# Budyko-Type Models and the Proportionality Hypothesis in Long-Term Water and Energy Balances

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## Abstract

**:**

## 1. Introduction

## 2. Budyko-Type Functional Relationships

#### 2.1. Budyko Hypothesis and Representation Diagrams

#### 2.2. Budyko-Type Functional Relationships

## 3. Derivation of Selected Budyko-Type Models

## 4. Budyko-Type Models in the P–Q Space

#### 4.1. Model TMPHCY

^{n}, P by P

^{n}, and Q by Q

^{n}.Figure 3a shows the situation when the variables are raised to the n-th power.

#### 4.2. Model SZ

#### 4.3. Model Z

#### 4.4. Model WT

#### 4.5. Non Proportionality Hypothesis Models

## 5. An Expolinear Model of the P–Q Relationship

## 6. Discussion

## 7. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

Symbol | Unit | Description |

P | mm | Precipitation |

E | mm | Actual evapotranspiration. The E symbol in hydrology is commonly used for Evaporation, but we change it in line with common notations in Budyko type models |

Q | mm | Runoff |

E_{0} | mm | Potential evapotranspiration. Symbol changed as used in common hydrology practice |

m, k, n, W, ε, ω, α, b, f, d | - | Fitting parameters of the Budyko type models |

## Appendix A

#### Appendix A.1. Proportionality Hypothesis of the CN-SCS Method

#### Appendix A.2. P-Q Space Limits

**Figure A2.**(

**a**) Water limit in the P-Q space and (

**b**) simple model of a single saturated bucket, with F = 0.

**Figure A3.**(

**a**) Water storage limit (energy limit) in P − Q space and (

**b**) simple model of a single empty bucket, with F > 0.

#### Appendix A.3. Case Ia = 0

**Figure A5.**Case of Ia = 0 for the analysis concerning the line Q = P and Q = P − S, equal triangles.

**Figure A6.**Case of Ia = 0 for the analysis concerning the line Q = P and Q = P − S, similar triangles.

#### Appendix A.4. Case Ia > 0 and Different Origins of P and S

#### Appendix A.5. Case Ia > 0 and Equal Origins of P and S

**Figure A8.**(

**a**) Case of Ia > 0, with S and P with equal origins and (

**b**) a simple model of one bucket with two reservoirs.

#### Appendix A.6. General Discussion

#### Appendix A.7. Historical Context

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**Figure 1.**Budyko-type diagrams showing the relationships (

**a**) Eo/P versus E/P; (

**b**) P/Eo versus E/Eo; and (

**c**) P/Eo versus Q/Eo.

**Figure 3.**Budyko-type space: (

**a**) P

^{n}/(Eo)

^{n}− Q

^{n}/(Eo)

^{n}and (

**b**) P

^{n}/(Eo)

^{n}− (P

^{n}− E

^{n})/(Eo)

^{n}.

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**MDPI and ACS Style**

Paz Pellat, F.; Garatuza Payán, J.; Salas Aguilar, V.; Velázquez Rodríguez, A.S.; Bolaños González, M.A.
Budyko-Type Models and the Proportionality Hypothesis in Long-Term Water and Energy Balances. *Water* **2022**, *14*, 3315.
https://doi.org/10.3390/w14203315

**AMA Style**

Paz Pellat F, Garatuza Payán J, Salas Aguilar V, Velázquez Rodríguez AS, Bolaños González MA.
Budyko-Type Models and the Proportionality Hypothesis in Long-Term Water and Energy Balances. *Water*. 2022; 14(20):3315.
https://doi.org/10.3390/w14203315

**Chicago/Turabian Style**

Paz Pellat, Fernando, Jaime Garatuza Payán, Víctor Salas Aguilar, Alma Socorro Velázquez Rodríguez, and Martín Alejandro Bolaños González.
2022. "Budyko-Type Models and the Proportionality Hypothesis in Long-Term Water and Energy Balances" *Water* 14, no. 20: 3315.
https://doi.org/10.3390/w14203315