# Comparative Evaluation of Algorithms for Spatial Interpolation of Atmospheric State Parameters Based on a Dynamic Stochastic Model Taking into Account the Vertical Variation of a Meteorological Field

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## Abstract

**:**

## 1. Introduction

## 2. Problem Statement and Features of the Proposed Approach

#### 2.1. General Problem Statement and Description of the Model

_{i}, y

_{i}), where i = 0, …, s, is given in the mesoscale range at the ground level. Regular upper-air measurements of meteorological variables (temperature and orthogonal wind velocity components) were conducted in the vertical profile of the atmosphere above points i = 1, …, s. It should be noted that such measurements are absent for point i = 0 with coordinates (x

_{0}, y

_{0}). The restoration of the meteorological field ξ

_{0}at point i = 0 implies estimating its values from the measurements obtained at points i = 1, …, s. According to [21,22], the resulting estimate of the field ${\xi}_{0}$ at the required point (x

_{0}, y

_{0}) is determined as the sum of the random ${\xi}_{0}^{\prime}$ and regular ${\overline{\xi}}_{0}$ components:

_{0}, y

_{0}) are unknown and need to be estimated. The coefficients $\alpha $, $\beta $, and $\gamma $ should be determined in advance during the preliminary analysis of the upper-air data obtained for the selected mesoscale range. This is because the meteorological fields can be considered to be homogeneous and isotropic for time intervals between upper-air observations and within the selected mesoscale range (maximum size 500 × 500 km).

#### 2.2. Kalman Filter Structure for the First Variant of the Interpolation Algorithm

_{0}, y

_{0}) for a given height level h at time k. Therefore, the transition matrix also consists of a single element:

#### 2.3. Kalman Filter Structure for the Second Variant of the Interpolation Algorithm

#### 2.4. General Matrix Expressions for the Synthesized Algorithms

_{0}, y

_{0}) at height h.

## 3. Results of the Experiment and Statistical Evaluation

#### 3.1. Description of the Experimental Data

_{0}is the ground height above sea level, and h is the height of the upper boundary of the layer), for atmospheric layers: 0–200, 0–400, 0–800, 0–1200, 0–1600, 0–2000, 0–2400, 0–3000, 0–4000, 0–5000, 0–6000 and 0–8000 m. Thus, the observations were based on the average values of temperature $<\mathrm{T}{>}_{{h}_{0},h}$, zonal $<\mathrm{U}{>}_{{h}_{0},h}$, and meridional $<\mathrm{U}{>}_{{h}_{0},h}$ wind velocity components for the layer, which were calculated using the actual measurements taken at standard isobaric levels in the height range from 0 to 8 km [21,22].

#### 3.2. Initial Conditions for LKF Initiation

#### 3.3. Comparative Analysis of the Simulation Results

#### 3.4. Additional Notes

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 2.**Vertical distributions of the root mean square (RMS) spatial interpolation (extrapolation) errors on air temperature, as well as zonal and meridional wind velocity components, based on a low-order parametric model with a vertical component for first (1) and second (2) variant of design, and the corresponding vertical profiles of the standard deviations (3), for Moscow (

**a**) and Kursk (

**b**) stations. Summer.

**Figure 3.**Vertical distributions of the RMS spatial interpolation (extrapolation) errors on air temperature, as well as zonal and meridional wind velocity components, based on the low-order parametric model with a vertical component for the first (1) and second (2) variant of design, and the corresponding vertical profiles of the standard deviations (3), for Moscow (

**a**) and Kursk (

**b**) stations. Winter.

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**MDPI and ACS Style**

Popov, Y.; Lavrinenko, A.; Krasnenko, N.; Popova, A.; Popova, K.; Shelupanov, A.
Comparative Evaluation of Algorithms for Spatial Interpolation of Atmospheric State Parameters Based on a Dynamic Stochastic Model Taking into Account the Vertical Variation of a Meteorological Field. *Symmetry* **2019**, *11*, 1207.
https://doi.org/10.3390/sym11101207

**AMA Style**

Popov Y, Lavrinenko A, Krasnenko N, Popova A, Popova K, Shelupanov A.
Comparative Evaluation of Algorithms for Spatial Interpolation of Atmospheric State Parameters Based on a Dynamic Stochastic Model Taking into Account the Vertical Variation of a Meteorological Field. *Symmetry*. 2019; 11(10):1207.
https://doi.org/10.3390/sym11101207

**Chicago/Turabian Style**

Popov, Yuriy, Andrey Lavrinenko, Nikolay Krasnenko, Avgustina Popova, Kseniya Popova, and Alexander Shelupanov.
2019. "Comparative Evaluation of Algorithms for Spatial Interpolation of Atmospheric State Parameters Based on a Dynamic Stochastic Model Taking into Account the Vertical Variation of a Meteorological Field" *Symmetry* 11, no. 10: 1207.
https://doi.org/10.3390/sym11101207