Mountain Snowpack

Image analysis determination of micro-structural parameters of snow andtheir correlation with snow tensile strength

DN Sethi, PK Srivastava· and P Mahajan!Snow & Avalanche Study Establishment, Chandigarh·, Indian Institute ofTechnology, Delhi!

Abstract: Deformation behaviour of snow under tensile stress, particularly the strength characteristics withrespect to material microstructure, deformation rate and temperature, is one of the key inputs to understand therelease mechanism of a slab avalanche. Present study investigates the micro-structural effect on the tensile strengthof sieved snow samples tested with a universal testing machine (capacity 10 KN, DARTEC make) in a coldlaboratory. Constant deformation rate tests under tension at the rate of 5 x 10.5 sec·! were conducted at temperaturesof-10°C and _6°C and image analysis was used to determine which critical micro-structural feature correlates bestwith failure strength. Dimensionless micro-structural index 11 (which reflects the influence of bond radius,coordination number, ice volume fraction and grain size) is appearing as the most significant parametercharacterizing snow tensile strength. The paper also discusses the measurement of several stereology basedstructural parameters such as surface area per unit volume, mean intercept length, volume fraction and bond radius.The implementations of these measurement techniques through image analysis are also described using typicalsurface section images. Such implementations made the measurement of micro-structural parameters very efficient.

Keywords: Snow microstructure, Image processing, Tensile strength

1. Introduction

Tensile fracture of snow across the crownsurface on a mountain slope plays an important rolein the release of a slab avalanche. Investigation of thedeformation behaviour and failure mechanism ofsnow under tensile stress is thus one of the keyexperiments to understand the slab avalancheformation. It is also well established that (Kry, 1975a, b; Gubler, 1978; Narita, 1983; Edens & Brown,1991) the mechanical properties of snow aredetermined by the types of bonds between the icegrains of the snow structure. Beside bonds, a numberof other micro-structural parameters, such as grainsize, coordination number, density, inter-granular slipdistance etc. are known to influence the behavior ofsnow. It would, therefore be, interesting if we couldrecognize some non-dimensional quantities thatcombine various micro-structural parameters andstudy their influence on mechanical behavior ofsnow. Agrawal and Mittal (1994) demonstrated that adimensionless variable [v = Sv / (NbvL/)] coulddescribe many of the mechanical properties of snow.In this study, we have grouped the density ratio (icevolume fraction), grain size and the contact area intoa single non-dimensional quantity and have tried to

·Corresponding author address: PK Srivastava,Snow & Avalanche Study Establishment, Him­Parisar, Sector-37 A, Chandigarh-160036, India; tel:91-172-699804; fax: 91-172-699802; email:pks103@rediffmail.com

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establish the relation of this quantity to tensilestrength of sieved snow samples. The basis for thisgrouping is established in the next section

To determine the micro-structural parameters ofsnow, we have used stereology and image analysis.Using this we can obtain values of ice volume.fraction, grain size, coordination number and bondradius from a 2-dimensional planar section of snow.For this study the two-dimensional information thatare of primary importance are point density Pp andintercept density PL. These two quantities along withbond information form the independent propertiesfrom which average values of 3-dimensional micro­structural parameters can be estimated. In the presentpaper we also discuss the implementation ofmeasurement techniques using automated imageanalysis to quantify micro-structural parametersmentioned above.

2. Formulation of non-dimensionalmicro-structural index

The strength of snow is related to the strength ofthe ice bonds connecting the snow particles. Thestrength of different types of snow besides dependingon strength of ice bonds would also be governed bythe micro-structural parameters mentioned above.

The basic idea for combining the density ratio,grain size and the contact area into a single non­dimensional quantity has been taken from the work ofMahajan and Brown (1993). They utilized probabilityprinciples along with principle of virtual work to

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3.1 Sieved snow

3. Experimentation in cold laboratory

3.2 Constant strain rate experiments in tension

The experiments were performed on a 10 KNcapacity DARTEC UTM with data acquisitioncapabilities. Snow samples are held between twoplates with rough surfaces (figure 1). Heatingelements were fixed inside the platen and constantvoltage supply was ensured. Platens were heated tillthey started melting the snow grains on touch. Properfreezing of the samples was then ensured to haveperfect grip with the platens. The contact surface ofthe platens had many grooves to increase the contactsurface area between the snow specimen and theplatens. Platens were left in contact with the samplefor one hour prior to testing. Strength measurementswere conducted only after a snow sample attainedthermal equilibrium with the chamber temperature.Samples of different density, prepared as explainedearlier, were now subjected to constant strain rate of5 x 10-5 sec·! in tension. Constant strain rateexperiments were performed at chamber temperaturesof-6°C and -10°C.

Analysis of the surface section allows differentgrains and their interaction with surrounding grains tobe observed. Prior to each tensile strengthexperiment, vertical sections of samples wereprepared for micro-structural analysis. A cube ofapproximately 20mm x 20mm x 20mm was cut fromthe specimen and transferred to pre-cooled steel tray.We used dimethyl phthalate, which has a meltingpoint of 2°C and supercools easily, as a fillingmaterial for void space of snow. Pore filler wasallowed to freeze inside the sample contained in

3.3 Surface section preparation and imageacquisition

Figure 1: Mounting of snow sample on UTM

(1)

(2)

The factor on R.H.S. within the square bracket isbasically the stress enhancement in the neck/bondregion above the macroscopic applied stress andreflects the influence of the bond radius, grain radius,coordination number and ice volume fraction y.Rather than treating all of these as separate variables,above equation shows that they can be grouped asone parameter and should be a fundamentaldimensionless parameter relating the microstructureto mechanical properties of the material. Baseq on thestress enhancement factor, we define a non­dimensional micro-structural index 11 as:

Snow samples were collected from different fieldlocations in Himachal Pradesh (India) and stored incold chamber at -20° C to remove its wetness. Tohave desired grain size (0.5 to 1.0 mm) in preparedsamples, stored snow was sieved using mechanicalsiever. To ensure the uniformity amongst the varioussnow sample prepared, weight of initial charge,intensity and time of sieving were kept the same forall samples. Using brush, this sieved snow was thentransferred to pre-cooled cylindrical aluminumsamplers of 65-mm mean diameter and 150 mmheight. The said transfer was done from a fixedheight and at a slow rate, so that no voids aregenerated inside the snow sample. Samplers werepr~lubricated with a very thin layer of silicon oil(freezing point - 70°C) to minimize friction betweensample and the wall of the sampler. Initial density ofthe sample was measured by weighing the samplerand by calculating its volume. Samples were left tosinter at -20°C for 6 days.

determine stresses at the contact points of ice grains.In a system of spherical ice grains of radius rg, icevolume fraction y, coordination number N3 andassuming bond as thin circular disc of radius rb, thefinal equation relating the stress vector in the bondregion an to the macroscopic stress vector on snow t;,is given as:

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sample tray. A mixture of snow and dimethylphthalate becomes sufficiently hard to flatten afterfreezing. A shining smooth surface was made with asharpened sledge microtome blade. The microtomedsurface was allowed to sublimate at chambertemperature for three minutes. The surface was thenpolished gently with felt black to enhance contrastbetween ice grains and pore phase.

Each polished specimen was placed under amicroscope (LIECA M-10), and its surface wasobserved by a 256 gray level CCD camera. Theimages are digitized for analysis with a frame grabber(Flash Bus) interfaced to a computer. A criticalcomponent in any image analysis based study ofsnow microstructure is the quality of imagescaptured Our main objective was to maximize thecontrast between the ice grains and the pore fillermatrix surrounding them, and to produce a pixelimage that preserves the information in the real imageto an acceptable resolution so that subsequentanalysis of the captured image is representative of thetrue relationship between the particles and the porespace. Images were captured with the zoom positionof microscope on 8 X, which resulted inapproximately 80 to 100 grains in a given image. Thesize of the resulting image is 640 x 480 pixels with256 gray levels. A ruled glass micrometer scale wasalso photographed for calibration and later reference.

3.4 Micro-structural parameters determinationusing image analysis

This section discusses the definition ofstereology based structural parameters such assurface area per unit volume Sv, mean interceptlength L3, ice volume fraction y, and bond radius rband their determination using imaging techniques.The analysis reported herein was carried out using acommercial image analysis software Image Pro. Thegeneral procedure for implementing themeasurements is summarized in figure 2.

The digitized surface section images are in theform of gray level images and are usually not veryclear. Image pre-processing or image enhancementinvolves a collection of techniques that are used toconvert the image to a form, which is better suited forhuman or machine interpretation. Depending on thespecimen illumination and capture conditions,different techniques, e.g., gray level histogrammodifications, smoothing of noisy images and imagesharpening was used to improve grain-poreboundaries.

Gray level histogram modifications are basedon remapping the gray levels within an image byapplying a linear or non-linear transformationfunction. Noise is the random variation in the pixel

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content caused by image acquisition and digitizationprocess. Image smoothing and image sharpening isachieved using spatial filters called low pass filterand high pass filter respectively. Their application isbased on the discreet convolution of the originalimage with a special mask.

Image pre-processing (case -dependent)

..I Binary images I..

Generate test probes I..Image processing to reveal quantities

desired: Pp, PL, L3, rb, y, Sv

Figure 2: General procedure to determine micro­structural parameters

The next step is to make it into a two-phaseblack and white image. The gray scale histogram ofthe image simply tells what fraction of the pixels haswhich gray scale value. If the density of the solidphase (ice) is known, then the porosity can also bedirectly measured from physical bulk densitymeasurement. If Pi is the solid ice density, and Psbulk density, then the porosity is simply equal to 1­pJpi. A threshold gray scale can then be chosen, sothat all pixels with gray levels above this thresholdare black (ice), and all below are white (pores), suchthat the correct porosity is achieved.

Another method is profile edge tracing, tracedprofile masking and image binarization. Figure 3aand 3b shows one of the original gray level imageand corresponding binary image which weresubsequently analyzed using the procedures givenbelow.

Figure 3a: Original gray level surface section image

International Snow Science Workshop (2002: Penticton, B.C.)

Figure 3b: Binary image corresponding to figure 3a

Surface area per unit volume (Sv)

For a material with two phases, one phase isseparated from the other by an inter-phase interfacei.e. each phase bounding surface. Knowledge of thedensity of this surface area is quite important. Basedon Underwood's (1970) analysis Sv can be obtainedby observing PL , the number of intersections with thesurface per unit length oftest lines as:

Figure 4: A set of parallel test lines

Figure 5: Profile edges ofbinary image(3)

This equation is valid for any mixture ofsurfaces in space, provided all surface interceptdirections have equal probability of occurring. Thetest probe for determining the number of intersectionsper unit length is a set of parallel test lines (figure 4).An image processing operation 'edge detection' isfirst used to identify the pixels that are on theboundary of the particles in the binary image offigure 3b. When edge detection is applied only theoutline (boundary pixels) of each feature are selectedwhich create a new binary image with lines that areone pixel wide representing the particle perimeters(figure 5). The intersections (figure 6) are simplyobtained by using logical AND operation on theimages shown in figure 4 and 5.

The total length of the test line is obtained fromthe coordinates of the two end pixels of each testlines. For a test line with two end pixels (xo,Yo) and(XbYl), the length of this test line in pixels is:

Figure 6: Intersections ofprofiles

(4)Figure 7: Grain intercept lengths

Mean intercept length (L3)

The number of intersections per unit length PL

is simply the ratio of the intersection count P to thetotal length of all parallel test lines TL.

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L3 provides a measure of the linear size of aphase (ice). It measures the average straight-linetraverse through ice phase. In case of a system ofuniformly sized spheres, of diameter d, L3 = (2 d / 3).The equation for L3, as derived by Underwood

(1970), is valid without any assumption of grainshape, size or distribution and is given by:

(5)

where LL is the length fraction of test line Lintersecting the ice phase. The test probe fordetermining mean intercept length is again a set ofparallel test lines. The intercept length of particles(figure 7) can be obtained by applying the logicoperation AND to the test lines (figure 4) and thebinary image (figure 3b). To calculate the meanintercept length, the number and the total length ofintercepts need to be determined. The number andtotal length of intercepts is again obtained bymeasuring the feature number parameter (which ispart of 'Count/size' routine of Image Pro) from figure6.

Mean Bond radius (rb)

The mean 3D-bond radius is a significantparameter of pressure sintering as well as bondstrength. Fullman (1953) derived the relation for rbwith the assumption that the grain bonds are circulardisks. Kry (1975a) had shown that this assumptionyields self-consistent results:

Jrrb =- (6)

4Ewhere E is the harmonic mean of 2-D bond

lengths. Currently no method is available which canidentify whether two neighbouring particles in asurface section are distinct or rather are connectedabove or below the section plane. There is ongoingdebate on whether bonds between grains can beidentified on the surface section at all. Though Eden'ssoftware (1997) uses a form of algorithm to estimatethe size and number of bonds, its validation is stillrequired for all types of natural snow. For this studybonds are marked manually as a one-pixel wide line,on the binary image based on the criteria given byKry (1975a). The 2D bond lengths (figure 9) can beseparated by applying the logic operation XOR to the?riginal binary image (see figure 3b) and the binarynnage with bonds marked as a one pixel wide line(figure 8). To calculate the total number andharmonic mean E of bond lengths, 'Count/size'routine of Image Pro is again used. Once E isdetermined, mean bond radius can be easilycalculated.

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r~';'!'·~J~.1.,. -.Jt,.~..,,~~ ...,.- .f.""~.••• ·~.1:1:. (.;• .~., I.~ ~ __~......_761

Figure 8: Bonds marked on binary image

Figure 9: Separation of2D bond lengths

Ice volume fraction (y)

Measurement of ice volume fraction isimplemented by calculating the ratio of black pixelsto total pixels. In 2D, we defme a function f(ij),where (ij) indicates the location of a pixel in the A =

M x N image, i = 1, ... , M and j = 1,... , N. Assumef(ij) = 1 for ice and f(ij) = 0 for pores. Then, icevolume fraction is given as:

'L!Ci,})r = (7)A

The remammg important micro-structuralparameter is coordination number N3 i.e. the meannumber of bonds per grain. Normally someassumption about grain shape is required forcalculating the coordination number. In this study N3

has been calculated by a method suggested by Gubler(1978) and Edens and Brown (1991) which requireno assumption of grain shape, other than for an initialguess. Using the above-mentioned microstructuralparameters, a statistical analysis is attempted tocorrelate them with snow tensile strength.

4. Results and discussion

The strength of snow is a function of its micro­structural parameters, ice volume fraction,temperature and strength of ice connections, (Jj. Thecomplex interaction between micro-structuralparameters and stress distribution within the ice

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network makes it very difficult to put a relationshipfor snow strength in the fonn of a precise expression.Guided by the discussion of section 2, it can be put ina functional fonn as:

It is well established that density of snow aloneis not a very good parameter of its mechanicalbehaviour and strength. Figure 10, in which icevolume fraction is plotted against strength, shows thisfor a limited density range used in our experiments at-10°C and ~oC. Table 1 also shows that R(correlation coefficient) for this data is 0.82 and 0.86respectively. Figure 11 shows the variation of (rJrg)2with strength at -10°C and _6°C. The linearcorrelation coefficients of as with (rJrg? and N3 arealso given in table 1.

as (y) 0.82 0.86

crs (rJr2 i 0.39 0.33

crs (N3) 0.83 0.76

crs (T]) 0.94 0.83

To analyze the effect of temperature statistically,multiple regression with Kelvin temperature T andnon-dimensional index T] as independent variableswas performed for failure strength. The multipleregression equation is given as:

logas =10gb + 1J log m1 +Tlogm2 (9)

where b = 9.92E25,m1 = 1.42E16,m2 = 0.80

4.1 Effect of temperature on tensile strength

Figure 12 shows the strength data (log-scale) atboth temperatures along with the multiple regressioncurves and coefficient of multiple correlation R. TheSE value shows the standard error in log crs .

The foregoing analysis indicates thatstereological calculations can provide some insightabout the strength characteristics of snow. The indexT] is appearing as the most significant parameterbased on the correlation analysis. An experimentalprogram is currently under progress to cover a widerrange of temperature and strain rate.

Figure 12 shows the tensile strength variationwith non-dimensional index T]. Table 1 indicates that,index T] is correlating best with the snow tensilestrength. As mentioned in section 2, The strength ofsnow is related to the strength of the weakest links inthe ice matrix i.e. ice bonds connecting the snowparticles. The non-dimensional index is related to thestress enhancement in the neck/bond region. Thetensile strength of snow is achieved when stresses inthe bond region are greater than the tensile strength ofice connections. The strength of ice connections isnot a constant quantity but again varies with appliedstrain rate and temperature.

(8)

0.60.50.40.3

•~ ... •• ••

• · .. ,.--• •

~

I. Strength al-l0 C• Strenath at ~ C

10.2

Table 1: Linear correlation coefficientsParameters Correlation coefficients

_10°C -6 C

100

Iill.

~.r:. 10C,t:ell..u;

Ice volume fraction

Figure 10: crs as a function of y at -10°C 5. Conclusion

Figure 11: crs as a function of (rJrgf at -10°C

••• -• • •... •. ·•I_ Strength at-l0 ci• Strength at ~ C

100

Iill.

~.r:. 10-Clt:CD..-II)

o 0.05 0.1 0.15(rJrg)2

0.2

The result of this preliminary study is animproved understanding of the lnicro-structuralfactors that are required for modeling the functionaldependence of snow tensile strength and are lilnitedto sieved snow, applied strain rate and temperaturestested. Our results confirm the theory of section 2 andfor tension tests the strength of snow has been relatedto the strength of ice through the index. Themeasurement techniques for the lnicro-structuralparameters were implemented using digital imageanalysis and proved to be efficient. Non-dimensionalmicro-structural index T] is appearing as reasonablygood parameter for representing tensile strengthcharacteristics of snow.

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R =0.82SE= 0.30-ns

Q.~-

1000 --p=======;-----.-----,---~--_____,• Strength at -10 C

....... Regression fit at -10 C

• Strength at -0 C~Regression fit at -0 C

100 +----t-----t----+----t---t-------i

10 +----------!::+--=---=+----t----+----+-------j

•

Mountain Snowpack

10.01 0.02 0.03 0.04 0.05

Non-dimensional Index0.06 0.07

Figure 12: Tensile strength as a function of11, together with results ofmultiple regression analysis

Acknowledgements

We are grateful to Maj Gen SS Sharma, KC,VSM, Director SASE for his valuable comments anddiscussion.

References

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Edens, M.Q. and R.L. Brown, 1991. "Changes inmicrostructure of snow under large deformations". JGlaciol., 37(126), 193-202.

Edens, M. Q., 1997. "An experimental investigationof metamorphism induced microstructure evolution ina 'model' cohesive snow", Montana State University,Ph.D. thesis.

Fullman, R. L. 1953. "Measurement of particle sizesin opaque bodies". AJME, 197,447-452.

Gubler, H., 1978. "Determination of the meannumber of bonds per snow grain and of thedependence of the tensile strength of snow onstereological parameters", Journal of Glaciology,Vol. 20, No, 83, pp. 329-341.

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Kry, P. R, 1975a. "Quantitative stereologicalanalysis of grain bonds in snow", Journal ofGlaciology, Vol. 14, No. 72, pp. 479 - 500.

Kry, P. R, 1975b. ''The relationship between theviscoelastic and structural properties of fme grainedsnow", Journal of Glaciology, Vol. 14, No. 72, pp.467 -477.

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Narita, H. 1983. "An experimental study on tensilefracture of snow". Contrib. Inst. Low Temp. Sci., Ser.A 32,1-37

Underwood, E.E., 1970. Quantitative stereology.Reading, MA, Addison-Wesley.