# Characterization of Temporal and Spatial Variability of Phosphorus Loading to Lake Erie from the Western Basin Using Wavelet Transform Methods

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## Abstract

**:**

## 1. Introduction

^{6}kg was achieved for the first time in 1982 and, since then, the target has been maintained except during extremely wet years. Even though the target load level has been maintained and TP started to decrease as a result of the point-source reduction and the effects of conservation practices, soluble reactive phosphorus (SRP) from several tributaries especially the western basin showed an increase. As a result, Lake Erie tended to be more eutrophic [16], which is indicated by the increase in cyanobacteria since the mid-1990s [17,18].

## 2. Theoretical Background

#### 2.1. Wavelet Analysis

_{n}using WT can be accomplished using the following function [35].

#### 2.2. Continuous Wavelet Transform (CWT)

_{0}(n) for any time series x

_{n}(n = 0, …, N − 1) for δt time spacing with a zero mean localized in time and frequency space. The choice and selection of the wavelet function depends on the analyzed series. The Morlet wavelet function is considered appropriate for hydrological time series because it strikes a proper balance between time and frequency [34]. The function is defined below.

_{o}are the non-dimensional time parameter and non-dimensional frequency, respectively. Readers can refer to [34] for detail about the CWT.

#### 2.3. Discrete Wavelet Transform (DWT)

_{0}symbolizes the location variable whose value is greater than zero. For practical reasons and in order to get the dyadic scales position, the values for s

_{0}and γ

_{0}are normally chosen to be 2 and 1, respectively [38,42]. The wavelet coefficients (C) of the DWT can then be calculated using the following equation [41], where scale s = 2j and location γ = 2jk.

## 3. Methodology

## 4. Study Area

#### 4.1. Continuous Wavelet Transform

_{0}= 6 was used in all CWT analysis because it gives a good balance between time and frequency localizations [34]. CWT computed the wavelet power spectrum and its amplitude for each of the analyzed time series. The significance level for each wavelet spectrum was tested against the red noise modelled as a univariate lag-1 autoregressive (AR-1) process. The wavelet variance as indicated in the GWPS was computed by incorporating the squared transform coefficients at different scales for all data. The dashed line (in a GWPS graph) shows the area of global wavelet significance, which depicts the 95% confidence limit above the line. The CWT and wavelet power spectra were determined using normalized daily time series.

#### 4.2. Discrete Wavelet Transform

## 5. Results and Discussion

#### 5.1. Wavelet Analysis

#### 5.1.1. Raisin Watershed—Phosphorus

#### 5.1.2. Raisin Watershed—Flow

#### 5.1.3. Maumee Watershed—Phosphorus

#### 5.1.4. Maumee Watershed—Flow

^{1}years (based on the dyadic scale). Significant variability was detected at 6 months (especially from 1975 to 1985) and 6 years (continuously from 1988 to 2002). In addition to the 1-year scale, the GWPS showed that the periodicity at around 4 years was also significant but with smaller variance (Figure 8a, right panel). The GWPS showed that the 4-year periodicity was significant but had a smaller variance (Figure 8a, right panel). Similarly, based on the DWT results, it appeared that the trends in streamflow were also influenced by the 4-year periodicity and more strongly by the 8-year periodicity (Figure 8b).

#### 5.1.5. Sandusky Watershed—Phosphorus

#### 5.1.6. Sandusky Watershed—Flow

#### 5.1.7. Vermilion Watershed—Phosphorus

#### 5.1.8. Vermilion Watershed—Flow

#### 5.1.9. Cuyahoga Watershed—Phosphorus

#### 5.1.10. Cuyahoga Watershed—Flow

#### 5.1.11. Grand Watershed—Phosphorus

#### 5.1.12. Grand Watershed—Flow

## 6. Summary and Conclusions

- The annual periodicity was highly significant and it contains the largest proportion of the variance of the time series. However, the variance varied considerably among the different sites.
- In addition to the annual periodicity, significant periodicities at higher scales were also revealed especially when long-term data were used. For the Maumee River, the 2-year and 8-year periodicities were also significant and, for the Sandusky River, the 2-year periodicity was significant. Periodicity of 2 and 8 year for Maumee River indicated the significant loading from Maumee River.
- The annual pattern of P loading was continuously significant for most sites. However, it was not significant in Grand and Cuyahoga Rivers.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**The locations of the sampling stations in Cuyahoga, Grand, Maumee, Vermilion, Raisin, and Sandusky watersheds.

**Figure 3.**Raisin River: (

**a**) Wavelet power spectrum of daily total phosphorus (TP), (

**b**) Plots of the sequential Mann–Kendall analyses on the Details (with added Approximation component) of the annual TP data. The top and bottom dashed lines are ±1.96, which are the confidence limits of the MK test using α = 5%. The solid and dotted lines show the sequential MK values for the original data and the Detail components, respectively. For this TP data, the D1 (2 years) and D3 (8 years) components were the most dominant periodicities for the trend.

**Figure 4.**Raisin River: (

**a**) Wavelet power spectrum of daily soluble reactive phosphorus (SRP), (

**b**) Plots of the sequential Mann–Kendall analyses on the Details (with added Approximation component) of the annual SRP data. The top and bottom dashed lines are ±1.96, which are the confidence limits of the MK test using α = 5%. The solid and dotted lines show the sequential MK values for the original data and the Detail components, respectively. For this SRP data, the D1 (2 years) component was the most dominant periodicity for trend.

**Figure 5.**Raisin River: (

**a**) Wavelet power spectrum of daily Streamflow. (

**b**) Plots of the sequential Mann–Kendall analyses on the Details (with added Approximation component) of the annual streamflow data. The top and bottom dashed lines are ±1.96, which are the confidence limits of the MK test using α = 5%. The solid and dotted lines show the sequential MK values for the original data and the Detail components, respectively. For this streamflow data, the D3 (8 years) component was the most dominant periodicity for trend.

**Figure 6.**Maumee River: (

**a**) Continuous wavelet transform of the daily total phosphorus (TP). (

**b**) Plots of the sequential Mann–Kendall analyses on the Details (with added Approximation component) of the annual TP data. The top and bottom dashed lines are ±1.96, which are the confidence limits of the MK test using α = 5%. The solid and dotted lines show the sequential MK values for the original data and the Detail components, respectively. For this TP data, the D1 (2 years), D2 (4 years), and D3 (8 years) components were the most dominant periodicities for the trend.

**Figure 7.**Maumee River: (

**a**) Continuous wavelet transform of the daily soluble reactive phosphorus (SRP). (

**b**) Plots of the sequential Mann–Kendall analyses on the Details (with added Approximation component) of the annual SRP data. The top and bottom dashed lines are ±1.96, which are the confidence limits of the MK test using α = 5%. The solid and dotted lines show the sequential MK values for the original data and the Detail components, respectively. For this SRP data, the D1 (2 years) and D3 (8 years) components were the most dominant periodicities for this trend.

**Figure 8.**Maumee River: (

**a**) Continuous wavelet transform on a daily basis. (

**b**) Plots of the sequential Mann–Kendall analyses on the Details (with added Approximation component) of the annual streamflow data. The top and bottom dashed lines are ±1.96, which are the confidence limits of the MK test using α = 5%. The solid and dotted lines show the sequential MK values for the original data and the Detail components, respectively. For this streamflow data, the D2 (4 years) and D3 (8 years) components were the most effective periodic oscillation.

**Figure 9.**Sandusky River: (

**a**) Wavelet power spectrum of the daily total phosphorus (TP). (

**b**) Plots of the sequential Mann–Kendall analyses on the Details (with added Approximation component) of the annual TP data. The top and bottom dashed lines are ±1.96, which are the confidence limits of the MK test using α = 5%. The solid and dotted lines show the sequential MK values for the original data and the Detail components, respectively. For this TP data, until the early 2000s, the D3 (8 years) components was the most dominant periodicity for this trend.

**Figure 10.**Sandusky River: (

**a**) Continuous wavelet transform of daily soluble reactive phosphorus (SRP). (

**b**) Plots of the sequential Mann–Kendall analyses on the Details (with added Approximation component) of the annual SRP data. The top and bottom dashed lines are ±1.96, which are the confidence limits of the MK test using α = 5%. The solid and dotted lines show the sequential MK values for the original data and the Detail components, respectively. For this SRP data, the D3 (8 years) component was the most dominant periodicity for this trend.

**Figure 11.**Sandusky River: (

**a**) Continuous wavelet transform of the daily streamflow. (

**b**) Plots of the sequential Mann–Kendall analyses on the Details (with added Approximation component) of the annual streamflow data. The top and bottom dashed lines are ±1.96, which are the confidence limits of the MK test using α = 5%. The solid and dotted lines show the sequential MK values for the original data and the Detail components, respectively. For this streamflow data, the D1 (2 years) and D3 (8 years) components were the most dominant periodicity for this trend.

**Figure 12.**Continuous wavelet power spectrum of (

**a**) daily total phosphorus, (

**b**) soluble reactive phosphorus, and (

**c**) streamflow for the Vermilion River.

**Figure 13.**Cuyahoga River: (

**a**) Continuous wavelet spectrum of daily total phosphorus (TP). (

**b**) Plots of the sequential Mann–Kendall analyses on the Details (with added Approximation component) of the annual TP data. The top and bottom dashed lines are ±1.96, which are the confidence limits of the MK test using α = 5%. The solid and dotted lines show the sequential MK values for the original data and the Detail components, respectively. For this TP data, the D1 (2 years) and D2 (4 years) components were the most dominant periodicities for this trend.

**Figure 14.**Cuyahoga River: (

**a**) Continuous wavelet transform of daily soluble reactive phosphorus (SRP). (

**b**) Plots of the sequential Mann–Kendall analyses on the Details (with added Approximation component) of the annual SRP data. The top and bottom dashed lines are ±1.96, which are the confidence limits of the MK test using α = 5%. The solid and dotted lines show the sequential MK values for the original data and the Detail components, respectively. For this SRP data, the D1 (2 years) and D2 (4 years) components were the most dominant periodicities for this trend.

**Figure 15.**Cuyahoga River: (

**a**) Continuous wavelet spectrum of daily streamflow. (

**b**) Plots of the sequential Mann–Kendall analyses on the Details (with added Approximation component) of the annual streamflow data. The top and bottom dashed lines are ±1.96, which are the confidence limits of the MK test using α = 5%. The solid and dotted lines show the sequential MK values for the original data and the Detail components, respectively. For this streamflow data, the D3 (8 years) component was the most dominant periodicity for this trend.

**Figure 16.**Grand River: (

**a**) Continuous wavelet power spectrum of daily total phosphorus (TP). (

**b**) Plots of the sequential Mann–Kendall analyses on the Details (with added Approximation component) of the annual TP data. The top and bottom dashed lines are ±1.96, which are the confidence limits of the MK test using α = 5%. The solid and dotted lines show the sequential MK values for the original data and the Detail components, respectively. For this TP data, the D2 (4 years) component was the most dominant periodicity for this trend.

**Figure 17.**Grand River: (

**a**) Continuous wavelet transform of daily soluble reactive phosphorus (SRP). (

**b**) Plots of the sequential Mann–Kendall analyses on the Details (with added Approximation component) of the annual SRP data. The top and bottom dashed lines are ±1.96, which are the confidence limits of the MK test using α = 5%. The solid and dotted lines show the sequential MK values for the original data and the Detail components, respectively. For this SRP data, the D3 (8 years) component was the most dominant periodicity for this trend.

**Figure 18.**Grand River: (

**a**) Continuous wavelet transform of daily streamflow. (

**b**) Plots of the sequential Mann–Kendall analyses on the Details (with added Approximation component) of the annual streamflow data. The top and bottom dashed lines are ±1.96, which are the confidence limits of the MK test using α = 5%. The solid and dotted lines show the sequential MK values for the original data and the Detail components, respectively. For this streamflow data, the D1 (2 years) and D2 (4 years) components were the most dominant periodicity for this trend.

Name | Watershed Area (km^{2}) | Land Use Distribution (%) | ||||
---|---|---|---|---|---|---|

Agriculture (Row Crops) | Grassland (Pasture) | Forest | Urban/Residential | Other | ||

Raisin | 2764 | 50 | 19 | 10 | 11 | 10 |

Maumee | 16,995 | 75 | 6 | 6 | 10 | 3 |

Sandusky | 3678 | 78 | 4 | 9 | 8 | 1 |

Vermillion | 694 | 73 | - | 25 | - | 2 |

Cuyahoga | 2095 | 9 | 12 | 35 | 38 | 6 |

Grand | 1826 | 25 | 13 | 44 | 9 | 9 |

**Table 2.**The Mann–Kendall values of the original and DWT annual data of the total phosphorus, streamflow, and soluble reactive phosphorus. Each DWT is composed of the D (details) components (D1–D3), the approximation (A3), and a set of combinations of the Ds and their respective A3. The most effective periodic components for trends are indicated in bold.

Data | Raisin River | Maumee River | Sandusky River | Cuyahoga River | Grand River |
---|---|---|---|---|---|

Total phosphorus | |||||

Original | −0.18 | −0.48 | 0.68 | 1.05 | −0.45 |

D1 | −0.06 | −0.22 | 0.02 | 0.15 | 0.61 |

D2 | −0.53 | −0.13 | 0.68 | −1.05 | −0.23 |

D3 | −0.26 | 0.73 | 1.16 | 0.66 | −0.15 |

A3 | 3.14 * | −0.73 | 2.03 * | −0.79 | −1.36 |

D1 + A3 | 0.61 | 0 | 1.38 | 0.34 | −0.68 |

D2 + A3 | 1.36 | 0.06 | 1.67 | 0.18 | −1.06 |

D3 + A3 | 0.73 | −0.45 | 1.81 | 1.25 | −2.27 * |

Soluble reactive phosphorus | |||||

Original | 2.51 * | 0.36 | 1.98 * | 0.96 | 2.35 * |

D1 | 0.06 | −0.5 | −0.05 | 0.28 | 0 |

D2 | −0.06 | 0.04 | 0.6 | −0.66 | 0.38 |

D3 | −0.53 | −0.48 | 0.39 | 0.5 | −1.36 |

A3 | 4.33 * | 1.22 | 2.54 * | −0.96 | 5.08 * |

D1 + A3 | 2.94 * | 1.04 | 1.65 | −0.02 | 3.33 * |

D2 + A3 | 3.97 * | 1.88 | 3.22 * | 0.02 | 3.11 * |

D3 + A3 | 4.09 * | 1.22 | 2.13 * | 1.12 | 2.65 * |

Streamflow | |||||

Original | 0.41 | 1.21 | 1.52 | 0.57 | 0 |

D1 | 0.14 | −0.27 | −0.29 | 0.05 | 0.3 |

D2 | 0.34 | 0.27 | 0.56 | 0.21 | 0.22 |

D3 | −0.61 | 0.22 | 0.65 | 0.6 | −1.29 |

A3 | 1.76 | 1.84 | 1.52 | 1.48 | 0.23 |

D1 + A3 | 1.17 | 1.86 | 1.43 | 1.31 | 0.61 |

D2 + A3 | 2.43 * | 1.35 | 2.13 * | 2.12 * | 0.68 |

D3 + A3 | 0.06 | 1.57 | 1.48 | 1.57 | −2.05 * |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Sharma, S.; Nalley, D.; Subedi, N. Characterization of Temporal and Spatial Variability of Phosphorus Loading to Lake Erie from the Western Basin Using Wavelet Transform Methods. *Hydrology* **2018**, *5*, 50.
https://doi.org/10.3390/hydrology5030050

**AMA Style**

Sharma S, Nalley D, Subedi N. Characterization of Temporal and Spatial Variability of Phosphorus Loading to Lake Erie from the Western Basin Using Wavelet Transform Methods. *Hydrology*. 2018; 5(3):50.
https://doi.org/10.3390/hydrology5030050

**Chicago/Turabian Style**

Sharma, Suresh, Deasy Nalley, and Naba Subedi. 2018. "Characterization of Temporal and Spatial Variability of Phosphorus Loading to Lake Erie from the Western Basin Using Wavelet Transform Methods" *Hydrology* 5, no. 3: 50.
https://doi.org/10.3390/hydrology5030050