#### 2.1. Study Area

The Trotuș basin is an upper mesoscale mountainous catchment (i.e., 4350 km

^{2}) located in the central-eastern part of the Eastern Carpathians (

Figure 1). Trotuș River is one of the major tributaries of Siret River, which is, in turn, the largest tributary of the Danube within the Romanian territory.

In terms of lithology, the Carpathian flysch accounts for the largest area of Trotuș drainage basin (~55% of the basin area), ensued by the pericarpathian molasse (~25%), composed of highly erodible rocks such as friable sandstones, clays and marls. The elevation ranges between 73 m above sea level (a.s.l.) at the junction with Siret River and 1672 m a.s.l. at the headwaters. The climate is typical of Carpathian mountainous areas with mean annual precipitation of 720 mm, ranging from 580 in the lowlands to 1150 mm at high altitudes. The share of the precipitation related to the total sum of maximum amounts recorded in 24 h above 100 L/m

^{2} has been rising continually: 8.3% between 1941 and 1960; 30.8% between 1961 and 1980; 47.5% between 1980 and 2000; and 67.7% after 2000 [

33]. As regards the land cover/land use, forests, pasturelands and meadows are prevalent in the higher regions of the upper and mid courses, whereas in the lowlands corresponding to the lower course agricultural lands and pastures are dominant. The hydrology of Trotuș River basin is regulated by the pluvio-nival regime, with spring flooding occurring typically in April–May as a result of snowmelt, high precipitation, or the overlapping of both. In June, July, and occasionally extending to August, summer floods can occur as a result of abundant precipitation, reaching very high amplitudes, as was the case with the floods of June–July 2005.

The total length of the Trotuș River is ~160 km. The median diameter (D

_{50}) of the bed material along the 160 km amounts to an average of 71.3 mm, with extreme values ranging from 130 mm (in the midcourse) to 20 mm (in the lower course) [

34,

35]. The mean gradient of the channel ranges between 0.17 mm

^{−1} in the upper course (Lunca de Sus) and 0.018 mm

^{−1} in the lower course (just downstream of Vrânceni) [

36]. Based on the average suspended sediment yield (263 t/km

^{2}/year) Trotuș drainage basin ranks among the category of rivers with moderate rates from the Romanian Carpathian area [

37,

38].

In this study, the assessment of the effective discharge was carried out at four gauging stations along the Trotuș River where long-term records of streamflow discharge and suspended sediment load are available. The location and key attributes of these stations are shown in

Figure 1 and

Table 1, respectively.

#### 2.2. Flow Data Frequency Analysis

Subsequent to the publication of the study by Wolman and Miller [

7], a whole array of studies approached the magnitude-frequency analysis (MFA) of flow discharge, which is an essential part of determining the effective discharge [

16,

39,

40]. Although Wolman and Miller [

7] suggested that a theoretical probability density function (PDF) can be employed to illustrate the flow regime, it was only after the 1990s that the theoretical MFA approach was perfected [

8,

19,

23,

40,

41,

42].

Three major approaches were used to determine flow frequency [

27,

43,

44], including (i) employing a fixed number of classes (e.g., 25) of equal width and magnitude [

16,

17]. This method is employed most often but is likely the most criticized [

6,

11,

32] because it depends, to a large extent, on the number of classes. Studies based on the MFA for assessing effective discharge used either an arithmetic scale for flow or a logarithmic scale for building histograms [

45]. The following elements should be considered during the process of producing flow-frequency histograms [

11,

43,

45]: the size of the class interval, the number of flow discharge classes, the time period for averaging the discharge, and the length of the period of record. Yevjevich [

46] suggested that the size of the class interval for flow discharge should not be larger than SD/4 (where SD represents the standard deviation of the flow for the considered sample), and the number of classes should range between 10 and 25 depending on the sample size. In regard to the length of the period of record, Biedenharn et al. [

43] recommended the use of data series 10 to 20 years in length. The second approach is (ii) representing the observed frequency of flow based on the theoretical flow frequency distribution that approximates it, such as the lognormal distribution. This conceptual approach was introduced by Wolman and Miller [

7] and later developed by Nash [

19]. The application of this method may create particular problems in the case of bimodal, multimodal, heavy-tailed, or heavily skewed flow frequencies [

27,

41]. The third approach is (iii) computing the amount of sediment transported for each flow class and determining the effective discharge from the steepest point of the cumulative sediment transport curve [

26].

#### 2.3. Effective Discharge Computation

The methods involving the use of the flow frequency distribution and a sediment rating curve for the assessment of effective discharge [

7,

19,

47] are regarded as traditional approaches (or deterministic approaches). Moreover, the methods derived from the methodology presented by Crowder and Knapp [

11] are considered mean approaches [

6,

25]. In other articles, the methods for estimating the effective discharge are ranked as class-based (magnitude–frequency) and model-based (analytical) approaches [

48].

The estimated value of the effective discharge is strongly influenced by the size and number of class intervals used in the flow frequency analysis [

6,

9,

11,

25,

27,

32,

39,

49,

50]. To remove some of the subjectivity generated by the empirical choice of the size of class interval (CI) or the number of flow discharge classes (N) [

25], four different methods were employed for determining the CI and N.

The first method was introduced by Yevjevich [

46]. According to the observations by Ma et al. [

25] on streams with large flow amplitudes, the criteria proposed by Yevjevich [

46] (CI ≤ SD/4 and N = 10–25), Biedenharn et al. [

39] and Crowder and Knapp [

11] (each class interval should contain at least one flow event) are difficult to apply. For example, at the Vrânceni gauging station, the minimum discharge is 2.2 m

^{3}/s, whereas the peak discharge is 2359 m

^{3}/s (and the second largest discharge is 1468 m

^{3}/s, which results in a difference of 891 m

^{3}/s). The SD/4 value is 12.36. Under these circumstances and by considering the previously mentioned recommendations, the number of classes either exceeds 25 or falls below 10 (i.e., four classes, where the effective discharge is placed into the first class, which is also not recommended). To eliminate these drawbacks, Ma et al. [

25] proposed to divide flow discharge records into classes by using equal arithmetic intervals corresponding to SD, 0.75 SD, 0.5 SD, and 0.25 SD. Considering the fact that the smaller the size of the class interval is, the more precise the results [

9], the class intervals corresponding to 0.25 SD/4, 0.5 SD/4, 0.75 SD/4, and SD/4 were employed for this study.

The second method for determining the CI and N consisted of using kernel density estimation (KDE), which is a non-parametric way to estimate the PDF of a stream flow [

23]. The kernel distribution histogram was built using the R statistical software, version 3.5.1. (R is a programming language and free software environment for statistical computing and graphics that is supported by the R Foundation for Statistical Computing). In the case of the two approaches (SD and KDE) employed to estimate the effective discharge of suspended sediment transport, several steps were necessary [

14]: (i) determining the flow-frequency distribution, (ii) determining the suspended sediment transport rating curve, and (iii) calculating the effective discharge by multiplying the suspended sediment transport rate for a certain discharge class with the frequency of the respective discharge. The discharge class that accounted for the maximum value of the product was defined as the effective discharge [

1].

The third method pertaining to the class-based approach used in this study is the one introduced by Sichingabula [

32], which is also known in the literature as the event-based class method (EBM). For this method, the discharge class width is equal in magnitude to the number of decimal places in the maximum value of the data series. For instance, for a maximum discharge ranging between 1 and 9.99 m

^{3}/s, the class width is 0.01 m

^{3}/s; between 10 and 99.99 m

^{3}/s, the class width is 0.1 m

^{3}/s; and above 100 m

^{3}/s, the class width is 1 m

^{3}/s [

12,

27]. In the case of the Trotuș River, the 0.1 and 1 m

^{3}/s class widths were used, which resulted in a total number of classes on the order of hundreds or thousands. For each discharge class, we determined the average magnitude for the class and subsequently used the sediment transport rating curve to evaluate the sediment transport for that class via the class-averaged discharge. The sediment transport rate for each class was multiplied by the flow frequency corresponding to the respective class. The effective discharge was considered to be either the discharge class with the largest value after multiplying the sediment transport and frequency or the peak in the plot of the frequency of sediment transport versus discharge [

12].

The fourth method introduces a new way of estimating the effective discharge based on the utilization of real suspended sediment load data instead of the transport rate, which is determined using the suspended sediment rating curve. In fact, this approach is a mixture of the computational versions proposed by Andrews [

1]; Sichingabula [

32]; Crowder and Knapp [

11]; Ma et al. [

25]; Tena et al. [

50]; and López-Tarazón and Batalla [

9]. The discharge classes were established according to the method introduced by Sichingabula [

32]. For simplicity, the representative discharge in each class was considered to be the midpoint of the corresponding interval. While Crowder and Knapp [

11] employed the average suspended sediment load for each discharge class, Tena et al. [

50] and López-Tarazón and Batalla [

9] used the real sediment rate for each flow class. In the study by Ma et al. [

25], the total suspended sediment load (SSL) transported by the flow discharge of each class interval was calculated by summing the suspended sediment loads of all sample points that fell within the corresponding class interval. The data processing included the following steps: summing the suspended sediment loads for each flow class from the data series, and dividing the value obtained for each class by the frequency (i.e., number of days) of the respective class to yield SSL/day (kg/s). The metric SSL/day (kg/s) was converted into the suspended sediment flux (SSF) (tons/day) [

51] for each class. By multiplying the SSF (tons/day) by the flow frequency characteristic for each class, the total suspended sediment load (TSSL) was obtained for each class. TSSL is equivalent to the product of transport rate for suspended sediments and flow frequency, which was yielded via methods based on the sediment rating curve. A much simpler version yielding the same result would imply converting SSL (kg/s) into SSF (tons/day) and summing all of the values obtained for each class, which would determine the real suspended sediment load (RSSL) corresponding to each flow class. The midpoint of the discharge class that transported the largest amount of suspended sediments was considered the effective discharge.

The assessment of effective discharge using an analytical approach was performed according to the indications provided by Nash [

19]; Vogel et al. [

8]; Goodwin [

41]; Quader and Guo [

42]; Klonksy and Vogel [

23]; and Sholtes et al. [

40]. According to this method, the sediment transport mechanics are represented by a power function:

where Qs represents the amount of suspended sediment load (kg/s), Q represents the flow rate (m

^{3}/s), and a and b are the fitting parameters.

If f(Q) represents the frequency distribution function of the flow series, combining f(Q) with Equation (1) results in the transport effectiveness curve, where the peak is taken as the effective discharge for maximum geomorphic work. The function f(Q) was determined by fitting the logarithmic function to the original distribution of the historical discharge data [

48]. According to Nash [

19], it was assumed that the daily river discharge (Q) follows a two-parameter lognormal (LN2) distribution, such as:

where μ and σ represent the mean and standard deviation, respectively, of ln(Q).

The effective discharge determined by multiplying the results of Equation (1) with those of Equation (2) can be derived directly by applying Equation (3):

where Q

_{eff} (Wolman and Miller) represents the effective discharge based on the Wolman and Miller [

7] approach (Q

_{effWM}) [

8].

Vogel et al. [

8] questioned the geomorphic significance of effective discharge and preferred instead to use the half-load discharge (Q

_{1/2}) (i.e., the discharge above which half of the total load is transported) to summarize the effectiveness of rare floods [

12]. Q

_{1/2} was determined according to the methodology presented by Klonksy and Vogel [

23].

The effective discharge was estimated at each gauging station using all the approaches described in this section. At each station the following effective discharges were determined: the effective discharge for the entire investigated time frame (1994–1995), the effective flow below the bankfull discharge, and the effective flow prior to and after the floods of 2005.