Since the aim is to identify the relevant ETC that caused a local impact at a specific time, we first define a time window around the time of impact with start and end timestamps ts and te, respectively. For the application of the algorithm to storm surge events in this paper, a time window of ±24 h was chosen around the beginning and end peaks of each storm surge event, yielding a minimum of 49 time steps per event. This search window is specifically balanced to ensure sufficient track continuity for the primary driver while minimising the inclusion of secondary, impact-irrelevant tracks that could introduce ambiguity during the automated calibration of post-processing parameters. This search window is highlighted in Fig. .

Within this time window, the algorithm identifies candidate locations for the centre of an ETC by searching for local MSLP minima in the entire CERRA domain . Across large spatial domains with high resolution grids, there are two issues: (1) the number of grid cells is large and simple local neighbourhood searches will generate multiple local minima, and (2) there may be multiple ETCs present within the domain at any given time. To handle the high resolution of the 5.5 km grid and avoid detecting excessive local noise, we employ a spatial partitioning approach rather than traditional grid coarsening or “upscaling”.

We first attempted using a kernel around each point to determine its status as a local minima, but this approach proved too computationally expensive. Instead, the goal is to upscale the original grid into coarser cells without missing any local minima due to overly large cells. The full domain is divided into non-overlapping square blocks where the diagonal length of each block is defined by the pruning radius, r. Using the Pythagorean theorem, we can derive the length of the squares from the diagonal, given by the pruning radius r as follows: l=r22. We proceed through the following steps (Fig.  shows the remaining potential candidate points after each step):

The algorithm determines the local MSLP minima within each spatial block, specifically identifying the individual 5.5 km grid cell with the lowest value. This ensures that the final coordinates remain faithful to the original data resolution without downsampling (Fig. a).

From these steps, we identify potential candidate points that represent the centre of an ETC. These will be used in the correspondence problem in the next step of the algorithm. The hyperparameters in these steps (r, maximum MSLP value at ETC centre, and minimum regional ζ500 intensity) can all be fine-tuned for a given application of the algorithm. For the storm surge examples in this paper, we use the values r=350 km, ζ500min=1.5×10-4s-1, and MSLPmax=1010 hPa. The choice of r determines the size of the atmospheric structures that are preserved. It has to be large enough to filter away grid-scale noise and minor vortices if these are not of interest, but small enough to separate relevant features. If the user targets synoptic-scale systems, the pruning radius can reasonably be set to 500–1000 km, while smaller values are necessary to identify, e.g., secondary lows. The value of r can have a large impact on the computational efficiency of the algorithm, which will be investigated further in Sect. . In general, smaller pruning radii yield more candidate points, which in turn affects the algorithm's computational efficiency.

In a few rare instances, we have observed that two nearby candidate points have identical MSLP values in the CERRA data, which means that the aforementioned step 2 cannot select a unique point within the pruning radius. In this case, we have chosen to keep both points as possible candidates for the correspondence problem in the following section, rather than making an arbitrary selection between them. In Fig. , we show an example of MSLP minima points and 1 hPa isobar levels for the 11 October 1997 event, visualising the data in which the algorithm operates.

The candidate points found in Sect.  for each individual time step must now be connected through adjacent time steps to determine the storm tracks – a task known as the correspondence problem in the storm tracking literature . The general objective is to minimise the overall distance between the pairs of connected candidate points at time ti and ti+1. An added complication is that the number of candidate points in each time step is not guaranteed to be identical. If we define n and m as the number of candidate points at time ti and ti+1, respectively, then the problem can be either balanced if n=m or unbalanced if n≠m. A theoretical solution to this problem can be found through global optimisation. However, due to concerns about the computational requirements of global optimisation, it is common in current storm tracking algorithms to use a heuristic approach. This is often based on the nearest-neighbour approach, where one iterates through the candidate points at ti and sequentially connects them to the candidate point closest in space at ti+1. Usually, a threshold for the maximum possible travel distance of a storm centre between two time steps, Dmax, is part of the heuristic algorithms. Heuristic-based approaches are not guaranteed to provide the correct solution, but they are computationally very fast.

In our tracking framework, we have implemented a global optimisation solution that avoids some of the pitfalls of nearest-neighbour heuristics. This type of correspondence problem is formally known as the “assignment problem” in optimisation theory, and it can be efficiently solved using the Hungarian Algorithm. The algorithm is well-studied in other fields, such as logistics and scheduling , and therefore we will only provide a general description here. The Hungarian algorithm begins with setting up a cost matrix, which here will be an n×m distance matrix, which represents the distance between all n candidate points at ti and all m points at ti+1. Equation () exemplifies how such a distance matrix is set up, with the leftmost matrix in Eq. () being n×m. The algorithm requires that the distance matrix be square, and since this is not guaranteed, the next step is to pad the matrix with phantom candidate points with a maximum cost (middle matrix in Eq. ). We then apply a maximum allowable distance for a connection to be “realistic”, Dmax. This parameter can be physically interpreted as a maximum allowable translation speed of the ETC centre, where reasonable estimates may be in the order of 100–150 km h⁻¹. However, the parameter can also be understood more conceptually as a “search radius” applied to the tracking variable (here MSLP). For the storm surge application presented in the Nordic region, we chose a value of 300 km, since complex terrain can cause the identified ETC centre to jump longer distances than what would be considered as physically reasonable. This maximum distance is implemented by censoring all matrix elements with values above Dmax (see rightmost matrix in Eq. ). 1258810583626402510239525411231926942577→258810583Dmax626402510Dmax2395254112Dmax31926942577Dmax→Dmax10583Dmax6DmaxDmaxDmaxDmax254112DmaxDmaxDmaxDmaxDmax

Once the distance matrix is set up, made square, and censored, the algorithm finds the optimal solution by minimising the sum of distances across all connections. This is done through row/column reduction, covering zeros with lines, and matrix adjustments in the following steps:

Identify the smallest element in the matrix and subtract it from all elements, thus obtaining a matrix with at least one zero.

The connections found by the Hungarian Algorithm represent the solution that minimises the total sum of the connections between ti and ti+1. Computationally, in terms of big-O notation, it scales at O(n3). If multiple optimal solutions with the same objective value exist, the algorithm will return only one of them. In the case of storm tracking, with costs defined as the exact geographical distance between points, we deem it unlikely to occur.

The framework also contains a second option for solving the correspondence problem with a standard nearest-neighbour (NN) heuristic, which we will benchmark HA against. For the NN approach, we employ a greedy algorithm, which iteratively selects the valid minimum distance connection until all pairings are exhausted or exceed a predefined distance threshold Dmax (see Algorithm ):

The implemented HA and NN algorithms optimise connections with information from just two adjacent time steps, ti and ti+1. The limitation of such a one-step-at-a-time approach is that it can lead to a track breaking in two ways. First, if one of the timestamps along the ETC path is missing a candidate point, i.e., no local minima could be found. Second, even if a local minimum exists for all time steps, it can also have downstream implications for the connections that can be made later at ti+2 onwards. Tracks can break if the “optimal” connection from ti chooses a point at ti+1 that is further away from the minimum in ti+2 than one of its competing points, thus potentially not allowing for a connection to be made between ti+1 and ti+2. For such a scenario to occur, the user would have to specify a Dmax value larger than the pruning radius, r, which would create the possibility of multiple candidate points within the Dmax distance. To mitigate this second possibility, we introduce an additional algorithmic step to handle cases where tracks break because the required connection exceeds Dmax.

In this study, MSLP is used to identify candidate points for the centres of ETCs. We chose MSLP over other possible tracking variables because of its close relationship to surface winds that drive storm surges. However, MSLP is a diagnostic variable, computed from the modelled surface pressure by estimating the sea level pressure under the assumption of hydrostatic balance . Over complex terrain, this assumption often breaks down, e.g., over the Scandinavian mountains. As a result, artefacts and inaccuracies can arise in the MSLP field, potentially causing tracks to break into several fragments. A recurring example of this arises when an ETC crosses a mountain range: Candidate centres often appear to “stall” on the upstream side of the mountains, while a new candidate centre simultaneously appears downstream in the same time step. In this case, there are multiple alternative candidate points in the same time step for the same ETC system. This causes the track to break, resulting in a single ETC being misidentified as two separate systems. A tracking variable at higher atmospheric levels might be less prone to such issues, but it would also be less closely related to the surface impacts. We therefore introduce a novel track reconciliation step to allow for surface level tracking with MSLP while mitigating this type of track fragmentation. The reconciliation step is based on a BLOB analysis technique that performs synoptic-scale spatial continuity checks using the MSLP field.

We observed that when breaks occur, the upstream and downstream candidate points typically remain part of the same synoptic-scale low-pressure system, i.e. in close spatial proximity and with similar MSLP values. BLOB analysis leverages this by identifying contiguous low-pressure “regions” within a given MSLP range.

The method works as follows: for a given candidate point, we define a pressure range (e.g. ±5 hPa around its MSLP value). The entire MSLP field is then binarised, with grid points within the range set to 1 (inside a BLOB) and those outside set to 0. The result is a BLOB that represents spatially coherent low-pressure areas. For each time step, we check whether other candidate points fall within the same BLOB as the one containing the current candidate point. Figure a shows an example of a single ETC track that has broken into two fragments as it passed over the Scandinavian mountains. Figure b shows a snapshot at the time step where the track broke. Here, the yellow point is the candidate point under investigation, the red points are other ETC candidate points in the same time step, and the shaded areas are the computed BLOBs where MSLP values in the domain sit within a range of ±5 hPa of the MSLP value of the yellow point. In this case, there are three separate BLOBs in the domain but the algorithm only looks for red candidate points within the BLOB that contains the yellow point. The rest are disregarded.

In certain edge cases, the BLOB's can become very large. To avoid false positives, we impose a maximum allowable BLOB bounding box of 3000 km and a threshold of 600 km for the maximum distance between two candidate points to be considered connected. When choosing the value of this second maximum BLOB candidate point distance parameter for a given use case, it is important to set it high enough to connect points within the same synoptic-scale system, e.g., in the range 500–1000 km. Its value should be larger than Dmax to work as intended, but values much larger than 500–1000 km could risk merging distinct synoptic systems.

In simple cases, the discontinuity occurs at the end of one track fragment and the start of the next, which can then be merged. In more complex cases, two tracks may overlap temporally: the upstream track continues for a few time steps while the downstream track has already begun. The task then becomes choosing whether and when the jump from the first track to the start of the second track should occur. In these instances, we identify the first time step in which the two candidate points fall within the same BLOB. From that point onwards, we monitor both the continuations of the tracks and select the one that persists for the longest number of time steps as the final coherent track (see Fig. ).

In the results section, we demonstrate the value of this track reconciliation step by showing how frequently it is used for the storm surge events in this study, how it affects the length of the final tracks, and how it prevents broken tracks from being filtered out in the post-processing steps described in the next section.

Within the impact time window, multiple ETC tracks are often present throughout the data domain. These tracks vary in length, duration, and spatial trajectory. Therefore, a crucial final step in any storm tracking algorithm is the filtering of tracks that are either insignificant or unrelated to the local impact.

In this study, we implement three options for post-processing tracks to identify the storm(s) that caused the local impact.

Impact-oriented calibration of standard post-processing parameters,

The full tracking algorithm contains many parameters and hyperparameters, some of which can be reasonably selected from knowledge of the physical system, while others need to be tuned for a given case area. In the literature on tracking algorithms, we often find a lack of (hyper-)parameter documentation, so here we present Table , which provides a full and transparent overview of the parameters in this algorithm.

Here we present results on the trade-off between solution quality and computational efficiency of the HA algorithm presented in Sect.  over a range of complexities of correspondence problems.

To explore the influence of problem complexity on NN and HA performance, we vary the pruning radius r used in the candidate point identification step. Smaller r-values yield denser point clouds and larger correspondence problems. We tested six radii: 100, 175, 250, 350, 500, and 700 km, evaluating the run time and three additional metrics that explore solution differences between HA and the greedy NN solution:

Number of correspondence problem solutions that yield different connections.

During this analysis of the performance of HA and NN for the 37 impact events, a total of 1966 correspondence problems were solved, as summarised in (Table ). Dates of the impact events are included in Table  in the Appendix.

As the radius decreases, the number of candidate points and assignment options increases, which leads to greater problem complexity. Below a radius of 250 km, there is a sharp increase in different matches between HA and NN as well as extra HA connections. This happens because the pruning radius is significantly lower than the Dmax value of 300 km, leading to multiple candidate points becoming eligible for connection. For larger radii and smaller problem sizes, the algorithms have very few legal moves, leading to similar solutions. At a 100 km radius, 13.9 % of correspondence problems yield different solutions using the NN method, and 3.97 % of the problems result in fewer valid storm track connections compared to HA.

Although the HA is significantly slower at small radii (e.g. 4.88 s at 100 km), it consistently achieves more connections between candidate points, which can be important for avoiding premature track termination. In applications where the domain is large and time steps are longer than the one-hour steps in CERRA (e.g. global reanalyses), similar complexity issues could potentially arise even with larger radii, making global optimisation even more relevant.

In our repository , we show an animation of the HA and NN algorithms connecting points through time for an illustrative event where the two solutions differ. In this specific example, the NN solution actually ends up with two tracks that cross each other in a single time step.

The Dmax parameter is important for how the correspondence problem is solved with both HA and NN, as it controls the maximum connection distance between two candidate points in time. We have carried out a minor sensitivity analysis for Dmax in order to assess its impact on the tracks that are formed. This can be found in the Appendix in Fig. . As expected, we observed that small Dmax-values led to an increased number of tracks and generally shorter track durations, as fewer connections could be made across time steps. The sensitivity analysis showed that there was a smooth trend in the resulting number of tracks and length of individual tracks as Dmax was varied. There was thus no obvious cut-off value to choose. We expect this to generally be the case for other areas and reanalysis time step lengths. However, to give some practical guidance for how to tune this parameter, we recommend that a good lower estimate for it would be a physically realistic value of how far the object being tracked can potentially travel in a single time step. The user can then increase the Dmax value to pragmatically account for other effects, such as noise in the tracking field, terrain-driven discontinuities, etc. This is inherently a tuning problem that requires some trial and error testing.

To evaluate the effect of the BLOB analysis technique for track reconciliation, we compare the full tracking system with and without the reconciliation step (see Table ). This comparison does not assess the physical skill of reproducing the reconciled tracks as such. Instead, it quantifies how often reconciliation is applied and how it influences properties and survivability in the final track dataset after post-processing filters. The evaluation metrics are as follows:

The number of impact events in which at least one storm track survived post-processing out of the 37 total events.

Using the pruning radius of 350 km as an example, we notice that out of 37 total events, we have 37 events where at least one track survived the post-processing when utilising gradient tracing (see Table ). The BLOB utilisation played a critical role in 19 events, affecting 19 surviving tracks. Across those 19 post-processed tracks, 28 reconciliations have occurred, meaning multiple surviving tracks have undergone more than one reconciliation. In summary, the use of BLOB analysis has ensured that a significantly higher number of tracks have survived post-processing. All 19 events in which BLOB analysis has been utilised have been manually investigated on an individual basis, and it has been found that the vast majority of the tracks in these events would otherwise not have represented reality satisfactorily.

Figure  shows the geographical locations where BLOB reconstructions have occurred, as well as the distribution of the distances between candidate points connected by BLOB reconstructions. How active the BLOB step is depends heavily on the user-specified value of the pruning radius parameter. We therefore present BLOB reconstructions for the three different pruning radius values of 700 km (a, b), 350 km (c, d), and 100 km (e, f). In all cases, Max BLOB candidate point distance is set to a large value of 2500 km to analyse how connections are formed. The results generally show that smaller pruning radii lead to more blob connections because they leave many candidate points at each time step. The histograms of all three experiments show that there are a few reconstruction distances in the range of 1000–1600 km, which intuitively seems high. The vast majority of distances lie between the applied pruning radius (vertical, dashed line) and around 800 km, which is more reasonable given its intent to fix jumps across mountainous terrain. For the Nordic countries, we recommend using values of 600–800 km for this parameter, but users will need to re-evaluate this for other areas where these jumps may be smaller or larger.

Figure  also shows cases where reconstructed connections happen over very short distances. This shows up as reconstruction distances below the pruning radius value in the histograms (see Fig. b, d, f). These are formed during time steps t when multiple candidate points have the exact same MSLP value within each other's pruning radius. As explained earlier, we keep both points as candidates in such cases, but when the HA and NN algorithms then have to connect candidate points through time, there is a chance that the track ends prematurely, if the connection from t-1 to t is to one of the points but the continuation onwards from t to t+1 continues from the other point. The BLOB reconstruction handles this issue and connects the two broken track fragments.

Figure  shows the results of the parameter sensitivity analysis with both metrics. Each subplot in the figure shows the sensitivity of ETC detection to one of the four post-processing parameters. Given the discretisation of the chosen grid search for each parameter, there are a total of 325 376 possible combinations of the four parameters. The plots show the median, the 5th and 95th percentiles of the score values across all combinations. The results show that ETC detection is most sensitive to the size of the AoR (Fig. a), time in AoR (Fig. b), and the minimum travel distance (Fig. c). Notably, there is the special case where minimum time in AoR equals 0, meaning that an ETC does not need to enter the AoR for detection, which leads to a large drop in performance (Fig. b). The large distance between the 5th and 95th percentiles, as indicated by the shaded areas, shows that the same parameter value can lead to very different post-processing performance depending on the values of the other post-processing parameters. Thus, there is a strong dependence between parameters; e.g., the size of the AoR and the minimum time in the AoR are closely related.

Figure  shows the marginal and joint distributions of the Storm Count Accuracy and Single Storm Score after the values of both scores have been min-max normalised to fit the same 0–1 range. The marginal distributions follow each other closely with small deviations in some ranges. The joint distribution confirms that the Single Storm Score yields normalised values similar to the Storm Count Accuracy. This, combined with the general agreement of the scores in Fig. , shows that the Single Storm Score is good at approximating the Storm Count Accuracy and thus suitable for impact-focused calibration when manual labelling is not possible.

For the presented case of ETCs that produce storm-surge events, we select the parameter set that yields the best Single Storm Score. Given the grid search discretisation, there are 325 376 possible combinations of the four post-processing parameters. In the case presented, 20 unique parameter sets yielded the same score. We selected the parameter set that was closest to the median of each parameter (see values in Table ), thus avoiding extremes or accidentally fortunate local minima. We note that this method for selecting parameter combinations is only suitable if the possible combinations are centred in nature. If they had, e.g. been clustered in two different areas of the parameter space, the user would have to devise another way to select the final set. It is worth noting that even though all 325 376 parameter combinations were simulated in a brute-force manner, this still only took ∼ 3 min to complete on a standard laptop. This highlights that calibrating post-processing parameters is a simple, fast step that should be performed.

To assess the quality of the identified ETC tracks, we present a set of representative storm surge cases here. Figure  shows six examples of well-defined ETC tracks that exhibit consistent spatial progression, relatively smooth trajectories, and alignment with the underlying MSLP field. These cases demonstrate the algorithm's ability to capture the temporal and spatial evolution of ETCs, even as they traverse complex regions with steep topography and land-ocean boundaries.

In contrast, Fig.  highlights six events that illustrate the challenges of the algorithm. In these subplots, tracks that are coloured red were deemed impact-irrelevant despite manual inspection of the events suggesting that this was the wrong decision. Other colours than red show tracks that were deemed impact-relevant and has an uncertainty estimate associated with it in the legend of each subplot. For the storm surge case of this paper, we chose 300 km as the value of Dmax but several of the events in Fig.  ends up with fragmented tracks because this value is too low. This is true for the two red track fragments in the 29 January 2001 event and the two red fragments in the 18 November 2004 event. The 8 January 2005 consists of two small red track fragments and two larger ones coloured in blue and orange, however, manual inspection revealed that these all belong the same ETC. It was again a too small Dmax value that caused the break between the second red fragment and the blue fragment, and again between the blue and orange fragments. If a user of the framework is particularly interested in a single-event analyses of these events, it would be straightforward to experiment with new Dmax values until the desired track is obtained. However, for batch processing of many events at once where manual single-event inspection is deemed unfeasible, it is a limitation of the framework that the user has to specify a single Dmax value for all cases. Based on performing many experiments with the algorithm ourselves, we find that no single parameter set performs perfectly at all events.

During the 12 January 2007 event, three separate ETC were present at the time of impact (coloured red, orange, and blue). While the red and orange tracks may look two fragments of a single track because the red ends close to where the orange starts, they were in fact two separate systems. At the time of impact, the red track was located west of the impact location, the orange track to the east, and the blue track far north between Iceland and Norway. Manual evaluation of the event suggests that the impact is caused by the combined effect of the red and orange tracks that together produce strong MSLP gradients (and related westerly winds) in the North Sea. The MSLP field in the still image of the figure shows that the impact location sits on a ridge in the field. 86 % of the initialised gradient tracing attempts traces the impact northwards on the eastern side of the Scandinavian mountains all the up to the blue ETC track, and not westwards towards the red track, which there is left undiscovered as an impact-relevant ETC. This example shows that the gradient tracing method in some special cases can be vulnerable to small-scale variations in the MSLP field, even when it initialises many tries in a 250 km radius around the impact location.

In the 26 December 2016 event shows a unique failure of the framework. Here, manual inspection of the event revealed that the two red fragments and the green fragment belonged to the same ETC, which originated in the North Atlantic and crossed the Scandinavian mountains in a northwest-southeast movement, terminating in Eastern Europe. However, the framework was not able to connect these fragments because there are several time steps where no ETC centre candidate points could be found as the ETC crossed the Scandinavian mountains. This happened because the MSLP field became very vauge without a well-defined minima. Between the second red and the green fragments there was a gap of four time steps without a minima, which incidentally was right at the time of impact of several of the impacted locations shown in the inset at the lower left corner of the 26 December 2016 event. Because the combined red-green fragmented track did not exist in the four time step gap, the gradient tracing algorithm traced the impacts to the orange and blue ETCs that were located further away.

A central question when choosing between a heuristic and a mathematically exact solution to the correspondence problem is whether the additional computational cost of global optimisation meaningfully affects overall algorithm performance. Table  shows the total computational run times for serial execution of ImpactETC1.0 on all 37 events. The runtime is divided into the main steps of the framework and tested across several pruning radii (i.e., r-values), which lead to different problem complexities to solve. This analysis shows that solving the correspondence problem with the HA accounts for only a small fraction of the total computational time, especially compared to more time-consuming components such as data loading and candidate point identification. Even at the highest level of correspondence complexity (100 km pruning radius), solving the correspondence problem remains a minor contributor to total run time compared to candidate point processing and BLOB analysis. The analysis in Table  shows that the differences between HA and NN drastically increase with the complexity of the problem (small pruning radii), and that HA enables additional connections. While we here investigate tracking of ETCs, which exist at large spatial scales, it is likely that HA would be even more suitable for tracking smaller atmospheric objectives and for tracking in noisier fields. This is something future research could target.

While the ImpactETC1.0 framework demonstrates robust performance in identifying impact-relevant ETC tracks, there are several avenues to further improve its performance and computational efficiency.

Computational efficiency. Data loading consistently dominates runtime across all pruning radii, and while optimised data handling could reduce this component, it will likely remain a significant fraction of total runtime. The candidate identification step is already quite scalable, but it might be possible to avoid excessive block comparisons with further developments. For track reconciliation, there are likely gains to be made, e.g. by adding a distance-based pre-check before BLOB initialisation, which would reduce unnecessary BLOB constructions. The current implementation of the framework is serial but future developments could exploit parallelisation on modern computing infrastructure. Since the framework is event-focused by design, it would be straightforward and efficient to analyse independent impact events in parallel. Specific algorithmic steps could further benefit from parallelisation, especially the candidate point identification and the BLOB construction, since they treat individual time steps independently. The track reconciliation based on the constructed BLOBs would still have to remain serial to avoid conflicts during merging of track fragments. The grid search employed for the impact-oriented calibration of post-processing parameters is embarrassingly parallel. The “nearest storm selection” and “gradient tracing” post-processing options treat each impact location independently and could similarly be parallelised.