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Hybrid model of the near-ground temperature profile

Abstract

The topic of the paper is modelling and prediction of atmospheric variables that are further used for prediction of the consequences of radioactive-material release to the atmosphere. Physics-based models of atmospheric dynamics provide an approximate description of the true nature of a dynamic system. However, the accuracy of the model’s short-term predictions and long-term forecasts, especially over complex terrain, decreases when the information at a micro-location is sought. Integration of a physics-based model with a statistical model for enhancing the prediction power is proposed in the paper. Gaussian Processes models can be used to identify the mapping between the system input and output measured values. With the given mapping function, we can provide one-step ahead prediction of the system output values together with its uncertainty, which can be used advantageously. In this paper, we combine a physics-based model with a Gaussian-process model to identify air temperature from measurements at different atmospheric surface layers as a dynamic system and to make short-term predictions as well as long-term forecasts.

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Acknowledgements

The authors acknowledge the project “Method for the forecasting of local radiological pollution of atmosphere using Gaussian process models”, ID L2-8174, and research core funding No. P2-0001, which were financially supported by the Slovenian Research Agency. We are grateful to the Krško NPP for the measurement data from their automatic measuring system. The discussions and technical assistance with data processing by Martin Stepančič are gratefully acknowledged.

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Correspondence to Juš Kocijan.

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Appendix: Performance measures

Appendix: Performance measures

We use the following statistical measures for the assessment:

  • The normalised root-mean-square error—NRMSE

    $$\begin{aligned} {{\mathrm {NRMSE}}}=1-\frac{\Vert {{\mathbf {y}}} -\mathbf {\mu }\Vert ^2}{\Vert {{\mathbf {y}}}-E({{\mathbf {y}}})\Vert ^2}, \end{aligned}$$
    (11)

    where \({{\mathbf {y}}}\) the vector of validation values, \(\mathbf {\mu }\) the vector of mean predicted values, \(E({{\mathbf {y}}})\) the mean value of \({{\mathbf {y}}}\).

    NRMSE is 1 for a perfect match and \(-\infty\) for a very bad match of the validation and mean predicted values.

  • The standardised mean-squared error—SMSE

    $$\begin{aligned} {{\mathrm {SMSE}}}=\frac{1}{N}\frac{\sum _{i=1}^N(E({\hat{y}}_i)-y_i)^2}{\sigma _y^2}, \end{aligned}$$
    (12)

    where \(\sigma _y^2\) the variance of the observations.

    SMSE is a frequently used standardised measure for the accuracy of predictions’ mean values with values between 0 and 1, where the value 0 is the result of a perfect model.

  • The Pearson’s correlation coefficient—PCC:

    $$\begin{aligned} {{\mathrm {PCC}}}=\frac{\sum _{i=1}^N(E({\hat{y}}_i) -E(\hat{{{\mathbf {y}}}}))(y_i-E({{\mathbf {y}}}))}{N\sigma _y\sigma _{{\hat{y}}}}, \end{aligned}$$
    (13)

    where \(E(\hat{{{\mathbf {y}}}})\) the expectation, i.e., the mean value, of the vector of predictions, \(\sigma _y\) the standard deviation of the observations, \(\sigma _{{\hat{y}}}\) the standard deviation of the predictions.

    PCC is a measure of associativity and is not sensitive to bias. Its value is between \(-1\) and \(+1\), with ideally linearly correlated values resulting in a value of 1.

  • The mean standardised log loss—MSLL (Rasmussen and Williams 2006):

    $$\begin{aligned} {{\mathrm {MSLL}}}= \,& {} \frac{1}{2N}\sum _{i=1}^N\left[ \ln (\sigma _i^2)+ \frac{(E({\hat{y}}_i)-y_i)^2}{\sigma _i^2}\right] \\&-\frac{1}{2N}\sum _{i=1}^N\left[ \ln (\sigma _y^2)+ \frac{(y_i-E({{\mathbf {y}}}))^2}{\sigma _y^2}\right] , \end{aligned}$$
    (14)

    where \(\sigma ^2_i\) the prediction variance in the i-th step, \(E({{\mathbf {y}}})\) the expectation, i.e., the mean value, of the vector of the observations.

    MSLL is a standardised measure suited to predictions in the form of random variables. It weights the prediction error more heavily when it is accompanied by a smaller prediction variance. The MSLL is approximately zero for the not very good models and negative for the better ones.

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Kocijan, J., Perne, M., Mlakar, P. et al. Hybrid model of the near-ground temperature profile. Stoch Environ Res Risk Assess 33, 2019–2032 (2019). https://doi.org/10.1007/s00477-019-01736-5

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  • DOI: https://doi.org/10.1007/s00477-019-01736-5

Keywords

  • Hybrid model
  • Vertical temperature profile
  • Physics-based model
  • Statistical modelling
  • Gaussian-process model