# Climate Extrapolations in Hydrology: The Expanded Bluecat Methodology

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## Abstract

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## 1. Introduction

## 2. Bluecat and Its Expansion to Deal with Climate Projections

- To appropriately correct the D-model bias, which may differ for different ranges of $Q$;
- To infer the model uncertainty in terms of confidence bands, whose width may also differ for different ranges of $Q$.

## 3. Data

## 4. Results

## 5. Discussion and Conclusions

- It appropriately corrects the D-model bias.
- It infers the model uncertainty in terms of confidence bands.

- If the conditioning value $Q$ lies within the range of “observations”, the conditional distribution is inferred using a subset of pairs $\left(q,Q\right)$ with adjacent $Q$ values. Namely, those pairs satisfying $Q-\Delta {Q}_{1}\le \underset{\_}{Q}\le Q+\Delta {Q}_{2}$ are chosen, where $\Delta {Q}_{1}$ and $\Delta {Q}_{2}$ are properly specified to form a sample of pairs $\left(q,Q\right)$ that allows the empirical estimation of the conditional probabilities ${F}_{q|Q}\left(q|Q\right)\approx P\left\{\underset{\_}{q}\le q|Q-\Delta {Q}_{1}\le \underset{\_}{Q}\le Q+\Delta {Q}_{2}\right\}$.
- Otherwise, an extrapolation is made by the following method: A constant $c\ne 1$ is chosen so that $cQ$ lie within the range of “observations” and the above method be applicable by replacing or $Q$ with $cQ$. Then, the conditional distribution is estimated by enrolling the relationship ${F}_{q|Q}\left(q|Q\right)\approx {F}_{q|Q}\left(q+a\left(1-1/c\right)cQ|cQ\right)$, where $a$ is a parameter representing a regression slope between $Q$ and $q$. This parameter is locally determined for the high or low values of $Q$, depending on the direction of the extrapolation.

- The scales of the application of the Bluecat’s S-model are not necessarily different from that of the climate D-model. They could be precisely the same, as in the present application to Italy. In this respect, Bluecat makes the D-model consistent with reality irrespective of spatial scale. Apparently, the case where the scales are different (e.g., smaller area of the S-model with respect to that corresponding to the D-model) is also served by the proposed methodology, without any change with respect to what is described above.
- The Bluecat methodology considers both the time sequence and the amplitude of the D-model and actual series. It does not make a lumped fitting on the entire set of past data to find unique parametric relationships to be applied to the adaptation of the future values (as usually conducted in downscaling techniques). Nor does it regard the future values as correct ones that only need downscaling. Rather, it treats past and future values produced by the D-model in the same manner—as representing a model and not the truth.
- In addition to modifying a D-model series, correcting it for bias, Bluecat advances it to a stochastic representation, thus characterizing the uncertainty that is illustrated in this paper in terms of confidence bands.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Comparison of the actual and D-model temperature data: (

**upper**) original monthly data; (

**middle**) after the aggregation to annual scale; (

**lower**) after the subtraction of monthly means to reduce the effect of periodicity.

**Figure 3.**Comparison of actual and D-model precipitation data: (

**upper**) original monthly data; (

**lower**) after aggregation to annual scale.

**Figure 4.**True values vs. D-model predictions for temperature, along with the S-model predictions (median and 90% confidence limits). The vertical dotted lines define the area out of which extrapolation is necessary.

**Figure 5.**True values and predictions by D-model and S-model (median and 90% confidence limits) of temperature in Italy: (

**upper**) entire period; (

**middle**and

**lower**) focus on 20-year periods.

**Figure 6.**As Figure 5, but with S-model results (median and 90% confidence limits) replaced by running averages over 12 months.

**Figure 7.**Modified Figure 5 for two hypothetical cases, in which the D-model values are reconstructed so as to be (

**upper**) very close to reality, by replacing the D-model series with the weighted sum of true and D-model values with weights of 0.75 and 0.25, respectively, and (

**lower**) completely irrelevant to reality, by randomly rearranging the time order the true values.

**Figure 8.**True values vs. D-model predictions for precipitation, transformed for normalization, along with S-model predictions (median and 90% confidence limits). The vertical dotted lines define the area out of which extrapolation is necessary.

**Figure 9.**True values and predictions by D-model and S-model (median and 90% confidence limits) of precipitation in Italy: (

**upper**) entire period; (

**middle**and

**lower**) focus on 20-year periods. (Nb. A normalizing transformation by Equation (10) with $\lambda =2$ mm/d is used, while the plotted values are back transformed).

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**MDPI and ACS Style**

Koutsoyiannis, D.; Montanari, A.
Climate Extrapolations in Hydrology: The Expanded Bluecat Methodology. *Hydrology* **2022**, *9*, 86.
https://doi.org/10.3390/hydrology9050086

**AMA Style**

Koutsoyiannis D, Montanari A.
Climate Extrapolations in Hydrology: The Expanded Bluecat Methodology. *Hydrology*. 2022; 9(5):86.
https://doi.org/10.3390/hydrology9050086

**Chicago/Turabian Style**

Koutsoyiannis, Demetris, and Alberto Montanari.
2022. "Climate Extrapolations in Hydrology: The Expanded Bluecat Methodology" *Hydrology* 9, no. 5: 86.
https://doi.org/10.3390/hydrology9050086