# Trend Detection in Annual Streamflow Extremes in Brazil

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{0}) of the test statistic, which is derived based on the assumption that the series is stationary. One uses the null distribution and the sampling estimate of the test statistic to decide whether the series is stationary (accept H

_{0}) or nonstationary (reject H

_{0}). There is no definite answer to this question because there is always a probability of mistakenly rejecting H

_{0}when the series is in fact stationary (Type 1 Error), which may result in overpreparedness or erroneously accepting H

_{0}when the trend is present (type 2 error), which may induce underpreparedness [11]. Traditionally, NHST procedures employ a significant level (e.g., α = 0.05) to control type 1 error, neglecting type 2 errors.

## 2. Materials and Methods

#### 2.1. Study Area

#### 2.2. Streamflow Data

^{2}with a median of 3220 km

^{2}. The majority (80%) of the gauges have drainage area less than 20,000 km

^{2}, and only 10% have drainage areas larger than 50,000 km

^{2}. Figure 3 presents the proportion and number of gauges for each hydrographic region stratified by classes of record length. The PRN hydrographic region has almost 30% (325) of all gauges used in the study, but the ASD has the most gauges (92) with a record length greater than 60. The PNB hydrographic region has the smallest number of gauges with only 16.

#### 2.3. Streamflow Indices

#### 2.4. Presence of Reservoirs

^{2}. About a quarter of them (66,372) are composed of natural water bodies with a total area of 128,165.89 km

^{2}. The remaining are artificial water bodies that account for a total area of 45,583.76 km

^{2}.

#### 2.5. Trend Detection Framework

#### 2.5.1. Mann–Kendall Trend Test

_{1}, x

_{2}, x

_{3}, . . ., x

_{n}} a time series of length n, then the MK test statistic S is given by:

_{0}for the test is ‘‘there is no trend in the time series”. If H

_{0}is true, then S is normally distributed with zero mean and variance ${\sigma}_{0}^{2}$,

_{i}is the number of ties of size i. A tied group is a set of sample data having the same values. In cases where the sample size n is greater than 10, S is approximately normally distributed, which justifies the use of the standard normal variate, Z, as the test statistic:

#### 2.5.2. Theil–Sen Slope Estimator

_{i}and x

_{j}are the data values at times $1\le i<j\le n$. The decadal relative TS slope estimator of β is given by:

#### 2.5.3. Adjustment for Autocorrelated Data

_{1}, given by:

_{1}, falls within the interval given by Equation (8), the sample data are assumed to be serially independent. Otherwise, the sample data are considered serially correlated.

_{1}has a downward bias (negatively) [92], we also use a bias-corrected estimator based on [101]:

- Estimate the magnitude of the trend, $\beta $, using Equation (7).
- Obtain a detrended series, ${X}_{t}^{d}$, by removing the estimated trend from of the original series, ${X}_{t}^{d}={X}_{t}-\widehat{\beta}{\overline{X}}_{t}t/10$, where t is the time interval.
- Estimate an unbiased sample autocorrelation (r
_{1}) of the detrended series, ${X}_{t}^{d}$. - If r
_{1}is not statistically different from zero, then the MK test is applied to the original series, ${X}_{t}$. Otherwise, the PW procedure is applied to the original series, ${X}_{t}$, to obtain a prewhitened series, ${Y}_{t}^{MTFPW}={X}_{t}-{r}_{1}{X}_{t-1}$. - Apply the MK test to ${Y}_{t}^{MTFPW}$ to check the significance of the trend.

#### 2.5.4. The Multiplicity Problem

## 3. Results and Discussion

#### 3.1. Impacts of Serial Correlation and Multiplicity of Tests

#### 3.2. Trend Analysis in Hydrographic Regions

#### 3.3. Analysis with Gauges Unaffected by Reservoirs

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**Location of the streamflow gauges used in the analysis. Red circles indicate gauges with 30–44 years of record length, yellow circles indicate gauges with 45–59 years, and blue circles represent gauges with at least 60 years of data.

**Figure 3.**Proportion and number of gauges located in the 12 hydrographic regions shown by classes of record length. Red, yellow, and blue bars indicate classes of record length, respectively, 30–44 years, 45–59 years, and at least 60 years of data.

**Figure 4.**DoR values for all river stretches of the national river network. Gauges with $DoR\le 0.02$ were considered unaffected by the presence of reservoirs.

**Figure 5.**Flowchart describing a series of steps for Brazilian streamflow trend analysis in this study.

**Figure 6.**Proportion (%) of gauges with significant trends for the eight streamflow indices using three different approaches: MK, MK-MTFPW, and MK -MTFPW-FDR.

**Figure 7.**Spatial distribution of trend detection results for Q7. Red (blue) circles represent gauges with decreasing (increasing) trends, while black dots represent gauges with no significant trends. Panels (

**a**–

**c**) present, respectively, the results obtained by the MK, MK-MTFPW, and MK-MTFPW-FDR approaches.

**Figure 8.**Trend detection using the MK-MTFPW approach applied to 1106 streamflow gauges in Brazil. Results are shown for each of the 12 hydrographic regions. Red, gray, and blue bars represent significant decreasing, no significant, and significant increasing trends, respectively. Panel (

**a**): Q7; panel (

**b**): Q

_{mean}; panel (

**c**): QX1d; panel (

**d**): QX30d.

**Figure 9.**Results of trend detection study based on the MK-MTFPW-FDR approach for Q7d (panel (

**a**)), Qmean (panel (

**b**)), QX1d (panel (

**c**)), and QX30d (panel (

**d**)).

**Figure 10.**Spatial distribution of gauges with significant trends for Q7 and QX1d for two different datasets. Panel (

**a**): Q7 for gauges unaffected by reservoirs upstream. Panel (

**b**): Q7 for all 1106 gauges. Panel (

**c**): QX1d for gauges unaffected by reservoirs upstream. Panel (

**d**): QX1d for all 1106 gauges.

Index | Definition |
---|---|

QX1d | Annual maximum daily flow |

QX5d | Annual maximum 5-day consecutive average flow |

QX30d | Annual maximum 30-day consecutive average flow |

Qmean | Mean annual streamflow |

Q7 | Annual minimum 7-day consecutive average flow |

Q30 | Annual minimum 30-day consecutive average flow |

Q7Wtri | Annual minimum 7-day consecutive average flow (wettest trimester) |

Q7Wsem | Annual minimum 7-day consecutive average flow (wettest semester) |

**Table 2.**Percentage of MK trend test results for the 1106 stations across Brazil (NS = nonsignificant and S = significant) for the eight streamflow indices stratified in three classes of record length and three classes of relative change per decade ($\widehat{\beta})$ .

Indice | Res | $\widehat{\mathit{\beta}}$ | $\widehat{\mathit{\beta}}$ | $-5\%\widehat{\mathit{\beta}}-10\%$ | $5\%\widehat{\mathit{\beta}}10\%$ | $0\%\widehat{\mathit{\beta}}-5\%$ | $0\%\widehat{\mathit{\beta}}5\%$ | Tot | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

30–44 years | 45–59 years | >60 years | 30–44 years | 45–59 years | >60 years | 30–44 years | 45–59 years | >60 years | 30–44 years | 45–59 years | >60 years | 30–44 years | 45–59 years | >60 years | 30–44 years | 45–59 years | >60 years | |||

QX1d | NS | 6% | 1% | 0% | 2% | 0% | 0% | 86% | 2% | 1% | 5% | 3% | 2% | 12% | 6% | 9% | 11% | 7% | 10% | 86% |

7% | 3% | 11% | 9% | 27% | 28% | |||||||||||||||

S | 4% | 1% | 1% | 1% | 0% | 1% | 0% | 0% | 1% | 1% | 1% | 2% | 0% | 0% | 0% | 0% | 0% | 0% | 14% | |

6% | 2% | 2% | 3% | 0% | 1% | |||||||||||||||

QX5d | NS | 7% | 1% | 0% | 2% | 0% | 0% | 9% | 3% | 1% | 5% | 3% | 2% | 10% | 6% | 9% | 11% | 6% | 10% | 87% |

8% | 2% | 13% | 10% | 26% | 27% | |||||||||||||||

S | 4% | 2% | 1% | 0% | 1% | 0% | 0% | 0% | 1% | 1% | 0% | 2% | 0% | 0% | 0% | 0% | 0% | 1% | 13% | |

6% | 1% | 2% | 3% | 1% | 1% | |||||||||||||||

QX30d | NS | 6% | 1% | 0% | 2% | 0% | 0% | 9% | 3% | 2% | 5% | 3% | 2% | 10% | 6% | 11% | 11% | 6% | 8% | 84% |

8% | 2% | 13% | 9% | 27% | 25% | |||||||||||||||

S | 5% | 2% | 1% | 0% | 0% | 0% | 1% | 0% | 2% | 1% | 0% | 3% | 0% | 0% | 0% | 0% | 0% | 0% | 16% | |

7% | 1% | 3% | 4% | 1% | 0% | |||||||||||||||

Qmean | NS | 8% | 2% | 0% | 2% | 0% | 0% | 7% | 3% | 2% | 6% | 2% | 1% | 10% | 8% | 10% | 9% | 5% | 7% | 81% |

11% | 2% | 11% | 8% | 28% | 21% | |||||||||||||||

S | 6% | 2% | 0% | 0% | 0% | 0% | 1% | 1% | 2% | 1% | 1% | 4% | 0% | 0% | 0% | 0% | 0% | 0% | 19% | |

8% | 1% | 3% | 6% | 0% | 0% | |||||||||||||||

Q7 | NS | 10% | 3% | 1% | 4% | 0% | 0% | 7% | 3% | 2% | 3% | 3% | 1% | 8% | 6% | 8% | 7% | 3% | 7% | 77% |

14% | 5% | 12% | 7% | 22% | 16% | |||||||||||||||

S | 7% | 2% | 1% | 2% | 1% | 2% | 1% | 1% | 1% | 0% | 0% | 3% | 0% | 0% | 0% | 0% | 0% | 0% | 23% | |

11% | 4% | 3% | 3% | 1% | 0% | |||||||||||||||

Q30 | NS | 8% | 3% | 0% | 4% | 1% | 0% | 7% | 4% | 2% | 4% | 2% | 1% | 9% | 6% | 9% | 6% | 3% | 6% | 77% |

12% | 5% | 13% | 7% | 24% | 16% | |||||||||||||||

S | 8% | 2% | 1% | 2% | 1% | 1% | 2% | 1% | 1% | 0% | 1% | 3% | 0% | 0% | 0% | 0% | 0% | 0% | 23% | |

11% | 4% | 4% | 4% | 0% | 0% | |||||||||||||||

Q7WTri | NS | 8% | 3% | 0% | 2% | 0% | 0% | 5% | 4% | 2% | 3% | 2% | 1% | 8% | 7% | 9% | 9% | 5% | 8% | 78% |

11% | 2% | 12% | 7% | 24% | 21% | |||||||||||||||

S | 7% | 5% | 2% | 1% | 0% | 0% | 0% | 1% | 2% | 0% | 0% | 2% | 0% | 0% | 0% | 0% | 0% | 0% | 22% | |

14% | 1% | 3% | 2% | 1% | 0% | |||||||||||||||

Q7WSem | NS | 7% | 2% | 0% | 3% | 0% | 0% | 8% | 4% | 2% | 5% | 2% | 2% | 9% | 5% | 8% | 7% | 4% | 7% | 76% |

9% | 3% | 13% | 9% | 23% | 19% | |||||||||||||||

S | 7% | 4% | 1% | 1% | 1% | 1% | 1% | 2% | 2% | 0% | 0% | 2% | 0% | 0% | 1% | 0% | 0% | 0% | 24% | |

13% | 2% | 5% | 3% | 1% | 0% |

_{mean}, QX1d, and QX30d are presented, but they provide a representation of the situation.

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**MDPI and ACS Style**

Souza, S.A.d.; Reis, Jr., D.S.
Trend Detection in Annual Streamflow Extremes in Brazil. *Water* **2022**, *14*, 1805.
https://doi.org/10.3390/w14111805

**AMA Style**

Souza SAd, Reis, Jr. DS.
Trend Detection in Annual Streamflow Extremes in Brazil. *Water*. 2022; 14(11):1805.
https://doi.org/10.3390/w14111805

**Chicago/Turabian Style**

Souza, Saulo A. de, and Dirceu S. Reis, Jr.
2022. "Trend Detection in Annual Streamflow Extremes in Brazil" *Water* 14, no. 11: 1805.
https://doi.org/10.3390/w14111805