Sediment trend analysis of the hylebos waterway: Implications for liability allocations

Abstract

The following provides a review, discussion, and description of how sediment transport pathways are obtained. It excludes the details of the mathematical proof, demonstrating the changes in grain size distributions that occur with transport as contained in McLaren and Bowles (1985). Sediment trend analysis (STA) requires for its data the grain size distributions of sediments collected on regular grid spacing over the aquatic site of interest. The sampled sediments are described in statistical terms (by the moment measures of mean, sorting, and skewness), and the basic underlying assumption is that processes causing sediment transport will affect the statistics of the sediments in a predictable way. For this purpose, a grain size distribution defines for any size class the probability of the sediment being found in that size class. Size classes are defined in terms of the well-known ϕ (phi) unit, where d is the effective diameter (diameter of the sphere with equivalent volume) of the grain in millimeters. Given that the grain size distribution g(s), where s is the grain size in phi units, is a probability distribution, then In practice, grain size distributions do not extend over the full range of s and are not continuous functions of s. Instead, discretized versions of g(s) with estimates of g(s) in finitesized bins of 0.5ϕ widths are used. Selection of the bin width is largely empirically derived. An increase in width can result in losing information contained in the distribution, whereas a decrease in width can produce an increasingly noisy distribution (a discussion of this dilemma is found in Bowles and McLaren [1985]). Three parameters related to the first 3 central moments of the grain size distribution are of fundamental importance in STA. They are defined here, both for a continuous g(s) and for its discretized approximation with N size classes. The 1st parameter is the mean grain size (m), defined as The 2nd parameter is sorting (s), which is equivalent to the variance of the distribution, defined as Finally, the coefficient of skewness (k) is defined as With the removal of r(s) from g(s), the remaining sediment (a lag) has a new distribution denoted by l(s) (Figure A1) where The function t(s) is defined as a sediment transfer function and is described in exactly the same manner as a grain size probability function except that it is not normalized. It can be thought of as a function that incorporates all sedimentary and dynamic processes that result in initial movement and transport of particular grain sizes. Data from flume experiments show that distributions of transfer functions change from having a high negative skewness to being nearly symmetrical (although still negatively skewed) as the energy of the eroding/transporting process increases. These 2 extremes in the shape of t(s) are termed low-energy and high-energy transfer functions, respectively (Figure A2). The shape of t(s) is also dependent not only on changing energy levels of the process involved in erosion and transport, but also on the initial distribution of the original bed material, g(s) (Figure A1). The coarser g(s) is, the less likely it is to be acted on by a high-energy transfer function. Conversely, the finer g(s) is, the easier it becomes for a high-energy transfer function to operate on it. In other words, the same process can be represented by a high-energy transfer function when acting on fine sediments and by a low-energy transfer function when acting on coarse sediments. The terms high and low energy are, therefore, relative to the distribution of g(s) rather than to the actual process responsible for erosion and transport. Sediment transport model to develop a lag deposit (see the text for a definition of terms). That t(s) appears to be mainly a negatively skewed function results in r(s), the sediment in transport, always becoming finer and more negatively skewed than g(s). The function 1 — t(s) (Figure A1) is, therefore, positively skewed, with the result that l(s), the lag remaining after r(s) has been removed, will always be coarser and more positively skewed than the original source sediment. McLaren and Bowles (1985) provide the mathematical proof for these statements. If t(s) is applied to g(s) many times (i.e., n times, where n is large), then the variance of both g(s) and l(s) will approach zero (i.e., sorting will become better). Depending on the initial distribution of g(s), it is mathematically possible for variance to become greater before eventually decreasing. In reality, an increase in variance in the direction of transport is rarely observed. Given 2 sediments whose distributions are, d1(s) and d2(s), and d2(s) is coarser, better sorted, and more positively skewed than d1(s), it might be possible to infer that d2(s) is a lag of d1(s) and that the 2 distributions were originally the same (Table A1, case A). Consider a sequence of deposits d1(s), d2(s), d3(s), … dn(s) that follows the direction of net sediment transport (Figure A3). Each deposit is derived from its corresponding sediment in transport according to the 3-box model shown in Figure A1. Each dn(s) can be considered a lag of each rn(s). Thus, dn(s) will be coarser, better sorted, and more positively skewed than rn(s). Similarly, each rn(s) is acted on by its corresponding tn(s), with the result that the sediment in transport becomes progressively finer, better sorted, and more negatively skewed. Any 2 sequential deposits (e.g., d1[s] and d2[s]) can be related to each other by a function X(s) (Eqn. 9). As illustrated in Figure A3, d2(s) can also be related to d1(s) by The function X(s) combines the effects of 2 transfer functions, t1(s) and t2(s) (Eqn. 10b). It could also be considered a transfer function in that it provides the statistical relationship between the 2 deposits and it incorporates all of the processes responsible for sediment erosion, transport, and deposition. The distribution of deposit d2(s) will, therefore, change relative to d1(s) according to the shape of X(s), which in turn is derived from the combination of t1(s) and t2(s) as expressed in Equation 10b. It is important to note that X(s) can be derived from the distributions of the deposits d1(s) and d2(s) (Eqn. 10a), and it provides the relative probability of any particular sized grain being eroded from d1, transported, and deposited at d2. With the use of empirically derived t(s) functions, it can be shown that when the energy level of the transporting process decreases in the direction of transport (i.e., t2[si] < t1[si]) and both are low-energy functions, then X(s) is always a negatively skewed distribution (Figure A4). This will result in d2(s) becoming finer, better sorted, and more negatively skewed than d1(s). Therefore, given 2 sediments (d1 and d2), where d2(s) is finer, better sorted, and more negatively skewed than d1(s), it might be possible to infer that the direction of sediment transport is from d1 to d2 (Table A1). Diagram showing the extremes in the shape of transfer functions t(s). In the event that t1(s) is a high-energy function and t2(si) > t1(si) (i.e., energy is decreasing in the direction of transport), the result of Equation 10b will produce a positively skewed X(s) distribution (Figure A4). Therefore, d2(s) will become coarser, better sorted, and more positively skewed than d1(s) in the direction of transport. When these changes occur between 2 deposits, it might be possible to infer that the direction of transport is from d1 to d2 (Table A1). Sediment coarsening along a transport path will be limited by the ability of t1(s) to remain a high-energy function. As the deposits become coarser, it will be less and less likely that the transport processes will maintain high-energy characteristics. With coarsening, the transfer function will eventually revert to its low-energy shape (Figure A2), with the result that the sediment must become finer again. Cases A and C produce identical grain size changes between d1 and d2 (Table A1). Generally, however, the geological interpretation of the environments being sampled will differentiate between the 2 cases. The above model indicates that grain size distributions of sedimentary deposits will change in the direction of net sediment transport according to either case B or case C (Table A1 and Figure A5). Thus, if any 2 samples (d1 and d2) are compared sequentially (i.e., at 2 locations within a sedimentary facies) and their distributions are found to change in the described manner, the direction of net sediment transport can be inferred. Sediment transport model relating deposits in the direction of transport. Summary diagram of t1 and t2 and corresponding X distribution (Eqn. 10b) for cases B and C (Table A1). Changes in grain size descriptors along transport paths. In reality, perfect sequential changes along a transport path as determined by the model and summarized in Figure A5 are rarely observed. This is because of a variety of uncertainties that can be introduced in sampling, in the analytical technique to obtain grain size distributions, in the assumptions of the transport model, and in the statistics used in describing the grain size distributions. These uncertainties are discussed in further detail (see Uncertainties section). One approach that appears to be successful in minimizing uncertainty is a simple statistical method whereby the case (Table A1) is determined among all possible sample pairs contained in a specified sequence. Given a sequence of n samples, (n2 - n)/2 directionally orientated pairs can exhibit a transport trend in one direction and an equal number of pairs in the opposite direction. When any 2 samples are compared with respect to their distributions, the mean can become finer (F) or coarser (C), the sorting can become better (B) or poorer (P), and the skewness can become more positive (+) or more negative (-). These 3 parameters provide 8 possible combinations (Table A2). In STA, if it is postulated that a certain relationship exists among the set of n samples and that this relationship is evidenced by particular changes in sediment size descriptors between pairs of samples, then the number of pairs for which the trend relationship occurs should exceed the number of pairs that would be expected to occur at random by a sufficient amount to state confidently that the trend relationship exists. Suppose that the probability of any trend existing between any pair of samples, if the trend relationships were established randomly, is p. Since there are 8 possible trend relationships among 3 sediment descriptors, and it is assumed that each of these is equally likely to occur, the p value is set to 0.125. To determine whether the number of occurrences of a particular case exceeding the random probability of 0.125, the following 2 hypotheses are tested: H0: p > 0.125 with no preferred direction H1: p < 0.125 and transport occurs in the preferred direction. The Z statistic is considered valid for N < 30 (i.e., a large sample). Thus, for this application, a suite of 8 or 9 samples is the minimum required to evaluate a transport direction. To assess the validity of any transport line, we use the Z score and an additional statistic, the linear correlation coefficient R2, defined as The value of R2 can range from 0 to 1. The definition of R2 is based on the use of a model to relate a dependent parameter y to 1 or more independent parameters (x1, x2, …). In this case, the model used is linear, which can be written as The data (y, x1, x2) are grain size distribution statistics, and the parameters (a0, a1, a2) are from the data with a The dependent parameter is defined as the and the independent parameters are the mean size and the An assumption is that distributions from samples along a transport if in in Figure would to be along a The of the line, which are the would on the of transport or there is no to a linear relationship among the 3 descriptors, there is also no any other of according to the of the relationship for the not to be except from a in of R2 with a high value of the Z score provide in the validity of the transport A low R2 can occur, when the Z score statistic is the of the of many sediment trend from many it appears that low R2 can result when sediments on an assumed transport path are, in reality, from and valid trend statistics the sediments are from a but the sequence is only a approximation of the actual transport and sediments been introduced the transport as in the case of R2, therefore, is and can provide information on the sediment transport The Z score and R2 statistics for each of the sample (Figure used to determine the sediment transport in the are in The McLaren and Bowles (1985) model requires that the grain size distributions of the sampled sediments be described in statistical terms (by the moment measures of mean, sorting, and The basic underlying assumption is that sequential deposits following the of net sediment transport will affect the statistics of the size distributions of the sediments in a predictable way. from this the size distributions of the sediments provide the data with which to for of net sediment transport. in the transport method is used to requires a model of the sediment transport The model is based on the assumption that are more than (i.e., the probability of transport, on a phi as grain size this it can be shown that erosion and of sediments will change the moments of their size distributions in a predictable in the direction of transport. as in transfer functions from sediment data in flume this assumption might not always be the transfer function over only a of the grain before to contained within the assumption is a further assumption that the probability of transport of 1 particular grain size must be independent of the transport of other grain sizes. as whereby the of can the transport of an increase in the of the finer or a decrease in the ability of the process to additional with all that the transport process is a function related to the sediment distribution and the of the erosion Thus, the of the the that the probability of transport must increase over a large range of in the deposits to produce the As and the technique to determine net transport pathways in a variety of and environments has been empirically the use of that an assumption be size distribution of a particular can be the result of sediment from and at It is assumed that is sampled is the of all the sediment derived from an number of The transport direction might not to that for a with a transport In STA, it is assumed that a sample provides a of a sediment facies) with no the to which the sample any that the sample in the for a of many Each might a particular transport and event at a might be from that of the transport as a The can be determined by the in a that a sufficient number of are the sample to the assumption that the sample an of the The distribution of all the the can be compared with a sample on the To provide d1 might be a sample an over whereas d2 of deposition. The trend analysis whether a possible sediment transport relationship or between the 2 deposits exists. might be to transport With between sample locations an of sediments by transport. in further detail in the indicates that to a continuous with samples, the must be sampled at the contained in the This would that for STA, sample could only transport over a in the of or over less than would as or could transport pathways the process of In practice, of a sample spacing must the number of environments likely to affect the the of the sediment and the shape and of the and samples will be by random These can in the the effects of and a sediment and random Sediment trend analysis is, in many to In the information is to a where a is that both the information as as The must be of the information from the noisy In sedimentary the information is the transport direction and the is the sediment The of is to the information from the noisy In the information can be by the from the an approach that in because the of the information and the are both This however, will be in because the of the information the is A large of analytical has been to in These 2 and For in the is as a a the all other than the The then for the to the original the the level of the to the to be the it easier to because of of the It is important to note that is being for is of importance in the would not be sufficient to the In STA, with no to the other of might to the in is with the use of that the level for the range of the If these of the the will the of the and the can be that increase the to The in is not as because the is not and the of the is only In this by can be because the can however, statistical might be to the In a sedimentary can be considered in 2 sample and in a sediment samples can be by One to this would be to many samples in and to produce a method that to this is For there has been on the use of a to sediments because it appears to provide a to many deposits and to the of STA, it has been shown that parameters of the distribution should change in the of erosion or and It that erosion and the parameters of the distribution to in particular when on the shape of the distribution and all however, are that distributions provide information (e.g., and and McLaren whether or is based on the assumption that sediments distributions. a to the sedimentary it is assumed that that do not on the are and are In this if in sediments do to the If do then the process as as a could be more noisy than the original In the approach of STA, the of are because only the data of each sediment grain size distribution are used from which the moments are sediment is over a can be To this of of samples could be used. of the by are, in reality, to in this manner (i.e., the Z as described or the of and These of of samples, which is not If the of the and the information is not the of samples can the information more than it the is to random by a number of samples from the same to a better distribution of that In STA, the assumption is that is and to the trend as the after these might in might provide more and In it is to the from the (i.e., a that over to the which the of the (i.e., the amount of the that is by all of the this is with a which the its the an is to the to the In sediment the rather than but exactly the same analysis can be In this case, the data grain size distributions of the sediment can be represented as a of with a the is a of the sedimentary deposits that how over For 1 would the of changes over a over a that the sample as discussed will set as to can be the in this the to be how is it is In the information is it it would be if not to in a simple analysis of the sedimentary data (e.g., the in the mean grain it is that a transport direction would be To the it is to an assumption as to is being It is then possible to the data to this and whether in a corresponding to the assumption is For a transport process that would produce the of sediments over a To this a can be and all with of less than could be An of the data would then over the The important of this approach in the approach discussed is the use of many sample to the transport direction. This the level of The however, is that it is to because the number of possible transport in a given can be large to The of a transport direction be and can only be to a level and information from other (e.g., In the Z score statistic, however, a transport trend can be determined whereby all possible pairs in a sample sequence are compared with each When either a case B or case C trend random probability within the sample the direction of net sediment transport can be inferred. As the grid spacing must be with the and the number of environments likely to be the shape of the and the statistical of the For it has been found for in sample spacing should not exceed 1 in spacing can be to For (e.g., to determine the transport for a sample spacing will be that a minimum number of samples can be to has also shown that samples should be over of (e.g., and in which the regular grid is to (e.g., and the direction of transport over an many sample this the sediment that should at the sample the with the Z score statistic from the grain size distributions of the and the assumed direction and the the is from the when a and of transport pathways is that or nearly of the samples, the assumption that information transport is contained in the grain size has been the to all the uncertainties that might be It must be that the actual processes responsible for the transport of along the derived pathways are They might in one be in a in the and in still the effects of one of the in the transport is to assess the processes that are likely to The shape of the X distribution is important in the of transport of the along a the of X is Consider a transport N X is then defined as in Equation d2 in one pair is d1 in and of d2 and d1 are with Equation that X is not defined as the of the mean value of d2 by the mean value of d1, the results of the 2 are For of d1, and X are before in there is no to that the of the X distribution should be X(s) can be thought of as a function that the relative probability of each being from d1 and deposited at d2. of X distributions from a large number of environments has shown that basic are when compared with the distributions of the deposits d1(s) and d2(s) (Figure 1. shape of the X distributions d1(s) and The relative probability of being transported, therefore, is a distribution to the actual Thus, the probability of a particular sized grain in the deposit is equal to the probability of its transport and (i.e., there must be a along the transport The bed is and is, therefore, in dynamic An X distribution dynamic can be found in either case B or case C transport, a fine between erosion and when environments are both case B and case C can be along the sample sequence. This is to as a and when this it is that the transport is also a state of dynamic of the 3 distributions are but the of X is finer than the of d1(s) and The of X can be thought of as the size that is the the of the deposits are coarser than these are more deposited than The therefore, must be in a state of net can only be in case B transport. Summary of the given to the of X distributions relative to the d1 and d2 the of the 3 distributions are but the of X is coarser than of d1(s) and This is the of net the size is coarser than the As a the deposits are erosion along the transport erosion can only be in case C transport. of the of d1(s) and d2(s), the X distribution more or less over the size range of the Sediment must fine in the direction of transport however, the bed is no it is a of sediment that with from transport The of is to sediments. X only in fine sediments when the mean grain size is a fine or the X distribution can be sediments are found from their and the of the X distribution that their is no related to size In other words, all an equal probability of being This of the X distribution 1st in the deposits of a and is described in McLaren the occur in fine in which any changes in the distributions are X distributions can be found in both case B and case C R2 is the correlation coefficient derived from the mean, sorting, and skewness of each sample pair a This is a relative of how the samples are related by transport. becoming finer, better sorted, and more negatively skewed in the direction of transport. becoming coarser, better sorted, and more positively skewed in the direction of transport. N is the number of possible pairs in the of X is the number of pairs a particular trend in a direction. Z is the Z score at the level are indicates transport in the direction. indicates transport in the direction. defines the dynamic of the sediments the of samples (i.e., 1 for a