# Bias Correction of RCM Precipitation by TIN-Copula Method: A Case Study for Historical and Future Simulations in Cyprus

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Study Area and Climate Characteristics

#### 2.2. Data

#### 2.3. Methodology

- (1)
- The monthly total and extreme precipitation values were estimated from the daily data series (observed and model). As a metric for extreme precipitation, the 99th percentile of the daily precipitation values for each month was selected, based on the literature [45]. Consequently, using the initial daily data series, the monthly data series were calculated for the two studied variables (total and extreme precipitation).
- (2)
- Bias correction with the TIN-Copula method starts with the TINs formation. Ten stations (Table 1: stations with *, Figure 3: red points) are selected attending the Cyprus area to be covered by triangles (Figure 3: green triangles). The TIN network formation was based on the Delaunay triangulation [42] and, as a consequence, the formatted triangles are non-overlapping. Bias correction with the TIN-Copula method can be applied to every model grid point within these triangles.
- (3)
- Two main calculations are carried out at the triangles’ vertices (10 default stations), using only the observed data. More specifically:
- (a)
- At every vertex (station) of the formatted triangles, more than 20 copula families (including the rotated versions) (Table 2) are tested in order to select the most appropriate one for the description of the studied variables dependence. The selection of the final copula family is made according to the AIC [46] and BIC [47] criteria. The robustness of the selection is higher when the length of the data series is greater.
- (b)
- Apart from the copula family selection, the mathematical distributions (marginals) that satisfactorily fit the studied variables are also selected. Six commonly used distributions are tested (normal, log-normal, Gamma, Pareto, generalized extreme value distribution (GEV), Weibull) and the final selection relies on the same criteria (AIC, BIC).

- (4)
- The second round of the estimations is focused on the model grid points—the x-points (Figure 3—blue points).
- (a)
- Initially, the distances between each x-point (model grids) and the vertices within the triangle are calculated (Figure 3: red points), resulting in a distance index (Wn, n = 1...3 n: the triangle vertices). The greatest value of the distance index is calculated for the vertex with the largest distance.

- (5)
- Combining the distance index (W) with the selected copula families of the respect vertices (n), a new function—a new copula family—was calculated at every x-point. Thus, the influence of every vertex copula family on the final new copula is inversely proportional to the respective distance.
- (6)
- A similar procedure (combination of distance index with the selected functions) is followed for the combination of the studied variables marginal distributions at the x-point.
- (7)
- Consequently, for every studied point (x-point) a unique function—a unique new copula family—is calculated. Similar calculations for the new marginal function at the x-point are followed.
- (8)
- The final step of the bias-correction procedure is the use of the x-point values (model values) as inputs in the corresponding new copula function. The output of this function is a normalized dataset, which is fitted by the estimated marginal function. The result is the final bias-corrected dataset.

## 3. Results

#### 3.1. Historical Period (1986–2000)

#### 3.1.1. Seasonal and Annual Total Precipitation

#### 3.1.2. Seasonal and Annual Extreme Precipitation

#### 3.2. Future Periods

#### Projections

#### 3.3. Climate Change Signal

## 4. Discussion and Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Location and topography of the Cyprus area (image provided by Google Earth: https://earth.google.com/web/). Map of area’s classification according to Principal Component Analysis PCA (bottom-left map). Red points—three points with different characteristics; one form each class.

**Figure 2.**Bagnouls–Gaussen’s ombrothermic diagram of Cyprus (period 1986–2000—average of 10 meteorological stations).

**Figure 3.**Location of the studied points. Red and yellow points are the available stations. The numbers of these points are in accordance with Table 1. Blue points are the location of the model’s grids. The green triangles present the designed triangular irregular network.

**Figure 4.**(Left maps) Observed seasonal total precipitation values. (Right maps) Biases between observations and the model’s estimations before and after bias correction (TIN-Copula).

**Figure 5.**Grid points with statistically significant biases, before and after the application of the TIN-Copula bias correction (t-test, level of significance 95%).

**Figure 6.**Monthly mean total precipitation values in Cyprus for the period of 1986–2000. Presentation with (

**a**) line plots, (

**b**) Taylor diagram and (

**c**) boxplots.

**Figure 7.**Line plots of the monthly and quantile–quantile plots of the annual total precipitation values of the three selected grid points, presented in Figure 1. (

**Grid 1**: coastal grid,

**Grid 2**: mountainous grid,

**Grid 3**: continental grid).

**Figure 8.**(Left maps) Observed winter extreme precipitation values. (Right maps) Biases between observations and model’s estimations before and after bias correction with the TIN-Copula method.

**Figure 9.**Grid points with statistically significant biases, before and after the application of the TIN-Copula bias correction (t-test, level of significance 95%).

**Figure 10.**Monthly mean extreme precipitation values in Cyprus for the period of 1986–2000. Presentation with (

**a**) line plots, (

**b**) Taylor diagram and (

**c**) boxplots.

**Figure 11.**Line plots of the monthly and quantile–quantile plots of the annual extreme precipitation values of the three selected grid points, presented in (Figure 1). (

**Grid 1**: coastal grid,

**Grid 2**: mountainous grid,

**Grid 3**: continental grid).

**Figure 12.**Projection for the annual total (

**A**—left panels) and extreme (

**B**—right panels) precipitation, before and after bias correction (model—TIN-Copula), for three selected grid points (Figure 1) and for two future periods.

**Figure 13.**Climate change signal (CCS) of total precipitation, before and after the use of the TIN-Copula method.

**Figure 14.**Climate change signal (CCS) of extreme precipitation, before and after the use of the TIN-Copula method.

Code | Station | Lon | Lat | Height (m) | Code | Station | Lon | Lat | Height (m) |
---|---|---|---|---|---|---|---|---|---|

1 | Agios | 34.88 | 33.03 | 995 | 18 | Meniko | 35.12 | 33.15 | 265 |

2 | Akrounta | 34.77 | 33.08 | 110 | 19 | Nicosia * | 35.16 | 33.35 | 160 |

3 | Alaminos | 34.8 | 33.43 | 70 | 20 | Ora | 34.87 | 33.2 | 520 |

4 | Amargeti | 34.83 | 32.58 | 420 | 21 | Pachna | 34.78 | 32.8 | 710 |

5 | Amiantos * | 34.93 | 32.92 | 1397 | 22 | Pafos * | 34.72 | 32.48 | 10 |

6 | Apesia | 34.78 | 32.98 | 470 | 23 | Panagia* | 34.92 | 32.63 | 871 |

7 | Dora | 34.78 | 32.75 | 605 | 24 | Panagia | 35.02 | 33.08 | 440 |

8 | Filousa | 34.85 | 32.72 | 440 | 25 | Pano | 34.93 | 32.92 | 1380 |

9 | Kapoura | 35.02 | 33.00 | 580 | 26 | Pedoulas | 34.97 | 32.83 | 1080 |

10 | Kato | 34.85 | 33.3 | 510 | 27 | Pera | 35.02 | 33.38 | 255 |

11 | Kato | 35.08 | 33.28 | 270 | 28 | Prodromos * | 34.95 | 32.83 | 1423 |

12 | Koilani | 34.85 | 32.87 | 820 | 29 | Psevdas | 34.95 | 33.47 | 160 |

13 | Larnaca * | 34.88 | 33.63 | 2 | 30 | Saittas * | 34.86 | 32.91 | 641 |

14 | Lefkara * | 34.9 | 33.29 | 391 | 31 | Stavros * | 35.02 | 32.63 | 810 |

15 | Limassol * | 34.66 | 33.02 | 31 | 32 | Tripylos | 35.00 | 32.68 | 1220 |

16 | Mantra | 34.953 | 33.23 | 640 | 33 | Vretsia | 34.9 | 32.65 | 560 |

17 | Mathiatis | 34.95 | 33.33 | 375 | 34 | Ypsonas | 34.7 | 32.97 | 80 |

Family Name | Function | Kendall τ | Parameter | Tail Dependence | |
---|---|---|---|---|---|

Elliptical Families | |||||

1 | Gaussian | C(u_{1}, u_{2}) = Φ_{ρ}(Φ^{−1}(u_{1})_{,} Φ^{−1}(u_{2})) | $\frac{2}{\pi}$arcsin(ρ) | ρ ∈ (−1,1) | 0 |

2 | Student-t | C(u_{1}, u_{2}) = t_{ρ,ν}(t_{ν}^{−1}(u_{1})_{,} t_{ν}^{−1}(u_{2})) | $\frac{2}{\pi}$arcsin(ρ) | ρ ∈ (−1,1), v > 2 | 2t_{v+1}(−$\sqrt{v+1}\sqrt{\frac{1-\rho}{1+\rho}}$) |

- Φρ denotes the standard bivariate normal distribution function and θ is the correlation coefficient. - tρ,ν denotes the standard bivariate Student-t distribution with correlation coefficient ρ and v degrees of freedom. | |||||

Archimedean Families | |||||

3 | Clayton | $\frac{1}{\theta}$(${t}^{-\theta}-1)$ | $\frac{\theta}{\theta +2}$ | θ > 0 | (${2}^{-\frac{1}{\theta}}$,0) |

4 | Gumbel | (−$\mathrm{log}t$)^{θ} | 1 −$\frac{1}{\theta}$ | θ ≥ 1 | (0,2−${2}^{\frac{1}{\theta}}$) |

5 | Frank | $\mathrm{log}\frac{{e}^{-\theta t}-1}{{e}^{-\theta}-1}$ | 1 −$\frac{4}{\theta}$ + 4$\frac{D1\left(\theta \right)}{\theta}$ | θ ∈ R\{0} | (0,0) |

6 | Joe | −$\mathrm{log}\left[1-{\left(1-t\right)}^{\theta}\right]$ | 1 + $\frac{4}{{\theta}^{2}}$ ${{\displaystyle \int}}_{0}^{1}tlog\left(t\right){\left(1-t\right)}^{2\left(1-\theta \right)/\theta}$dt | θ > 1 | (0,2−${2}^{\frac{1}{\theta}}$) |

7 | ΒΒ1 (Clayton + Gumbel) | ${\left({t}^{-\theta}-1\right)}^{\delta}$ | 1 −$\frac{2}{\delta \left(\theta +2\right)}$ | θ > 0, δ ≥ 1 | (${2}^{-\frac{1}{\theta \delta}}$, 2 − ${2}^{\frac{1}{\theta}})$ |

8 | BB6 (Joe + Gumbel) | ${\left(-\mathrm{log}\left[1-{\left(1-t\right)}^{\theta}\right]\right)}^{\delta}$ | 1 + $\frac{4}{\delta \theta}$ ${{\displaystyle \int}}_{0}^{1}\left(-\mathrm{log}\left(1-{\left(1-t\right)}^{\theta}\right)\times \left(1-t\right)(1-{\left(1-t\right)}^{-\theta}\right))$dt | θ ≥ 1, δ ≥ 1 | (0, 2−${2}^{\frac{1}{\theta \delta}}$) |

9 | ΒΒ7 (Joe + Clayton) | ${\left(1-{\left(1-t\right)}^{\theta}\right)}^{-\delta}-1$ | 1 + $\frac{4}{\delta \theta}$ ${{\displaystyle \int}}_{0}^{1}(-{\left(1-{\left(1-t\right)}^{\theta}\right)}^{\delta +1}\times \frac{{(1-{\left(1-t\right)}^{\theta})}^{-\delta}-1}{{\left(1-t\right)}^{\theta -1}}$dt | θ ≥ 1, δ > 0, | (${2}^{-\frac{1}{\theta}},$2−${2}^{\frac{1}{\theta}}$) |

10 | ΒΒ8 (Joe + Frank) | −$\mathrm{log}\left[\frac{1-{\left(1-\delta t\right)}^{\theta}}{1-{\left(1-\delta \right)}^{\theta}}\right]$ | 1 + $\frac{4}{\delta \theta}$ ${{\displaystyle \int}}_{0}^{1}\left(-\mathrm{log}\left(\frac{{\left(1-t\delta \right)}^{\theta}-1}{{\left(1-\delta \right)}^{\theta}-1}\right)\times \left(1-t\delta \right)(1-{\left(1-t\delta \right)}^{-\theta}\right))$dt | θ ≥ 1, δ ∈ (0, 1] | (0, 0) |

The version of the families rotated by 90, 180 and 270 degrees: | C_{90} (u_{1}, u_{2}) = u_{2} – C ( 1− u_{1}, u_{2})C _{180} (u_{1}, u_{2}) = u_{1} + u_{2} –1 + C (1 − u_{1},1 − u_{2})C _{270} (u_{1}, u_{2}) = u_{1} – C (u_{1},1 − u_{2}) | ||||

− D_{1} (θ) = ${{\displaystyle \int}}_{0}^{\theta}\frac{c/\theta}{\mathrm{exp}\left(x\right)-1}$ dx is the Debye function. |

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## Share and Cite

**MDPI and ACS Style**

Lazoglou, G.; Zittis, G.; Anagnostopoulou, C.; Hadjinicolaou, P.; Lelieveld, J.
Bias Correction of RCM Precipitation by TIN-Copula Method: A Case Study for Historical and Future Simulations in Cyprus. *Climate* **2020**, *8*, 85.
https://doi.org/10.3390/cli8070085

**AMA Style**

Lazoglou G, Zittis G, Anagnostopoulou C, Hadjinicolaou P, Lelieveld J.
Bias Correction of RCM Precipitation by TIN-Copula Method: A Case Study for Historical and Future Simulations in Cyprus. *Climate*. 2020; 8(7):85.
https://doi.org/10.3390/cli8070085

**Chicago/Turabian Style**

Lazoglou, Georgia, George Zittis, Christina Anagnostopoulou, Panos Hadjinicolaou, and Jos Lelieveld.
2020. "Bias Correction of RCM Precipitation by TIN-Copula Method: A Case Study for Historical and Future Simulations in Cyprus" *Climate* 8, no. 7: 85.
https://doi.org/10.3390/cli8070085