## Appendix 1

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

## Authors

### Patrick McLaren,

Corresponding author- E-mail address: patrick@geosea.ca

- GeoSea
^{®}Consulting (Canada), 7236 Peden Lane, Brentwood Bay, British Columbia V8M 1C5

- GeoSea
^{®}Consulting (Canada), 7236 Peden Lane, Brentwood Bay, British Columbia V8M 1C5

### R Paul Beveridge

- Heller Ehrman LLP, 701 Fifth Avenue, Suite 6100, Seattle, Washington 98104–7098, USA

- First published: Full publication history
- DOI: 10.1002/ieam.5630020310View/save citation
- Cited by: 0 articles

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.

(1)

Given that the grain size distribution *g*(*s*), where *s* is the grain size in phi units, is a probability distribution, then

(2)

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

(3)

The 2nd parameter is sorting (s), which is equivalent to the variance of the distribution, defined as

(4)

Finally, the coefficient of skewness (k) is defined as

(5)

Consider a sedimentary deposit that has a grain size distribution *g*(*s*) (Figure A1). If eroded, the sediment that goes into transport has a new distribution, *r*(*s*), that is derived from *g*(*s*) according to the function *t*(*s*), so that

(6)

where *g*(*s*_{i}) and *r*(*s*_{i}) define the proportion of the sediment in the *i*th grain size class interval for each of the sediment distributions and *k* is a scaling factor that normalizes *r*(*s*) to get Equation 7. (The scaling factor *k* is actually more complex than a simple normalizing function, and its derivation and meaning is the subject of further research. It appears to take into account the masses of sediment in the source and in transport and might be related to the relative strength of the transporting process.)

(7)

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

(8)

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.

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, *d*_{1}(*s*) and *d*_{2}(*s*), and *d*_{2}(*s*) is coarser, better sorted, and more positively skewed than *d*_{1}(*s*), it might be possible to infer that *d*_{2}(*s*) is a lag of *d*_{1}(*s*) and that the 2 distributions were originally the same (Table A1, case A).

Consider a sequence of deposits *d*_{1}(*s*), *d*_{2}(*s*), *d*_{3}(*s*), … *d*_{n}(*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 *d*_{n}(*s*) can be considered a lag of each *r*_{n}(*s*). Thus, *d*_{n}(*s*) will be coarser, better sorted, and more positively skewed than *r*_{n}(*s*). Similarly, each *r*_{n}(*s*) is acted on by its corresponding *t*_{n}(*s*), with the result that the sediment in transport becomes progressively finer, better sorted, and more negatively skewed. Any 2 sequential deposits (e.g., *d*_{1}[*s*] and *d*_{2}[*s*]) can be related to each other by a function *X*(*s*) (Eqn. 9).

(9)

As illustrated in Figure A3, *d*_{2}(*s*) can also be related to *d*_{1}(*s*) by

(10)

(11)

The function *X*(*s*) combines the effects of 2 transfer functions, *t*_{1}(*s*) and *t*_{2}(*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 *d*_{2}(*s*) will, therefore, change relative to *d*_{1}(*s*) according to the shape of *X*(*s*), which in turn is derived from the combination of *t*_{1}(*s*) and *t*_{2}(*s*) as expressed in Equation 10b. It is important to note that *X*(*s*) can be derived from the distributions of the deposits *d*_{1}(*s*) and *d*_{2}(*s*) (Eqn. 10a), and it provides the relative probability of any particular sized grain being eroded from *d*_{1}, transported, and deposited at *d*_{2}.

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., *t*_{2}[*s*_{i}] < *t*_{1}[*s*_{i}]) and both are low-energy functions, then *X*(*s*) is always a negatively skewed distribution (Figure A4). This will result in *d*_{2}(*s*) becoming finer, better sorted, and more negatively skewed than *d*_{1}(*s*). Therefore, given 2 sediments (*d*_{1} and *d*_{2}), where *d*_{2}(*s*) is finer, better sorted, and more negatively skewed than *d*_{1}(*s*), it might be possible to infer that the direction of sediment transport is from *d*_{1} to *d*_{2} (Table A1).

Case | Relative change in grain size distribution between deposit d_{2} and deposit d_{1} | Interpretation |
---|---|---|

A | Coarser | d_{2} is a lag of d_{1}. No direction of transport can be determined. |

Better sorted | ||

More positively skewed | ||

B | Finer | 1) The direction of transport can be from d_{1} to d_{2}. |

Better sorted | 2) The energy regime is decreasing in the direction of transport. | |

More negatively skewed | 3) t_{1} and t_{2} are low-energy transfer functions. | |

C | Coarser | 1) The direction of transport can be from d_{1} to d_{2}. |

Better sorted | 2) The energy regime is decreasing in the direction of transport. | |

More positively skewed | 3) t_{1} is a high-energy transfer function, and t_{2} is a high- or low-energy transfer function. |

In the event that *t*_{1}(*s*) is a high-energy function and *t*_{2}(*s*_{i}) > *t*_{1}(*s*_{i}) (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, *d*_{2}(*s*) will become coarser, better sorted, and more positively skewed than *d*_{1}(*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 *d*_{1} to *d*_{2} (Table A1).

Sediment coarsening along a transport path will be limited by the ability of *t*_{1}(*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 *d*_{1} and *d*_{2} (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 (*d*_{1} and *d*_{2}) 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.

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, (*n*^{2} - *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:

*H*_{0}: *p* > 0.125 with no preferred direction

*H*_{1}: *p* < 0.125 and transport occurs in the preferred direction.

With the Z score statistic in a 1-tailed test (Spiegel 1961), *H*_{1} is accepted if

(12)

where *x* is the observed number of pairs representing a particular case in 1 of the 2 opposing directions and *N* is the total number of possible unidirectional pairs given by (*n*^{2} - *n*)/2. The number of samples in the sequence is *n*; *p* = 0.125; and *q* = 1.0 - *p* = 0.875.

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 *R*^{2}, defined as

(13)

The value of *R*^{2} can range from 0 to 1. The definition of *R*^{2} is based on the use of a model to relate a dependent parameter *y* to 1 or more independent parameters (*x*_{1}, *x*_{2}, …). In this case, the model used is linear, which can be written as

(14)

The data (*y, x*_{1}, *x*_{2}) are grain size distribution statistics, and the parameters (*a*_{0}, *a*_{1}, *a*_{2}) are estimated from the data with a least squares criterion. The dependent parameter is defined as the skewness, and the independent parameters are the mean size and the sorting. An implicit assumption is made that distributions taken from samples along a transport pathway, if plotted in skewness/sorting/mean space (as in Figure A5), would tend to be clustered along a straight line. The slopes of the straight line, which are the fitted parameters, would depend on the type of transport (fining or coarsening). Although there is no theoretical reason to expect a linear relationship among the 3 descriptors, there is also no theory predicting any other kind of relationship, so according to the principle of Occam's razor, the simplest available relationship was chosen for the model. (Occam's razor: Entities ought not to be multiplied except from necessity. Occam was a 14th Century philosopher who died in 1349.) High values of *R*^{2} (≥0.8) together with a significantly high value of the *Z* score provide confidence in the validity of the transport line.

^{a}Case B trend.^{b}Case C trend.
| ||||||||

1^{a} | 2 | 3 | 4 | 5 | 6 | 7^{b} | 8 | |

Mean | F | C | F | F | C | F | C | C |

Sorting | B | B | P | B | P | P | B | P |

Skewness | − | − | − | + | + | + | + | − |

A low *R*^{2} can occur, even when the *Z* score statistic is acceptable. On the basis of the empirical evaluation of many sediment trend analyses from many different environments, it appears that low *R*^{2} values can result when 1) sediments on an assumed transport path are, in reality, from different facies and valid trend statistics occurred accidentally; 2) the sediments are from a single facies, but the chosen sequence is only a poor approximation of the actual transport path; and 3) extraneous sediments have been introduced into the natural transport regime, as in the case of dredged material disposal. *R*^{2}, therefore, is assessed qualitatively and can provide extra useful information on the sediment transport regime under study.

The *Z* score and *R*^{2} statistics for each of the sample sequences (Figure 4) used to determine the sediment transport patterns in the Hylebos Waterway are provided in Appendix 2.

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 skewness). The basic underlying assumption is that sequential deposits following the pathway of net sediment transport will affect the statistics of the particle size distributions of the sediments in a predictable way. Following from this assumption, the size frequency distributions of the sediments provide the data with which to search for patterns of net sediment transport.

*Assumptions in the transport model*—Whatever method is used to describe sediments, STA requires a model of the sediment transport process. The STA model is based on the assumption that smaller grains are generally more easily transported than larger grains (i.e., the probability of transport, on a phi scale, monotonically increases as grain size decreases). Under this assumption, it can be shown that erosion and deposition of sediments will change the moments of their particle size distributions in a predictable way in the direction of transport. However, as seen in transfer functions obtained from sediment data in flume experiments, this assumption might not always be strictly true. More often, the transfer function monotonically increases over only a portion of the available grain sizes before returning to 0. Furthermore, contained within the assumption is a further hidden assumption that the probability of transport of 1 particular grain size must therefore be independent of the transport of other grain sizes. Factors such as shielding, whereby the presence of larger grains can impede the transport of smaller grains; an increase in the cohesion of the finer grains; or a decrease in the ability of the eroding process to carry additional fines with increasing load all suggest that the transport process is a complicated function related to the sediment distribution and the strength of the erosion process.

Thus, the mathematics of the theory demand the somewhat unsatisfactory assertion that the probability of transport must increase monotonically over a sufficiently large range of sizes present in the deposits to produce the predicted changes. As Gao and Collins (1994) pointed out, the technique to determine net transport pathways in a wide variety of different marine and coastal environments has been empirically validated through the use of alternative approaches, indicating that such an assumption cannot be too unreasonable.

*Temporal fluctuation*—The particle size distribution of a particular facies can be the result of sediment arriving from several different directions and at different times. It is assumed that what is sampled is the average of all the sediment derived from an unknown number of directions. The average transport direction might not conform to that developed for a specific particle population associated with a single transport pathway.

In STA, it is assumed that a sample provides a representation of a specific sediment type (or facies) with no direct time connotation. Nor does the depth to which the sample was taken contain any significance provided that the sample does, in fact, accurately represent the facies.

Consider, for example, a beach face composed of many lamina. Each lamina might represent a particular transport and depositional event that, at a small scale, might be locally different from that of the beach transport regime as a whole. The latter can be determined by sampling the beach face in such a way that a sufficient number of lamina are incorporated into the sample to allow the assumption that the sample now represents an average of the beach face facies. The average distribution of all the lamina making up the beach face can now be compared with a similar sample taken elsewhere on the beach face. To provide another example, *d*_{1} might be a sample representing an accumulation over several tidal cycles, whereas *d*_{2} represents several years of deposition. The trend analysis simply determines whether a possible sediment transport relationship or pathway between the 2 deposits exists.

*Sample spacing*—The sampling interval might be too great (frequency too low) to detect relevant transport directions. With increasing distance between sample locations comes an increasing possibility of collecting sediments unrelated by transport. Communications theory (discussed in further detail in the *Communications analogy* section) indicates that to represent accurately a continuous signal with samples, the signal must be sampled at twice the highest frequency contained in the signal (Shannon 1948). This would imply that for STA, sample sites placed *x* km apart could only reliably detect transport directions occurring over a distance in the order of 2*x* km or more. Directions occurring over distances less than 2*x* km would appear as noise or could create spurious transport pathways through the process of aliasing.

In practice, selection of a suitable sample spacing must take into account 1) the number of sedimentological environments likely to affect the area under specific study, 2) the desired spatial scale of the sediment trends, and 3) the geographic shape and extent of the study area.

*Random environmental and measurement uncertainties*—All samples will be affected by random errors. These can include unpredictable fluctuations in the depositional environment, the effects of sampling and subsampling a representative sediment population, and random measurement errors.

Sediment trend analysis is, in many ways, analogous to communications systems. In the latter, information is transmitted to a distant location, where a signal is received that includes both the desired information as well as noise. The receiver must be capable of extracting the information from the noisy signal. In sedimentary systems, the information is the transport direction and the received signal is the sediment samples. The goal of STA is to extract the information from the noisy signal. In theory, the information can be recovered by simply subtracting the noise from the signal, an approach that works well in communications systems because the nature of the information and the noise are both well known. This approach, however, will be difficult in STA because neither the nature of the information nor the noise is well understood.

A large body of analytical techniques has been developed to extract signals in communications systems. These techniques generally fall into 2 categories: Signal coding and noise reduction. For example, in FM radio transmission, the signal is coded as a time-varying frequency about a carrier frequency. At the receiver, rejecting all frequencies other than the carrier frequency reduces noise. The receiver then looks for the time-varying frequency component to extract the original signal. Reducing the noise increases the level of the signal to noise ratio, enabling the signal to be detected. Coding the signal simply makes it easier to find because of prior knowledge of the information. It is important to note that knowing what is being looked for is of critical importance in communications systems: merely analyzing the incoming signal would not be sufficient to interpret the signal correctly.

In STA, with no opportunity to code the signal, other aspects of communications theory (noise reduction) might have applications pertinent to the technique. Typically, noise in communications systems is reduced with the use of filters that selectively reduce the signal level for frequencies outside the frequency range of the signal. If these frequency components contain parts of the signal, the filter, too, will reduce them. Knowing the nature of the noise and the signal, filters can be designed that optimally increase the signal to noise ratio.

The situation in STA is not as straightforward because the noise is not well understood and the nature of the signal is only incompletely understood. In this situation, noise reduction by filtering can be problematic because the filtering can remove significant signal components; however, some statistical communications techniques might be applicable to improve the situation.

In a sedimentary system, noise can be considered in 2 areas, sample noise and spatial noise.

*Sample noise*—Even in a uniform sediment deposit, individual samples can be corrupted by noise. One way to address this noise would be to take many samples in close proximity and average them to produce a characteristic sample. Another method that implicitly attempts to reduce this noise is curve fitting. For example, there has been considerable research on the use of a log-hyperbolic curve to describe sediments because it appears to provide a good fit to many naturally occurring deposits (Barndorff-Nielsen 1977; Bagnold and Barndorff-Nielsen 1980). Similar to the concepts of STA, it has been shown that parameters of the log-hyperbolic distribution should change in deterministic ways under the influence of erosion or deposition (Barndorff-Nielsen and Christiansen, 1988). It was proposed that erosion and deposition cause the location shape–invariant parameters of the log-hyperbolic distribution to vary in particular ways when plotted on the shape triangle of the log-hyperbolic distribution (Barndorff-Nielsen et al. 1991; Hartmann and Christiansen 1992). Not all researchers, however, are convinced that log-hyperbolic distributions provide superior information (e.g., Wyrwoll and Smyth 1988; Hill and McLaren 2001),

Curve fitting analysis, whether log-normal or log-hyperbolic, is based on the assumption that sediments follow specific distributions. By fitting a curve to the sedimentary data, it is assumed that points that do not fall on the curve are noise and are removed. In theory, this works if in fact sediments do conform to the proposed curve. If they do not, then the curve-fitting process removes signal as well as noise; accordingly, a fitted curve could be more noisy than the original sample. In the present line-by-line approach of STA, the pitfalls of curve fitting are avoided because only the raw data of each sediment grain size distribution are used from which the log moments are calculated.

*Spatial noise*—As sediment is transported over a distance, noise can be introduced. To reduce this noise, average values of groups of samples could be used. Many of the techniques proposed by researchers are, in reality, efforts to reduce noise in this manner (i.e., the 1-dimensional *Z* score, as described above, or the vector approaches of Gao [1996] and Le Roux [1994]). These procedures generally involve some form of averaging of samples, which is not strictly valid. If the nature of the noise and the information is not known, the averaging of samples can reduce the information content more than it reduces the noise levels. (An exception is to reduce random noise by averaging a number of samples from the same local environment to generate a better single distribution representative of that environment.)

In STA, the assumption is that noise is randomly distributed and therefore averages to 0, leaving the true trend as the residual after averaging. Although these techniques might in fact reduce noise, signal-processing techniques might provide more refined and controllable methods.

In communications theory, it is often convenient to transform the signal from the time domain (i.e., a signal that varies over time) to the frequency domain, which shows the frequency spectrum of the signal (i.e., the amount of the signal that is carried by all of the individual frequency components). Mathematically, this is performed with a Fourier transform, which converts the signal into its frequency components. After removing the undesirable (noise) components, an inverse transform is performed to transform the signal back to the time domain. In sediment analysis, the signal varies across distance rather than time, but exactly the same analysis can be performed. In this case, the data (the grain size distributions of the sediment samples) can be represented as a sum of distance-varying sinusoids with a 2-dimensional Fourier transform. What the transform produces is a characterization of the sedimentary deposits that shows how they vary over different distance scales. For example, 1 component would indicate the intensity of changes over a 100-m range, another over a 1-km range, etc. (Note that the sample spacing, as discussed above, will set limits as to what distance ranges can be considered.) Having the signal in this form allows the unwanted components to be removed. However, how is it known what is undesirable? In communications systems, the information is known (if it was not, it would be difficult, if not impossible, to find anything). By analogy, in performing a simple analysis of the sedimentary data (e.g., mapping the variation in the mean grain size) it is highly unlikely that a transport direction would be discovered. To extract the relevant signal, it is necessary to make an assumption as to what is being looked for. It is then possible to filter the data to highlight this and detect whether in fact a signal corresponding to the assumption is actually present. For example, assume a transport process that would produce the fining of sediments over a 5-km distance. To extract this process, a 2-dimensional Fourier transform can be calculated, and all frequency components associated with variations of less than 5 km could be removed. An inverse transformation of the data would then highlight variations over the proposed distance scale.

The important feature of this approach (which, in fact, approximates the line-by-line approach discussed previously) is the use of many sample sites to detect the dominant transport direction. This effectively reduces the level of noise. The problem, however, is that it is difficult to mechanize because the number of possible transport directions in a given area can be much too large to try them all. The choosing of a trial transport direction cannot be easily analytically codified and can only be reduced to a manageable level through experience and information from other sources (e.g., bathymetric data).

In using the *Z* score statistic, however, a transport trend can be determined whereby all possible pairs in a sample sequence are compared with each other. When either a case B or case C trend exceeds random probability within the chosen sample sequence, the direction of net sediment transport can be inferred. As suggested before, the grid spacing must be compatible with the area under study and take into account the number of sedimentological environments likely to be involved, the geographic shape of the study area, and the desired statistical certainty of the pathways. For practical purposes, it has been found that, for regional studies in open ocean environments, sample spacing should not exceed 1 km; in estuaries, spacing can be reduced to 500 m. For site-specific studies (e.g., to determine the transport regime for a single marina), sample spacing will be reduced so that a minimum number of samples can be taken to ensure adequate coverage. Experience has also shown that extra samples should be taken over sites of specific interest (e.g., dredged material disposal sites) and those areas in which the regular grid is insufficient to accommodate specific bathymetric features (e.g., bars and channels).

At present, the line-by-line approach is undertaken as follows:

- 1)Assume the direction of transport over an area comprising many sample sites;
- 2)From this assumption, predict the sediment trends that should appear at the sample sites;
- 3)Compare the prediction with the
*Z*score statistic obtained from the grain size distributions of the samples; and - 4)Modify the assumed direction and repeat the comparison until the best fit is achieved.

Following from the communications analogy, when a final and coherent pattern of transport pathways is obtained that encompasses all, or nearly all, of the samples, the assumption that information (the transport pathways) is contained in the signal (the grain size distributions) has been verified, despite the inability to define accurately all the uncertainties that might be present.

It must be emphasized that the actual processes responsible for the transport of particles along the derived pathways are unknown. They might in one environment be breaking waves in a littoral drift system; in another, the residual tidal currents; and in still another, the incorporated effects of bioturbation. Nevertheless, one of the great values in obtaining the transport patterns is to assess the probable processes that are likely taking place to achieve such patterns.

The shape of the *X* distribution is important in defining the type of transport (dynamic behavior of the bottom sediments) occurring along a line (erosion, accretion, total deposition, etc.); thus, the computation of *X* is important. Consider a transport line containing *N* source/deposit (*d*_{1}/d_{2}) pairs. *X* is then defined as in Equation 14.

(15)

Often, *d*_{2} in one pair is *d*_{1} in another pair, and vice versa. Mean values of *d*_{2} and *d*_{1} are computed with Equation 15.

(16)

Note that *X* is not defined as the quotient of the mean value of *d*_{2} divided by the mean value of *d*_{1}, even though the results of the 2 computations are often almost identical. For ease of comparison, *d*_{1}, *d*_{2}, and *X* are normalized before plotting in reports, although there is no reason to expect that the integral of the *X* distribution should be unity. *X*(*s*) can be thought of as a function that describes the relative probability of each particle being removed from *d*_{1} and deposited at *d*_{2}.

Examination of *X* distributions from a large number of different environments has shown that 5 basic shapes are most common when compared with the distributions of the deposits *d*_{1}(*s*) and *d*_{2}(*s*) (Figure A6).

*1. Dynamic Equilibrium*—The shape of the *X* distributions closely resembles *d*_{1}(*s*) and *d*_{2}(*s*). The relative probability of grains being transported, therefore, is a similar distribution to the actual deposits. Thus, the probability of finding a particular sized grain in the deposit is equal to the probability of its transport and redeposition (i.e., there must be a grain-by-grain replacement along the transport path). The bed is neither accreting nor eroding and is, therefore, in dynamic equilibrium.

An *X* distribution signifying dynamic equilibrium can be found in either case B or case C transport, suggesting a fine balance between erosion and accretion. Often when such environments are determined, both case B and case C trends can be significant along the selected sample sequence. This is referred to as a “mixed case,” and when this occurs, it is believed that the transport regime is also approaching a state of dynamic equilibrium.

*2. Net Accretion*—The shapes of the 3 distributions are similar, but the mode of *X* is finer than the modes of *d*_{1}(*s*) and *d*_{2}(*s*). The mode of *X* can be thought of as the size that is the most easily transported. Because the modes of the deposits are coarser than *X*, these sizes are more readily deposited than transported. The bed, therefore, must be in a state of net accretion. Net accretion can only be seen in case B transport.

*3. Net Erosion*—Again the shapes of the 3 distributions are similar, but the mode of *X* is coarser than of *d*_{1}(*s*) and *d*_{2}(*s*). This is the reverse of net accretion; the size most easily transported is coarser than the deposits. As a result, the deposits are undergoing erosion along the transport path. Net erosion can only be seen in case C transport.

*4. Total Deposition (type 1)*—Regardless of the shapes of *d*_{1}(*s*) and *d*_{2}(*s*), the *X* distribution more or less increases monotonically over the complete size range of the deposits. Sediment must fine in the direction of transport (case B); however, the bed is no longer mobile. Rather, it is accreting under a rain of sediment that fines with distance from source. Once deposited, transport ceases. The occurrence of total deposition is usually confined to cohesive, muddy sediments.

*5. Total Deposition (type 2, horizontal X distributions)*—Occurring only in fine sediments when the mean grain size is a very fine silt or clay, the *X* distribution can be essentially horizontal. Such sediments are usually found far from their source, and the horizontal nature of the *X* distribution suggests that their deposition is no longer related strictly to size sorting. In other words, all sizes now have an equal probability of being deposited. This form of the *X* distribution was 1st observed in the muddy deposits of a British Columbia fjord and is described in McLaren et al. (1993). Because the trends occur in very fine sediments, in which any changes in the distributions are extremely small, horizontal *X* distributions can be found in both case B and case C trends.

Definitions:

- 1)
*R*^{2}is the multiple correlation coefficient derived from the mean, sorting, and skewness of each sample pair making up a significant trend. This is a relative indication of how well the samples are related by transport. - 2)Case B: Sediments becoming finer, better sorted, and more negatively skewed in the direction of transport.
- 3)Case C: Sediments becoming coarser, better sorted, and more positively skewed in the direction of transport.
- 4)
*N*is the number of possible pairs in the line of samples. - 5)
*X*is the number of pairs making a particular trend in a specific direction. - 6)
*Z*is the*Z*score statistic. Only trends at the 99% level are accepted. - 7)“Down” indicates transport in the down-line direction. “Up” indicates transport in the up-line direction.
- 8)Status defines the dynamic behavior of the sediments making up the line of samples (i.e., Net Erosion, Net Accretion, Dynamic Equilibrium, etc.) See Appendix 1 for a complete explanation.

Line Nr^{a} | Case | Direction | R^{2} | N | X | Z^{b} | Interpretation of dynamic behavior |
---|---|---|---|---|---|---|---|

^{a}See Figure 4.^{b}Trends are significant at the^{*}95% and^{**}99% level.
| |||||||

1 | B | Down | 1 | 6 | 4 | 4.01^{**} | Total Deposition (type 1) |

Up | 6 | 1 | 0.31 | ||||

C | Down | 6 | 0 | −0.93 | |||

Up | 6 | 1 | 0.31 | ||||

2 | B | Down | 1 | 15 | 11 | 7.12^{**} | Total Deposition (type 1) |

Up | 15 | 3 | 0.88 | ||||

C | Down | 15 | 0 | −1.46 | |||

Up | 15 | 1 | −0.68 | ||||

3 | B | Down | 0.99 | 15 | 8 | 4.78^{**} | Total Deposition (type 1) |

Up | 15 | 3 | 0.88 | ||||

C | Down | 15 | 0 | −1.46 | |||

Up | 15 | 2 | 0.1 | ||||

4 | B | Down | 0.99 | 21 | 13 | 6.85^{**} | Total Deposition (type 1) |

Up | 21 | 2 | −0.41 | ||||

C | Down | 21 | 3 | 0.25 | |||

Up | 21 | 0 | −1.73 | ||||

5 | B | Down | 0.98 | 36 | 22 | 8.82^{**} | Total Deposition (type 1) (see Figure 7) |

Up | 36 | 7 | 1.26 | ||||

C | Down | 36 | 3 | −0.76 | |||

Up | 36 | 1 | −1.76 | ||||

6 | B | Down | 0.3 | 15 | 2 | 0.1 | Net Erosion |

Up | 15 | 1 | −0.68 | ||||

C | Down | 15 | 5 | 2.44^{**} | |||

Up | 15 | 3 | 0.88 | ||||

7 | B | Down | 0.95 | 21 | 0 | −1.73 | Net Erosion (see Figure 6) |

Up | 21 | 4 | 0.91 | ||||

C | Down | 21 | 11 | 5.53^{**} | |||

Up | 21 | 3 | 0.25 | ||||

8 | B | Down | 0.94 | 21 | 0 | −1.73 | Net Erosion |

Up | 21 | 3 | 0.25 | ||||

C | Down | 21 | 8 | 3.55^{**} | |||

Up | 21 | 6 | 2.23^{*} | ||||

9 | B | Down | 0.7 | 15 | 7 | 4.00^{**} | Net Accretion |

Up | 15 | 0 | −1.46 | ||||

C | Down | 15 | 4 | 1.66^{*} | |||

Up | 15 | 0 | −1.46 | ||||

10 | B | Down | 1 | 3 | 2 | 2.84^{**} | Net Accretion |

Up | 3 | 0 | −0.65 | ||||

C | Down | 3 | 0 | −0.65 | |||

Up | 3 | 0 | −0.65 | ||||

11 | B | Down | 0.95 | 45 | 31 | 11.44^{**} | Dynamic Equilibrium |

Up | 45 | 5 | −0.28 | ||||

C | Down | 45 | 2 | −1.63 | |||

Up | 45 | 4 | −0.73 | ||||

12 | B | Down | 36 | 9 | 2.27^{*} | Dynamic Equilibrium | |

Up | 36 | 3 | −0.76 | ||||

C | Down | 0.89 | 36 | 16 | 5.80^{**} | ||

Up | 36 | 3 | −0.76 | ||||

13 | B | Down | 55 | 8 | 0.46 | Net Erosion | |

Up | 55 | 5 | −0.76 | ||||

C | Down | 0.93 | 55 | 33 | 10.65^{**} | ||

Up | 55 | 3 | −1.58 | ||||

14 | B | Down | 0.95 | 91 | 24 | 4.00^{**} | Mixed case |

Up | 91 | 8 | −1.07 | ||||

C | Down | 0.94 | 91 | 33 | 6.85^{**} | ||

Up | 91 | 14 | 0.83 | ||||

15 | B | Down | 0.93 | 120 | 34 | 5.24^{**} | Mixed case |

Up | 120 | 23 | 2.21^{*} | ||||

C | Down | 0.97 | 120 | 42 | 7.45^{**} | ||

Up | 120 | 10 | −1.38 | ||||

16 | B | Down | 0.98 | 66 | 38 | 11.07^{**} | Dynamic Equilibrium |

Up | 66 | 8 | −0.09 | ||||

C | Down | 66 | 1 | −2.7 | |||

Up | 66 | 4 | −1.58 | ||||

17 | B | Down | 0.98 | 66 | 32 | 8.84^{**} | Dynamic Equilibrium (see Figure 8) |

Up | 66 | 10 | 0.65 | ||||

C | Down | 66 | 2 | −2.33 | |||

Up | 66 | 6 | −0.84 | ||||

18 | B | Down | 0.98 | 55 | 32 | 10.24^{**} | Total Deposition (type 1) |

Up | 55 | 9 | 0.87 | ||||

C | Down | 55 | 6 | −0.36 | |||

Up | 55 | 5 | −0.76 | ||||

19 | B | Down | 0.97 | 28 | 12 | 4.86^{**} | Dynamic Equilibrium |

Up | 28 | 2 | −0.86 | ||||

C | Down | 28 | 7 | 2.00^{*} | |||

Up | 28 | 4 | 0.29 | ||||

20 | B | Down | 0.99 | 28 | 4 | 0.29 | Net Erosion |

Up | 28 | 5 | 0.86 | ||||

C | Down | 28 | 13 | 5.43^{**} | |||

Up | 28 | 3 | −0.29 | ||||

21 | B | Down | 1 | 36 | 6 | 0.76 | Net Erosion |

Up | 36 | 3 | −0.76 | ||||

C | Down | 36 | 19 | 7.31^{**} | |||

Up | 36 | 4 | −0.25 | ||||

22 | B | Down | 0.91 | 28 | 13 | 5.43^{**} | Total Deposition (type 1) |

Up | 28 | 5 | 0.86 | ||||

C | Down | 28 | 2 | −0.86 | |||

Up | 28 | 2 | −0.86 | ||||

23 | B | Down | 0.88 | 28 | 18 | 8.29^{**} | Total Deposition (type 1) |

Up | 28 | 2 | −0.86 | ||||

C | Down | 28 | 0 | −2 | |||

Up | 28 | 2 | −0.86 | ||||

24 | B | Down | 0.85 | 36 | 19 | 7.31^{**} | Total Deposition (type 1) |

Up | 36 | 2 | −1.26 | ||||

C | Down | 36 | 0 | −2.27 | |||

Up | 36 | 2 | −1.26 | ||||

25 | B | Down | 0.98 | 66 | 34 | 9.58^{**} | Total Deposition (type 1) (see Figure 9) |

Up | 66 | 2 | −2.33 | ||||

C | Down | 66 | 0 | −3.07 | |||

Up | 66 | 11 | 1.02 | ||||

26 | B | Down | 0.99 | 55 | 34 | 12.69^{**} | Total Deposition (type 1) |

Up | 55 | 4 | −1.17 | ||||

C | Down | 55 | 6 | −0.36 | |||

Up | 55 | 2 | −1.99 | ||||

27 | B | Down | 0.88 | 55 | 20 | 5.35^{**} | Total Deposition (type 1) |

Up | 55 | 9 | 0.87 | ||||

C | Down | 55 | 8 | 0.46 | |||

Up | 55 | 4 | −1.17 | ||||

28 | B | Down | 0.71 | 78 | 23 | 4.54^{**} | Total Deposition (type 2) |

Up | 78 | 14 | 1.46 | ||||

C | Down | 78 | 5 | −1.63 | |||

Up | 78 | 1 | −3 | ||||

29 | B | Down | 0.74 | 91 | 38 | 8.44^{**} | Total Deposition (type 2) (see Figure 10) |

Up | 91 | 7 | −1.39 | ||||

C | Down | 91 | 3 | −2.65 | |||

Up | 91 | 6 | −1.7 | ||||

30 | B | Down | 0.7 | 78 | 39 | 10.01^{**} | Total Deposition (type 2) |

Up | 78 | 6 | −1.28 | ||||

C | Down | 78 | 3 | −2.31 | |||

Up | 78 | 2 | −2.65 | ||||

31 | B | Down | 0.75 | 120 | 53 | 10.49^{**} | Total Deposition (type 2) |

Up | 120 | 9 | −1.66 | ||||

C | Down | 120 | 7 | −2.21 | |||

Up | 120 | 12 | −0.83 | ||||

32 | B | Down | 0.97 | 45 | 19 | 6.03^{**} | Total Deposition (type 2) |

Up | 45 | 8 | 1.07 | ||||

C | Down | 45 | 7 | 0.62 | |||

Up | 45 | 5 | −0.28 | ||||

33 | B | Down | 0.93 | 210 | 65 | 8.09^{**} | Total Deposition (type 2) |

Up | 210 | 35 | 1.83^{*} | ||||

C | Down | 210 | 19 | −1.51 | |||

Up | 210 | 20 | −1.3 | ||||

34 | B | Down | 0.93 | 66 | 29 | 7.72^{**} | Total Deposition (type 2) |

Up | 66 | 14 | 2.14^{*} | ||||

C | Down | 66 | 10 | 0.65 | |||

Up | 66 | 2 | −2.33 | ||||

35 | B | Down | 1 | 6 | 0 | −0.93 | Dynamic Equilibrium |

Up | 6 | 0 | −0.93 | ||||

C | Down | 6 | 3 | 2.78^{**} | |||

Up | 6 | 2 | 1.54 |

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