Digital grain-size analysis based on autocorrelation algorithm

    Digital Grain-Size Analysis Based On Autocorrelation Algorithm Zhixuan Cheng, Haijiang Liu PII: DOI: Reference:

S0037-0738(15)00158-X doi: 10.1016/j.sedgeo.2015.07.008 SEDGEO 4886

To appear in:

Sedimentary Geology

Received date: Revised date: Accepted date:

12 March 2015 14 July 2015 15 July 2015

Please cite this article as: Cheng, Zhixuan, Liu, Haijiang, Digital Grain-Size Analysis Based On Autocorrelation Algorithm, Sedimentary Geology (2015), doi: 10.1016/j.sedgeo.2015.07.008

This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

ACCEPTED MANUSCRIPT DIGITAL GRAIN-SIZE ANALYSIS BASED ON AUTOCORRELATION ALGORITHM Zhixuan Cheng1, and Haijiang Liu1 Ocean College, Zhejiang University, 866 Yuhangtang Road, Hangzhou, Zhejiang 310058, China,

RI P

[email protected]

T

1

SC

Abstract:

Grain size is one of the most important parameters in geology and coastal engineering. However,

NU

all traditional methods are time consuming, laborious, and expensive. In this study, the

MA

autocorrelation technique, which was first expounded by Rubin (2004), was extended to estimate the size of well-sorted sediments and the grain-size distribution of mixed-size sediments. Long

PT ED

and intermediate axes of well-sorted sediments ranging from 1 to 20 mm obtained from applying the autocorrelation method are compared with the corresponding results measured using a vernier caliper. Using the autocorrelation technique, the sediment mean size was calculated and

CE

was found to compare better with point counts than sieving. Regarding the mixed-size sediment, a nonlinear programming method, which is different from the conventional ‘least-squares with

AC

non-negativity’ method, the kernel density method, and the maximum entropy method, was used to obtain the representative grain sizes and associated sediment inherent parameters, such as mean diameter, median diameter, sorting, skewness, and kurtosis. Image pre-processing was used in the present analysis to enhance the contrast of the recorded image, and a conversion method applied to take into account the difference between the two-dimensional digital image method and the three-dimensional sieving method. Using the modified fitting points and the improved Gaussian function fitting method, the cumulative grain-size distribution curve and the probability density curve of the mixed-size sediments were obtained. The enhanced autocorrelation technique that was developed from the traditional ‘look-up-catalogue’ approach provided a more

1

ACCEPTED MANUSCRIPT accurate estimation of the grain-size distribution, as well as the relevant physical parameters of

RI P

T

the mixed-size sediment.

Keywords: Well-sorted grains, Mixed-size grains, Digital image, Autocorrelation algorithm,

NU

SC

Grain size, Grain-size distribution

MA

1. Introduction

The movement of sediment particles, such as erosion, transport, and deposition, are governed

PT ED

fundamentally by grain size. Sieve analysis is the most conventional and convenient method for analyzing particle size distribution. Manual sieving and mechanical sieving are both laborious

CE

and time consuming, and other traditional methods, such as settling and laser diffraction, have their limitations as well. It is difficult to apply the settling method on coarse-grained systems,

equipment.

AC

and the laser diffraction method has a limitation on the size range as well as requiring expensive

The digital grain-size analysis has been developed to overcome these problems. To quantify the grain size, a high-resolution digital camera has been used to take photographs by an increasing number of researchers. Ibbeken and Schleyer (1986) were the first to propose digital grain-size analysis. They used the ‘photo-sieving method’ first to quantify the grain size of coarse-grained, unconsolidated bedding surfaces. There are two main types of modern digital analysis methods. The first is the edge detection method (e.g., Butler et al., 2001; Sime and Ferguson, 2003; Graham et al., 2005a, b, 2010; Baptista et al., 2012; Chang and Chung, 2012;

2

ACCEPTED MANUSCRIPT Karunatillake et al., 2013), which is a geometrical approach to determine grain-size properties. This approach is a combination of two basic image-processing steps: gray-scale thresholding to

T

create a binary image and watershed segmentation to grow edges on the binary image to identify

RI P

individual grains. Graham et al. (2005a, b) presented its application for measuring exposed fluvial gravels and other coarse-grained sediments. An improved automated image-processing

SC

algorithm was proposed by Baptista et al. (2012) and its accuracy and robustness were validated

NU

in both the laboratory and sandy shores. However, the edge detection method has limitations such as particle overlapping and indistinguishable grains in the image (Graham et al., 2010).

MA

Another limitation is the lower limit of clast detection at approximately 23 pixels, producing less accurate results (Graham et al., 2005a). On the other hand, to quantify the grain size, Rubin

PT ED

(2004) proposed the second method using an autocorrelation algorithm. This technique is a statistical approach that uses the photograph texture in an image. To derive the mean grain size of a mixed-size sand sample, Rubin et al. (2007) had validated its utility. Barnard et al. (2007)

CE

demonstrated that the autocorrelation technique works well on high-energy dissipative beaches.

AC

The autocorrelation method was further developed by Buscombe (2008) and Buscombe and Masselink (2008), and the grain-size distribution was successfully estimated. This method showed its validity in both coarse sand (0.7 mm) and gravel (up to ~20 mm). Warrick et al. (2009) presented an application of the autocorrelation method on a mixed sand and gravel beach. Pina and Lira (2009) demonstrated that the autocorrelation method is more similar to the sieve analysis than laser diffraction. In the case of large-scale airborne imagery of landforms, the derivative technique, using local semivariance, proved to be effective (e.g., Carbonneau et al., 2004, 2005). To characterize the mean grain size, Buscombe et al. (2010) recently used the spectral decomposition of sediment images. For its utility in field settings, especially in remote

3

ACCEPTED MANUSCRIPT or inaccessible areas or long-term deployments, the new non-calibration method facilitates the development of a fully transferable method. In addition, the “autocorrelation” approach

T

including subsequent wavelet method has been successful in obtaining grain-size distribution and

RI P

associated parameters (Buscombe and Rubin, 2012; Buscombe, 2013; Buscombe et al., 2014). Although Buscombe and Rubin (2012) explored a framework for the simulation of natural

SC

well-sorted granular material based on the principle of Voronoi tessellation, most former

NU

researchers generally focused on the mixed-size sediment on site, and for the well-sorted sediment detailed validation of autocorrelation method was scarce. However, well-sorted

MA

granular material is very common in nature, and the first step would be an appropriate estimation of its size characteristics before applying this technique to the mixed-size sediment case. For the

PT ED

mixed-size natural grains, most researchers failed to provide realistic grain-size distributions (Rubin et al., 2007; Barnard et al., 2007; Warrick et al., 2009). Buscombe (2008) pointed out that both the least squares method and the least-squares with non-negativity method produce

CE

unrealistic grain-size distribution. For more accurate grain-size distribution, Buscombe applied

AC

the nonparametric kernel density estimation method. To estimate grain-size distribution, Gallagher et al. (2011) adopted the maximum entropy method based on autocorrelation analysis. Both of the aforementioned methods ensure that each grain-size fraction is positive and the proportion summation is unity, and these two constraints improve the grain-size distribution estimation. However, Buscombe (2008) found that the kernel method as yet is ineffective for skewness estimation. Gallagher et al. (2011) stated that the maximum entropy method was more suitable for bi-modal grain-size distribution. Nevertheless, many researchers realized the systematic bias that digital image analysis was a two-dimensional approach but no effort was made to rectify it (Buscombe, 2008; Pentney and Dickson, 2012). In addition, Rubin’s method

4

ACCEPTED MANUSCRIPT requires the sample image containing many grains, which decreases the image resolution, producing less accurate results.

T

Based on Rubin’s autocorrelation algorithm, an improved digital grain-size analysis method

RI P

was proposed in this study to estimate the grain size of well-sorted sediments and the grain-size distribution of mixed-size sediments. This method offers accurate estimations of the grain size

SC

(e.g., the long axis, the intermediate axis, and the mean size) of the well-sorted surface sediment

NU

ranging from 1 to 20 mm. Regarding the mixed-size sediment, a newly developed method using a nonlinear programming technique, which is different from the ‘least-squares with

MA

non-negativity’ method, the kernel density estimation method, and the maximum entropy method, provides accurate grain-size distributions and associated sediment inherent parameters.

PT ED

This paper is organized as follows. Section 2 describes the image recording technique and the preparation of sediment samples. Section 3 presents the detailed description of methodology and validation of the well-sorted sediments. Section 4 discusses digital grain-size analyses of several

CE

representative mixed-size sediment parameters, such as mean diameter, median diameter, sorting,

AC

skewness, and kurtosis; in addition, an improved Gaussian function fitting method was introduced to obtain the cumulative grain-size distribution curve and the probability density curve of the mix-sized sediments. Finally, section 5 summarizes and concludes this paper.

2. Image recording and sediment samples In this study, photographs were recorded using a Cannon EOS 60D digital camera, which was set on a tripod, to ensure that the camera is at approximately 0.48 m above the sample plate with a shooting angle being orthogonal to the sample plate. The recorded image had a resolution

5

ACCEPTED MANUSCRIPT of 5184  3456 pixels. Using a plate, the grain surface was flattened before recording the image. Each photograph samples a 15  10 cm area, and the image spatial resolution is approximately

T

0.03 mm per pixel. For each grain sample, three photographs were recorded for grain-size

RI P

analysis. Sediments in a grain sample were redistributed by shaking the sample plate before

SC

taking each photograph. All photographs were taken under the same lighting condition. Natural sediments were sieved into 11 well-sorted groups after 12 sieves (i.e., 1, 1.25, 1.43,

NU

1.6, 2.0, 2.5, 4.0, 5.0, 6.0, 8.0, 10.0, and 12.0 mm). These samples are regarded as well-sorted grains. The size of well-sorted sediments ranges from 1 to 20 mm, covering coarse sands, gravels,

MA

and pebbles. However, mixed-size grain samples were obtained by a mixture of the well-sorted sediment components in terms of a designed mass mixing proportion. In total, 11 mixed-size

PT ED

samples were considered. The designed mixing proportions of these samples are listed in Table 1, and six representative digital images (images of three well-sorted grains as well as three mixed-size grains) are shown in Figure 1. As shown in Table 1, mixed-size sediments consist of

CE

coarse sands and gravels. MS. 1 is mixed with two grain sizes leading to a bimodal size

AC

distribution; MS. 2–7 consists of five continuous well-sorted sediment components; and MS. 8–11 consists of eight components.

3. Digital grain-size analysis of well-sorted sediments 3.1 Methodology In this paper, we used the autocorrelation technique to quantify the size of well-sorted sediments. This approach is based on the spatial autocorrelation function that is sensitive to the dimensions of the grains in the images of sediments (Rubin 2004). The spatial autocorrelation r

6

ACCEPTED MANUSCRIPT between two regions in an image is given by, i

i

 y)

(1)

i

 ( xi  x)2

 ( yi  y)2 i

RI P

i

T

r

 ( x  x)( y

where xi and yi are the intensities of corresponding pixels in the two regions, andx andy are the

SC

mean intensities of pixels in their own regions. The region of x moves with a certain amount of pixel shift (i.e., offset) and obtained the region of y. Grain sample of a certain size has its own

NU

autocorrelation curve, and the curve of fine grains is always steeper than coarse sand under the

MA

same image resolution. Autocorrelation r approaches 1 when the offset between the two regions is small relative to grain size, and it decreases to zero when the offset approaches the size of the

PT ED

largest grains (Rubin 2004). However, this study shows that, different from the previous description, the calculated spatial autocorrelation can be smaller than zero, and then increase to a peak value as highlighted in Figure 2. This phenomenon is caused by the self-organization in

CE

sediments. Ortoleva et al. (1987) proved that geochemical self-organization is widespread in rocks. All types of rocks are capable of displaying a number of mineralogical or textural patterns

AC

that are not inherited, and therefore, some type of spontaneous self-organization can be ascribed to such patterns. Hence, the surface of uniform-size grains has a similar texture. If the sample image contains hundreds (or less) of sediment particles, then there is a high possibility of the two research areas (i.e., region of x and y) having similar textures, leading to the occurrence of a peak autocorrelation value. Accordingly, the actual length indicated by the offset at the peak value (OPV) of the autocorrelation curve can be regarded as the grain size, which works on the principle similar to the Buscombe et al. (2010) viewpoint, who proposed that the frequency of “typical” features (r = 0.5) in an image was directly related to the mean grain size. Measurements of grain size from the photographs, and in the field, provided consistent

7

ACCEPTED MANUSCRIPT estimates of the clast axis lengths. Warrick et al. (2009) suggested that surface grains are dominantly oriented with the short axis in the vertical direction. Digital photographs were found

T

to provide accurate estimates of the long and intermediate axes of the surface sediment. In this

RI P

study, we extended Rubin’s approach of autocorrelation analysis from only one direction to eight directions in one image, and chose the largest and the smallest OPVs to estimate the long and

SC

intermediate axes of the sample grains, respectively. Here, the autocorrelation curves of sediment

NU

sample with size ranging between 5 and 6 mm are selected as an example. The autocorrelation curves of all the eight directions show the occurrence of peak value, and results with the largest

MA

and smallest OPVs (corresponding to the long and intermediate axes) are presented in Figure 2. The mean size of the well-sorted sediments is represented by taking the average of the eight

PT ED

results.

Based on the above viewpoint, a MATLAB script was developed to estimate the long axis, the intermediate axis, and the mean size of the well-sorted sediments. The calculation flowchart

CE

is presented in Figure 3. Five samples containing 10 grains in each were randomly selected for

AC

method validation from the target sediment group, and the long and intermediate axes of these grains were measured using a vernier caliper (referred to as the “point counts” method). However, the mean size of sediment sample was also obtained through sieving. As Qian and Wan (1983) mentioned, the geometric mean diameter of the two neighboring sieves can be used to represent the mean size after sieving. Here, the two methods were applied (point counts and sieving) to validate the mean size of the well-sorted grains.

3.2 Validation Using a vernier caliper, the model validation was performed by comparing the

8

ACCEPTED MANUSCRIPT image-identified grain size with the physical measurement, covering from the sand to pebble with the grain size changing from 1 to 20 mm. Figure 4(A) illustrates the comparison between

T

the image analysis and point counts measurement of the well-sorted sediments. The diagonal in

RI P

the figure represents a perfect agreement, while the two dashed lines represent the agreement in a factor of 2. The long and intermediate axes measured from these two different techniques are

SC

well correlated (the correlation coefficient, r2=0.986, 0.982, respectively) for the entire grain-size

NU

range. Sediment mean sizes estimated through digital images are also inconsistent with the point counts measurement (r2=0.990).

MA

Based on a large number of samples from different sites, Warrick et al. (2009) suggested that photographs provide good estimates of the long and intermediate axes, but relatively poor

PT ED

estimates of the short axis. However, their analysis is based on the unburied sediments. In Figure 4(A), we can find that image-based intermediate axes between 4 and 12 mm are underestimated. This is ascribed to the fact that small grains, generally, are not exactly oriented with the short

CE

axis in the vertical direction when they are overlapped. Many surficial particles may rest with an

AC

angle of slope leading to a short intermediate axis if digitally recording from the top. However, small sediment particles (grain size <4 mm) are not significantly affected because their three axes are rather uniform. Nevertheless, considering the existence of cohesiveness between small sediment particles, the long axis is slightly overestimated by digital image analysis. Considering the mean size, coarse sand and gravel (between 5 and 11 mm) are slightly underestimated in digital image analysis, because the mean size measured by the image resulted from the average value of eight directions and some of them are underestimated (e.g., the intermediate axis). Actually, except the long axis, the other seven results achieved by photographs are generally smaller than their real sizes because of the oblique laying of the surficial grains.

9

ACCEPTED MANUSCRIPT In addition, the mean size was also measured by sieving. Figure 4(B) shows the comparison of grain size estimated between the autocorrelation and sieving. The results of these two methods

T

are also highly correlated (r2=0.970). However, the mean size measured by sieving is smaller

RI P

than the results from the digital image analysis, because sediment mean size obtained by digital image analysis is calculated by averaging eight values no smaller than the intermediate axis,

SC

while the sieving method measures a result of the intermediate axis. This leads to an

NU

overestimation of the mean grain size using image analysis in comparison to the mean grain size using sieving, which is in agreement with the findings of Barnard et al. (2007) and Warrick et al.

MA

(2009). Considering Figures 5(A) and (B), it was found that mean size estimates compare better with “point counts” measurements than mean sizes calculated through sieve analysis. This is also

PT ED

consistent with the findings from other researchers (e.g., Barnard et al., 2007; Buscombe, 2008;

CE

Buscombe and Masselink, 2008; Warrick et al. 2009; Buscombe et al., 2010; Buscombe, 2013).

AC

4. Digital grain-size analysis of mixed-size sediments 4.1. Methodology for estimating representative grain sizes and associated parameters

The autocorrelation method is sensitive to the texture in an image. Put differently, the accuracy of this method is determined by the serial correlation of numerical values represented by the sediment images. We focused on the grain texture and used the offset at the peak value of the autocorrelation curve to quantify the size of the well-sorted grains. Nevertheless, digital grain-size analysis of mixed-size grains is more sophisticated. A detailed flowchart of digital grain-size analysis of mixed-size grains is presented in Figure 5. The external texture of each

10

ACCEPTED MANUSCRIPT sediment particles and the pores among them together contribute to the autocorrelation curve. In general, a calibration catalogue was set up based on the autocorrelation curves of a number of

T

well-sorted sediment fractions having similar external textures. Regarding the mixed-size

RI P

sediments, the existence of dissimilar textures and unusual patterns, especially for the large-size grains, renders the relevant autocorrelation curve different from well-sorted sediment-based

SC

autocorrelation curve. Therefore, it is inappropriate to add the autocorrelation curve of the “raw”

NU

mixed-size sediments directly into the well-sorted sediment-based catalogue to calculate the representative grain sizes. To obtain a better estimation of the median size and other physical

MA

parameters of the mixed-size grains, a preprocessing procedure (image enhancement) was implemented. A comparison between images with and without image enhancement is shown in

PT ED

Figure 6. Image enhancement was applied with two objectives: (1) to highlight the image contrast of pores or gaps among the sediment particles and (2) to eliminate the noise owing to the dissimilar textures and unusual patterns appearing on the surface of sediments. Improvement in

AC

4.3.2.

CE

digital grain-size analysis after introducing the image enhancement is further presented in section

After preprocessing, autocorrelation curves of well-sorted sediments are corrected to build up the calibration catalogue (Figure 7). Each autocorrelation curve is the mean curve of three images of its own size fraction, and it is representative of a specific grain size. In total, eight grain-size fractions were considered for size distribution estimation of the mixed-size samples. The differences between three different images of the same grain size were very small in comparison to the differences of autocorrelation curves of eight different grain sizes. Taking an average of five autocorrelation curves, in this study, derived from five different regions of the image, the autocorrelation curve of the mixed-size sediments was obtained, representing a more

11

ACCEPTED MANUSCRIPT general sediment mixing feature rather than considering the autocorrelation curve from only one specified region in an image.

T

To solve the proportions of calibrated sizes in the mixed-size sediments, Buscombe (2008)

RI P

modified Rubin’s general linear equations describing the problem as

a1,1 x1  a1,2 x2  ...  a1,m xm  b1

SC

a2,1 x1  a2,2 x2  ...  a2, m xm  b2 . .

(2)

NU

.

MA

an ,1 x1  an ,2 x2  ...  an ,m xm  bn where x1…xm are the proportion of size fraction 1:m in the sample, an,1…an,m are the numerical

PT ED

signature values for lags 1:n of the calibration samples 1:m, and b1…bn are the observed numerical values for lags 1:n in the mixed-size sample. A least-squares fit, or a nonnegative least-squares fit, was used to solve the linear equations (2) and obtain the grain-size distributions

CE

in previous studies (Rubin, 2004; Barnard et al., 2007). However, these methods have no constraint for the proportion of each size fraction xi and may produce unrealistic grain-size

AC

distribution. Later, Buscombe (2008) used the kernel density method and Gallagher et al. (2011) applied the maximum entropy method to include constraints. In this study, constraints on xi were introduced by using a nonlinear programming method. This method consists of three components: the design variables, the objective function, and the constraints. The design variables are x=[x1, x2,…,xm]T. The objective function is n

min. f  x     Ai x  bi 

2

(3)

i 1

where Ai=[ai,1,ai,2…ai,m], b=[b1, b2…bn] T. This function corresponds to a least-squares fit to solve the linear equations (2) and obtain the grain-size distribution. The constraints include two parts.

12

ACCEPTED MANUSCRIPT The first part is an equality constraint. This means the sum of each sample proportion is equal to unity:

T

x1  x2  ...  xm  1

(4)

RI P

The second part is an inequality constraint. This constraint requires the portion of each

0  xi  1

SC

well-sorted fraction to be larger than zero, but smaller than 1:

(5)

NU

The nonlinear programming method is aimed at minimizing the objective function, Eq. (3), and

MA

obtaining the design variables under the two constraints mentioned previously, Eqs. (4) and (5). During programming, a criterion was set to ensure that the nonlinear programming method

PT ED

always converges to a solution, and the value of the objective function is smaller than 10-2. The result x1, x2…xm is only the area proportion of each fraction in the image of the mixture of grains. Buscombe (2008) stated that there is a bias because an image contains grains that are

CE

overlapping in a two-dimensional plane, and the sieving technique is three dimensional. Barnard et al. (2007) argued that photographic images only sample surface grains, whereas hand-grab

AC

samples include subsurface grains, and this leads to a standard error of 6% between these two approaches. To achieve a closer agreement between the digital method and sieving, many former researchers suggested increasing the sampling images of mixed-size sediments and taking an average of all the results(Rubin, 2004; Barnard et al., 2007; Buscombe, 2008; Buscombe and Masselink, 2008; Pentney and Dickson, 2012). However, these researchers ignored the fact that grain-size distribution and associated parameters are obtained from the cumulative mass distribution curves by graphical method and it is extremely important to convert the image method to a three-dimensional method. In this study, a conversion step was applied to take into account the difference between the two-dimensional image method and the three-dimensional

13

ACCEPTED MANUSCRIPT sieving technique. Two assumptions need to be made before converting the 2D area proportion to a 3D mass proportion: (1) the sediment grains are treated as spherical balls and (2) sediment

T

grains have the same density. Under such assumptions, the area proportion of each size fraction

RI P

in an image, xi, could be calculated as,

(6)

NU

SC

r ni ( i )2 2 xi  r r1 2 r2 2 n1 ( )  n2 ( )  ...  nm ( m ) 2 2 2 2

where ni represents the amount of the grain-size fraction i in the recorded image, and ri is the

MA

diameter of the hypothetical spherical ball of the grain-size fraction i. Knowing the values of xi and ri, ni could be obtained from Eq. (6). Considering the aforementioned two assumptions, the

PT ED

volumetric (or the mass) proportion of each size fraction, yi, in a mixed size sample, could be calculated as,

4 ri 3 ( ) 3 2 yi  r  r r 4 n1 ( 1 )3  n2 ( 2 )3  ...  nm ( m )3   3 2 2 2 

(7)

CE

ni

AC

Subsequently, the area proportion of grain size xi was converted to the mass proportion yi, and the difference between the 2D digital image method and the 3D sieving method was investigated. After the mass proportion of each fraction was quantified, the representative grain-size distribution and associated parameters were obtained using the conventional graphical method. The detailed validations are presented in section 4.3.

4.2. Methodology for obtaining the cumulative grain-size distribution curve and the probability density curve Dean and Dalrymple (2002) pointed out that the distribution of sand size has shown to almost

14

ACCEPTED MANUSCRIPT obey a log-normal probability law. Zhang (1989) noted that both the cumulative grain-size distribution curve and the probability density curve of grain size obey the log-normal probability

T

law. Obviously, Gaussian function fitting method is suitable to obtain the accurate cumulative

RI P

grain-size distribution curve. In this study, two additional fitting points, representing the start and the end of the cumulative grain-size distribution curve, are added in the grain-size distribution

SC

analysis, in view of the limited number of the calibration samples (generally <10) and to enhance

NU

the statistical reliability of the cumulative grain-size distribution curve. The modified fitting points and the corresponding cumulative mass percentages for analyzing the gain size

MA

distribution are presented in Table 2. According to the conventional sieving technique proposed by Qian and Wan (1983), mass proportion of each grain-size fraction is modified because nearly

PT ED

half of the sediment particles in this fraction are smaller than its mean size. The general Gaussian function is given as, x b 2 ) c

(8)

CE

y  ae

(

where y is the value of cumulative mass percentage and a, b, c are design variables. The

AC

improved Gaussian function fitting method is implemented with two inequality constraints on the design variables. They are given by,

0  a 1

(9)

rm  b  1.5rm

(8)

Constraint (9) means the maximum value of cumulative grain-size distribution curve, a, should be smaller than 1. Constraint (10) requires the maximum grain size of the cumulative distribution curve to be between the actual maximum size (rm) and the added artificial maximum size (1.5rm). These two constraints are introduced to produce a more precise cumulative grain-size distribution curve. The probability density curve of the mixed-size sample is obtained

15

ACCEPTED MANUSCRIPT by taking the derivation of the cumulative distribution curve. Corresponding validations of these

T

two curves are presented in section 4.4.

RI P

1.1 4.3 Validation of representative grain sizes and associated parameters Buscombe (2008) suggested that the calibration catalogue must be truncated at a relatively

SC

short lag (cut-off value) because closely sized fractions of sediments are difficult to differentiate

NU

beyond this lag, which is in agreement with Barnard et al. (2007) and Rubin et al. (2007). However, Buscombe found that short correlograms threaten the statistical reliability of the

MA

grain-size distribution, and a detailed discussion of this cut-off value was not mentioned. The results of well-sorted grains were applied in the present study. As mentioned previously, the

PT ED

autocorrelation curve of fine sediment is always steeper than that of the coarse sediment and presents an early occurrence of the peak value under the same image resolution. Accordingly, the lag value was adopted as the offset corresponding to the peak value of the smallest-size grain in

CE

the calibration catalogue.

AC

After the nonlinear programming method, the area proportion of each size fraction in the mix-sized sediments was obtained and converted into mass proportion. Blott and Pye (2001) concluded that the parameters used to describe a grain-size distribution fell into four principal groups: (a) the average size, (b) the spread (sorting) of the sizes around the average, (c) the symmetry or preferential spread (skewness) to one side of the average, and (d) the degree of concentration of the grains relative to the average (kurtosis). These parameters could be obtained by the graphical methods. Here, the most widely used formulas proposed by Folk and Ward (1957) were followed. Mean diameter, median diameter, standard deviation (sorting coefficient), skewness, and kurtosis of the mixed-size sediments are discussed individually as follows.

16

ACCEPTED MANUSCRIPT

4.3.1 Mean diameter

T

Mean diameter of the mixed-size grains is estimated according to its definition: m

RI P

i 1 m

i i

y i 1

i

( 11 )

SC

Md 

yr

On the other hand, mean diameter of mixed grains could also be estimated using the

NU

representative  values as proposed by Folk and Ward (1957):

MA

1 M z  (16  50  84 ) 3

( 12 )

PT ED

These two formulas were adopted to validate the correctness of mean diameter from the digital image analysis. Figure 8 (A, B) shows that the agreement between the digital image analysis and the sieving method using both mean size calculation formulas is satisfactory. In the

CE

case of the mean diameter obtained by Eq. (11), the image-based result shows a slight overestimation when the sand size is <4 mm. In digital image analysis, the autocorrelation

AC

method ignores some fine sediment because of the shading of fine grains by coarse particles in the recorded image. The mean diameter of autocorrelation technique thus becomes larger by the overestimation of mass proportion of coarse sand fraction. Conversely, if the mixed-size sediment contains a few fine grains, then the mean diameter is underestimated by image analysis. This is because the mean size is slightly underestimated by the image method for coarse sand and gravel size fraction, as explained in Figure 4(A). Regarding the mean diameter calculated by Eq. (12), a similar result was observed. These two systematic errors, that is, the overestimation of fine grain size and the underestimation of coarse grain size in digital image analysis, also affect the results of the following associated parameters.

17

ACCEPTED MANUSCRIPT

4.3.2 Median diameter

T

Figure 8(C) shows the comparison of median size estimated by image analysis and the

RI P

sieving method. Because of the shading of fine sand, the cumulative grain-size distribution curve moves toward the coarse fraction, and the median diameter of fine mixed-size samples is

SC

overestimated in the image analysis. However, in the case of mixed-size samples mainly

NU

consisting of coarse sands or gravels, the cumulative grain-size distribution curve shifts toward the fine fraction, which leads to an underestimation of the median size of the coarse mixed-size

MA

samples.

Figure 8(D) shows the comparison of median diameter estimates between sieving and image

PT ED

analysis (with and without image enhancement). It is confirmed that image-based estimates with image enhancement generally agreed better with the sieving data than with results without image enhancement. To quantitatively demonstrate the improvement in digital grain-size analysis after

CE

applying the image enhancement, Table 3 summarizes the correlation coefficients between image

AC

analysis and sieving method for both mean and median size estimates. It is clear that estimates of median and mean (using Eq. 11 or 12) diameters with image enhancement show higher correlation coefficients (better agreements) with the sieving results than with results without image enhancement.

4.3.3 Standard deviation (sorting coefficient) Standard deviation is calculated by the following formula: 1 4

 I  (84  16 ) 

1 (95  5 ) 6.6

(9)

Because of the presence of two errors mentioned previously, the cumulative grain-size

18

ACCEPTED MANUSCRIPT distribution curve when using the image analysis method always becomes steeper than when it is measured by the sieving method. This means that the autocorrelation method places more weight

T

on the central portion of the grain-size curve and less on the tails. Therefore, sediment size

RI P

distribution is more concentrated, because of which the sorting coefficient is generally underestimated for mixed-size samples as shown in Figure 9(A). Considering the

SC

bimodal-distributed sample MS. 1, two tails are both underestimated and the grains are

NU

concentrated on a nonexistent middle size, which causes a large underestimation of sorting coefficient in the digital image method. On the other hand, for well-sorted sediments, the

MA

autocorrelation method was not effective because other size fractions were accounted for certain proportions in the mixed-size image method even they were not present. Accordingly, the digital

PT ED

image method presents large sorting coefficients (standard deviation) of well-sorted grains. In general, a more uniform size distribution (a small sorting coefficient) is obtained in digital image analysis for mixed-size samples, whereas a more nonuniform size distribution (a large sorting

CE

coefficient) is obtained for well-sorted samples.

AC

4.3.4 Kurtosis

Kurtosis is calculated by the following formula: Ku 

(95  5 ) 2.44(75  25 )

( 10 )

Kurtosis reflects the degree of concentration of the grains relative to the average. According to Folk and Ward (1957), kurtosis between 0.90 and 1.11 is mesokurtic. The result is platykurtic in the case of kurtosis between 0.67 and 0.90 and leptokurtic in the case of kurtosis between 1.11 and 1.50. Extremely, the sample grains are very platykurtic and very leptokurtic for kurtosis smaller than 0.67 and larger than 1.50. As Figure 9(B) shows, most of data points are in accordance with a factor of 2 and around the value of 1, which is in agreement with the opinion 19

ACCEPTED MANUSCRIPT of Folk and Ward (1957) that natural sediment has an averaged kurtosis around 1.

T

4.3.5 Skewness

1 (16  84  250 ) 1 (5  95  250 )  2 (84  16 ) 2 (95  5 )

( 11 )

SC

SK 

RI P

Skewness is calculated by the following formula:

NU

Buscombe (2008) argued that although estimates of median size closely match the real sample, derived measures such as skewness is often very inaccurate using these distribution

MA

estimation methods. Nevertheless, skewness derived from the present image analysis is generally in good agreement between the two methods as demonstrated in Figure 10(A). According to the

PT ED

divisional points recommended by Folk and Ward (1957), the result of skewness calculated in Eq. (15) is symmetrical between −0.1 and 0.1 as marked in a red box in Figure 10(A). Results from 0.1 to 0.3 belong to the positive skewed and results from –0.3 to –0.1 belong to the negative

CE

skewed as marked in blue boxes in Figure 10(A). Very positive-skewed and very negative-skewed results are from 0.3 to 1.0 and from −1.0 to −0.3 as marked in green boxes in

AC

Figure 10(A). Data points located in these solid-line boxes demonstrate good agreements between the two methods. Subsequently, data in the black dashed-line boxes represent the relatively good agreement between the autocorrelation technique and the sieving method. In general, Figure 10(A) shows the skewness obtained by the digital image method in accordance with the corresponding result measured by sieving. The autocorrelation method provides a poor estimate of skewness for bimodal-distributed sample MS. 1 ascribing to the two systematic errors mentioned previously. Positive values of skewness indicate that sample grains have a “tail” of fine grains, whereas the negative values indicate the sample grains have a ‘tail’ of coarse grains. Figure 10(B) shows 20

ACCEPTED MANUSCRIPT the relation among skewness, mean diameter, and median diameter of all samples. Positively skewed data points are marked using stars, and negatively skewed data are marked using circles

T

in Figure 10(B). It shows that the mean diameter is larger than the median diameter for negative

RI P

skewness, and vice versa. This is in agreement with the understanding of skewness properties. Friedman (1961) stated that for a negatively skewed sand size distribution, the mean size has a

SC

smaller phi value than the median size, indicating that the metric mean diameter is larger than the

NU

metric median diameter.

4.4 Validation of the cumulative grain-size distribution curve and the probability

MA

density curve

As mentioned in section 4.2, an improved Gaussian function fitting method was used to

PT ED

obtain the modified cumulative grain-size distribution curve and the probability density curve. Figure 11 shows the comparison of the cumulative grain-size distribution curve obtained from the image analysis and sieving methods. Figure 11(A) shows “percent coarser” and the value at a

CE

particular diameter represents the total sample percentage by weight that is coarser than that

AC

diameter. In general, this plot is presented with the phi diameter on the abscissa (Dean and Dalrymple, 2002). Because of the shading of fine grains in recorded image, for the mixed-size sample, the autocorrelation method ignores some fine particles. Accordingly, the fitting points of fine grain size move toward the coarse size, as well as the bottom part of the cumulative grain-size distribution curve. However, when considering coarse fractions, the image-based cumulative grain-size distribution curve moves toward the fine size because the mean diameter of the coarse fraction is underestimated. Consequently, the cumulative grain-size distribution curve from the image analysis method always becomes steeper than when it is measured by the sieving method, as demonstrated from results of different mixed-size samples, for example, MS.

21

ACCEPTED MANUSCRIPT 6 and MS. 7 in Fig. 11. The same results could be drawn by changing the phi diameter to the metric diameter in the abscissa as shown in Fig. 11(B).

SC

RI P

（x  ）

1 2 y  e 2  2

T

The Gaussian function is expressed as ( 12 )

Eq. (16) has a similar expression as in Eq. (8). The standard deviation

are important and affect the profile of the cumulative distribution curves. In fact, is in accordance with parameter b in Eq. (8) and reveals the

MA

the mathematical expectation

NU

expectation

and mathematical

mean diameter of sample grains. The bigger the parameter

is in accordance with the parameter c in Eq. (8) and reflects the sorting

PT ED

are. Parameter

(or b) is, the coarser the sediments

coefficient of sample grains. Sample grains with small sorting coefficients have a small standard deviation and a steeper shape in the cumulative grain-size distribution curve. Fine sediments

CE

account for a large proportion in MS. 6 (a small value of parameter

or b); the cumulative

distribution curve of MS. 6 is therefore on the left of that of MS. 7 in Fig. 11(B). However, these (or c).

AC

two mixed-size samples present similar curve shapes because they have a similar

Figure 12 shows the comparison of the probability density curve. The difference in distribution of various mixed-size samples could be identified from the probability density curve. For example, the MS. 4 has more mass proportion of the fraction (2–2.5 mm), and the corresponding probability density curves of the image-based method and sieving method both are more leptokurtic than those of MS. 2, which has equal mass mixing proportions of each fraction as shown in Table 1.

22

ACCEPTED MANUSCRIPT 5. Conclusion Following Rubin’s (2004) autocorrelation technique, this study proposed an enhanced

RI P

distribution of mixed-size sediments from digital image.

T

autocorrelation algorithm to estimate the size of well-sorted sediments and the grain-size

SC

Accurate size estimation of well-sorted grains is fundamental and important to establish the calibration catalogue in analyzing the grain-size distribution of mixed-size sediments. Using the

NU

self-organization theory, a peak value in the autocorrelation curve was identified, and the corresponding offset value was used to estimate the grain size of well-sorted grains. To choose

MA

the largest and smallest offset results of the peak autocorrelation as the corresponding length of the long and intermediate axes of the sediment particle, the autocorrelation analysis was

PT ED

extended from only one direction to eight directions in the image. Validation demonstrates that grain-size features estimated from the digital image show fairly good agreements with those

CE

measured using vernier caliper, for example, the lengths of long/intermediate axes and the mean size of samples. The digital method is more in accordance with the result achieved by point

AC

counts measurement in comparison to sand size measured from sieving. The improved autocorrelation algorithm offered accurate and rapid grain-size estimation of the well-sorted sediments.

Regarding the mixed-size sediment, a nonlinear programming method, which is different from the conventional “least-squares with non-negativity” method, the kernel density estimation method, and the maximum entropy method, was used to obtain accurate representative grain sizes and associated parameters, such as mean diameter, median diameter, sorting, skewness, and kurtosis. In the present analysis, image preprocessing was used to enhance the contrast of the recorded image. A concrete improvement in the digital grain-size estimation was confirmed after

23

ACCEPTED MANUSCRIPT the image enhancement. In addition, a conversion method was added to determine the difference between the two-dimensional digital image method and the three-dimensional sieving method.

T

The modified cumulative grain-size distribution curve and the probability density curve were

RI P

obtained using the improved Gaussian function fitting method. A series of validation tests were conducted by comparing the image-based results with those obtained from the sieving method. In

SC

general, good agreement was confirmed between these two approaches. Two systematic errors

NU

were identified in the digital image analysis, that is, overestimation of fine grain-size fraction (due to the shading of fine grains in the recorded image) and underestimation of large grain-size

MA

fraction (due to the oblique resting of surficial coarse grains). The enhanced autocorrelation technique offers a more accurate estimation of the grain-size distribution and the associated

PT ED

inherent parameters of the mixed-size sediment, which is developed from the conventional

CE

“look-up-catalogue” approach.

AC

Acknowledgments

This study was financially supported by the Natural Science Foundation of Zhejiang Province (No. LR14E090002).

References Baptista, P., Cunha, T.R., Gama, C., Bernardes, C., 2012. A new and practical method to obtain grain size measurements in sandy shores based on digital image acquisition and processing. Sedimentary Geology 282, 294-306.

24

ACCEPTED MANUSCRIPT Barnard, P.L., Rubin, D.M., Harney, J., Mustain, N., 2007. Field test comparison of an autocorrelation technique for determining grain size using a digital ‘beachball’ camera versus

T

traditional methods. Sedimentary Geology 201 (1-2), 180-195.

RI P

Blott, S. J., Pye, K., 2001. GRADISTAT: a grain size distribution and statistics package for the analysis of unconsolidated sediments. Earth surface processes and Landforms 26(11),

SC

1237-1248.

NU

Buscombe, D., 2008. Estimation of grain-size distributions and associated parameters from digital images of sediment. Sedimentary Geology 210 (1-2), 1-10.

MA

Buscombe, D., 2013. Transferable wavelet method for grain-size distribution from images of sediment surfaces and thin sections, and other natural granular patterns. Sedimentology 60(7),

PT ED

1709-1732.

Buscombe, D., Masselink, G., 2008. Grain size information from the statistical properties of digital images of sediment. Sedimentology 56, 421-438.

CE

Buscombe, D., Rubin, D.M., Warrick, J.A., 2010. A universal approximation of grain size from

AC

images of noncohesive sediment. Journal of Geophysical Research 115, 1-17. Buscombe, D., Rubin, D. M., 2012. Advances in the simulation and automated measurement of well-sorted granular material: 2. Direct measures of particle properties. Journal of Geophysical Research,117, F02002, doi:10.1029/2011JF001975. Buscombe, D., Rubin, D. M., Lacy, J. R., Storlazzi, C. D., Hatcher, G., Chezar, H., Sherwood, C. R., 2014. Autonomous bed‐sediment imaging‐systems for revealing temporal variability of grain size. Limnology and Oceanography: Methods 12(6), 390-406. Butler, J.B., Lane, S.N., Chandler, J.H., 2001. Automated extraction of grain-size from gravel surfaces using digital image processing. Journal of Hydraulic Research 39, 519-529.

25

ACCEPTED MANUSCRIPT Carbonneau, P., Lane, S.N., Bergeron, N., 2004. Catchment-scale mapping of surface grain size in gravelbed rivers using airborne digital imagery. Water Resources Research 40, W07202.

T

Carbonneau, P., Bergeron, N., Lane, S.N., 2005. Automated grain size measurements from

RI P

airborne remote sensing for long profile measurement of fluvial grain sizes. Water Resources Research 41, W11426.

SC

Chang, F.J., Chung, C.H., 2012. Estimation of riverbed grain-size distribution using

NU

image-processing techniques. Journal of Hydrology 440-441, 102-112. Dean, R.G., Dalrymple, R.A., 2002. Coastal Processes with engineering applications. Cambridge

MA

University Press, 475 p.

Folk, R.L., Ward, W.C., 1957. Brazos River bar: a study in the significance of grain size

PT ED

parameters. Journal of Sedimentary Petrology 27, 3-26. Friedman, G.M., 1961. Distinction between dune, beach, and river sands from their textural characteristics. Journal of Sedimentary Petrology 31, 514-529.

CE

Gallagher, E. L., MacMahan, J., Reniers, A. J. H. M., Brown, J., Thornton, E. B., 2011. Grain

AC

size variability on a rip-channeled beach. Marine Geology 287(1), 43-53. Graham, D.J., Reid, I., Rice, S.P., 2005a. Automated sizing of coarse-grained sediments: image-processing procedures. Mathematical Geology 37, W07020. Graham, D.J., Rice, S.P., Reid, I., 2005b. A transferable method for the automated grain sizing of river gravels. Water Resources Research 41, W07020. Graham, D.J., Rollet, A.J., Piegay, H., Rice, S.P., 2010. Maximizing the accuracy of image-based surface sediment sampling techniques. Water Resources Research 46, W02508. Ibbeken, H., Schleyer, R., 1986. Photo-sieving: a method for grain-size analysis of coarse-grained, unconsolidated bedding surfaces. Earth Surface Process and Landforms 11,

26

ACCEPTED MANUSCRIPT 59-77. Karunatillake, S., McLennan, S.M., Herkenhoff, K.E., Husch, J.M., Hardgrove C., Skok,

RI P

sedimentology, Part 1: Algorithm. Icarus 229, 400-407.

T

J.R.,2014. A martian case study of segmenting images automatically for granulometry and

Ortoleva, P., Merino, E., Moore, C., Chadam, J., 1987. Geochemical self-organization I:

SC

reaction-transport feedbacks and modeling approach. American Journal of Science 287,

NU

979-1007.

Pentney, R.M., Dickson, M.E., 2012. Digital grain size analysis of a mixed sand and gravel

MA

beach. Journal of Coastal Research 28(1), 196-201.

Pina, P., Lira, C., 2009. Sediment Image Analysis as a Method to Obtain Rapid and Robust Size

PT ED

Measurements. Journal of Coastal Research 56 (2), 1562-1566. Qian, N., Wan, Z.H., 1983. Mechanics of sediment transport. Science Press of China, 687p. (in Chinese)

CE

Rubin, D.M., 2004. A simple autocorrelation algorithm for determining grain size from digital

AC

images of sediment. Journal of Sedimentary Research 74 (1), 160-165. Rubin, D.M., Chezar, H., Harney, J.N., Topping, D.J., Melis, T.S., Sherwood, C.R., 2007. Underwater microscope for measuring spatial and temporal changes in bed-sediment grain size. Sedimentary Geology 202, 402-408. Sime, L.C., Ferguson, R.I., 2003. Information of grain sizes in gravel-bed rivers by automated image analysis. Journal of Sedimentary Research 73(4), 630-636. Warrick, J.A., Rubin, D.M., Ruggiero, P., Harney, J.N., Draut, A.E., Buscombe, D., 2009. Cobblecam: grain-size measurements of sand to boulder from digital photographs and autocorrelation analyses. Earth Surface Processes and Landforms 34, 1811-182.

27

ACCEPTED MANUSCRIPT Zhang, R.J., 1989. River and sediment dynamics. China Water and Power Press, 230p. (in

AC

CE

PT ED

MA

NU

SC

RI P

T

Chinese)

28

ACCEPTED MANUSCRIPT Table 1

Designed mixed-size grain samples and the corresponding mass mixing proportions of each component.

1.6–2

8 1 8 1

8 1 7 2

1 1 2 1 1 5 1 4 2 6 3

1 1 1 1 4 2 4 2 5 4

NU

2.5–4

4–5

5–6

1 1 2 1 3 3 2 4 4 5

1 1 1 1 2 4 2 4 3 6

1 1 1 1 2 1 5 1 8 2 7

1 8 1 8

PT ED

MA

2–2.5

RI P

1.43–1.6

SC

1.25–1.43

CE

MS. 1 MS. 2 MS. 3 MS. 4 MS. 5 MS. 6 MS. 7 MS. 8 MS. 9 MS. 10 MS. 11

1–1.25

AC

samples (MS)

T

Well-sorted sediment components in a mixed-size sample (mm)

Mixed-sized

29

ACCEPTED MANUSCRIPT Table 2

The modified fitting points and corresponding cumulative mass percentages. Mean size

Cumulative mass Mass proportion

Modified mass proportion

0.5r1 (added point 1)

-

-

r1

y1

r2

y2





rm

ym

1.5rm(added point 2)

-

percentage (fitting points)

T

(grain size of each

0.5 y1

SC

0.5 y1

RI P

calibration samples)

0.5y2

0.5 y2

NU



0.5ym -

0.5 y1

y1+0.5y2  y1+ y2+…+ 0.5ym 1

AC

CE

PT ED

MA

0.5ym

0

30

ACCEPTED MANUSCRIPT Table 3

The comparison of grain-size estimation with and without image enhancement. Without image enhancement 0.668

Mean size calculated using Eq. (11)

0.675

Mean size calculated using Eq. (12)

0.675

0.794 0.800 0.764

AC

CE

PT ED

MA

NU

SC

Median size

With image enhancement

T

image analysis and sieving

RI P

Correlation coefficient between

31

ACCEPTED MANUSCRIPT Fig. 1. well-sorted sand 4mm
mixed-size sand MS. 1

mixed-size sand MS. 4

well-sorted sand 8mm
mixed-size sand MS. 10

MA

NU

SC

RI P

T

well-sorted sand 1.43mm
AC

CE

PT ED

Fig.1. Digital image samples of different sediments. The top row: well-sorted sediments obtained by sieving. The bottom row: mixed-size sediments achieved by a mixture of the well-sorted fractions according to a designed mass mixing proportion. The red scale bar in each image represents 10 mm.

32

ACCEPTED MANUSCRIPT

NU

SC

RI P

T

Fig. 2.

PT ED

MA

Peak value

AC

CE

Fig.2. Autocorrelation curves of a well-sorted grain sample between 5 and 6 mm.

33

ACCEPTED MANUSCRIPT Fig. 3.

RI P

T

Image recording of well-sorted grains

MA

NU

Autocorrelation algorithm

SC

Image pre-processing (Gray-scale process)

PT ED

Autocorrelation curve

Convert from pixel to mm

Long axis, intermediate axis and mean size

AC

CE

Fig.3. Flowchart of digital grain-size analysis of well-sorted grains.

34

ACCEPTED MANUSCRIPT Fig. 4. B

NU

SC

RI P

T

A

AC

CE

PT ED

MA

Fig.4. Comparison of grain-size estimation. The error bar represents the standard deviation. (A) Comparison of grain size determined by point counts measurement and image analysis. (B) Comparison of the mean size measured by sieving and image analysis.

35

ACCEPTED MANUSCRIPT Fig. 5.

NU

SC

Image preprocessing (Gray-scale process/ Image enhancement)

RI P

T

Image recording of mixed-size grains

MA

Autocorrelation algorithm

Image recording of well-sorted grains

Autocorrelation curve

PT ED

Calibration catalogue

CE

Nonlinear programming method

AC

Convert from 2D to 3D

Representative grain sizes and associated parameters

Improved Gaussion function fitting method

Cumulative grain-size distribution curve and probability density curve Fig.5. Flowchart of digital grain-size analysis of mixed-size grains.

36

ACCEPTED MANUSCRIPT Fig. 6. B

MA

NU

SC

RI P

T

A

AC

CE

PT ED

Fig.6. Images of mixed-size grains before (A) and after (B) the image enhancement.

37

ACCEPTED MANUSCRIPT

PT ED

MA

NU

SC

RI P

T

Fig. 7.

AC

CE

Fig.7. Autocorrelation curves of eight sieved grain-size fractions.

38

ACCEPTED MANUSCRIPT Fig. 8. A

NU

SC

RI P

T

B

C

AC

CE

PT ED

MA

D

Fig.8. Comparison of the mean size and median size measured by sieving and image analysis. (A) Mean size calculated using formula (11). (B) Mean size calculated using formula (9). (C) Median size. (D) Median size measured by image analysis with and without image enhancement.

39

ACCEPTED MANUSCRIPT Fig. 9. A

NU

SC

RI P

T

B

MA

Fig.9. Comparison of the sorting (standard deviation) (A) and kurtosis (B) measured by sieving and

AC

CE

PT ED

image analysis.

40

ACCEPTED MANUSCRIPT Fig. 10. A

MA

NU

SC

RI P

T

B

Fig.10. Comparison of the skewness (A) and the relation between skewness and diameter (B). (A) Skewness calculated using formula (15). (B)The relation between skewness and median/mean

AC

CE

PT ED

diameter.

41

ACCEPTED MANUSCRIPT Fig. 11. A

NU

SC

RI P

T

B

MA

Fig.11. Comparison of the cumulative distribution measured by sieving and image analysis. (A) Cumulative distribution curve with abscissa in phi value. (B) Cumulative distribution curve

AC

CE

PT ED

with abscissa in metric value.

42

ACCEPTED MANUSCRIPT

PT ED

MA

NU

SC

RI P

T

Fig. 12.

AC

CE

Fig.12. Comparison of the probability density measured by sieving and image analysis.

43