# Development of an Objective Low Flow Identification Method Using Breakpoint Analysis

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{10}flow (corresponding to the 10th percentile of the flow), also referred to as Q

_{90}if the cumulative distribution function is used [33] or the 7Q

_{10}criterion (as the 10th percentile of the 7-day average flows) [34], the development of modern methods of numerical modeling introduces another issue: discretization and uniqueness of data.

_{10}as the threshold will represent statistical information; however, in some cases, for example with the National Water Model (NWM) retrospective data, minimum streamflow values are often repeated for extended periods [38]. In this case, the FDC flattens out on the lower flows, with the 10th percentile being equivalent to higher percentile values (i.e., 15th or 20th percentile). In this case, the use of percentiles as thresholds leads to the separation of values from the respective environmental information or even false statistical results if the threshold is equal to all of the other lowest flows in the data series. Such episodes will have zero volume based on the TLM definition, which further translates into issues in parameterizing low flow episodes.

_{10}method, and Section 5: conclusions. Additionally, Supplementary Materials are available, which include information about how to obtain, install, and use the developed algorithm (a Python module termed objective_thresholds), which will allow researchers to directly apply the method without the need to reconstruct the methodology.

## 2. Data

^{3}/s, which, from the perspective of drought analysis, introduced the risk of a misrepresentative calculations for both threshold level and drought event statistics. The study dataset, therefore, consisted of 73,891 nodes, from which daily mean flow values were calculated from the hourly model values. Additional criteria of no more than 5% of zero or null data were introduced to avoid computational bias, which resulted in 61,948 nodes.

## 3. Methods

#### 3.1. Breakpoint Approach to Low Flow Identification

#### 3.2. Breakpoint Algorithm Selection

_{15}was considered the middle probability). Since all series were considered part of the same data distribution, a dependent T-test was used. The differences between the objective thresholds, median values, and middle-range flow percentiles were shown to be statistically significant, meaning the series did not result in the same values most of the time (Table 2). This shows that although the FJ method showed a tendency to define the breakpoint of the FDC as a value close to the median, the resulting Q

_{obj}remained statistically significant in its difference from the median.

#### 3.3. Objective Threshold Approach Description

- 1.
- Determination of the number (n) of points in the daily flow series needed to calculate the breakpoint based on the lower FDC range (by default: below the 35th percentile, as described further in the Results section):Q: {Q ∈ R + | Q ≤ Q
_{35}}

- 2.
- Implementation of the Fisher–Jenks algorithm to define the breakpoint [52] by minimizing deviation of each class from the class mean, while maximizing the deviation of each class from the means of the other classes:
- a.
- Order flow data series in increasing order and assigning weights (w):w: i ∈ {1,…,n}

- b.
- Compute the diameter matrix Di,j to store the distance between all pairs of n observations, such that:1 ≤ i ≤ j ≤ n

- c.
- Populate the error matrix with variance of n observations when classified into two classes (one class for atmospheric driven resources, representing mean flow conditions (FDC part above breakpoint), and second for the drought conditions and baseflow (FDC part below breakpoint)):$$E\left[{P}_{i,L}\right]=\mathrm{min}\left({D}_{1,j-1}+E\left[{P}_{j-1,L-1}\right]\right)$$

- d.
- Locate the optimal partition from the error matrix by maximizing inter-class variance and minimizing intra-class variance:$$E\left[{P}_{n,2}\right]=E\left[{P}_{j-1,1}\right]+{D}_{j,n}$$

- 3.
- Application of the defined breakpoint (Q
_{t}) as the low flow threshold for further analysis of low flow distribution, streamflow droughts, or for water management systems at the alert point, according to the following relation:$${Q}_{lf}=\{\begin{array}{c}0,ifQ{Q}_{t}\\ Q,ifQ\le {Q}_{t}\end{array}$$

- Q
_{t}—threshold flow determined by Fisher-Jenks algorithm, - Q
_{lf}—flow identified as low flow.

## 4. Results

_{10}and Q

_{obj}can be represented by a linear relation (Figure 6), with R

^{2}values of around 0.998 for the study area rivers. This relationship reveals that, on average, Q

_{obj}is 1.17 times higher than the Q

_{10}threshold. Higher threshold values relate to an increase in low flow parameters; however, because the values fall in the 10th–30th percentile range, they remain in the range of “shallow” and “deep” streamflow drought as indicated in Section 3. Less than 1% of the cases in the tested data sample had threshold values lower than the associated Q

_{10}, although in about 90% of the nodes, the increase did not exceed 100% of the Q

_{10}threshold value (Figure 6). In a few cases, the ratio of Q

_{obj}to Q

_{10}exceeded three; however, these cases corresponded to situations when the threshold value determined by the Q

_{10}was low (~0.01–0.03 m

^{3}/s) due to the flattening of the FDC at the lower range (multiple repetitive values), while the threshold determined by the objective breakpoint method was around 0.05–0.10 m

^{3}/s.

_{10}method and the objective breakpoint method, a different distribution of the density function occurred (Figure 7). Concerning the Q

_{10}method, in most cases the duration of low flow events averaged around 30–35 days, with a low variance around this value (Table 2). For low flow duration based on Q

_{obj}, the distribution has a higher mean and variance. In most cases, the average number of low flow days each year is about 60; however, due to a more normal distribution of values in the series, it is possible to better capture the specific environmental conditions occurring in each catchment area individually. Both distributions are left skewed, indicating that there are nodes with a lower number of days with low flow. For Q

_{10}, the kurtosis of the distribution was 4.0, while for Q

_{obj}, it was 2.6 (Table 3), implying a much more leptokurtic distribution for Q

_{10}(as shown in Figure 7).

_{obj}method relative to Q

_{10}, the basic parameters of low flow (e.g., number of events, duration, and volume) change accordingly. In the case of the number of low flow events determined using Q

_{10}, for each of the analyzed nodes, about 50–100 low flow events were observed during the 42-year study period (upper range was 175). These values increased substantially when using Q

_{obj}, where both the mean and the median increased by about 50 (Figure 8). The maximal number of episodes increased from 200 for Q

_{10}to 300 for Q

_{obj}, which translates to an average of 8.4 days per episode for Q

_{obj}and 6.3 days per episode for Q

_{10}per year. The low flows identified by the objective method are longer, which allows for the inclusion of periods occurring in streamflow, even when additional criteria, of a minimal time of 7 days, are applied [12,38].

_{10}, in most cases the low flows did not last more than 2000 days in total over the study period. However, the mean and median values are close to the lower and upper IQR limits, respectively (Figure 8), which indicates that, while outliers shift the median towards the upper limits, the considerable number of low values (around 1000 days) shifts the mean to the lower limit. This introduces inconsistency to the spatial distribution of low flows (Figure 8). When using Q

_{obj}, the range of values is higher, but this corresponds to the percentile range indicated earlier (convergence in relation to the 15–20th percentile) such that as the percentile value is doubled, the duration of low flows is doubled. The mean and median of the distribution are closer to each other and oscillate within the center of the IQR, as in the cases of normally distributed series (Figure 8).

_{obj}better captures the diversity of environmental conditions leading to the formation of outflow deficiencies with varying intensity.

_{obj}, there is a clear distinction between low flow volumes between these two rivers, while Q

_{10}shows similar volume ranges for both rivers (Figure 9). A similar pattern exists for the distribution of total low flow duration time, where the Mobile River has shorter durations, and the Apalachicola River has longer durations distributed along the reach. For Q

_{10}, the duration of low flows is similar among the two rivers, albeit with some outliers showing no distinct spatial pattern along the reach. The distribution of low flow volumes in rivers of order seven is similar for both methods, with four rivers having higher volumes when using Q

_{10}(Figure 9); however, the length of low flows is different with Q

_{10}, resulting in no spatial differentiation (with some outliers), while Q

_{obj}varies spatially. In general, most rivers have longer total low flow durations in their upper reaches that decrease downstream, which reflects the natural tendency of smaller tributaries to have a faster response in river levels to environmental events that drive streamflow. This pattern becomes more pronounced at lower Strahler stream orders, where the biggest differences are noted in the spatial distribution of low flow. Along these river reaches, the highest low flow volumes and times occur within the eastern part of the study area in North and South Carolina, as well as central parts of Georgia and Alabama. This relation is, however, not reflected in the Q

_{10}method, where the spatial distribution of low flow volumes and times is relatively equal throughout the study area.

_{obj}results in a wider spread of values relative to Q

_{10}, representing a greater difference in environmental conditions. This means that either the change in duration times does not affect the volumes, or changes in the volumes are not reflected in the changes in duration. Additionally, Q

_{obj}results in a lower number of nodes with volumes close to 0. It is worth mentioning that for nodes with higher Strahler stream orders (e.g., seven and eight), the relationship changes between the two thresholds. For Q

_{obj}, the volumes are usually close when there are small changes in duration time, while for Q

_{10}, the durations are close when volumes are prone to change. This is a direct result of the statistical character of Q

_{10}and the consequences defined earlier. The strength of the relationship between low flow time and volume depends on the stream order; however, when considering the mean correlation values, they are higher for the objective method by approximately 0.22 (Q

_{obj}$\overline{r}=0.57$ and Q

_{10}$\overline{r}=0.35$).

_{10}is unable to accurately represent spatial relations and differences, and due to its statistical nature, results in constant, undifferentiated low flow patterns across the study area (with some randomly occurring outliers). At the same time, Q

_{obj}is able to distinguish spatially varying river characteristics, such that low flows identified by this threshold vary spatially and along the course of individual rivers. Q

_{obj}allows for the accurate capture of the natural character of events like streamflow droughts and introduces the environmental aspects to the analysis, taking into account the specificity of a given river in the studied node. As the objective threshold (Q

_{obj}) fell within the 10th–30th percentile range for all nodes used in this study, it is important to investigate the relationship between not only Q

_{obj}and Q

_{10}but also between Q

_{obj}and Q

_{30}to better understand the pattern of the objective threshold values relative to the static statistical criteria. As shown in Figure 11, while the correlations between Q

_{obj}and both Q

_{10}and Q

_{30}have a strong linear relationship, the slope of the resulting regression lines shows opposite values relative to the 1:1 trend line. In other words, for Q

_{obj}compared to Q

_{30}, instead of exceeding the objective threshold value for a given percentile, there is a decrease in value relative to the percentile. This is expected as statistical thresholds inherently maintain a constant frequency of events and always result in the same part of the dataset considered as an event (for Q

_{10}this will be 10% of data and for Q

_{30}, 30% of data, regardless of the environmental aspects of the river).

## 5. Conclusions

_{10}, Q

_{30}, 7Q

_{10}, etc.). This article presents a new way of defining the low flow threshold based on an objective approach, utilizing a breakpoint method derived from a given streamflow time series, which is more representative of environmental criteria.

_{obj}and the widely used Q

_{10}threshold reveals that Q

_{10}is unable to differentiate spatial patterns, resulting in a similar range of defined low flow events, with skewed, widely spread distributions of low flow parameters. Based on the same data, Q

_{obj}is able to better capture the natural characteristics of rivers, allowing for spatial recognition of the drivers responsible for streamflow drought occurrence. The objective threshold approach outperforms set statistical criteria (e.g., 10th percentile) in terms of spatial pattern recognition by introducing environmental factors into the threshold definition. Additionally, low flow parameters such as duration and volume are closer to a normal distribution when defined using Q

_{obj}, with fewer outliers and volumes close to zero. The correlation between low-flow duration and volume depends on the stream order. On average, stream order to T and V correlation is higher by 0.22 for Q

_{obj}, compared to Q

_{10}.

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**Example of standardized flow values (Q

_{s}) for the lower range (to 35th percentile) of the FDC for a sample series with high discretization (

**a**: unique values constitute less than 0.01% of the lower FDC range), moderate discretization (

**b**: unique values constitute 1% of the lower FDC range), and close to natural distribution (

**c**: unique values constitute around 90% of the lower FDC range).

**Figure 3.**Trend change method for finding the threshold (Q

_{gen}) based on Tomaszewski’s method [48], on the example series of minimum annual flow values (Q

_{min}) sorted increasingly (

**a**) and the example series for which no clear trend change is present (

**b**).

**Figure 4.**(

**a**) Average difference (d) between threshold values returned by specific algorithms and the mean from all the methods, and (

**b**) the mean time (t) of algorithm execution per 100 nodes.

**Figure 8.**Box and whisker plots of the number of low flow events. (

**a**): Total duration of low flows and (

**b**) total volume of low flows (

**c**) over the study period; graphs do not include outliers.

**Figure 9.**Spatial variability of volumes (V) and duration time (t) of low flows at each node, according to stream order and the threshold method used.

**Figure 10.**Duration time (T) with volume (V) for low flow relations in the studied rivers, according to their order and the threshold method used.

**Table 1.**Characteristics of the lowest 10% of values from the study data. n—number of data points; n

_{u}—ratio of unique values; Q

_{m}—mean flow [m

^{3}/s]; std—standard deviation [m

^{3}/s]; var—variance; Cv—coefficient of variation; IQR—inter-quartile range [m

^{3}/s].

n | n_{u} | Q_{m} | std | Cv | IQR | |
---|---|---|---|---|---|---|

mean | 527.7 | 0.554 | 0.079 | 0.014 | 0.252 | 0.022 |

std | 333.8 | 0.353 | 0.231 | 0.039 | 0.196 | 0.062 |

var | 111438.8 | 0.125 | 0.053 | 0.002 | 0.038 | 0.004 |

Cv | 0.633 | 0.637 | 2.917 | 2.825 | 0.779 | 2.831 |

IQR | 665 | 0.703 | 0.032 | 0.007 | 0.133 | 0.011 |

**Table 2.**Results of T-tests for the comparison of series representing objective thresholds defined using the FJ algorithm (Q

_{obj}), median value, and middle percentile flow (Q

_{p}) for multiple FDC ranges.

Relation | Q_{obj}–Median | Q_{obj}–Q_{p} | Median–Q_{p} |
---|---|---|---|

FDC range | 20% | ||

statistics | −54.0508 | −59.3941 | 4.6209 |

p-value | 0.0000 | 0.0000 | 0.0000 |

FDC range | 25% | ||

statistics | −51.2043 | −58.6077 | 4.4995 |

p-value | 0.0000 | 0.0000 | 0.0000 |

FDC range | 30% | ||

statistics | −39.3186 | −44.1184 | 4.0386 |

p-value | 0.0000 | 0.0000 | 0.0001 |

FDC range | 35% | ||

statistics | −22.8904 | −25.4409 | 4.0794 |

p-value | 0.0000 | 0.0000 | 0.0000 |

FDC range | 40% | ||

statistics | 8.5917 | 13.9009 | 3.4002 |

p-value | 0.0000 | 0.0000 | 0.0007 |

FDC range | 45% | ||

statistics | 38.5965 | 41.6176 | 3.2249 |

p-value | 0.0000 | 0.0000 | 0.0013 |

FDC range | 50% | ||

statistics | 49.8975 | 50.6962 | 3.1834 |

p-value | 0.0000 | 0.0000 | 0.0015 |

**Table 3.**Descriptive statistics for the distribution of the annual number of days with low flow for Q

_{obj}and Q

_{10}methods: μ—mean [m

^{3}/s], m—median [m

^{3}/s], σ—standard deviation [m

^{3}/s], β2—kurtosis, S

_{kp}—skewness, n

_{σ}

_{1,2,3}—percent of values within one, two and three σ from μ.

μ | m | σ | β2 | S_{kp} | n_{σ}_{1} | n_{σ}_{2} | n_{σ}_{3} | |
---|---|---|---|---|---|---|---|---|

Q_{obj} | 59.64 | 62.05 | 16.03 | 2.635 | −1.315 | 76.76 | 94.49 | 97.45 |

Q_{10} | 32.40 | 36.45 | 8.816 | 3.979 | −2.012 | 85.62 | 92.74 | 96.49 |

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**MDPI and ACS Style**

Raczyński, K.; Dyer, J.
Development of an Objective Low Flow Identification Method Using Breakpoint Analysis. *Water* **2022**, *14*, 2212.
https://doi.org/10.3390/w14142212

**AMA Style**

Raczyński K, Dyer J.
Development of an Objective Low Flow Identification Method Using Breakpoint Analysis. *Water*. 2022; 14(14):2212.
https://doi.org/10.3390/w14142212

**Chicago/Turabian Style**

Raczyński, Krzysztof, and Jamie Dyer.
2022. "Development of an Objective Low Flow Identification Method Using Breakpoint Analysis" *Water* 14, no. 14: 2212.
https://doi.org/10.3390/w14142212