The Second Order Surfaces

Кривошапко, С. Н.; Иванов, В. Н.

doi:10.1007/978-3-319-11773-7_35

Cited by 5 publications

(3 citation statements)

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“…These are also clearly interpretable, e.g., the main axis yields the tilt of the streamer. For 3-D objects, we considered using subsets of quadrics (second order surfaces, see Krivoshapko and Ivanov, 2015) but recognized that the extension of 2-D fitting to 3-D data is a highly nontrivial task. To explore the geometrical structure, but also to exploratorily analyze the presented algorithm and the data set itself, interactive 3-D visualizations proved to be an essential tool for comprehensive depictions.…”

Section: Discussionmentioning

confidence: 99%

A novel method for objective identification of 3-D potential vorticity anomalies

Fischer

Fink

Schömer

et al. 2022

Preprint

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Abstract. Potential vorticity (PV) analysis plays a central role in studying atmospheric dynamics and in particular in studying the life cycle of weather systems. The three-dimensional (3-D) structure and temporal evolution of the associated PV anomalies, however, are not yet fully understood. An automated technique to objectively identify 3-D PV anomalies can help to shed light on 3-D atmospheric dynamics in specific case studies, as well as facilitate statistical evaluations within climatological studies. Such a technique to identify PV anomalies fully in 3-D, however, does not yet exist. This study presents a novel algorithm for the objective identification of PV anomalies in gridded data, as commonly output by numerical simulation models. The algorithm is inspired by morphological image processing techniques and can be applied to both two-dimensional (2-D) and 3-D fields on vertically isentropic levels. The method maps input data to a horizontally stereographic projection and relies on an efficient computation of horizontal distances within the projected field. Candidates for PV anomaly features are filtered according to heuristic criteria, and feature description vectors are obtained for further analysis. The generated feature descriptions are well suited for subsequent case studies of 3-D atmospheric dynamics as represented by the underlying numerical simulation, or for generation of climatologies of feature characteristics. We evaluate our approach by comparison with an existing 2-D technique, and demonstrate the full 3-D perspective by means of a case study of an extreme precipitation event that was dynamically linked to a prominent subtropical PV anomaly. The case study demonstrates variations in the 3-D structure of the detected PV anomalies that would not have been captured by a 2-D method. We discuss further advantages of using a 3-D approach, including elimination of temporal inconsistencies in the detected features due to 3-D structural variation, and elimination of the need to manually select a specific isentropic level on which the anomalies are assumed to be best captured. The method is made available as open-source for straightforward use by the atmospheric community.

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Section: Discussionmentioning

confidence: 99%

A novel method for objective identification of 3-D potential vorticity anomalies

Fischer

Fink

Schömer

et al. 2022

Preprint

View full text Add to dashboard Cite

show abstract

“…These are also clearly interpretable; e.g., the main axis yields the tilt of the streamer. For 3-D objects, we considered using subsets of quadrics (second-order surfaces; see Krivoshapko and Ivanov, 2015) but recognized that the extension of 2-D fitting to 3-D data is a highly non-trivial task. To explore the geometrical structure, but also to exploratorily analyze the pre-sented algorithm and the data set itself, interactive 3-D visualizations proved to be an essential tool for comprehensive depictions.…”

Section: Discussionmentioning

confidence: 99%

A novel method for objective identification of 3-D potential vorticity anomalies

et al. 2022

View full text Add to dashboard Cite

Abstract. Potential vorticity (PV) analysis plays a central role in studying atmospheric dynamics and in particular in studying the life cycle of weather systems. The three-dimensional (3-D) structure and temporal evolution of the associated PV features, however, are not yet fully understood. An automated technique to objectively identify 3-D PV features can help to shed light on 3-D atmospheric dynamics in specific case studies as well as facilitate statistical evaluations within climatological studies. Such a technique to identify PV features fully in 3-D, however, does not yet exist. This study presents a novel algorithm for the objective identification of PV anomalies along the dynamical tropopause in gridded data, as commonly output by numerical simulation models. The algorithm is inspired by morphological image processing techniques and can be applied to both two-dimensional (2-D) and 3-D fields on vertically isentropic levels. The method maps input data to a horizontally stereographic projection and relies on an efficient computation of horizontal distances within the projected field. Candidates for PV anomaly features are filtered according to heuristic criteria, and feature description vectors are obtained for further analysis. The generated feature descriptions are well suited for subsequent case studies of 3-D atmospheric dynamics as represented by the underlying numerical simulation. We evaluate our approach by comparison with an existing 2-D technique and demonstrate the full 3-D perspective by means of a case study of an extreme precipitation event that was dynamically linked to a prominent subtropical PV anomaly. The case study demonstrates variations in the 3-D structure of the detected PV anomalies that would not have been captured by a 2-D method. We discuss further advantages of using a 3-D approach, including elimination of temporal inconsistencies in the detected features due to 3-D structural variation and elimination of the need to manually select a specific isentropic level on which the anomalies are assumed to be best captured. These advantages, as well as the suitability of the implementation to process big data sets, also open applications for climatological analyses. The method is made available as open-source for straightforward use by the atmospheric community.

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“…Because both Equations 3 and 4 are second-order equations, they define second-order surfaces. 52,53 If we disregard the degenerate cases of these surfaces, then Equation 3 will describe only one surface, that is, a hyperbolic paraboloid all sections of which with horizontal planes are hyperbolas with perpendicular asymptotes (Figure 2A). A more general model (4) defines 2 possible response surfaces-a general hyperbolic paraboloid with a random arrangement of the asymptotes created by its horizontal sections, or an elliptic paraboloid 52,53 ( Figure 2B).…”

Section: Introductionmentioning

confidence: 99%

Using Various Nonlinear Response Surfaces for Mathematical Description of the Type of Combined Toxicity

et al. 2018

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The article considers the problem of characterizing the type of combined action produced by a mixture of toxic substances with the help of nonlinear response functions. Most attention is given to second-order models: the linear model with a cross-term and the quadratic model. General propositions are formulated based on the data on combined toxicity patterns previously obtained by the Ekaterinburg nanotoxicology team in animal experiments and analyzed with the help of the linear model with a cross-term. It is shown now that the quadratic model features these general characteristics in full measure, but interpretation of combined toxicity types based on isobolograms obtained by the quadratic model is more difficult. This suggests that where both models ensure a comparable quality of combined toxicity type identification, it would be enough to use the linear model with a cross-term.

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The Second Order Surfaces

Cited by 5 publications

References 0 publications

A novel method for objective identification of 3-D potential vorticity anomalies

A novel method for objective identification of 3-D potential vorticity anomalies

A novel method for objective identification of 3-D potential vorticity anomalies

Using Various Nonlinear Response Surfaces for Mathematical Description of the Type of Combined Toxicity

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