DOI 10.1007/s00348-006-0195-9 RESEARCH ARTICLE Solid transport measurements through image processing A. Radice Æ S. Malavasi Æ F. Ballio Received: 13 September 2005 / Revised: 20 July 2006 / Accepted: 24 July 2006 Ó Springer-Verlag 2006 Abstract The paper describes a novel technique which enables evaluation of sediment fluxes on the upper layer of a granular bed by means of separate measurements of concentration and velocity of the moving particles. Specific elaboration techniques based on digital image processing were applied to films of the solid fluxes, allowing for automatic measurement. Sediment concentration was measured via a technique based on image subtraction and following proper filtering procedures, while grain velocity was measured by Particle Image Velocimetry. The method was applied in laboratory experiments of one dimensional bed load; the solid discharges measured by the proposed image processing technique were compared to those obtained by manual count of the grains passing over a fixed plate used as a sight. After calibration of the automatic technique, dimensionless solid discharges ranging from 1.0 10 3 to 1.2 10 1 were measured with a maximum error as large as 25%. The technique proposed also enables measurement of the time variation of the quantities and the two dimensional direction of sediment motion, for a complete characterization of grain kinematics in solid transport processes. 1 Introduction The transport of solids by a water stream, as bedload or in suspension, has topical importance in a variety of technical problems (river stability, reservoir silting, A. Radice (&) Æ S. Malavasi Æ F. Ballio D.I.I.A.R., Politecnico di Milano, Piazza Leonardo da Vinci, 32, 20133 Milano, Italy e-mail: alessio.radice@polimi.it channel self-cleaning). Laboratory experimentation is a fundamental tool for the study of such phenomena, and the discharge of transported sediments is typically one of the key variables measured (for examples see Sumer and Deigaard 1981; van Rijn 1984; Wijetunge and Sleath 1998). Bed load can be measured with sediment traps, which generally enable reliable data to be collected in a rather straightforward manner. Traps allow an integral measure to be made of the transported sediment volume in a certain time interval. As a consequence, they are widely used in field conditions to set sediment balances over a river reach. Yet the measure with a sediment trap is intrusive, which makes its use less effective in laboratory conditions. Moreover, if a twodimensional flux is measured, the direction of sediment motion is not resolved. Finally, the bed elevation must be almost constant, in order to prevent uncovering or sinking of the trap. Acoustic concentration profilers have been also used (see for example Admiraal et al. 2000), but the applicability of such instruments is limited to certain ranges of grain diameters and requires a deep active layer. The application of visualization techniques in study of sediment movement on a plane bed has grown apace in recent years. Image techniques are virtually nonintrusive; moreover, instantaneous measures of the quantities can be obtained, so that it is possible to study sediment dynamics at the particle scale in order to develop models of water/sediment interaction (once correspondent data for the water motion are available). Sediment transport can be observed either from the side or from the top. The two approaches have different advantages and disadvantages. The view from the side is used when the active sediment layer is
deeper than one grain diameter, but it only allows observation of the movement near the lateral wall, which in many cases is not fully representative of the global transport through the cross section, owing to the wall effects on the shear stress distributions and on the transversal grain sorting. This point of view is typical of studies on grain saltation (see, for example, Lee and Hsu 1994; Hu and Hui 1996; Niño and Garcìa 1996, 1998; Lee et al. 2000). By contrast, observation of the sediment motion from above provides a good representation of the upper layer, but the deeper ones may be hardly visible, in particular if the bed load rate is considerable. Yet the variability of sediment transport with lateral position can be observed to obtain global information. The view from above is generally used in analyses of incipient motion (Pilotti et al. 1997; Keshavarzy and Ball 1999; Sechet and Le Guennec 1999) or of diffuse transport (Drake et al. 1988; Papanicolaou et al. 1999). The two approaches also differ substantially in terms of the problems with which the image processing has to deal: in general, the moving grains observed from the side are easily distinguished from the background (which is typically constituted by the water), while the identification of grains moving on a background of similar ones is not straightforward if bed load is observed from the top. In the above-cited works, this problem has often been overcome by creating an artificial contrast between the moving particles and the background. Pilotti et al. (1997), on analyzing the inception of grain movement in laminar flow, either released dark particles on a light smooth fixed bed or highlighted the particle contours by means of careful arrangement of the light sources. Sechet and Le Guennec (1999) released natural sand particles on a smooth fixed bed and measured their motion by means of a Particle Tracking Velocimetry technique. Papanicolaou et al. (1999) tracked the motion of green glass beads over a layer of fixed transparent ones. In similar cases, a simple thresholding of the movie frames (or a colour recognition) is sufficient to distinguish the transported particles from the background. It should be noted, however, that releasing coloured sediments on a fixed bed (either smooth or constituted by fixed identical grains) enables the study of the kinematics of the single particles, but obviously does not permit analysis of the dynamics of the whole sediment bed. Yet such analysis may be important for the study of complex sediment transport phenomena, for example the evolution of bed forms or local scour processes. Therefore, the need for contrast greatly restricts the phenomena that can be studied. Finally, such a need is obviously difficult to satisfy in field studies. In literature studies without contrast between the moving and the still particles, at least part of the process of particle identification has been run manually. Drake et al. (1988) conducted field tests with natural sediments, performing all the elaboration by hand. This analysis did not suffer from the absence of contrast, but an extremely long time was required to run all the operations. Keshavarzy and Ball (1999) used natural sediments, and ran part of the elaboration automatically and part manually; their work will be discussed in more detail in Sect. 3. In this study, we focus exclusively on bed load observed from above. We will describe two image processing techniques which, jointly applied, enable the automatic measurement of solid fluxes on a granular bed also in the absence of artificial contrast between the moving grains and the background. In Sect. 2 the experimental equipment is described. In Sect. 3 the measuring technique is outlined and validated against the experimental data. In Sect. 4 some phenomenological features of one dimensional bed load are described in order to highlight the technique capabilities. 2 Experimental setup and bed load tests The experiments reported in what follows were run at the Hydraulic Engineering Laboratory of the Politecnico di Milano, Italy. We used a rectangular channel, 15 m long and 0.60 m wide. The channel was filled with nearly uniform sediments with mean diameter d 50 = 1.9 mm. The standard deviation of the lognormal grain size distribution was r g =(d 84 /d 16 ) 0.5 = 1.3 and the specific gravity was D =(q s q)/q = 1.57, where d i is the particle diameter corresponding to the i-percentile, q and q s are the densities of water and sediments, respectively. Both water and sediments were recirculated, using two different circuits. Water came mostly from the main laboratory circuit. The sediments were recirculated through a pump placed in a tank at the end of the channel; thus, part of the water was recirculated together with the sediments. Flow depths were measured by piezometric probes at various stations along the channel; water discharge was measured by a magnetic flowmeter as regards the part coming from the main circuit, while the other part was known after preliminary determination of the characteristic curve of the sediment pump. We performed 13 bed load experiments, with water discharges ranging from 16 to 22 l/s and water depths from 6 to 7 cm. Uncertainties in the measurement of the above quantities can be quantified as 1.5 l/s and 0.5 cm, respectively.
Table 1 Measured quantities and related uncertainties for the experimental tests. Symbol e qsx1000 denotes the uncertainty on the average solid discharge after one thousand independent measurements Test d (s) q sp (cm 2 /s) C sa ( ) u s (cm/s) q sx (cm 2 /s) CONV (pix/cm) Dx (cm) e Dx (%) N ( ) e d50 (cm) e CsA (%) e us (%) e qsx (%) e qsx1000 (%) 05 10 2.8 10 3 0.001 8.7 0.003 49.3 3.0 2.0 0.5 0.07 77.0 23.5 42.8 1.4 04 10 7.3 10 3 0.003 17.9 0.011 49.4 3.0 2.0 1.2 0.05 48.5 11.5 43.2 1.4 06 10 2.4 10 2 0.005 21.5 0.025 49.1 3.0 2.0 2.6 0.03 32.8 9.7 33.8 1.1 02 17 3.1 10 2 0.007 19.6 0.027 49.0 3.0 2.0 3.1 0.03 29.9 10.6 31.9 1.0 01 30 3.4 10 2 0.009 19.2 0.033 49.3 3.0 2.0 4.0 0.02 26.3 10.8 29.5 0.9 14 26 1.4 10 1 0.018 35.3 0.119 47.6 3.1 2.1 9.1 0.02 17.7 6.4 21.2 0.7 13 10 1.0 10 1 0.021 30.1 0.122 48.8 3.0 2.0 10.1 0.02 16.8 7.2 19.8 0.6 12 10 1.0 10 1 0.027 26.5 0.137 49.5 3.0 2.0 12.7 0.01 15.1 7.9 18.6 0.6 10 23 1.7 10 1 0.026 28.3 0.137 48.7 3.0 2.1 12.4 0.01 15.2 7.6 18.8 0.6 09 10 1.5 10 1 0.031 27.9 0.162 49.1 3.0 2.0 14.5 0.01 14.1 7.7 17.5 0.6 11 10 2.5 10 1 0.047 24.5 0.220 50.0 2.9 2.0 21.5 0.01 11.7 8.5 15.4 0.5 08 10 2.6 10 1 0.047 24.9 0.224 49.8 3.0 2.0 21.7 0.01 11.7 8.4 15.3 0.5 07 10 4.0 10 1 0.061 26.6 0.308 50.0 2.9 2.0 27.8 0.01 10.4 7.8 13.9 0.4 Hydraulic control parameters corresponded to Shields numbers ranging from 0.07 to 0.11. The test section was placed roughly midway along the channel. During each experiment, sediment motion was filmed using a black and white CCD camera with a frame rate of 100 fps and a resolution of 491 656 pixel, and a PC equipped with a proper frame grabber. Test duration varied from 10 to 30 s (see Table 1). The limited duration was due to the availability of memory space for image storage. A metallic plate was placed on the granular bed. The plate and the area upstream of it were filmed (Fig. 1) to observe the motion of sediments over both the plate and the granular bed. The plate width was 8 cm; the focus area of the CCD was about 13 10 cm 2. In order to avoid image distortion due to the irregularity of the free surface, an acrylic window was laid on the water surface at the test section to provide clear images; the acrylic sheet spanned the entire width of the channel and its length was 1 m. The window had to be slightly dipped into the water to prevent the entrainment of air bubbles that might compromise the image quality, particularly during the tests with high solid transport rates. As a result, the flow under the sheet was pressurized. Moreover, some scour occurred under the window, resulting in a nonstationary bed load, although the preliminary setting of the channel slope typically allowed the imposition of steady conditions in the flume. This problem did not affect the calibration and validation of the measuring technique, as discussed below. Mean solid discharge for each test was evaluated by observing all the images of a film and manually counting the number of particles travelling over the plate during the test. With N p denoting the total number of particles passing over the plate, w the plate width and d the test duration, the average solid discharge per unit width on the plate q sp was calculated as q sp = N p ÆW g / (dæw), where W g is the volume of a spherical particle of diameter equal to d 50. We then measured the same quantity using the image processing technique described below. Comparison between the two values of the bed load rate did not suffer from the unsteadiness of the solid transport, since simultaneous measures were made. 3 The measuring technique 3.1 Problem definition and support scales In this paper we consider sediment transport phenomena where only the upper sediment layer is active, Fig. 1 Example of two successive frames in a movie (test 09). Temporal order is from a to b; mean flow is from right to left
so that the process can be fully described by observation from above. In these conditions some of the sediments in the surface layer are still, while others are entrained by the stream and travel a certain distance, either saltating or rolling, before they stop. Since individual grains are discrete particles, their behaviour is typically discontinuous. One of the typical expressions of the resulting solid discharge is the following (see, for example, Fernandez Luque and van Beek 1976): q _ s ¼ C_ sav _ s d 50 ; ð1þ where q _ s is the average solid discharge per unit width, C _ sa is the average aerial concentration of the bed load, _ vs is the aerial average velocity of bed load particles and d 50 is the representative particle diameter. The solid discharge in Eq. 1 is not defined as a mass flux through a transversal surface, but rather as an aerial average quantity. All variables therefore refer to a given support area; specifically, C _ sa; _ v s and d 50 are spatially averaged values over that area. Quantities in Eq. 1 are defined over a finite area at a given instant. Yet their measurement requires choice of a proper time interval. Movements are detected by comparing subsequent images separated by a finite time lag. As a consequence, we cannot say whether or not a certain grain is moving in a particular instant: all we can determine is whether the particle has changed its position during the time interval that separates two successive frames. We therefore substitute the instantaneous concentration C _ sa in Eq. 1 with the concentration of the displaced particles within a time interval, named C sa. Likewise, we can only measure the sediment velocity averaged over a time window corresponding to the temporal scale; this will hereafter be denoted as v s. The expression of the unit width solid discharge q s at the same scale is consequently: q s ¼ C sa v s d 50 : ð2þ The measurement support time interval must be compared against the temporal scales of sediment motion. Following a typical approach in the literature, the lagrangian motion of bed particles can be described as a series of movements and periods of rest. Nikora et al. (2002) identified three ranges (local, intermediate and global) with reference to particle diffusion. According to these authors, the local range corresponds to ballistic particle trajectories between two successive collisions with the bed; the intermediate range corresponds to particle trajectories between two successive rests or periods of repose; finally, the global range corresponds to particle trajectories consisting of many intermediate trajectories, just as intermediate trajectories consist of many local trajectories. With reference to the measurement of the quantities in Eq. 2, given a temporal interval Dt, the values of both C sa and v s are means over the same measuring scale. If the temporal measuring scale is within the local/intermediate range, the values of the quantities can be identified with the instantaneous ones. If the temporal scale lies within the global range, the values measured will be dependent on the time scale itself: larger temporal scales will yield larger concentration values (since many grains move successively) and smaller sediment velocities (since the real instantaneous velocity is averaged with the zero velocity of the periods of rest). It should also be noted that large temporal measuring scales will yield less discontinuous temporal series of the measured quantities, since the discontinuity of the solid medium is smoothed by the temporal mean. The dependence of the measured quantities on the temporal support scale is analyzed in detail in Radice (2005). As will be discussed in Subsect. 3.5, in this work we used a temporal scale within the local range of grain motion. The following sub-sections will describe two separate techniques whose application furnishes the measure of the solid discharge by independent evaluation of the primitive quantities in Eq. 2. 3.2 The measure of concentration In order to measure the aerial concentration of displaced sediments C sa, the number N of displaced grains within a specified area and between two frames must be determined. The two quantities are obviously related, in that: C sa ¼ NW g Dx Dyd 50 ; ð3þ where Dx and Dy are the dimensions of the spatial observation scale in the longitudinal and transversal directions, respectively. Figure 1 shows an example of two successive frames in the film taken during one of the bed load tests described in Sect. 2 (the metallic plate is visible in the left part of the image). The number of grains displaced within the time interval separating the frames is measured by comparing the two images. Keshavarzy and Ball (1999) showed that the occurrence of a movement can be highlighted by subtracting the consecutive frames. Ideally, the difference image contains zero values where no motion has occurred,
while it has non-zero values (within the interval 255 to 255) in the position occupied by the displaced grains in the two frames. Figur shows the subtraction image, where levels were adjusted to the typical range of grayscale images (0 255) for representation purposes only. In this image, the negative part of the subtraction image appears in dark tones and the positive part in light ones; the no-movement areas appear as a uniform grey, with an intensity value of 255/2. After obtaining images like that in Fig. 2, Keshavarzy and Ball (1999) manually identified the displaced grains. At this point, it is crucial to distinguish significant movements (i.e. actual grain displacements) from both the vibration of particles around a stable position and the variation in intensity induced by optical disturbances. A subjective evaluation (as in Keshavarzy and Ball 1999) may yield a satisfactory judgement on a case-by-case basis, but it requires a very long processing time. Moreover, it typically yields underestimated values of the concentration of displaced sediments, as we will show in the following sub-sections. In our case, we devised a measuring technique that worked automatically after the first calibration. The subtraction duplicates the images of a displaced grain, which typically appears as a pair of dark and light points. This effect can be easily removed by filtering either the positive or the negative values to zero. If the positive part of the difference image is retained, a particle moving on a background of darker ones will be retained as it appears in the final position, while moving particles which appear darker than the background will be identified in their initial position. The opposite will happen if the negative part of the difference image is retained. In our experiment, the upper moving particles were typically more exposed to illumination than the others, so that they behaved as light particles on a dark background. We tested both possibilities, finding that the choice of retaining the negative or the positive values had very little influence on the measured concentration values; we chose to use positive values, obtaining images like that in Fig. 3. The positive difference image was first thresholded to obtain a binary one (Fig. 4), which appears as a uniformly black background, with white blobs identifying the displaced particles. This image is affected by noise in terms of white blobs of very small size, not corresponding to significant particle displacements. A proper filtering technique is necessary to remove such blobs. The blobs in the binary image were initially identified and counted, and then their average area was determined. The pixel connection criterion played an important role in this phase, because a different number of blobs would be detected if a 4-point or 8-point connection was considered. Next, the smaller blobs had to be removed. We used for this purpose a relative filter based on the ratio between the blob area (in pixels) and the average one: all the blobs with an area smaller than a certain fraction C of the average one were removed. This yielded the final image in the majority of cases. In cases with no moving particles only vanishing blobs were present, and the relative filter did not remove all of them. For this reason, an absolute aerial filter was added: the size under which the blob should be removed was a fraction D of the area of a grain in an original image. As said, the second filter was ineffective in the majority of cases, since the Fig. 2 Subtraction image obtained from the two frames shown in Fig. 1 Fig. 3 Positive part of the subtraction image shown in Fig. 2
The complete filtering technique involves four parameters: the threshold value BIN, the connection criterion CONN, the coefficients for determination of the relative filter dimension C and of the absolute filter dimension D. The parameters were calibrated in order correctly to measure the solid discharges for our bed load tests (the comparison is set out in Sect. 3.4); for the results shown: BIN = 77, CONN = 4, C = 0.2, D = 0.08. After the displaced sediments have been identified, the aerial concentration can be calculated with Eq. 3. We considered measuring cells of a size equal to 3 3cm 2 (Fig. 6). 3.3 The measure of grain velocity Fig. 4 Binary image obtained from Fig. 3. Blob number in the whole image is 197 second filtering dimension was typically smaller than the first one. The final filtered image is shown in Fig. 5. Note that the different blob sizes in the final image do not necessarily reflect different real dimensions of the sediments. The final area of a blob depends on both the lighting conditions and the threshold level BIN. In the case of the image shown in Fig. 5, the average pixel size of a blob is 35 pixels, while the actual size of a grain is 70 pixels. This consideration also explains our decision to measure the number of displaced grains on the basis of the blobs instead of, for example, considering the percentage of white pixels in the binary image divided by a fixed representative particle size. Once the final image has been obtained, all the blobs are counted as if they have the same area. We wanted to measure the velocity of the moving particles alone. Taking up the suggestion of Keshavarzy and Ball (1999), therefore, a particle image velocimetry (PIV) technique was applied to pairs of consecutive difference images containing the representation of such particles alone, without any further elaboration. The original frames were divided into a regular grid, so that a velocity vector was assigned to each cell. The grid shown for the evaluation of C sa (Fig. 6) was also used for the velocity measurements, with no overlapping. In order to find the most probable sediment displacement in a grid cell, all the possible displacements were considered. For each possible displacement, two areas, namely the interrogation and the research area, were fitted into the grid cells so that the displacement corresponded to the vector connecting the centres of the areas; obviously these areas where Fig. 5 Filtered binary image. Blob number in the whole image is 148 Fig. 6 Partition of the image into equal cells
smaller than the measuring cell, with smaller size as a larger tentative displacement was considered (see also Sect. 3.5). Next, considering the part corresponding to the interrogation area in the first image and the part corresponding to the research area in the second image, the S parameter was calculated: N x N y S ¼ P Nx P Ny i¼1 j¼1 j Ii; ð jþ Ri; ð jþ ð4þ j; where I and R are the intensities in the interrogation and research areas, respectively, and N x and N y are the pixel dimensions of the two areas. The most probable displacement maximizes the value of S. In cases where the number of displaced particles is small, the algorithm may choose unrealistic displacements; a control on the ratio between the maximum S value and the average value checks vector reliability. If the above ratio is smaller that a certain fixed value, the velocity vector is eliminated and no velocity value is measured. We did not use a post-processing algorithm for interpolation of the missing velocity values. The required calculation parameters were the maximum possible displacements to be tested and the minimum ratio between the peak and average values of S. The limits for the displacements were set after visual estimation of the maximum velocities in some sample images, and they were equal to one fifth and one tenth of the measuring cell side in the longitudinal and transversal directions, respectively. The measured velocity values depended to a very minor extent on these limits provided the latter were large enough to avoid cutting the maximum velocity values. The minimum peak/average S ratio was equal to 1.1. Once the displacement vector has been measured, the velocity vector can be calculated as follows: light areas. Figure 8 shows the surface in the space (d x, d y, S), whose maximization gives the most probable displacement for the bottom left cell in Fig. 7. 3.4 Calibration validation of the measuring technique Each film of our experimental campaign was processed to produce the temporal series of sediment concentration and velocity upstream of the metallic plate. We used a Pentium IV computer with a RAM as large as 1 Gb; typical calculation times for a 10 s film were about 2 min and almost 2 h for measurement of sediment concentration and velocity, respectively. The temporal series of the solid discharge were obtained via Eq. 2. Finally, we averaged the data to obtain the mean solid discharge per unit width for each test. The averaged result was compared to the solid discharge obtained via the manual count of the grains crossing the plate during the filmed period. u s ¼ d x FPS CONV ; v s ¼ d y FPS CONV ; ð5aþ ð5bþ Fig. 7 Example of sediment velocity field for test 09. Measured velocity in the bottom left cell was equal to 16.6 cm/s where (d x, d y ) is the most probable displacement in pixels, FPS is the sampling frequency and CONV is the parameter converting a pixel distance in the image into a real length. Figure 7 shows a single velocity field in the filmed area upstream of the plate. The sediments mainly moved parallel to the mean stream direction, even if lateral displacements were measured as well. The image on which the vector field is superimposed was obtained in the same way as that in Fig. 2; in this representation, grains motion is typically from dark to Fig. 8 S surface corresponding to the bottom left vector in Fig. 7
Parameter tuning is more important for measurement of sediment concentration than for measurement of sediment velocity, because only the calculation parameters for concentration have a significant effect on the final measured solid discharges. We therefore focused on calibrating the parameters for the grain concentration. There are two options in this case: first, values of the calculation parameters can be set by comparing the measured concentrations for some sample images and those obtained, as in Keshavarzy and Ball (1999), by visual evaluation of the number of displaced grains on the same images; second, the parameters can be set so that they match the solid discharge measurements manually made at the plate. The second option is objective, but requires ad hoc experimentation. We used both methods, adopting a trial and error procedure: first, tentative values were determined with the first method and solid discharges were calculated and compared to those from the manual count of the grains on the plate. We found that visual calibration underestimated the sediment concentration for the highest transport rates, since low values of the solid discharge were obtained, with a minimum underestimation factor of 0.5. We then tuned the calculation parameters to achieve a best fit with the solid discharge values from the plate, obtaining the parameter set quoted in Sect. 3.2. The comparison between the solid discharges for the plate and for the automatic measurements with the two combinations of parameters is shown in Fig. 9, which shows the Fig. 9 Comparison of dimensionless solid discharges from the manual count and from image processing dimensionless solid discharge per unit width U = q sx / (gdd 3 50 ) 0.5 (in the last expression, q sx is the longitudinal component of the solid discharge and g is the gravitational acceleration). For the best fit calibration, the same set of parameters was kept constant for dimensionless solid discharges varying by more than two orders of magnitude (from 1.0 10 3 to 1.2 10 1 ), with most points lying within the ±25% bounds. A significant underestimation of the highest transport rates was made for the visual calibration. Our results are consistent with those of Drake et al. (1988), who made solid discharge measurements in a natural stream with non-uniform sediments and a bed shear stress equal to twice the threshold for incipient motion of the median diameter particles. They manually tracked the motion of individual sediments to obtain the average solid discharge value, which was compared to that obtained by a sampler placed at the test reach. The authors found a significant underestimation of the transport rate for the smallest sediments; by contrast, bed load rates measured with the two methods were in good agreement for the larger sediments. Drake et al. (1988) dealt with a single transport condition with non-uniform grains, while we have several transport conditions with rather uniform grains; underestimation of the solid discharge was found for the higher shear stresses in our case, and for the smallest sediments in the former, that is, again for high shear stresses. Keshavarzy and Ball (1999) worked in similar conditions as ours, but they did not validate their measures against solid discharge data. Yet their study was concerned with the correlation between the temporal variation of sediment concentration and the sweep events of the turbulent boundary layer, and therefore they were not greatly involved in quantitative verification of the absolute values of solid discharge. Moreover, because they are focused on the incipient motion of sediments they would probably suffer little from the problems of grain identification. No universality of the calculation parameters for our technique can be expected; typically, they vary with the sediment characteristics and the lighting conditions. Calibration is therefore necessary. Two considerations can be made in this regard: first, the method is robust, in that relatively large variations of the solid discharge do not require variation of the calculation parameters; second, simple visual calibration enables determination of the order of magnitude of the solid discharge that, in some cases, may be a first indication of the transport levels. The average values of sediment velocity were computed so that they could be compared to previous literature findings. Figure 10 compares the measured
the grain to make non-negligible displacements between a frame and the next, thereby imposing an upper limit on the sampling frequency. The measurement of grain concentration requires grain displacement to be larger than a certain fraction a of the particle diameter d 50 ; the maximum sampling frequency increases with sediment velocity u s and can be quantified as: FPS max ¼ u s : ð6þ a d 50 The measure of particle velocity requires grain displacement to be larger than a certain b number of pixels. We thus obtain another quantification of the maximum sampling frequency: Fig. 10 Sediment velocity for the experimental tests and experimental points presented by previous literature works average sediment velocity to experimental points quoted in Niño and Garcìa (1998) and in Lee et al. (2000). The sediment velocity is normalized by the shear velocity u* =(/ÆgÆDÆd 50 ) 0.5, where / is the Shields parameter. In the absence of a direct measurement of the bed shear stress, we evaluated the Shields parameter of our tests using the Meyer-Peter and Muller equation U =8Æ(/ / c ) 1.5, where / c is the threshold value for incipient sediment motion (taken equal to 0.05). Owing to the wide dispersion of literature data we can only conclude that the order of magnitude of the measured velocities is consistent with previous literature results. It is interesting to note that grain velocity is only weakly dependent on the bed shear stress, indicating that the variation of the solid discharge is mainly due to analogous variation of the sediment concentration. From a phenomenological point of view, this means that an increase in the shear stress increases the number of entrained particles but, once a particle has moved, its velocity is little dependent on the shear stress. 3.5 Choice of the support scales and application ranges This sub-section furnishes some criteria for the selection of proper temporal and spatial measuring scales. We discussed in the Sub-sect. 3.1 that the measured values of the quantities are good approximations of the instantaneous ones when the temporal support scale lies within the local/intermediate range of grain motion. This would suggest keeping the temporal scale as small as possible. Yet the measuring technique requires FPS max ¼ u s CONV ; ð7þ b depending also on the pixel/cm conversion CONV. Obviously, the maximum sampling frequency must be computed with reference to the minimum velocity values to be measured. For a sediment velocity equal to 2 cm/s and considering our typical values of CONV (about 50 pixel/cm, see Table 1), if we assume a = 0.1 and b = 1 pixel, we obtain a maximum frequency of 105 Hz from Eq. 6 and of 100 Hz from Eq. 7. We therefore worked with the largest sampling frequency that we could maintain. Measuring with a smaller frequency enables identification of the relation between the temporal scales and the dynamic ranges of particle motion addressed in Sect. 3.1. We measured the solid discharge of our test considering only the odd frames in each film (as the sampling frequency was equal to 50 Hz), and we found solid discharge values very similar to those for the 100 Hz sampling. We concluded that the independence of the measured values on the sampling frequency indicated that our time scales lay within the local range of motion. The minimum side of the measuring cells depended on the requirements of our PIV algorithm. The sizes of the interrogation and research areas decrease as larger tentative displacements are considered; if c x and c y are the ratios between the cell side and the maximum displacement in the longitudinal and transversal direction, respectively, then the minimum ratio between the size of the interrogation (or research) area and that of the measuring cell will be: N x N y Dx p Dy p min min ¼ c x 1 c x c y 1 c y ; ð8þ
where Dx p and Dy p are the pixel dimension of the measuring cells. It is good practice for the dimension of the above areas to represent a considerable fraction of that of the measuring cell (in other words, the c values should not be too small). Minimum cell dimension in the x direction can be evaluated as: Dx min ¼ u s c x FPS ; ð9þ where FPS is the sampling frequency. Obviously, Dx min must be computed with reference to the maximum velocity values to be measured. If we consider maximum u s values as large as 60 cm/s and a c x value of 4, we obtain Dx min = 2.3 cm for the frequency of 100 Hz. In our measurements, by maintaining a cell length of 3 cm we were able to set a c x value of 5 (as stated in Sect. 3.3). Then, in order to increase the significance of the interrogation and research areas, we also set c y = 10. In this way, our interrogation and research areas were at least 72% of the cell dimension. Conceptually, there are no upper limits on the cell size. It should be borne in mind, however, that the larger the cell size, the less the method is able to measure the spatial variability of sediment motion. Furthermore, as is well known, the PIV technique finds the most probable displacement of a group of particles by assuming a coherent translation of the same grains. Using larger measuring grids increases the likelihood that the grain displacements will be different, which compromises the PIV measurement. There are no analogous difficulties in measurement of concentration. Finally, our technique may encounter practical difficulties when the sediment concentration is large, owing to particle accumulation that hinders particle identification by the algorithm presented. Concentration values larger than 15 20% may impede the making of measurements. Such values would correspond to shear stresses almost three times above the threshold for sediment motion. In cases of this kind, a more sophisticated technique for particle identification may be necessary. Yet it is worth bearing in mind that typically more sophisticated algorithms are less general in their applicability. 4 Results: one dimensional solid discharge By means of the image elaboration we obtained the time series of the aerial concentration of displaced sediments C sa and of the longitudinal and transversal components of the grain velocity value (u s and v s, respectively) for each cell of the measuring grid. The temporal series of the aerial concentration of displaced grains and of the longitudinal component of sediment velocity are shown in Fig. 11a, b, with reference to test 09 and the bottom left cell of Fig. 6. Both signals show that the quantities are not constant in time; concentration levels fluctuate around a mean value l CsA = 3.5%, with coefficient of variation CV CsA = r CsA / l CsA = 0.39, where r CsA is the mean square of the concentration. The velocity time series has a mean value l us = 0.28 m/s and CV us = 0.44. Concentration values lay between 0.01 and 0.07. This means that for each investigation area a number of particles varying between 2 and 35 were displaced during a single time interval. The physical quantities C sa and v s are consistently defined by Eqs. 3 and 5 even if a single particle is moving, but the outcoming signals are highly irregular, given the above considerations on the discontinuous nature of the solid medium. The aim of the techniques described here was to measure sediment fluxes, not to identify the kinematics of individual particles (velocity, duration of entrainment and distrainment periods, saltation lengths,...). Hence the use of larger interrogation areas and/or larger sampling intervals would increase the number of displaced particles, and would therefore generate more regular series of values. This would cause measurement accuracy to degenerate because of the above difficulties of the PIV velocity measurement when the grain displacement is not coherent within the measuring cell. Since variables are defined also for limited numbers of moving particles, we preferred to smooth the output signal of Fig. 11 by means of moving averages over relatively long time windows (0.3 s). The smoothed curves (thick lines in Fig. 11) show that C sa and u s follow similar trends (which is reasonable, since both can be assumed as measures of the stream power). Analogous successions of peaks and caves can be detected: they correspond, respectively, to more intense transport events and to periods of relatively weak sediment activity. The duration of a transport event is variable, but typically of the order of 1 s. Solid discharge could be evaluated for each time interval using Eq. 2. The temporal series is shown in Fig. 11c (l qsx = 0.19 cm 2 /s, CV qsx = 0.58). Again, a succession of peaks and caves can be observed which is similar to those obtained for C sa and u s. The population of solid discharge vectors can be visualized by means of quadrant analysis in a (q sx, q sy ) plot. Solid discharges for test 09 and for the bottom left cell of Fig. 6 are plotted in Fig. 12, where each point corresponds to the tip of a velocity vector, and the time averaged solid discharge is highlighted by the grey diamond. Use of the image processing method to capture the two dimensional features of grain motion
Fig. 11 Temporal variation of sediment concentration, velocity and solid discharge for test 09, bottom left cell of Fig. 6 yields the finding that, although a steady bed load was imposed in the channel, solid discharge vectors can be angulated with respect to the average flow direction. Yet the preferential direction of motion is clearly identifiable, since the average transversal component of the velocity vector is a negligible fraction of the longitudinal one. This preliminary results show that the technique presented has some advantages in respect to the use of the more traditional sediment traps: first, it is not intrusive; second, it enables measurement of the primitive quantities that concur in the determination of the solid flux (the primitive quantities are often related to the characteristics of the turbulent boundary layers in water sediment interaction models); finally, it permits to have adequate spatial and temporal resolution, while a sediment trap furnishes only an integral measure. Furthermore, the availability of two dimensional velocity data may be not essential for study of one dimensional bed load, but it makes our technique a powerful tool for analysis of more complex solid transport phenomena (such as local erosion processes). 5 Conclusions The paper has discussed the measurement of sediment fluxes in solid transport phenomena. As in most studies on this topic, the kinematics of the solid particles have been analyzed using an eulerian approach; solid discharges can be calculated once the aerial concentration of transported sediments and the velocity of the same particles have been measured. Digital films of the ongoing phenomena were taken during the laboratory tests. Two separate image processing techniques for independent measurement of
Fig. 12 Quadrant analysis of solid discharge for test 09, bottom left cell of Fig. 6. The grey diamond corresponds to the average value the primitive quantities stated above were then devised. The aerial concentration of transported grains was measured by means of a subtraction between successive frames and using filtering procedures. Grain velocity was measured by applying PIV to two consecutive difference images. It was found that the most critical measurement is that of sediment concentration, because of the difficulties in particle identification. Hence greater effort was devoted to setting the first part of the technique. Differently from literature studies with naturally coloured sediments, the measuring procedure was completely automated. Moreover, it was found that visual individuation of the displaced particles may yield a significant underestimation of the resulting solid discharge, particularly for the higher shear ratios, even if the orders of magnitude can be correctly evaluated. Calibration of the method is therefore desirable, to obtain reliable quantitative determination of the sediment fluxes. The work yielded the following main conclusions: Image processing enables more refined measurements than those with sediment traps, in that the measure is not intrusive and the primitive quantities that concur in determination of the sediment flux can be quantified separately. The technique developed permits quantification of sediment fluxes when the transport process is characterized by the movement of the upper layer. Dimensionless solid discharges varying from 1.0 10 3 to 1.2 10 1 (corresponding to shear stresses ranging from one to more than twice the threshold for sediment motion) were measured with errors smaller than 25%. The same parameter set was kept constant across the entire range of our experiments. Proper temporal and spatial measuring scales must be chosen with reference to the expected values of the quantities to be measured. The possibility of particle identification can be limited by concentration values corresponding to the higher shear ratios. Separate measurements of the temporal variation of sediment concentration and velocity enable observation of the main features of one dimensional solid transport. Moreover, our technique enables local measurements of the solid discharge, allowing analysis of the spatial distribution of sediment fluxes. The availability of two dimensional grain velocity measurements and complete automation of the measure allow for complete characterization of the sediment motion and makes the technique useful for analysis of more complex phenomena. Acknowledgments This study was supported by the Italian Ministry of University and Research (MIUR) under the Project Erosione d alveo in prossimità di ponti fluviali (Local scour at river bridges). We gratefully acknowledge the anonymous reviewers for their comments and suggestions, that enabled us to greatly improve the quality of the manuscript. Appendix: Solid discharge measurement uncertainty In what follows we consider only the longitudinal component of the vector solid discharge; analogous expressions hold for the transversal one. Solid discharge is evaluated by the following equation, that is, the x component of Eq. 2: q sx ¼ C sa u s d 50 : Thus measure uncertainty is given by: e qsx ða1þ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi @q 2 sx ¼ e @u 2 u s þ @q 2 sx C s @C sa þ @q 2 sx d sa @d 50 50 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ ðc sa d 50 Þ 2 u s þ ðu s d 50 Þ 2 C sa þ ðc sa u s Þ 2 d 50 : ða2þ
Preliminary determination of the measure uncertainties of both sediment concentration and velocity is required in order to evaluate the measuring uncertainty of solid discharge. Sediment concentration is given by Eq. 3 and, explicating the sediment volume W g : s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi @Dy 2 e Dy ¼ @CONV e2 CONV sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ Dy 2 p CONV 2 e2 CONV ¼ Dy p CONV 2 e CONV; ða5bþ e CsA sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi @C 2 sa ¼ N @N þ @C 2 sa d @d 50 þ @C 2 sa Dx 50 @Dx þ @C 2 sa Dy @Dy sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p d 2 2 50 ¼ N 6 Dx Dy þ p Nd 2 50 d 3 Dx Dy 50 þ p Nd 2 2 50 6 Dx 2 Dx Dy þ p Nd 2 2 50 6 Dx Dy 2 Dy: ða6þ C sa ¼ NW g ¼ p Nd 2 50 Dx Dyd 50 6 Dx Dy : ða3þ Sediment velocity is given by Eq. 5a; thus, velocity measurement uncertainty is: e us sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi @u 2 s ¼ d @d x þ @u 2 s FPS x @FPS þ @u 2 s CONV @CONV sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi FPS 2 ¼ d 2 x d CONV x þ FPS CONV þ d x FPS 2 CONV 2 CONV : ða7þ The physical dimensions of the measuring cells Dx and Dy are given by: Dx ¼ Dy ¼ Dx p CONV ; Dy p CONV : One thus obtains: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi @Dx 2 e Dx ¼ CONV @CONV sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ Dx 2 p CONV 2 e2 CONV ¼ Dx p CONV 2 e CONV; ða4aþ ða4bþ ða5aþ On the basis of these equations, the measuring uncertainty of solid discharge depends on the actual values of sediment concentration and velocity, and on some fixed quantities, which are now enumerated. Pixel dimensions of the measuring cells are Dx p = Dy p = 147 pixels. Uncertainty e dx of the pixel sediment displacement is considered equal to 1 pixel. This is rather conservative if compared to the most usual estimates in the PIV literature, and it is so because we did not pursue the best refinement of the velocity measurement algorithm. Uncertainty e CONV of the pixel/cm conversion is considered equal to 1 pixel/ cm. Sampling frequency is equal to 100 ± 1 Hz. The measuring error of the number of displaced grains is zero, since once the filtered image has been created the measure is a simple count of the white blobs. The uncertainty of the average diameter of the moving grains within a cell e d50 must be specifically addressed: it is not due to uncertainty in the sieve mesh, but rather