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.

Weather systems and extreme weather events result from non-trivial three-dimensional (3-D) interactions in the atmosphere. The potential vorticity
(PV) perspective on atmospheric dynamics provides an often used conceptual framework to understand such interactions

Of particular importance to the PV perspective is the tropopause, as it separates air masses with typically low PV values in the troposphere from
typically high-PV air in the stratosphere.

PV is typically positive in the Northern Hemisphere and negative in the Southern Hemisphere. low-PV air here refers to low absolute values and high-PV air to high absolute values. For notational convenience we consider the northern hemispheric situation only.

A strong PV gradient across the tropopause has motivated the definition of the so-called dynamical tropopause as an upper-level PV isosurface in the literature. Typically the PV values used range between 1.5–4The tropopause slopes from higher isentropic levels in the tropics towards lower isentropic levels at the poles. On a given isentropic level,
poleward excursions of the tropopause denote an anomaly of low PV, whereas equatorward excursions denote an anomaly of high PV. A relatively smooth
meridional undulation of the tropopause signifies quasi-linear Rossby wave dynamics. In the context of extreme weather, however, much attention has
been given to highly nonlinear PV features, in particular zonally narrow and meridionally extended “tongues” of high PV that intrude equatorwards

In the present study, we consider the objective identification of PV structures. Objective identification of atmospheric features from numerical
simulation output has proven beneficial for a number of applications in both atmospheric research and operational meteorology. Typical uses include
statistical analysis such as the computation of climatologies of feature occurrence

All techniques discussed by

We note that in 2-D analysis, the concepts of both PVSs and PV cutoffs have been widely established. They are clearly defined and widely
used. However, a 3-D anomaly stretching over multiple isentropes can exhibit both types, PVS and PV cutoff, depending on the considered
level. Therefore, this 2-D classification is challenging for evaluations and especially does not hold for an analysis of 3-D features. In this study
and for 3-D analysis, we hence name these features PVAs. These anomalies can exhibit characteristics of both streamers and cutoffs when
considered on a single isentropic level only. 3-D cutoffs, i.e., fully isolated areas of stratospheric air surrounded by tropospheric air, are only
rarely a result of detachment from the main reservoir and mostly appear in the lower troposphere due to latent heat release

An identification of 3-D PV structures has previously been performed for subsets of anomalies only, although the importance of their vertical
structure has been demonstrated in the literature

In summary, while many previous studies contributed to identifying different aspects of PV features on single isentropic levels, the objective
identification of fully 3-D PV structures is still an open challenge and motivates our work. We expect that a 3-D identification method will yield the
opportunity for novel analyses, including new aspects within climatological studies and comprehensive case studies of 3-D atmospheric dynamics. In
the article at hand, we hence present a novel algorithm for the identification of PV anomalies, which can be applied not only to 2-D but also to 3-D
fields. In the latter case, the algorithm operates on the complete 3-D fields instead of on individual isentropic levels. To the best of our
knowledge, this is the first approach aiming for a strategy that identifies the full spectrum of PVAs in 3-D. We use image processing techniques for
the identification, more specifically numerical solutions that are based on morphological operators. These operators, which originate in analytical
geometry, have been adapted to suit the requirements of a meteorological application. This adaptation mainly revolves around using physical units to
ensure interpretability, while considering the properties of the used projection. Parameters of the identification are adjustable and the algorithm is
usable at different resolutions. A low-dimensional and human-readable feature vector is computed for each identified PVA consisting of quantifiable
measures, e.g., centroid, intensity, or best-fit geometry. This memory-efficient feature representation can serve as a basis for statistical analyses
or as potentially relevant predictors. The identified 3-D PVA features are also well suited for studying atmospheric dynamics by means of interactive
3-D visual analysis

This article is structured as follows. Section

The identification algorithm is motivated by so-called morphological image processing techniques. These operations, often used on binary data, process
images based on the characteristics of their shape

A sketch of our identification strategy applied to a PVS-like structure.

Figure

Following the erosion, the dilation step (Fig.

This approach uses an abstract concept of binary masks in discrete environments, and care is required to maintain interpretability and to design the
algorithm with meaningful parameters. Figure

A visual example (same PVS-like structure as in Fig.

Using this concept in real-world environments unfortunately is non-trivial. The distance measure required must follow the domain instead of using a
direct spherical distance, as seen in Fig.

Depiction of the stereographic projection used in our identification strategy. This figure shows that circles on the sphere are mapped to circles on the projection plane. Therefore, this projection is conformal, a vital property for this strategy. Note that despite having the same shape, the sizes of the circles on the sphere vary by distance from the projection center.

For the identification algorithm described in this study we choose a pole-centered stereographic projection (Fig.

We choose this type of projection due to several advantages.

The PV field encircles the earth and is centered around the poles. Using the pole as center of projection leads to a clear depiction of the features to be detected.

Contrary to equirectangular projections, this stereographic projection avoids singularities at the North Pole. PV intrusions over poles can occur, and their analysis is highly difficult in projections where a singularity is involved. Additionally, the stereographic projection solves issues handling the antimeridian boundary.

A stereographic projection is conformal, and the only known true perspective projection with this property

The main drawback of this strategy is the introduction of a singularity at the projection center, i.e., the opposing pole. To avoid this problem while mapping global data, we create one stereographic projection centered around the North Pole and one centered around the South Pole, both extending towards the Equator. The singularity at the pole shifts to a boundary at the Equator. Since the dynamical tropopause changes its sign at the Equator and generally lacks proper definition at lower latitudes, this area is not regarded as a region of interest for this present study.

As introduced in the previous section, our novel algorithm requires a way to calculate distances within the projection following a given field, as
illustrated in Fig.

As the next step, we define the distance map

However, most of these approaches suffer from metrication errors. They construct the path on segments parallel to the grid dimensions. Directions
invariant to the grid orientation will always lead to errors induced by the

This shortest path problem can also be formulated as the path space integral. We search the shortest distance

Generally, this formulation can be used with arbitrary cost functions, but replacing it with our distance field

This first-order nonlinear differential equation is used widely in scientific applications, mainly in wave propagation problems as a front propagation
approach. In these cases, the cost function on the right-hand side is typically expressed as a propagation velocity at every point in the domain. To
our knowledge, we are the first to formulate this equation in a sense to compensate for distorted distances in a projection instead, leading to a
formulation for distances from a given boundary for conformal projections. The field of shortest distances

Solving the above differential equation yields a far more accurate result than graph-based approaches on a discrete field, resolving metrication
errors. There are approaches to solving the eikonal equation, mainly the fast marching method (FMM) introduced by

Based on the above introduced strategies, we outline our identification technique for 2-D PVSs on a given isentropic level. Figure

Illustration of the algorithm to identify PVSs.

Pseudo-code of the identification process, both for 2-D and 3-D application. The 2-D and 3-D identifications differ with respect to distance measure (scale analysis for 3-D) and filtering strategy (additional area-based filtering for 3-D); cf. Sect.

objects

For the 2-D analysis, we use isentropic levels from both the ERA-5 reanalysis

Before executing the identification strategy, the distance field

In this study, we use a polar-centered stereographic grid with a projection pixel spacing of 75

Furthermore, as additional metric, the area map

Inputs for the algorithm are the field of PV data and the width threshold

The next step is called

After that, areas that are

In morphology, the

These extracted areas resemble the identified PVS (shaded areas in Fig.

Sensitivity of the identification process to the width threshold

Finally, the identified anomalies are filtered. We separate the filtering into two steps.

Here, some elementary measures for each of the identified PVSs are introduced. This list is not exhaustive and can be extended with more metrics. For
an identified object

the object's area

Besides as a filtering metric, the object area is a component of some of the subsequently mentioned metrics.

the length

Our identification strategy yields a useful and robust length measure, while other identification strategies only measure the length along the
2

the object's average PV, weighted by the area measure at each point:

the object's centroid as weighted average of the position over the area

The centroid gives a quantifiable estimate of the object's center. For example, in Fig.

the main axes of the object. Evaluating the eigenvalues and eigenvectors of the covariance matrix of the second-order moments define the main
axes of the shape

A 200–500

Before extending the idea to 3-D data sets, the introduced algorithm will be evaluated. For this purpose, the identification strategy by

The PVS identification strategy by WS07 is motivated by the thin and elongated nature of PVSs. They identify the outermost 2

Comparing the algorithm parameters, our width threshold

Comparison of the identified PVSs based on the 340

Same as Fig.

Figure

Figure

These structures are not identified by the approach from WS07. By only tracing the outermost 2

As introduced, we filter PVSs that do not satisfy a length threshold of 1000

For a set test case, the computation of anomalies takes a similar amount of time. Regarding our strategy, this time also includes the computation of feature vectors (e.g., best-fit geometry and centroids) for all detected structures. Nevertheless, due to different approaches in preprocessing, different programming languages and different use of parallelism, a quantifiable comparison of the run times proves to be difficult. For our algorithm, 85 % of the computational time is required to project the data onto a stereographic grid. Analyzing bigger data sets, data level parallelism (e.g., single instruction, multiple data (SIMD) processing) can highly improve the efficiency of projecting data from multiple isentropic levels and multiple time steps. Projecting a 3-D data set with 50 isentropic levels takes only twice the time compared to a single isentropic level, signifying a very high speedup.

Similar to WS07, we currently do not consider the origin of a PVA as a filtering strategy. For example high PV of non-stratospheric origin (e.g., in
tropical cyclones

For the 3-D identification, ERA-5 data on model levels (here we use levels 40–137) are interpolated to isentropes with a vertical spacing of
2

There are several possibilities to extend the algorithm introduced in Sect.

Three-dimensional visualizations of the dynamical tropopause (defined by the 2

Figure

Our algorithm is designed in a manner that it can be executed on 2-D and 3-D data in a similar way. Note that the basic morphological operations
(erosion and dilation; see Sect.

The identification method introduced in Fig.

Figure

Effect of filtering anomaly candidates with respect to the extent threshold

The same filtering process as in 2-D is applied: the goal is primarily to keep as many detected features as possible. The user can then decide which
of these features are of importance for a specific use case and filter these by their feature descriptions accordingly. The centroid of an anomaly is
computed in 3-D (see Eq.

However, in 3-D, we have to introduce an area-based filtering strategy (cf. Sect.

Existing identification techniques (see an overview of them in

Our feature identification strategy is implemented in a Python framework for general identification and tracking of meteorological structures. This
framework is part of the enstools Python package

On a test system (AMD Ryzen 5 2600X), the identification process for a 3-D data set with 46 isentropes (290 to 380

Comparison of the location of the anomalies defined by the 200

We evaluate our identification method along a case study. The selected extreme precipitation event affected northeastern Vietnam in late July and
early August 2015. This event has been analyzed in detail by

Between 25 July and 3 August 2015, record-breaking precipitation amounts of more than 1000 mm were observed at the coast of northeastern
Vietnam. According to the analysis by

To examine the precursors of the extreme event, we visualize the 3-D PV environment over eastern Asia for five different time steps in
Fig.

The synoptic–dynamic development associated with the event. Here, five time steps are visualized:

On 21 July 2015, a wide trough is located over eastern China, which is below the thresholds for our identification strategy in 2-D and 3-D to be
detected as a PVA. Over the following days, a high tropopause area moves from the Arabian Peninsula towards the Tibetan Plateau, further
illustrated in the supplementary animation

Comparison with the results of

As outlined in Sect.

Comparison of the 340 and 355

To give an understanding of identifying 3-D structures over their 2-D counterpart, Fig.

The case study also shows that distinguishing between the concepts of a PVS and PV cutoff in 2-D is not applicable to the 3-D view. The vast majority of cutoffs are attached vertically to the main stratospheric reservoir, while PVSs are connected typically both horizontally and vertically. Furthermore, a structure could often be interpreted as a streamer or a cutoff depending on the isentropic cross-section one is investigating. Visually, it can become completely unclear, which type a 3-D structure belongs to more. This supports our suggested nomenclature to take a more abstract approach and name these intrusions anomalies (PVAs).

In this study, a novel automated identification strategy for PVAs along the dynamical tropopause has been presented. It is based on sequential
distance measures in the PV field to extract anomaly-like structures. The method is capable of identifying these structures in both 2-D and 3-D data
sets robustly. It is based on image processing operations, which are able to detect disturbances in a multidimensional field and has been adapted to
suit the needs and requirements of the present application. Specifically, the adaptation keeps the interpretability of the process by using physical
units and compensates for the distortions introduced by the particular flattened projections used. Here, we use the eikonal equation and reinterpret it
in a novel fashion to efficiently compute accurate distances in the distorted field, taking advantage of the properties of the projection. The
strategy has proven well scalable for processing big data sets. The presented algorithm takes a width parameter

A drawback of existing 2-D identification techniques is that their decisions are solely based on the outermost 2

In 3-D analysis, the algorithm provides an independent configuration for all seasons and use cases and is therefore suitable for climatologies. In 2-D analysis, on the other hand, care has to be taken to choose a suitable isentrope for analysis during a season or for a specific event or depending on which regimes one is interested in. Also, more interesting features can be detected by considering the full 3-D neighborhood instead of focusing on a single 2-D level. The 3-D structures are more cohesive and consistent in space and time compared to their 2-D counterparts. Furthermore, vertical gaps introduced by stacking results of the 2-D identification are prevented.

A case study has been investigated to show up advantages of the 3-D analysis. A trough detected using single-level geopotential height or vertically
averaged PV as in

We put effort into describing the identified object in an abstract geometric fashion, similar to ellipses in 2-D. There, most structures can be well
approximated by one (or in degenerative cases a few) simple geometric shapes. 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

In the future, this technique can be applied both to more individual use cases and to big data sets. Analyses and climatologies help to exploratorily examine the properties and behavior of PVAs in certain 3-D environments. Computed feature descriptions could be a basis for finding correlations and clusters in data, classifying anomalies into distinct categories. Since the algorithm is well scalable, it can be applied to ensemble forecasts as well. However, the performance bottleneck for big data analysis is the computation of high-resolution isentropic data out of model levels. Therefore, processing large data sets might require a trade-off by using a different vertical dimension even though this leads to a more challenging interpretation, especially when evaluating horizontal cross-sections. Although this study focuses on the identification of anomalies along the dynamical tropopause, the algorithm in a broader sense can be applied to identify anomalies along any 2-D or 3-D spatial boundary. Therefore, other identification tasks in meteorology can also benefit from these ideas and strategies.

Further research is required on certain aspects of this identification, which might lead to improved results. For individual use cases, the tracking
of these anomalies could be done by hand, whereas for climatologies automated tracking techniques are required. Ideas for further work include simple
overlap tracking

The automated 3-D PV identification, tracking, and its application to many meteorological cases opens an avenue to study which characteristics of the
PV objects are related to the genesis and improved forecast using statistical and machine learning approaches. For example,

The implementation of our framework is available at

A video supplement showing 3-D visualizations of the synoptic evolution of our case study can be freely accessed at

ES, AHF, and MRi proposed, supervised, and administrated this study. CF designed the algorithm, implemented the software, performed algorithm analyses, did visualizations, and wrote the publication. RvdL performed meteorological analyses on the case study and provided the corresponding results. MRa contributed helpful input and comments regarding Met.3D and visualizations in general. MMG provided expertise regarding the dynamics of PV. All authors provided feedback on and critical review of the paper.

The contact author has declared that neither they nor their co-authors have any competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The research leading to these results has been accomplished within the project C3 “Predictability of tropical and hybrid cyclones over the North Atlantic Ocean” of the Transregional Collaborative Research Center SFB/TRR 165 “Waves to Weather” funded by the German Science Foundation (DFG). This work uses S2S data. S2S is a joint initiative of the World Weather Research Programme (WWRP) and the World Climate Research Programme (WCRP). The original S2S database is hosted at ECMWF as an extension of the TIGGE database. The GPM (IMERG) data were provided by the NASA/Goddard Space Flight Center and archived at the NASA GES DISC. Thanks to the Institute for Atmospheric and Climate Science at ETH Zurich, in particular to Michael Sprenger and Heini Wernli, for sharing the source code of their PVS identification technique

This research has been supported by the Deutsche Forschungsgemeinschaft (project C3 in Transregional Collaborative Research Centre SFB/TRR 165 “Waves to Weather”; grant no. 257899354 – TRR 165).This open-access publication was funded by Johannes Gutenberg University Mainz.

This paper was edited by Juan Antonio Añel and reviewed by two anonymous referees.