**1. Introduction**

The need for knowledge of the amount of sediment reaching specific points of streams and river segments became evident from the early 20th century [1–3]. As a consequence of that, the investigation of the sediment transport processes and mechanisms emerged as a high significance research topic for hydrologists, physicists and engineers in the years that followed. Sediments constitute an integral part of river flows, relentlessly forming the shape of fluvial systems and variously affecting everything in their path [4,5]. Water-quality issues, changes in the wet cross-section, increased flooding risk and obstruction of navigation, as a result of excessive depositions, effects on the aquatic ecosystems, decline of macrophyte growth, clogging of spawning gravel, pressures inflicted on coastal zones, effective diminution of dams' storage volume, due to excessive sedimentation, and extreme erosion rates in the case of sediment-starved water (usually below storage dams—theory of hungry water) [6–11], are some of the effects of sediments, which constitute the driving force behind the investigation of sediment transport processes, as well as modeling and quantification efforts. Moreover, knowledge about the interrelated interactions among water-biota-sediment in natural rivers is one of the central issues in today's sustainable river management [12].

The total sediment load results as the sum of the suspended load and the bed load, with the suspended load being the largest part of it. According to the literature, bed and bank erosion, in rivers, can be considered as a percentage of 10–20% of the total load [13–15], although this largely depends on whether they are sandy-bed or gravel-bed rivers [16]. Naturally, the finer the bed material is, the more easily it is entrained and transported downstream. Hence, the bed load ratio—as a fraction of the total load—increases, as the bed material becomes finer.

The result of decades of intensive research on river sedimentology and sediment transport is an amplitude of formulas, models, and theoretical concepts, aiming at the estimation of sediment load in natural streams. Depending on their target, these models can be divided into three principal classes: (a) bed-load models [17,18], (b) suspended-load models [19,20], (c) total-load models [21,22]. Despite most of the above-cited models were developed half a century—or more—ago, their theoretical basis and fundamental equations are so powerful, that even today they dominate the stream sediment transport research. The models for total sediment load can be further categorized as follows [23]: (a) stochastic models and regression models [24–26], (b) energy models [22,27,28], (c) shear stress models [20,29,30].

Yang first introduced his unit stream power theory for the determination of total sediment concentration, in open channels, in 1972 [27]. This new theory questioned the assumption, made by conventional sediment transport equations, that sediment transport rate could be determined on the basis of water discharge, average flow velocity, energy slope, or shear stress [31]. Yang [22], primarily, implemented his unit stream power theory for sandy-bed open channels, and thus developed a formula applicable for bed material with particle size less than 2 mm. In 1984, Yang [32] extended his unit stream power equation from sand transport to gravel transport, for gravel beds with particle sizes between 2 mm and 10 mm. Yang's unit stream power theory has been extensively applied in the literature, and with more than 2000 citations, it constitutes one of the most esteemed formulas for the determination of total sediment yield.

Fuzzy logic has proved a particularly useful tool in the hands of engineers, and its use in recent decades has been widespread in hydrology, hydraulics and sediment transport [33–35]. Fuzzy linear regression provides a functional fuzzy relationship between dependent and independent variables [36], where uncertainty manifests itself in the coefficients of the independent variables.

Fuzzy logic has been utilized in a variety of cases to study the sediment transport processes, as well as to estimate the total sediment concentration. As a recent paradigm, Chachi et al. [36] introduced a fuzzy regression method based on the Multivariate Adaptive Regression Splines (MARS) technique, to estimate suspended load, based on discharge and bed-load transport data, using fuzzy triangular numbers. The comparison of the model's results with real data and two other fuzzy regression models (fuzzy least-absolutes and fuzzy least-squares regressions) showed that the fuzzy regression model performs well for predicting the fuzzy suspended load, by discharge, as well as the fuzzy bed load transport data. In 2018, Spiliotis et al. [37] transformed the threshold—expressed by a dimensionless critical shear stress—for incipient sediment motion into a fuzzy set, by means of Zanke's formula [38], for the computation of the dimensionless critical shear stress, by using fuzzy triangular numbers instead of crisp values. The fuzzy band produced included almost all the used experimental data with a functional spread. The same group of researchers carried out similar studies, with an adaptive fuzzy-based regression and data from several gravel-bed rivers from mountain basins of Idaho, USA [39], and with conventional fuzzy regression analysis and a goal programming-based fuzzy regression using experimental data [40]; the results were satisfactory in both cases. In 2015, Özger and Kabata¸s [41] successfully applied fuzzy logic and combined wavelet and fuzzy logic techniques (WFL) to predict suspended sediment load data which, then, were compared with monthly measured suspended sediment data from Corukhi River and miscellaneous East Black Sea basins. Ki¸si, in 2009 [42], and Ki¸si et al. [43] efficiently elaborated evolutionary fuzzy models (EFMs) and triangular fuzzy membership functions for suspended sediment concentration estimation using data from the US Geological Survey (USGS). Lohani et al. [44] applied Zadeh's [45] fuzzy rule-based approach to derive

stage-discharge-sediment concentration relationships. Firat (2010) [46] used an Adaptive Neuro-Fuzzy Inference System (ANFIS) approach as a monthly total sediment forecasting system.

The present study aims to redefine the coefficients of the stream sediment transport formula of Yang [22] with a fuzzy regression, using the very same experimental data that Yang used for the original equation. Basically, it is intended to build a functional "fuzzy twin" of the original equation, which will provide a fuzzy band for the total sediment concentration for natural sandy-bed rivers. The study initiated with the collection, analysis and processing of the primary experimental data, which, by itself, was a painstaking process. Finally, the 93.3% of the original experimental data was possible to be collected. Based on this data, a fuzzy "duplicate" of Yang's equation was built, by means of the fuzzy regression model of Tanaka [47]. In addition, the original sediment transport equation was reconstructed, by means of classic multiple linear regression, in order to validate the quality of data by comparing the calculated coefficients with the original ones. Apart from the coefficients, an efficiency assessment was carried out on the basis of comparison between the measured crisp total sediment concentrations and the calculated concentrations with a fuzzy band. It was shown that all the elaborated methods produced successful results for both the classic and the fuzzy multiple regressions.

It is the authors' belief that fuzzy logic efficiently deals with the uncertainties that naturally envelop the complex sediment transport processes, by providing a fuzzy band for the final result—whichever this might be.

#### **2. Unit Stream Power Theory of Yang for Sediment Transport in Natural Rivers**

In 1972, Yang [27], with the introduction of the unit stream power theory, fundamentally questioned the applicability of most sediment transport models which until then argued that sediment transport rate could be determined on the basis of physical magnitudes, such as discharge, flow velocity, energy slope or shear stress.

Yang defines the unit stream power as the velocity-slope product. The rate of energy per unit weight of water available for transporting water and sediment in an open channel of reach length *x* and total drop *Y* is [31]:

$$\frac{dY}{dx} = \frac{dx}{dt}\frac{dY}{dx} = VS = \text{unit stream power} \tag{1}$$

where *Y* is the elevation above a datum which also equals the potential energy per unit weight of water above a datum; *x* is the longitudinal distance; *V* is the mean flow velocity; *S* is the energy slope; and *VS* is the unit stream power.

To determine total sediment concentration, Yang regarded a relation between several physical quantities of the following form:

$$
\varphi(\mathcal{C}\_t, \mathcal{V}\mathcal{S}, \mathcal{V}\_\*, \mathbf{v}, \omega, d\_{50}) = 0 \tag{2}
$$

where *Ct* is the total sediment concentration (ppm), with wash load excluded; *V*<sup>∗</sup> is the shear velocity (m/s); ν is the water kinematic viscosity (m2/s); ω is the fall velocity (m/s); and *d*<sup>50</sup> is the median particle diameter (m).

By means of Buckingham's π theorem, the total sediment concentration can be expressed as a function of dimensionless parameters, as follows:

$$\mathcal{L}\_t = q'(VS/\omega, V\_\*/\omega, \omega \cdot d\_{50}/\nu) = 0\tag{3}$$

Yang added a critical unit stream power in the formula, to account for incipient motion of sediment, and after dimensional analysis, he derived the following equation for the total sediment concentration:

$$\begin{array}{l} \log \mathcal{C}\_{\text{F}} = 5.435 - 0.286 \log \frac{\omega d \xi\_{0}}{v} - 0.457 \log \frac{V\_{\ast}}{\omega} \\ + \left( 1.799 - 0.409 \log \frac{\omega d \xi\_{0}}{v} - 0.314 \log \frac{V\_{\ast}}{\omega} \right) \log \left( \frac{V\_{\ast}}{\omega} - \frac{V\_{\ast} S}{\omega} \right) \end{array} \tag{4}$$

*Water* **2020**, *12*, 257

$$\frac{V\_{cr}}{\omega} = \frac{2.5}{\log(V\_{\ast}d\_{\overline{\varepsilon}0}/v - 0.06)} + 0.66, \qquad \text{if} \qquad 1.2 < \frac{V\_{\ast}d\_{\overline{\varepsilon}0}}{v} < 70 \tag{5}$$

$$\frac{V\_{cr}}{\omega} = 2.05, \qquad \text{if} \qquad \frac{V\_{\ast}d\_{50}}{v} \ge 70 \tag{6}$$

where *CF* is the calculated total sediment concentration (ppm); and *VcrS* is the critical unit stream power, derived as the product of mean critical flow velocity and energy slope.

Equation (4) is the dimensionless unit stream power equation that can be used to calculate the total sediment concentration, in ppm by weight, in both laboratory flumes and natural sandy-bed rivers, with median particle size less than 2 mm. Knowing the discharge and the geometric characteristics of the channel, and with simple calculations, the aforementioned sediment concentration can easily be transformed into any form of sediment load, sediment yield, or sediment discharge.

Yang's unit stream power theory has been applied in a plethora of cases in literature, both continuously [48,49] and event-based [50,51]. Quaintly, nonetheless successfully, it has also been applied for estimating overland flow erosion capacity [52,53]. Because of the fact that Yang's equations for total load [22,54] were built with data in the sand-size range, their application should be limited only in sandy rivers. However, Moore and Burch [52] proved that Equation (4) can be applied equally well to predict the sediment transport rate in sheet and rill flows, when soil particles are in ballistic dispersion. It should be mentioned, however, that Moore and Burch used a constant value, of 0.002 m/s, for the critical unit stream power [31].

#### **3. "Fuzzy Twin"—The Physical Meaning**

As mentioned above, the ultimate goal of this research is to build a functional "fuzzy twin" of the unit stream power formula of Yang. In an effort to explain the physical meaning of the term "fuzzy twin", it is considered meaningful to separately analyze "fuzzy" and "twin".

While a portion of the engaging parameters, such as the flow velocity, the flow depth, the bed slope and the water temperature, can be determined with a fairly high precision in natural streams, still the overall uncertainty that blankets the stream sediment transport processes, let alone the determination of the in-stream sediment concentration, is appreciably high. This is not only associated with the bed morphology and the grain size distribution, but also with the constantly altering flow conditions that prevail in natural rivers. Yet, any uncertainty due to measurement errors seems to be small compared to uncertainties in the computational part. This is due to simplifications and assumptions made by sediment transport formulas, in which reality is usually poorly reflected. Hence, apart from the measurement errors, the fuzzy band is even more meant to deal with uncertainties in the computational part, namely uncertainties that have to do with the representation of all the involved physical processes in a formula. Just to give an example, fall velocity, for instance, can be measured with much greater accuracy than it can be computed by any existing formula. Obvious reasons for this are that in all fall velocity formulas the particle is considered a sphere, and is usually represented by the median particle diameter, d50, and not by its actual diameter, as well as the disregard of turbulence. Going further, the uncertainty raises by the subjectivity in the estimation of the incipient motion criterion [54] and the turbulence impact on sediment transport. To better identify the source of uncertainty in Yang's formula, the assumption of one-dimensional, uniform and steady flow (especially, in the case of natural rivers), as well as the regression analysis between sediment transport rate and stream discharge, which partly neglects the physical mechanisms of the sediment transport phenomenon, must be considered, as well. The uncertainties would significantly be reduced in the case of an analytical physically based model. The complex nature of sediment transport and the associated uncertainties have been very well documented in literature [54–59]. In terms of uncertainty, Kleinhans (2005) [57] compares the notoriety of the sediment transport problem with that of the roughness problem and he stresses the necessity of calibration. In such cases, fuzzy regression, contrarily to conventional solutions such as classic regression, offers an efficient and applicable solution, by producing a fuzzy band within which the measured values are most likely included. Indeed, Azamathulla et al. [60] state that classic regression

does not efficiently cope with the uncertainties that dominate both input and output data and instead they use a Fuzzy Inference System (FIS) as a prediction model. Hence, "fuzzy" is justified by the fact that the computed sediment concentration is not a crisp value, as it would be if the classic formula of Yang Equation (4) had been used, but a range of values which is expected to contain the observed data.

As already mentioned, and as it is thoroughly presented in Section 4, the construction of the fuzzy total sediment concentration formula is based on the exact same datasets that Yang used 47 years ago to derive his unit stream power formula. Hence, both formulas were built upon the same foundation and this makes them "twins".
