearest neighbor distances, ra, to the expected mean of the nearest neighbor distances if the same number of particles are randomly distributed, re (Davis 1986). R is a measure of the degree of clustering or ordering of particles in the measurement window. Natural samples with values of R larger than 1.0 are ordered (closely packed), and those with values of R approaching or smaller than 1.0 are clustered (loosely packed) (Jerram et al. 1996). Characterization of Pore Space The pore space in unconsolidated sand can be seen as a continuous, irregularly shaped, three-dimensional interconnected network of capillary tubes, which is only partly represented by two-dimensional digital images. To determine the size characteristics of the pore network a technique originally based on erosion and dilation of pore pixels was used (Doyen 1988; Ehrlich et al. 1991; Ehrlich et al. 1984; Fens 2000). Erosion and dilation are binary image-processing operations for either peeling off or adding to one or more layers of pixels on the boundary between grains and pore space. A continuous chain of pore space can be separated into constrictions (pore throats) connected to wider chambers (pore bodies) with an increasing number of erosion cycles, independently of shape characteristics. Instead of using erosion and dilation cycles, the ultimate eroded points of features were used to determine number and size of pore bodies (Russ 1998). The ultimate eroded points image of a pore network consists of a number of small pixel clusters, located at the centers of pore bodies. Each pixel cluster contains information on the smallest distance from the center of a pore body to the nearest boundary between grains and pore space (Fig. 4A). Assuming that pore bodies are spherical, this distance can be seen as the radius of the largest inscribed circle within a pore body. Recording the number of occurrence and distance information in the ultimate eroded points image produces a size distribution of the pore bodies (Fig. 4D). By dilating the ultimate eroded points a number of times corresponding to their distance information and overlaying this image with the original image of grain sections (Boolean OR operation), the pore throats in the network are revealed (Fig. 4B). Computing their equivalent area diameter results in a size frequency distribution of pore throats (Fig. 4E). The preferential orientation of the pore network was studied by ‘‘skeletonizing’’ the original image of pore space. Skeletonization reduces pore space to a network of lines by progressively eroding pore pixels to singlepixel-wide lines connecting the ultimate eroded points (Fig. 4C) (Russ 1998). Overlaying the skeletonized pore space and the ultimate eroded points image with a Boolean XOR operation shows that the skeleton branches are the shortest connection, taking into account the shape of the pore network, between the centers of pore bodies. This can be seen as an
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indication that the known effect of erosion and/or dilation operations on shape and orientation of features (Russ 1998) is limited with skeletonization. The distribution of branch lengths was determined and orientation parameters similar to those used for the orientation analysis of grains were computed (Fig. 4F, G). Pore connectivity is defined as the degree to which the pore space is multiply connected in three dimensions (Dullien 1991), which is expressed in the coordination number z (the average number of pore throats meeting at a pore body). In this study the apparent two-dimensional coordination number z*, defined as the ratio of the number of throats to bodies, was used (Fens 2000). The total porosity wt was subdivided into that part of the total porosity that takes up the earlier differentiated pore throats, wpth, and the other part that is occupied by pore bodies, wpb (Table 1). OPTIMIZING SAMPLING STRATEGY
Before applying the automated image-analysis technique, the optimum area covered by the measurement window (sample size) and the resolution perpendicular to stratification (sampling interval) were evaluated. Sample Size As Buchter et al. (1994) pointed out, the area sampled by a measurement window should be large enough for the measured parameters to have a statistically stable value. In this context, parameters obtain a stable value in a heterogeneous medium if an increase in sample area does not significantly change their value. The minimal area at which a parameter stabilizes is called a representative elementary area (REA). A too small window introduces random noise in the measured values, a feature known as inadequate sample support (Anguy et al. 1994). The representative elementary areas of mean dg, mean dpb and mean dpth, were evaluated for the glass beads sample by increasing the square measurement window by increments of 100 pixels (0.54 mm) in the horizontal and vertical direction (Fig. 5). Mean dg and mean dpth attain stable values for a side length of approximately 1300 and 1700 pixels, respectively (7.0 and 9.2 mm). The heterogeneity of mean pore body size, however, appears to be too large for a stable value at the maximum sample size (side length of 2000 pixels, 10.8 mm). Considering the representative elementary area principle, mean dpb may probably attain stable values at larger measurement window sizes. The fact that the three parameters stabilize at different representative elementary areas is a feature also observed by Bear and Braester (1972). In order to predict the representative elementary area of natural sand, the ratio of the mean grain size to the representative elementary area found for the reference sample is used. When particle size increases, the corresponding representative elementary area should also increase in order to maintain stable measurement values. By assuming a constant ratio of grain size to the representative elementary area over the whole sand size range, the representative elementary area for material with known grain size can be estimated. The representative elementary area of any sand sample (REAsample) containing particles with mean grain size (Mean dg, sample), is estimated by REA sample
[
]
2
REA glass beads Res glass beads 5 Mean d g,sample · , Mean d g,glass beads Res sample
(1)
where REAglass beads is the representative elementary area of the reference sample (1300 3 1300 pixels) and Mean dg, glass beads is the mean size of the glass spheres (350 mm). The term in brackets corrects for differences in pixel resolution between the mosaics. For samples I21 and B2, with mean grain sizes of 281 and 197 mm respectively, this would mean a representative elementary area of 1575 3 1575 pixels (6.3 3 6.3 mm) for
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FIG. 4.—Example images of the reference glass-beads sample as used in the characterization of the pore network and characteristic output. A) The image containing the ultimate eroded points, B) image of reconstructed pore bodies, C) the image of skeletonized pore network, D) the histogram of pore-body size distribution, E) the histogram of pore-throat size distribution, F) the histogram of pore skeleton length, and G) the orientation histogram of the individual branches of the pore network together with derived (statistical) parameters.
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analysis of the succession of laminae, which decomposes the succession of laminae into cyclic components. These components are defined as the thickness within which a cycle of parameter variation takes place (Bracco Gartner 2000; Davis 1986). Sampling on regularly spaced intervals may obscure the lower-frequency pattern by also sampling the higher-frequency components. This aliasing error on the lower-frequency pattern is caused by all frequencies that have wavelengths less than twice the sampling interval. This relationship is based on the Nyquist frequency (Davis 1986), which states that the shortest wavelength (and thus the highest frequency) that can be detected is two times the sampling interval. In order to resolve a characteristic laminae succession this implies for our study that: sample interval 5 thickness of thinnest laminae succession/2
(2)
The thinnest laminae succession in I21 is approximately 1.4 mm, which can be characterized effectively if a sampling interval of 700 mm or 180 pixels is used. The thickness of the thinnest succession in B2 is 1.7 mm, which means that for effective characterization the sampling interval should be less than 850 mm or 70 pixels. The width of the rectangular measurement window for both samples is restricted to 2000 pixels, which implies in accordance with the areas calculated with Equation 1, that the heights of the REA should ideally be 1240 pixels (4.9 mm) for the I21 and 88 pixels (1.1 mm) for B2. Especially for sample I21, these values do not agree with the smaller window heights based on Equation 2. Consequently, the measured parameters may contain additional noise due to the sampling procedures, but this random component is considered small (Fig. 5) compared to variations observed between laminae (Fig. 6). RESULTS
Application of the digital image-analysis procedure to the two natural sand samples resulted in 123 measurements over the eolian sample and 83 measurements over the fluvial sample. In this section only parts of both profiles are presented, because it is our intention to show whether and how the technique deals with stratification. Profiles
FIG. 5.—Course of parameter values (A, mean grain size; B, mean pore body size; C, mean pore throat size) with increasing window size measured in the reference sample. The images used in Figures 3 and 4 correspond to the square measurement window with 1800 pixels side length (9.7 mm).
the eolian sample and 420 3 420 pixels (5.3 3 5.3 mm) for the fluvial sample. Sampling Interval When the thickness of sedimentary lamination is less than the height of the measurement window, information on individual laminae is averaged with surrounding laminae. Furthermore, the regular sampling interval causes the measurement window to sample over irregularly spaced laminae. The resulting loss of information on individual laminae cannot be avoided completely, but the sampling interval can be optimized for maximum resolution. A way of finding the optimum sampling interval is by using Fourier
The transect over the thin section of eolian sands shows a sharp increase in mean grain size (from 180 mm to 250 mm) approximately halfway the profile indicated in Figure 6A by line A–A9. The line follows a curved bedding plane. A distinct lamina is found at position B in which mean grain size decreases approximately 40 mm compared to the material above and below. All measured parameters except Std dg, wt , and z* show similar abrupt changes at these positions. The binary image of the fluvial sands shows a number of laminae cycles, which consist of a rapid alternation of fine and coarse material (Fig. 6B). The apparently coarse lamina crossing the image at an angle, shown in Figure 6B by line C–C9, is a band containing iron oxide. Variations in all other variables, except Std dg, wt , R, and z*, show the same pattern as the variations in the mean grain size. Verification of Image-Analysis Results The image-analysis results of the two natural samples are overall average values of the complete transects. Despite the fact that both techniques measure different characteristics of grains, that the sample material was taken at slightly different locations, and that several stereological corrections should be made, the resulting size distributions are compared directly (Table 3). The difference in mean grain size between the two techniques for the reference material and the two sand samples ranges from 30% to 26% of the value determined in the laboratory. The standard deviation and higher statistical moments are systematically underestimated with image-analysis. The total porosity of the glass-beads sample, measured with image-analysis, corresponds well with that determined in the laboratory. Image-analysis of
FIG. 6.—Details of transects over the thin sections of A) eolian sand and B) fluvial sand and corresponding measurement results of several parameters. Note the generally high correlation between the mean grain size with other parameters. The labels A–A9 and B in binary image I21 indicate a bedding plane and a lamina, respectively. Label C–C9 in binary image B2 indicates a narrow band with iron-oxide precipitation.
140 E.H. VAN DEN BERG ET AL.
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TABLE 3.—Grain-size characteristics (in f units) and total porosity determined by laboratory experiments and by image-analysis. Glass beads Parameter
Lab.
I.A.
Mean dg (f units) (mm) Std dg Skew dg Kurt dg wt
1.51 350 0.71 9.86 117.5 0.40*
1.64 320 0.27 0.44 1.20 0.40
Eolian sand, I21 Diff.
Lab.
I.A.
9% 62% — — 0%
1.85 281 0.89 5.56 49.20 0.41
2.38 192 0.53 0.26 1.24 0.36
Fluvial sand, B2 Diff.
Lab.
I.A.
Diff.
30% 40% — — 12%
2.34 197 1.50 3.73 17.7 0.42
2.26 208 0.58 0.09 1.13 0.40
26% 61% — — 5%
* Average of three measurements, with a standard deviation of 0.04.
the natural sandy samples, however, underestimates the experimentally determined porosity values. Other parameters, for which no independent laboratory measurements are available, compare well with general values found in the literature (Doyen 1988; Dullien 1991; Fens 2000; Jerram et al. 1996).
where they are related to w pb and wpth. Packing is incorporated together with the grain shape characteristics into the third factor. The second major difference between factor matrices is the factor in which total porosity is incorporated. Total porosity, which is unrelated to other parameters except for wpb in the eolian sample, shifts from the second factor in the eolian sample to the sixth factor in the fluvial sample.
Statistical Data Analysis The large data sets that are generated with image-analysis can be used as a basis for a wide variety of studies. As an example of how the data can be analyzed we performed standard factor analysis on both data sets in order to discover and simplify the interrelations between individual parameters. With factor analysis the coordinate system of the original parameters is rotated to such a position that the first axis (or factor) explains the highest common variance in the data set and subsequent factors explain a decreasing amount of common variance (Davis 1986). Using SPSS software, only factors whose eigenvalues (explained variance) were greater than 1 were extracted. It was observed that the median, skewness, and kurtosis of the parameters were not related to any of the other parameters, and for this reason were left out of the factor analysis. The rotated factor matrices for both profiles are presented in Tables 4 and 5. For the eolian sample, six factors explain 85.3% of the variance, whereas for the fluvial sample six factors explain 86.0% of the variance. There are, however, a number of differences in the parameter composition of the factors, especially in the first three. The first difference between the eolian and fluvial sample is found in factor number 1, which contains grain-sizerelated characteristics. In the fluvial sample the mean and standard deviation of dpth and R are absent. They can be traced back in the second factor, TABLE 4.—Rotated factor matrix1 extracted from the reduced data set of the eolian sample, I21. Factor/Parameter
1
Mean dg Std dg Mean dpb Std dpb Mean dpth Std dpth Mean q Std q ug Rg up Rp Mean lp Std lp wt wpb wpth R z* Eigenvalues % of explained variance
20.89 0.31 0.91 0.86 0.92 0.88 0.21 20.19
2
3
4
5
20.17 0.33 0.31
0.12 0.18 0.10
0.18 20.14 20.15
0.88 0.82 20.12
20.11 20.21 20.14 0.13 20.22
20.64 20.71 7.62 40.1
0.95 0.99 0.25 20.13 0.16 2.82 14.8
20.56 0.21 2.07 10.9
0.15
0.96 0.99 0.96 0.93 0.95 0.97 0.83 0.74 0.80 0.82 0.77 0.85 0.96 0.96 0.99 0.99 0.95 0.79 0.82 — —
0.88 0.13 0.85 0.88
20.13 20.14 0.27
0.96 0.96 0.11
hi 2#
0.24 0.35
0.85 0.21 0.24
6
0.94
1.54 8.08
1.19 6.31
0.48 1.97 5.14
1 Factor loadings (21 to 1) indicate the degree of interparameter association contained within a factor. Those shown are only loadings larger than 0.10 and smaller than 20.10. In bold are factor loadings larger than 0.60 and smaller than 20.60, which are used in the interpretation. # The communality, hi 2, gives the common percentage of variance of the ith parameter in the matrix of all factors.
DISCUSSION AND CONCLUSIONS
Two important issues on the performance and implications of the digital image-analysis procedure need to be addressed. The first is related to the general aim of this study: does the image-analysis technique generate reliable data on textural characteristics? Direct comparison between imageanalysis measurements and experimental results of laboratory measurements provides a quantitative answer, but several technical, sampling, and interpretation aspects of the image-analysis procedures lie at the basis of the generally observed discrepancies. Of particular importance is the proper segmentation of the images into binary format. Viewing thin sections with transmitted-light microscopy, measurements of total porosity systematically underestimate experimentally determined values, because of the shelving effect and underdetection of sub-pixel-size pore space (Fens 2000; Ruzyla 1992). The relatively good agreement between the reference sample and box core B2 (0 to 6% deviation) can be explained by the more intense blue staining of the resin, which increases the ability of segmenting the images more uniquely. Technical aspects that may cause deviation from the experimentally determined grain-size characteristics are errors made by the automated separation procedure and the difference in detection limits between both techniques. Van den Berg et al. (2002) noted that size distributions determined after automatically separating touching grains overestimate the number of smaller and larger grain-size intervals while underestimating the number of particles in the mean grain-size range. For higher TABLE 5.—Rotated factor matrix1 extracted from the reduced data set of the fluvial sample, B2. Factor/Parameter Mean dg Std dg Mean dpb Std dpb Mean dpth Std dpth Mean q Std q ug Rg up Rp Mean lp Std lp wt wpb wpth R z* Eigenvalues % of explained variance
1
2
20.77 0.89 0.87 20.36 0.12
20.26 20.60 5.36 28.4
4
0.33 20.24
20.16
20.23
5 0.25 20.11 0.38 0.32
6
hi 2#
0.41 0.94 0.12 0.11
0.94 0.98 0.96 0.94 0.99 0.99 0.87 0.84 0.97 0.84 0.98 0.87 0.96 0.96 0.94 0.98 0.99 0.87 0.83 — —
0.99 0.99 0.83 0.88 0.21
0.16
0.96 0.87 0.11 0.11
3
20.91 0.99
3.74 19.7
20.40 0.47 0.20 0.84 20.38 3.54 18.6
20.10 0.85 0.91 20.11 0.24
0.18 20.39 1.34 7.07
20.12 0.76 0.29
0.19
0.14 0.36 1.25 6.61
1.07 5.66
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E.H. VAN DEN BERG ET AL.
statistical moments, differences are caused mainly by the fact that imageanalysis results are based on size distributions between 30 to 2000 mm whereas those of the laser particle sizer range between 0.15 and 2000 mm. Furthermore, differences in sampling volume and location and the omission of stereological corrections (the two-dimensional to three-dimensional effect) are factors to be considered when comparing the two analytical techniques (Russ 1998). Avoiding a number of these limitations (except the two-dimensional to three-dimensional effect), the glass-bead sample shows a relatively good agreement for mean grain size. Higher statistical moments are largely underestimated, because of the inability of the laser particle sizer to correctly interpret a noise-containing signal which is inherently generated by single-size particle aggregates. In relation to the first question, we also evaluated whether the imageanalysis technique can detect sedimentological variability across laminae. Quantitative reference data are not available, but a visual comparison between the stratification observed in thin sections and parameter variability provided qualitative proof of its ability for capturing sedimentological variability. This is one of the most promising aspects of image-analysis: it provides quantitative estimates of parameters that could not be determined using conventional laboratory techniques at this fine scale, because of the increased chance of disrupting the internal structure and fabric in unconsolidated sediment. One aspect, however, that influences its ability to determine variability is the size of the measurement window. It is important for future studies to ensure local homogeneity by extending the sampling window parallel to sedimentary lamination. Factor analysis applied to the eolian and the fluvial data sets gives insight into the correlation structure between parameters. The mean grain size describes the largest variance in both data sets. For the eolian sands, the porebody and pore-throat size characteristics, packing, and the apparent coordination number are highly correlated with mean grain size. Porosity appears to be unrelated to other parameters in the data set. Because the variations in porosity should be related to variations in other textural characteristics, it can be concluded that parameters that do influence porosity (e.g., particle roughness) are not included in the present image-analysis procedure (Pryor 1973). Other reasons may be the insufficient sensitivity of the image-analysis techniques in combination with deviation from the ideal sample locations and inadequate sample support. For the fluvial sample similar relations exist but the throat-size characteristics and packing are unrelated to grain-size characteristics. This difference is most probably due to localized iron oxide precipitation, which takes place preferentially in pore throats and at grain contacts in the finer-grained laminae (Berner 1980). Furthermore, factor analysis teaches us that because mean grain size is highly related to other textural characteristics, these additional parameters could be predicted using empirical relationships, making these measurements largely redundant. Because the procedure presented here is a versatile tool to quantify textural variations across sedimentary structures at the millimeter scale, imageanalysis of thin sections can now be used to evaluate the effect of smallscale sedimentary heterogeneity on the magnitude and directional characteristics of physical properties. ACKNOWLEDGMENTS
This study is part of the HydroSed project at the Vrije Universiteit, Faculty of Earth Sciences. We would like to thank R.L. van Dam, B.M. van Breukelen, H. Jonker, V.E.A. Post, L.H. Broekema, P.W. Bogaart, and R.F. Houtgast for their discussion on the subject, their help in classifying the images, and commenting on earlier versions of this manuscript. J.A.M Kenter and J. Groen are acknowledged for initiating this project and their constructive review. E. Seyhan and S.J. Purkis are thanked for their advice on the factor analysis. The work has benefited greatly from reviews by P. Francus, T.W. Fens, J.H.S. Macquaker, and J.B. Southard. REFERENCES
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