Bedload Sediment Rate Prediction for the Sand Transport Along Coastal Waters in Ocean Management Strategy

Interactions among different landforms and varied complicated physical processes cause sediment transport in coastal regions being the interest of ocean management planning studies. In coastal zones, the derivation of the bedload sediment transport rate and the flow velocity distribution is done by entropy theory which assumes the modified spatiotemporal disorder power index (MSTDPI) and the time-averaged flow velocity (as a random variable). Studying the deposition trend of bedload sediment transport rate for the sand particle (BLSTRS) and estimating the coastal erosion rate as a case study, the Makran coast is selected. To analyze the spatiotemporal patterns, the disorder power index (entropy-power) method is applied in this study where the monthly data of six Makran coastal sections from January 1970 to December 2015 are used. The studied data are mainly focused on the correlation of the flow rate of the sediment from the Makran River, and the spatiotemporal patterns of BLSTRS. Despite their meaningful spatiotemporal variability, it is not very easy to explain how the abovementioned variables perform together; the entropy-power index allows a better understanding of the combined performance of such parameters as the flow velocity and sediment transport by showing clearer signals for the assessment of coastal engineering issues at very large (coastal) scales.


Introduction
Although land erosions within the continents are having an increasing trend throughout the world during recent decades, unfortunately, the ocean sediment input effects along the coastal regions are not studied at the same importance level. Noting that, water quality, deltas, and coastal zones, morpho-dynamics of estuaries, navigation/harbor capability, and coastal recreational areas are highly influenced by the sediment transport/distribution at the land-ocean interface. Modeling, laboratory tests, field surveys, and space observations can help in achieving scientific objectives, deepening the knowledge on these processes, refining the sediment budget (bedload/suspension) between compartments, and improving our observations/modeling capacities. Chemical/biological processes can influence sediment transport which is essentially enhanced by such phenomena as stratification, wind, currents, turbulence, tides, and so on in canyons, estuaries, or on the shelfs.
Classically, the sediment transport is conceived as occurring in two principal modes: bedload transport and suspended transport. Fig. 1 presents an introductory overview of the sediment transport modes on a beach. Bedload transport is characterized by particles that move by rolling, sliding and/or jumping (saltating) along the bottom.
Since previous studies have not sufficiently addressed the hydraulic characteristics of coastal regions, the present research aims at developing a general, reliable, robust relationship based on many data sets (including steady/unsteady flow) that can be used to calculate the bedload transport for different coastal conditions (marine, river, etc.) where the entropy theory is introduced as a novelty as a simple, efficient and applicable method in numerous engineering applications fields like immediate flood predictions and ocean management plans.

Modified spatiotemporal disorder power index
As a novelty, the authors in this paper have developed an entropy-based modified spatiotemporal disorder power index (MSTDPI) similar to that favorably used by Djebou et al. (2014) to analyze the wave height variability. The main difference between MSTDPI and disorder index (DI) is that they are based on joint-entropy and marginal entropy re-Υ i j spectively implying that the former applies not only to the marginal pattern but also to interactive ones as well whereas the latter applies only to the marginal pattern. If the coastline zone variable varies in space s and time t (based on the probability matrix ), MSTDPI will be: is a 2D actual joint-entropy (space-time) and is the maximum joint-entropy (for a uniform distribution).

∈
In this study, base 2 logarithm has been used (as in other papers), t is time in month [t (Jan., Feb.... Dec.)], and MSTDPI is in binary units (bits) (but should be relativized). In fact, the range of the spatiotemporal variations for a known variable at a specified point in comparison with other points (in a coastline) is shown by MSTDPI.

Bedload sediment transport rate on coastal zone
Generally, if the transport rate of the bedload sediment is added to that of the net suspended sediment, the result will be the net transport rate of the coastal sediment; it is the sediment transport (occurring as bedload or suspended load) that enables the realization of the link between the beach hydrodynamic forcing and the morphological response. Although the suspended load is believed to be more critical in modeling (in the surf area), especially when the energy is considerable, consensus does not exist on the issue because bedload measurement under field conditions is not an easy task. But the purpose of this study, with respect to its title of bedload sediment rate prediction for the sand transport along coastal waters in ocean management planning, is to focus on calculating the bedload sediment rate. It is noteworthy that the process of calculating bedload and suspended load by using entropy theory is entirely different.
Therefore, it is assumed that uniformly vertical beaches (to elongated coastlines) are subject to wave motions over a constant level of water. There are two terms in the expression of modeling sediment rates using the entropy theory; one expresses the transport rate in the catchment area and the other shows that in the coastal zone. h If the general and coastal water depths ( and , respectively) are sampled at individual points, their distributions are found using and (a probability density function for catchment basin and coastal area). Then the entropy can be written as: (2) 2.1.1 An equation for generation of entropy in river section h R In this study, Eq. (3) gives the dimensionless (Shannon entropy random variable) for a river section (Shannon and Weaver, 1949;Shannon, 1948): is the fixed maximum depth, and is the fixed minimum depth at boundaries.

Υ(η) Υ(η)
In the sector of sediment transport in the coastal areas, a non-uniform probability distribution "Throwing" is used. The following function for : Most experimental data have random errors that follow this "normal" probability density distribution, so it is a required distribution in modeling sedimentation problems. The average value of x is , with a standard deviation about this value . Note, that here, x ranges from to . To "throw" a Gaussian probability distribution in x, one would have to solve the following integral: For A(x), there is no analytic solution to this integral. However, there is an excellent "trick" to produce a Gaussian probability. If one throws a two-dimensional Gaussian probability distribution, the integral can be solved analytically.
This algorithm is derived by tossing a two-dimensional Gaussian probability in the plane. If we "toss" a random point in the plane that has a Gaussian probability density in and y about the origin with standard deviation for both and as , the probability density is Υ(η, ε) The expression is an area density, with units of 1/m 2 . The complete expression for probability is  (Nielsen, 1991).
Saeed KHORRAM China Ocean Eng., 2020, Vol. 34, No. 6, P. 840-852 841 Υ h C In this study, Gaussian probability distribution as a nonuniform probability distribution for the dimensionless random variable is given as for coastal area section: is the fixed maximum depth, and is the fixed minimum depth at boundaries. If Eqs. (7) and (8) are substituted into the entropy function will be: Since water depths may be positive or negative (below or above the mean water level, respectively), if and are the equilibrium bedload rates varying uniformly from a maximum at the offshore edge to a minimum depth (at the shoreline) along , Eq.
(2) can show the uncertainty in functions and .
To determine them, one can use ME (maximum entropy) proposed by Jaynes (1957aJaynes ( , 1957bJaynes ( , 1982 with some constraints on the flow velocity dip position. The MER (maximum entropy requires) that entropy functions and should be maximized subject to some specific constraints to obtain the minimum biased probability of the random variables; an appropriate definition of the constraints will satisfy the total probability theory for probability density functions and . The first constraint is: 10) and the other is the mean of ξd as: and are the coastal and river profile depths and their mean value is given by Eq. (11).
After neglecting the signs in the integration to find the minimum biased probability density function, the Lagrange multiplier is used to maximize Eq. (16) by ME subject to constraints in Eqs. (10) and (11): where, and are Lagrange multipliers (found from Eqs. (10) and (11)). The Euler-Lagrange equation is used in Eq. (12) to find that maximizes (13) f (ψ C ) A rearrangement of Eq. (13) will yield as follows for the flow velocity dip position that contains Lagrange multipliers: (14) f (ψ C ) The cumulative distribution function (CDF) is found as follows using Eq. (14): It is worth noting that if the depth probabilities are ≤ , will satisfy the following geometric equation: where varies from to and . The cumulative distribution is generated if is applied in Eq. (7). ψ It should be noted that since the samples' spatial positions are unknown, shows the discrete samples only randomly. If h R is related to the bedload rate, dependence on spatial coordinate has to be introduced. Since the equilibrium bedload rate (the water depth) varies uniformly along the -axis ranging from a minimum (at the shoreline ) to a maximum (at the offshore edge ), if Eq. (14) is substituted into Eq. (16), the SEF (Shannon entropy function) is found as follows: where and can be found as: As Eqs. (16) and (17) both show cumulative probability of , they should be identical meaning that: If Eq. (12) is substituted into Eq. (20) and the prime in is omitted for simplicity, the equilibrium bedload rate will be: By letting , Eq. (21) can be rewritten as: Only considering the coastal zone's underwater portion, having and , Eq. (22) is reduced to: (η, ε) θ It can be "toss" in Eq. (9) the point using the polar coordinates r and with the same probability density. The relationship between and (r, ) (in the polar coordinates) is as follows:

η=rcosθ, ε=rsinθ and dηdε=rdrdθ
Using these equations to transform into polar coordinates yields So if It will be "throw" r with the probability density (in r) of and with a non-uniform distribution probability density is uniform between 0 and 2π, then the point (r, ) on the x-y plane will have an x-coordinate which has a Gaussian probability distribution. The y-coordinate also has a Gaussian probability distribution as well. However, it cannot be used both the x and y points in the same simulation. One does not obtain two independent Gaussian probability distributions from the "r-toss".
To obtain r, first, throw r 1 with a uniform probability between zero and 1. Then r is found by solving If r 2 is placed between 0 and 1 (with uniform probability), .

An equation for generation of entropy in coastal area
If Gaussian probability distribution entropy function Eq. (2) is maximized by the maximum entropy subject to Eqs.
(3) and (8) (constraints), we will have: where, and are found from Eqs. (10) and (11). If the Euler-Lagrange equation is used in Eq. (12), (that maximizes L 0 ) will be found: Rearranging Eq. (12) will yield for flow velocitydip position containing Lagrange multipliers: will satisfy the following geometric relationship: where varies from to and . Putting in Eq. (14) will yield (the cumulative distribution).
It is to be noted that since samples' spatial positions are unknown, shows discrete samples only randomly. If is to be related to the bedload rate, dependence on spatial coordinate has to be introduced. Since the equilibrium bedload rate varies uniformly along the -axis ranging from a minimum (at the shoreline ) to a maximum (at the offshore edge ), if Eq. (14) is substituted into Eq. (9), the Shannon entropy function is found as follows: and are found as follows: Since Eqs. (16) and (17) both show cumulative probability of , they should be identical implying that: If Eq. (5) is substituted in Eq. (20) and the prime in is omitted for simplicity, the equilibrium bedload rate will be as follows: (20) can be rewritten as: Only considering the coastal zone's underwater portion, having and , Eq. (22) reduces to:

Coastal velocity profile
The coastal velocity profile (wave-induced longshore current) entropy probability density function (u) that assumes the entropy flow velocity profile occurs in the coastal profile is defined as: where is the water depth (depth of below water surface); is the vertical depth; M is the entropic parameter; is the maximum flow velocity along ith vertical; y is the distance of the point of flow velocity from the bed; N v is the number of verticals sampled across the bedload section; and is the position of the i-th sampled vertical from left coastal side. The value of M can be readily found by (u m , u max ) pair of the existing flow velocity dataset at the coastal zone through the entropy-based MSTDPI as follows: where u max is the maximum flow velocity in the coastal zone and u m is the mean flow velocity. u maxv can be found as a function of surface flow velocity for every vertical as follows even if it occurs below the water level: where . Specifically, if , it follows that and hence,

If
, it can be concluded that and hence, .
α Eq. (36) is similar to Eq. (37). The increase in the flow velocity is from the coastal zone ( ) to the water surface; it is uniform and has vertical distance.

Study area
To observe the effects of the coastline area in the study area (Makran zone), the authors have proposed the speed, wind, flow velocity and BLSTRS as three effective factors. This zone, which is a semi-desert coastal strip (in the Persian Gulf), consists of a unique mountain range formed during different geological processes. Minab fault in the west and Ornach Nal fault in the east are the borders of the zone which extends nearly 900 km from west (Iran) to east (Pakistan) (Figs. 2a and 2b).

Methodology applied
To determine the time-dependent variables, use has been made, in this study, of the Chi squared statistics (χ 2 ), likelihood ratio, and time homogeneity tests. When χ 2 of a known variable is found for its joint-distribution matrix, the first dimension is a representation of time t (in month) and the second is that of the variable's discrete classes. Accordingly, if it is more than the standard p-value (0.05 in our case), the result is time-homogeneity rejection (null hypothesis).
The time-homogeneity test results (2000−2013) found from the flow velocity data sets and the BLSTRS are given in Table 1. Table 2 and Fig. 3a show sediment transport rate and the stream-flow time series for every relative stream gauge; the authors have made use of z-scores (instead of actual values) and normalized the values to ensure comparable, realistic tests analyses because the variation ranges of BLSTRS are vast and heterogeneous.
In Fig. 4, the histograms for 12-hour (maximum) annual rainfall distribution and the probability density function (PDF) are shown for six stations. The frequency distributions are smooth for this duration data because the area is along the coastal region of the Persian Gulf. Since the coastal vicinity is an influential parameter for the climatic differences the Persian Gulf moderating moisture having a  high influence on this pattern. Fig. 3. The stream ordering and selection of the location of the stream gauges have been done on the basis of the Strahler-Horton's procedure and the values in parentheses show the BLSTRS time-series joint-entropy values based on the main outlet (Station 1). As shown, the closer a station to the outlet, the higher its joint-entropy which is especially true for second-order Stations where the spatial patterns are important. Accordingly, the authors made use of the k-mean method and grouped the spatial Stations in two clusters: one consisting of Stations 1, 2, 5, and 7 and one containing Stations 3, 4, and 6 ( Fig. 5). As shown, the k-mean results comply well with the joint-entropy because one cluster has the four larger values and the other contains the three smaller values; the same is the case with the structure of the stream network because one group contains the second-order stream stations while the other involves the first-order ones.

Flow velocity patterns
In this study, we formed a 154-cell grid (with monthly flow velocity and BLSTRS time series) by defining the coastline region inside its boundaries, used the cells' time series, and found the BLSTRS and flow velocity marginal patterns; the spatial patterns of the BLSTRS/flow velocity versus coefficients of variations are shown in Fig. 6.   Fig. 6b (BLSTRS vs. CV) shows averages that are seemingly incompatible and visual analyses cannot describe them easily; nonetheless, when a high correlation is achieved as shown in Table 2 resulted from analyses, their statistical similarity becomes clear. Fig. 7 shows that the flow velocity time-series monthly variation coefficients have consistent relationships (R 2 =0.89). Regardless of such relationships, coastline regions usually have heterogeneous flow velocity distributions which can cause confusion if CV-based spatial clusters are applied. This explains the reason why we have studied spatial clusters separately by applying k-mean to flow velocity and BLSTRS time series  and have determined sub-regions in 81 cells with satisfactory continuity of the two variables' variability (V 1 with Be 1 , V 2 with Be 2 , and V 3 with Be 3 (Fig. 6) are the three studied clusters). The flow velocity-BLSTRS relationship is clear in Fig. 6, but it is not in Fig. 5. In Fig. 7, there are similarity patterns (in flow velocity/BLSTRS time series) that strengthen the application of spatiotemporal analyses.
4.3 Analyzing joint patterns using entropy power and spatiotemporal disorder power indices The spatial clusters explained earlier were used to find the spatiotemporal probability distributions of the combined flow velocity and BLSTRS over the coastline region. Table 3 and Fig. 6 show respectively the entropy-power MSTDPI and DI estimates and comprehensive monthly matches of different scenarios (indexes are similar). DI and MSTDPI are different; the latter is found based on the jointentropy while the former is based on the marginal entropy, meaning that MSTDPI is related to both interactive and marginal patterns, but DI deals only with marginal patterns.
Actually, analyses of the MSTDPI variations show that the hydrological states throughout the coastal region are contrasting and the three variables are seemingly interdependent specifically at such a great scale. The MSTDPI curves (Fig. 8) show considerable likenesses meaning that the three variables reach their MSTDPI peaks in December. and there is another peak in June (for BLSTRS) and in August (for flow velocity). Unlike the flow velocity, this second peak is related to a decrease in the MSTDPI. Although the coastline components tend to interact (which is obvious from the signals in Fig. 6), what explains the rate of this tendency is the climatic conditions and the local ecosystem domain. Fig. 8 shows the all-year-round monthly averages for the coastline region when MSTDPI curves show considerable wave disorders that indicate strong interactions. Studying the coastline features mentioned before concludes that the main parameter that creates the BLSTRS variability in the coastline region is the flow velocity. Nevertheless, other parameters are perhaps interrelated and can be accounted for a better explanation of the effects of the flow velocity/ BLSTRS time-lags specifically for MSTDPI peaks for BLSTRS and flow velocity in June.
If hourly/daily time scales were used, the MSTDPI curves could perhaps show the coastline region concentration-time; however, since this time, according to Kirpich relation found based on the kinematic wave theory, is 3.25 days, this is not applicable at Makran because its coastline zone is small and the existing time scale is monthly. A concurrent monthly time scale linkage between flow velocity and BLSTRS is not only acceptable but also reasonable because their time scale resolution in this study is monthly making the lag detection impossible. Time-lags do exist among the patterns of the flow velocity profiles, and since waves in the coastal, marine and hydrological ecosystem domains follow previous velocity sequences at time t (timelag), the estimated value for l can be taken equal to one month.
Since the MSTDPI curves result analyses are useful in the determination of the variations of the coastal/marine/ hydrological ecosystems, the simultaneous and precise studying of the hydrological/climatic changes in the coastline area is quite acceptable. Again, since MSTDPI evaluations press on the BLSTRS-flow velocity relationship, it is necessary to potentially include the MSTDPI in the framework of the coastal, marine, and hydrological ecosystems.

Validation
To estimate model efficiency, use has been made of the following statistical coefficients: R 2 (coefficient of determination), D (deviation of observed flow from simulated flow), NSE (Nash-Sutcliffe Efficiency), RBIAS, RRMSE (relative root mean square error), the formulations of which are presented in the following lines wherein C obs and C cal show the observed, RSR (ratio of the root mean square error and standard deviation of measured data), and computed con- centrations, respectively. Molnar (2011) believed that NSE is an indication of the adjustment between simulated and observed data in the 1:1 line varying from −∞ to 1; he suggested the following ranges for it using daily simulation step: NSE > 0.8, the model is excellent; 0.6 < NSE < 0.8, it is very good; 0.4 < NSE < 0.6, it is good; 0.2 < NSE < 0.4, it is satisfactory; and NSE < 0.2, it is insufficient. Moriasi et al. (2007) believed that if NSE > 0.5, the model is qualified for simulation. Moriasi et al. (2007) reported intervals of values and performance evaluations for the recommended statistics and established guidelines for the evaluation of flow simulation models, sediment transport, and nutrients.
Accordingly, in addition to graphic techniques, they suggested NSE, RBIAS and RRMSE (3 quantitative statistics) be utilized to estimate models; model simulation, in general, can be acceptable if and and if for sediments.

Laboratory and field experiments
Sediment rate results from Eq. (23) was tested and verified by use of laboratory and field experiments, and two classical bedload transport formula, Bagnold (1966) and van Rijn (2007) with MSTDPI Entropy-based bedload sediment rate model. A significant milestone in the fieldwork was the experimental campaign completed at the Duck pier between 1995 and 1998 in the USA (Miller, 1999;Downing, 1983;Krumbein and Graybill, 1965). The bedload sediments consisted of fine to medium sand (0.15-0.3 mm) and the velocities and concentrations were measured by instruments mounted on the lower boom of a tower crane on the research field deck pier. The most significant study in Europe was the COAST3D-field experiment in the period from 1995 to 2000 (Grasmeijer, 2002).
While field studies of bedload sediment rate (sand transport) in the coastal waters of sandy beaches have been conducted over many decades. The pioneering work in the USA was done by Kana (1979), Kraus et al.,(1989), Noda (1969), Bijker (1971) and Bhattacharya (1971) using mechanical samplers employed by the authors themselves standing in the water. Table 4 shows a summary of the characteristics of different experimental, field data and two classical bedload transport formulas used in this study for coastal environment bedload sediment transport.
Although each device in the mentioned studies has its own limitations, the integrated presented data provide an overall view of the bedload sediment movement in coastal regions.

Comparisons with observed data and other models
Vertical profiles of bedload sediment rate were found through Eq. (23) using one set of the mean and reference values for each data set and to control the accuracy of the obtained results, Eqs. (30)−(35) were used to determine NSE, R 2 , D, RBIAS, RSR, and RRMSE, respectively. Five lab data sets each from Miller (1999), Grasmeijer (2002), van Rijn et al. (2002), Downing (1983) and Krumbein (1944) were analyzed by the model derived in Eq. (23) and the results were compared. Fig. 7 shows a comparison between bedload transport predicted by the new formula (Eq. (23)) and laboratory and field experiments data. The model prediction accuracy was checked by 5 field data sets, lab data sets and two classical bedload transport formulas as presented in Table 4 and Fig. 8. The measured and calculated data conform well because the three approaches yielded similar results. Data from Miller (1999), Grasmeijer (2002), van Rijn et al. (2002, Downing (1983) and Krumbein (1944) have been compared using the derived model. After calculating NSE, Error, R 2 , D, RBIAS, RSR, and RMSE, it was shown that D varied between 40.10% and 376.72%, Error variation range was −9.94 to 64.79, NSE varied from 0.56 to 4.38, R 2 variations were from 0.94 to 0.99, RBIAS varied between −0.436 and −0.018, RSP varied from 0.18 to 0.91, and RRMSE variation range was 0.109−0.613.
In general, the bedload sediment rate random walkbased model (entropy) (Eq. (23)) can yield favorable results as shown in Fig. 9. To show the validity of this claim, the values of NSE, R 2 , D, RBIAS, RSR, and RMSE have been computed and shown in Table 5 for all the experimental, field observations and classical bedload transport formula used in this research. 4.5 Spatial and temporal distribution of bed sediment load concentration through entropy-power and MSTDPI The spatial clusters discussed earlier, were used to determine the wave height-BLSTRS combination spatiotemporal probability distributions in the coastline area. The BLSTRS annual variability distribution was studied for a 45-year period (1970 to 2015) by the entropy-power MSTDPI as a variability index for different months/seasons of a year (a larger index means a larger variation during the time series). To study the temporal variations, the entropypower MSTDPI was first found for every station and the results were then compared to find the BLSTRS variability spatial distribution. Each specific year's/station's degree of variability is found according to a threshold for temporal/ spatial variation. Table 6 and Fig. 10 respectively show the entropy power MSTDPI and DI results and the comprehensive monthly variability scenarios (index is similar for the entropy-power DI). Nevertheless, MSTDPI and DI differ; the former is found based on marginal entropy, but the latter based on the joint-entropy meaning that MSTDPI relates to the interactive as well as the marginal patterns, but DI is related only to marginal patterns.
The MSTDPI variation analyses show that the hydraulic and hydrological states are contrasting along the entire shoreline area and the three variables always seem interdependent at such large scales. The MSTDPI curve analyses show considerable likenesses in Figs. 11a−11c meaning that Fig. 9. comparison between bedload transport predicted by the new formula (Eq. (23)) and laboratory and field experiments data. the three variables attain their highest MSTDPIs in December in addition to another peak in each curve in June (for BLSTRS and wave height) and in August (for sediment concentration); the second peak in the BLSTRS curve is related to an MSTDPI reduction. The signals in Fig. 8 show the coastline elements' interaction tendencies, but the local ecosystem and climatic conditions explain the extent. Great wave disorders (strong interactions), shown in the MSTDPI curve (Fig. 8), reveal that the driest month of the year throughout the coastline zone is December. Coastline features explained before clearly show that the main factor that defines the BLSTRS variability in the coastline zone is the wave height, but what makes the effects of the wave and sediment concentration-time lags understandable is the MSTDPI peaks' BLSTRS-wave height probable inter-relation in December and May, respectively.
After the pictures were taken, the information on the obtained spatiotemporal coastline locations was modified considering the effects of the tidal variations based on the mean water level and sea level deviations. The 2015 airborne laser depth measurement-based foreshore slope was used for coastline location modifications; the spatial coastline variations (from 1970) are shown in Fig. 4. Fig. 10 shows that between 1970 and 2015, the mean coastline moved forward by almost 25 m due to a sediment increase. Although the increase continued for 14 years (1970−1984), it seems that the coastline has not been able to match it appropriately. According to Fig. 11, between 1984 and 1993, the coastline reduced, but it started moving forward again in 1993 which explains why, compared with   1970, the location of the mean coastline was almost 75 m toward the sea in 2005. However, since the aerial photos showed only instant values, the coastline location offered both short-term and seasonal variations. According to the beach profiles in Fig. 12, since the SD of the 1-year daily variations in the Makran coastline location has been nearly 15 m, its long-term changes have been sufficiently large which means that the sediment increase has moved the coastline forward considerably. The use of hourly/daily time scales could possibly enable the MSTDPI curves to show the concentration-time of the coastline zone, but since the latter is not sufficiently large and time scale is monthly, this cannot be done in Makran area. In short, the MSTDPI curves' analytical results are quite useful, and probably important in studying the area because they can be used to evaluate the effects of the hydrological/marine/coastal ecosystems' changes and study the hydrological/climatic changes along the coastline zone simultaneously and precisely. Besides since MSTDPI evaluations confirm that the relations between waves and BLSTRS are delayed or associated, it seems necessary that it should be potentially included in the frameworks of the hydrological, marine and coastal ecosystems.

Conclusions
In this study, the coastal region is a complicated hublike system with components depending on spatiotemporal interactions that influence the variability of the features of the related marine and hydrological ecosystems. The influential variables dealt with in this study are the BLSTRS and flow velocity, and the related methodology not only shows the relevant coastline area patterns, but it also helps researchers to find the spatiotemporal variability of the mentioned variables as a parameter that stimulates the BLSTRS variations. Although entropy-power MSTDPI analyses have limited capacities to deal with differing interactions, they can usually result in an all-inclusive explanation of the interactions between BLSTRS and flow velocity. The time-lag detection, for instance, that shows vegetation-wave reactions at the scale of a coastline area considering the temporal data resolution, is possible through the joint analyses of the MSTDPI curves. This has been applied in the present work wherein the concentration-time of the coastline zone can be estimated with finer BLSTRS/flow velocity-time resolutions. The authors suggest finer time resolution scales for better MSTDPI applicability investigations because theirs is monthly. Application of the entropy-power MSTDPI is quite advantageous; it is a powerful index that can show the marginal as well as the interactive parameter effects in the coastline area, it addresses the coastline region as a hub like system facilitating its application for management studies, it shows spatiotemporal variability trends at a coastline region scale, it shows different dimensions of inter-parameter interactions of various coastline areas, and, finally, it can simulate the sediment transport rate under different flow and granulometric conditions and concentrations.