Three-dimensional properties of laboratory sand waves obtained from two-dimensional autocorrelation analysis

Transcription

1 Three-dimensional properties of laboratory sand waves obtained from two-dimensional autocorrelation analysis H. Friedrich, B.W. Melville, S.E. Coleman, T.M. Clunie The University of Auckland, New Zealand V.I. Nikora University of Aberdeen, Engineering Department, Aberdeen, Scotland, UK D.G. Goring National Institute of Water and Atmospheric Research, Christchurch, New Zealand ABSTRACT: A comprehensive data set of 3-D sand waves developing from a flat bed to equilibrium stage (known as the research program SWAT.nz - Sand Waves and Turbulence New Zealand) was obtained during investigations in a 5-ft-wide flume at The University of Auckland. The recorded sand bed is treated as a continuous field of sand-bed elevations, rather than subdivided into discrete sand waves. Aiming at 3-D interpretation of the random field of sand-bed elevations, 2-D autocorrelation analyses are introduced. It is shown that the 2-D autocorrelation function approach is a useful tool to depict 3-D characteristics of sand waves and their further potential is outlined. 1 INTRODUCTION Best (2004) gives an up-to-date description of the complexity of the field of underwater bed morphology, and refers to the lack of knowledge of the interaction of 3-D dune shape and the surrounding flow field. He notes that a better understanding of the 3-D behavior of sand forms in uniform flow conditions is needed to provide information for future research aimed at combining the complex morphodynamic and hydraulic features, which are inherent during the development of sand waves. In order to gain advances in the understanding of 3-D sediment and flow-field interaction, progress must be made in describing 3-D characteristics of subaqueous bed forms. Allen (1968) provided an earlier classification scheme for the 3-D nature of bed forms. He observed visually the position of the crest line in a plane parallel to the water surface (planimetric view) and distinguished the crest line characteristics as straight, sinuous, caternary, linguoid, cuspate, or lunate. Ashley (1990) proposes to use the term dune for all large-scale subaqueous bed forms, the wide variety which are subject to effects such as sediment size, water levels, unsteady and reversing flows and channelization. As one of the characteristics to describe the term with more detail, the dunes were distinguished according to their shape into 2-D dunes, which develop at lower flow speeds, and 3-D dunes, which occur at higher speeds. In contrast to previous technology that did not allow 3-D dune information to be gained very easily, the advances in the last decades now enable us to record and analyze 3-D dune fields. 2-D sandwaves can be comfortably analyzed using one transect parallel to the flow, mostly recorded along the centre line of the channel. For 3-D dunes, an analysis technique covering the whole surface area should provide enlightening new information about the pattern. Even though technological advances have enabled us to record 3-D surfaces, few researchers have previously tried to follow up this extended area of interpretation of submerged sediment waves. One of the most recent works was by Venditti (2003). He describes features of the planimetric view of sand waves with the help of video analysis. He used the non-dimensional span Λ NDS as a parameter (see Figure 1) to describe the 3-D characteristics of bed forms. Figure 1: a) Ratio Λ of λx (the stream wise distance that the lobes extend downstream) and λy (the cross-stream distance

2 between lobes of a crest line), after Allen (1968), and b) the non-dimensional span Λ NDS for closing crest lines Lc and Ly, after Venditti (2003). A critical value of 1.2 for the non-dimensional span is proposed to distinguish between 2-D and 3-D bed forms. His work is based on the pioneering work of Allen (1968) nearly 40 years earlier. The use of spatial recognition software prevents this method gaining universal status, along with difficulties associated with identifying certain crest lines, which is a very subjective procedure (see Figure 2). A 5-MHz Seatek ultrasonic ranging system, comprising 31 transducers (see Figure 3), was employed for the measurements of the sand bed. The moving probe arrangement, which was traveling up-anddown the flume on a moving carriage and which measured the 3-D elevations of a developing sand bed, was introduced in Friedrich et al. (2005), a methodology paper describing experiments undertaken in a narrow 440-mm-wide flume. This paper utilizes data obtained during experiments in a larger 5-ft-wide flume. Brief explanatory notes illustrating additional to Friedrich et al. (2005) 5-ft-flume methodology are given below. The 31 transducers were interrogated sequentially (starting from sensor 1 and finishing with sensor 31), with the same sequence repeated every 0.2-s. The sensors were mounted on a skewed grid (see Figure 4) to allow for the movement of the carriage in the time interval between readings of the components of the sensor array. This ensured that measurements were obtained on an equivalent rectangular recording grid (Figure 5). With a carriage velocity = 0.33-m/s, the rectangular recording grid shown in Figure 4 was generated. Figure 3: Arrangement of the moving probes across the 5-ft flume. Four Acoustic Doppler Velocimeters (ADVs) are placed in front of the acoustic sensors in the centerline of the flume to measure the flow field at different water depths. Figure 2: Identification of crest lines in order to determine the non-dimensional span, after Venditti (2003). In the present paper it is proposed to treat a recorded 3-D sand-bed elevation field as a random field rather than discrete sand waves, following Nikora et al. (1997) who used longitudinal and transverse spectra, correlation and structure functions to describe statistical sand wave dynamics. 2 EXPERIMENTAL INVESTIGATION Figure 4: Moving probe arrangement for the 5-ft flume. Figure 5: Schematic display of sand bed elevation recording grid in the 5-ft flume. Moving carriage is traveling downstream (recording) and upstream (sensors out of the water) along the flume over a length of m.

3 Each mobile-bed experiment (flat sand bed to equilibrium bed forms) utilizing moving probes was carried out over several hours. Experimental parameters for these experiments are given in Table 1. The sequence of operation was as follows. Initially, the moving carriage with submerged sensors traveled downstream for 56-s. Then the carriage was stationary for a waiting period of 4-s, which was used to move the sensors up out of the water, in order to reduce the water surface disturbance when traveling upstream. Afterwards, the carriage traveled upstream for 56-s. After a further waiting period of 4-s, and submerging the sensors again, the carriage moved downstream into the second cycle. Each cycle duration was two minutes. The experiment was continued with repeated cycles until appropriate bed-form development had occurred. Recording of the bed profiles occurred only during the downstream-moving portion of each cycle, based on the configuration of the recording grid. The frequency of recording of bed profiles was one profile every two minutes. As one can see from Figure 4, sensor no. 31 was inverted and used to record the water surface elevation, meaning that only 30 sensors were used to record bed elevations. Overall, ten longitudinal profiles with a transverse resolution of 150- mm were recorded over each sweep of a length of m. Run Name d 50 D T S e Run Duration Equilibrium U avg Fr Re [mm] [m] [ o C] [-] [hh:mm:ss] [m/s] [-] (x10 3 ) wsc07b :50:00 no wsc85a 04:44:00 yes wsc85b 05:34:00 yes wsc10a 05:02:00 yes wsc10b 05:16:00 yes wsc115 03:58:00 yes wsc115 03:56:00 yes wdc25a 07:02:00 no wdc25b 05:34:00 no wdc30a :18:00 yes wdc33a 06:54:00 yes wdc33b 02:26:00 no wdc35a 06:14:00 yes wdc35b 04:04:00 yes Note: Kinematic viscosity ν= m 2 /s; Specific gravity s=2.65; Critical shear velocity u *c (d 50 =0.85- mm)=0.0215m/s d 50 Median grain size, D Flow Depth, T Water Temperature, S e Flume Slope, U avg Average Flow Velocity, Fr Froude Number, Re Reynolds Number Table 1 Experimental parameters. 3 METHODS 3.1 Data Processing Firstly, the ten longitudinal profiles recorded for each sweep along the flume were considered individually. Filtering methods were applied to remove faulty signals and to take into account the detection of suspended particles. Afterwards, interpolation procedures were used to fill the gaps in the profiles. The sand-bed elevation surfaces were detrended linearly for every individual snapshot. Then the following autocorrelation procedures were applied with the help of matlab routines. The sand-bed elevations were considered as a 3-D random field D Autocorrelation functions As a simple introduction to correlation functions, one can say: everything is related to everything else, but like things are more related than unlike things. For a vivid description of one case of correlation functions, the spatial auto-correlation function showing the correlation of a variable with itself through space), one can say: if there is any systematic pattern in the spatial distribution of a variable, it exhibits spatial autocorrelation; nearby or neighboring areas being more alike gives positive spatial autocorrelation; negative autocorrelation arises for patterns in which neighboring areas are unlike, and fully random signals or fields exhibit no spatial autocorrelation. A full-dimensional correlation function R(Δx,Δy,τ) (spatial and temporal) is practically difficult to evaluate. In the past researchers used 1-D autocorrelation functions (spatial and temporal) to determine the wavelength of bed forms and the duration how long it takes for a bed form to move past a stationary object. Often a spatial or temporal longitudinal centerline flume profile of a sand bed has been used for the analysis (Andreotti et al. 2006; Callander 1978; Coleman and Eling 2000). The normalized spatial 1-D autocorrelation function R at Δx=kδx between two values of sand-bed elevation z at distances x i and x i+k along the flume (i = 1 to N-k, with N equal to the number of points in the data vector z along x) is defined as: N k i= 1 ( z( xi ) z)( z( xi+ k ) z) R ( Δx,0,0) = 2 σ (1) With z being the mean value of the data vector z along the flume, and σ being the standard deviation. The wavelength can be generally obtained by the position of the first peak of the autocorrelation function. 1-D spatial autocorrelation functions (see Figure 6) and the corresponding bed profiles (see Figure 7) are shown below for selected times of run wdc35b. Having recorded a random field along and across the flume, one can apply the 2-D spatial autocorrelation function defined as:

4 N k M l ( z( x, y ) z)( z( x, y ) z) i j i+ k j+ l i= 1 j= 1 R ( Δx, Δy,0) = (2) 2 σ Mirroring the information on the x and y axes results in the typical 2-D autocorrelation function, as is shown in Figure 9 for minute 240 of run wdc35b. with z being a 2-D data matrix along x and y, and the correlation being done at Δx=kδx and Δy=lδy between measurement points (i,j) and (i+k, j+l). As a result, one obtains the 1 st Quadrant ((Δx>=0, Δy>=0) autocorrelation function of Figure 8. Figure 9 Typical 2-D spatial autocorrelation function displayed for all four quadrants. Minute 240 of run wdc35b is shown. Axes dimensions in m. Figure 6: 1-D spatial autocorrelation function for selected times of run wdc35b centreline, developing from flattened sand bed. Figure 7: Corresponding centerline longitudinal sand bed profiles for the autocorrelation function of Figure 6. The locations of hills and troughs in all directions of the 2-D autocorrelation plot shows the 3-D character of the recorded sand bed. For the evaluation of 2-D autocorrelation functions, the restricted width of the flume prevents a sufficient number of data points across the flume to see a fully developed autocorrelation function with pronounced fluctuating characteristic in all directions. For the start of sand wave development, starting from a flat sand bed, which is exposed to a unidirectional water flow, the sediment transport s main direction is downstream and therefore the crests of the developing sand waves keep roughly perpendicular to the flume wall (Figure 10). Once a certain development is reached, either the flume walls or the flow depth act as a restricting factor in further growth of sand waves. As one expects, a random sand-bed field that has crest lines parallel to the flow (see Figure 10), as happens at the beginning of sandwave development or in flumes with a small width, will reveal a 2-D autocorrelation function as seen in Figure 11. Figure 10: Recorded 6.25-m long and m wide dune bed. Red indicates crests, and yellow troughs. The crest lines are parallel to each other and perpendicular to the flume wall. These measurements in a small 440-mm wide flume were discussed in Friedrich et al. (2005). Figure 8: Contour plot of 2-D spatial autocorrelation function for 1 st quadrant (Δx>=0, Δy>=0) for min 240 of run wdc35b. See colorbar for correlation values. Axes dimensions in m.

5 Figure 11: Schematic display of 2-D spatial autocorrelation function for a 2-D dune bed as shown in Figure 10. The autocorrelation function has a very pronounced 2-D character. The 2-D autocorrelation function in itself shows a very 2-dimensional character, as the correlation across the flume is very similar at every transect across the flume. Only minor changes between the transects are displayed in an uneven outline across the autocorrelation function. As soon as the crest lines start to bend and lose their parallel character, with individual transects across the flume showing more differences compared to each other, the typical peak and trough characteristics of the 2-D autocorrelation function (e.g. Figure 9), revealing 3-D characteristics of the random field. In an ideal case, the first peak of the contours along and across the flume could be used to determine the 3-D characteristics of a bed-form field, including the dominant wavelength along the flume and separation between individual bed forms across the flume. As the development of the bed forms across the flume is restricted by the flume walls, the bed forms are not free to spread out in that direction. Therefore one can not use the first peak in all directions to determine 3-D characteristics of bed forms. Nevertheless, by looking at the surface of the 2-D autocorrelation function (see Figure 9) one can see, that the centre cone shows distinctive features, which relate to 3-D characteristics and changes in the texture of the observed sand bed field. Based on the assumption that the slopes of the centre cone are constant between the correlation value of 1 and the first negative peak (see Figure 6), a horizontal cut at any correlation level between roughly 0.7 and 0.2 results in obtaining an ellipse shape along the horizontal cut (see Figure 12). The main objective for choosing the cut-off value is the closure of the ellipse, which is describing the centre cone. After the cut-off value is determined, ellipses describing bed-form parameters can be calculated, which provide information of the 3-dimensionality of the random field of sand bed elevations as well as information about the direction of the bed-from field. Figure 12: Ellipse shape of the horizontal cut of the centre cone of the 2-D autocorrelation functions (run wdc35b, min 240, correlation level 0.4). Taking into account the uniformity of the slope of the cone, the ellipticity of the surface at the contour level will provide measured 3-dimensionality of the bed-form field. 3.3 Determination of uniformity To determine the appropriate contour levels where the ellipse will be closed, contour plots such Figure 13 were visually evaluated. Figure 13 Typical contour plot at a certain time of a recorded sand-wave bed (min 240 of run wdc35b). From Figure 6 one can see that a correlation level of 0.4 ensures information on longitudinal levels even for early times in the development of sand waves. Furthermore, this gains closed ellipses around the centre cone of 2-D autocorrelation functions in all directions. For the early minutes of development from a flat sand bed, there might however be no information about the shape of the centre cone, as the chosen level does not close completely around the centre cone for the 2-D dunes at that time. Figure 14 shows that contours at the threshold level also exist outside the main centre cone, these contours being caused by the fluctuating nature of the 2-D autocorrelation function. In order to access the ellipticity of the centre cone, a routine was developed in Matlab to delete secondary cut-off con-

6 tour-level information and yield only the shape of the central contour (see Figure 15). Figure 14: Determination of contours at a correlation level of 0.5. The fluctuating nature of the autocorrelation function causes several contours to display, where only the centre cone is important for the present calculations (run wdc35b, minute 10 of development). Figure 16: Schematic display of geometrical information describing the shape of the ellipse obtained when cutting through the centre cone of the 2-D autocorrelation function at a certain correlation level. 3-D properties of the shape of the centre cone of the 2-D autocorrelation functions are principally described by the ratios x0/y0 and a/b, as well the angle of rotation of the ellipse, theta. Owing to the constant near-origin slopes of a correlation surface, the ratios and theta should not significantly depend on the threshold correlation levels selected. Indeed test have shown this to be the case. 4 RESULTS AND DISCUSSIONS Figure 15: Filtering method applied for multiple correlation peaks as displayed in Figure 14. Afterwards only the outline of the centre cone is left (run wdc35b, minute 10 of development). Determination of information about the ellipticity of the centre cone at a certain correlation level is done accordingly to Goring et al. (1999). Following their procedure for 2-D structure functions, we determined for the central ellipse (see Figure 16): x0, the longitudinal distance defined by the chosen correlation value. y0, the transverse distance defined by the chosen correlation value. a, the major axis of the ellipse at the chosen correlation value. b, the minor axis of the ellipse at the chosen correlation value. theta, the angle of rotation of the ellipse describing the cone shape. 4.1 General interpretation for all runs The experimental runs of Table 1, were analyzed using the 2-D autocorrelation function. Sand-bed elevation fields of a length of m and a width of 1.35-m exist for every second minute over the run durations (from flat bed conditions) given in Table 1. For each 3-D bed surface recorded, the ellipticity of the centre cone of the 2-D autocorrelation surface was calculated. Generally, three different cases of ellipticity can be distinguished according to their x0/y0 ratio (see Figure 17): a) With a ratio of 1, the shape of the cone is circular and isotropy exists. This stage of isotropy is rarely achieved, as it indicates a uniform spread of the bed forms along as well as across the flume. b) A ratio of less than 1 typically occurs early on in development, when neither the flume width nor the water depth restrict sandwave development. This shape indicates the formation of 2-D dunes.

7 They are characterized by crest lines perpendicular to the flow across the whole width of the flume. After a certain stage of development is reached, which is indicated by reaching a minima in the a/b ratio for the ellipse shape, the ellipse is growing again and 3-D dunes develop. With growing sandwave length, the flume walls act as a restricting factor and limit sandwave separation across the flume, keeping the ratio x0/y0 <1. The flow field is constricted by the walls to emphasize uni-directional flow downstream. c) A ratio larger than 1 indicates that the flow depth, and not the flume walls, act as the restricting factor of the 3-D development of bed forms. Sand waves are broken up across the flume and a very pronounced 3-D sand bed field can develop. function, results can be used to see the transition from 2-D to 3-D dunes. Furthermore, for bed-form development in the later stages, restrictions in the development of bed forms can be easily associated to either the flow depth or the flume walls. Both scenarios will be discussed in the following D to 3-D dune changes for deep flows For the deep flows of Table 1, 2-dimensionality of bed form development in the early stages can be observed. According to Figure 11, 2-dimensionality would result in a ratio of less than 1 for x0/y0 and a ratio of larger than 1 for a/b. Figure 19and Figure 20 show the development of both ratios over time for selected runs. Figure 19 shows results for run wdc25b in detail. One can see, that at around minute 60, both ratios, x0/y0 and a/b, level off and a stage is reached where the bed forms have the most uniform character along and across the flume. It is proposed that at around that time, the transition from 2-D to 3-D dunes proceeds. Figure 17: Different cases of ellipticity. One also needs to take into account the rotation of the ellipse (see) as well as the values of the size of the ellipse. The rotation theta indicates the main average direction of the crest lines, and the values of a and b indicate the relative major and minor magnitudes of the sandwave lengths. Figure 19: Ellipse ratios for run wdc25b see level off at minute 60, with fluctuating levels from there on. Figure 18: Rotation of the ellipse for cases b) and c) from Figure 17 indicates the main direction of the average crest line. Rotation theta is relative to the y axis for x0/y0 <1 and the x axis for x0/y0 >1. Therefore, with the help of analyzing the elliptical shape of the centre cone of the 2-D autocorrelation Figure 20: Ratios for the whole development all deep flows. When one has a look at the complete development process for all runs (see Figure 20), after the leveling-off stage, both ratios continue to fluctuate away from unity and even grow away again for some runs

8 from a uniform distribution. This indicates the bedsurface development of a more 3-dimensional bed surface causing the crest lines to break up at the flume walls with occurrence of local scour. The growth of bed forms along the flume still continues. Figure 21 shows a typical sand bed after development for one of the deep flows. Figure 21 Typical developed bed forms for the deep flow. The main crest lines are still stretching over most of the flume width, with minor separation across the flume. Figure 22 shows contour plots of the sand-bed elevation of run wdc33a at different stages. At minute 12 of the bed-form development, one can see that the crest lines are still very perpendicular to the flume walls, and no major separation occurs across the flume. At minute 30, the crest lines are still mainly orthogonal to the flume walls, but minor separation across the flume starts. The transition from 2-D to 3-D sand waves proceeds. This is the stage where according to Figure 20 the most uniform stage of bed form development along and across the flume appears for the whole development. After this stage, the sand waves continue to grow along the flume, but are restricted in their growth across the flume by the flume walls, which is shown here in the snapshot of minute 230 of development. 4.2 Differences in bed-form development associated with water depth As one can see from Table 1, the sand bed was exposed to two water depths, 0.15-m and 0.52-m. Results of the 2-D autocorrelation functions can be used to show the differences in development for these two different water depths. For the shallow flow, the water depth acts as the restricting force in the development of the dunes, with the bed surface losing its one-directional character early on in the development stage. The shallow depth and the wide flume causes a more pronounced separation across the flume of bed forms early on. This is reflected in ellipse shapes with ratios of x0/y0 larger than 1. Even for early stages, as soon as the flat sand bed is exposed to the water flow, bed forms separate across the flume, leading to 3-D dune development. For the deep flow, the flume walls act to restrict the development of the dunes, with no major separation of bed forms across the flume taking place and a transition from 2-D to 3-D dunes is the result. Figure 23 shows ratios x0/y0 for all runs over time. As one can see, for the shallow flows the ratio x0/y0 essentially never goes to less than 1 over the whole time of development. Minor changes here are caused by having the sand bed exposed to different flow velocities. The rotation theta of the ellipse (see Figure 23) shows either no rotation at all or a skewed fluctuation to a positive rotation. Both ellipse shape properties together indicate that the main average crest line keeps orthogonal to the flow with a slight skewness relative to the main flow direction. Figure 23: Ratio of x0/y0 for all runs at correlation value 0.4. Figure 22: Snapshots (length = m, width=1.35-m) of sand bed elevations at a) min 12, b) min 30 and c) min 230 for run wdc33a. Figure 24: Angles of rotation of ellipse theta for all runs at correlation value 0.4.

9 On the contrary, for the deep flow, the ratio x0/y0 approaches unity very fast at the beginning of development, indicating a degree of 2-dimensionality for the early stage. Additionally the degree of rotation is close to 0 at the beginning of development, which indicates 2-D dunes, or 3-D dunes perpendicular to the flow. Theta increases slightly with increasing flow strength. The crest lines at the beginning of bed-form development remain predominantly perpendicular to the flume wall, whereas with the 2-D to 3-D transition, the crest lines begin to skew across the flume. 5 CONCLUSION In this paper we have presented a new approach to gain 3-D sand-wave characteristics for sand beds developing from flat bed to equilibrium stage. The 2-D autocorrelation function has been applied on random fields of sand-bed elevations, each of a length of m and a width of 1.35-m. The contours of the centre cone of the 2-D autocorrelation function become generally elliptical and the shape and degree of rotation varies as the bed develops. For a shallow flow (ratio of flow depth to flume width less than 0.1), the major axis of the ellipse is oriented with the main direction of the flow field, which indicates 3-D development of the bed forms, from early on in the development stage. For a deeper flow (ratio of flow depth to flume width of 0.35), the major axis of the ellipse is orientated perpendicular to the main direction of the flow field, throughout the bed-form development. This indicates that the flume width is the restricting factor when it comes to the 3-D development of sand waves. As a deeper flow allows the sand waves to grow larger along the flume, this increase in growth potential is not supported across the flume with the flume walls acting to restrict development. An automatic method to determine 2-D to 3-D dune transition is introduced. This paper s findings are preliminary. From the first tests, as introduced above, we suggest the following: a) x0/y0 < 1 indicates that the 3-D development of sandwaves is wall-influenced; x0/y0 >1 indicates that the 3-D development of the sandwaves is flow-depth-influenced; b) theta tends generally to zero for 2-D wall influenced and 3-D sandwaves. Some high theta values can be obtained for 3-D flow-depthinfluenced runs, when ratio x0/y0 drops briefly below 1. For 3-D wall-influenced dunes, theta is skewed away from zero. c) for the deep flow, once the ratio (x0/y0)/(a/b) > 0.9, the 2-D to 3-D dune transition proceeds. Generally it can be said, that the introduced technique requires more testing on a more varied data set, which is currently done at The University of Auckland, using additional data sets similar to Friedrich et al. (2005). The application of 2-D autocorrelation functions of sand-bed surfaces requires further analysis, although the present results highlight that insights into 3-D characteristics of sand waves can be readily and objectively gained from this approach. 6 ACKNOWLEDGEMENT The research was partly funded by the Marsden Fund (UOA220) administered by the New Zealand Royal Society. 7 REFERENCES Allen, J. R. L. (1968). Current Ripples - Their relation to patterns of water and sediment motion, North-Holland Publishing Company, Amsterdam. Andreotti, B., Claudin, P., and Pouliquen, O. (2006). "Aeolian Sand Ripples: Experimental Study of Fully Developed States." Physical Review Letters, 96(2), Ashley, G. M. (1990). "Classification of large-scale subaqueous bedforms: a new look at an old problem." Journal of Sedimentary Petrology, 60(1), Best, J. (2004). "The dynamics and morphology of river dunes: synthesis and future research directions." Marine Sandwave and River Dune Dynamics, Enschede, The Netherlands, ppi- V. Callander, R. A. (1978). "River Meandering." Annu. Rev. Fluid Mech., 10, Coleman, and Eling. (2000). "Sand wavelets in laminar open-channel flows." Journal of Hydraulic Research, Vol.38(5). Friedrich, H., Melville, B. W., Coleman, S. E., Nikora, V. I., and Clunie, T. M. (2005). "Three-Dimensional measurement of laboratory submerged bed forms using moving probes." Proceedings of XXXI International Association of Hydraulic Engineering and Research Congress, Seoul, Korea, pp Goring, D. G., Nikora, V. I., and McEwan, I. K. (1999). "Analysis of the Texture of Gravel- Beds using 2-D Structure Functions." I.A.H.R. Symposium on River, Coastal and Esturine Morphodynamics, Genova, Italy, Nikora, V. I., Sukhodolov, A.N., Rowinski, P.W. (1997). "Statistical sand wave dynamics in one-directional water fows." Journal of Fluid Mechanics, Vol. 351, pp Venditti, J. G. (2003). "The Initiation and Development of Sand Dunes in River Channels," Doctoral Thesis, University of British Columbia, Vancouver.