Cell tracking of convective rainfall: sensitivity of climate-change signal to tracking algorithm and cell deﬁnition (Cell-TAO v1.0)

. Lagrangian analysis of convective precipitation involves identifying convective cells (“objects”) and tracking them through space and time. The Lagrangian approach helps to gain insight into the physical properties and impacts of convective cells and, in particular, how these may respond to climate change. Lagrangian analysis requires both a ﬁxed deﬁnition of what constitutes a convective object and a reliable tracking algorithm. Whether the climate-change signals of various object properties are sensitive to the choice of tracking algorithm or to how a convective object is deﬁned has received little attention. 5 Here we perform ensemble pseudo global warming experiments at convection-permitting resolution to test this question. Using two conceptually different tracking algorithms, Lagrangian analysis is systematically repeated with different thresholds for deﬁning a convective object, namely minimum values for object area, intensity and lifetime. We ﬁnd that the tracking method has no impact on the detected climate-change signal. The criteria for identifying a convective object, however, can have a strong and statistically signiﬁcant impact on the magnitude of the climate-change signal, for all analysed object properties. For 10 the case considered in our study, this insight reveals that projected changes in the characteristics of convective rainfall vary considerably between cells of differing intensity, area and lifetime; for example, an increase in the area of moderate-intensity cells alongside a decrease for the most intense cells. Our results suggest that for Lagrangian analysis of precipitation in climate models, sensitivity analysis of the climate-change signal in relation to how an object is deﬁned is a useful enhancement.


Climate simulations
ulations from the EURO-CORDEX experiment (Jacob et al., 2014). The three EURO-COREX runs were downscaled from CMIP5 (Taylor et al., 2012) simulations of the MPI-ESM-LR (r1; Giorgetta et al., 2013), EC-EARTH (r12; Hazeleger et al., 2012) and CNRM-CM5 (r1; Voldoire et al., 2013) global models. A 31-day running mean of the resulting climate-change sig-110 nal (Fig. 1c) is added to the initial and lateral boundary conditions of our 0.11 • simulations for all variables (e.g. temperature, specific humidity, pressure and winds). Deep convection is explicitly resolved by the model, while shallow convection is parametrized based on a modified Tiedtke 115 scheme (Tiedtke, 1989). All model settings are taken from the standard configuration of the German Weather Service and precipitation output is saved every 5 min. Aside from the added value of the COSMO-CLM, and CPMs in general, discussed in the introduction, shortcomings in the COSMO-CLM do still remain. Keil et al. (2014) reported insufficient convective triggering under conditions of weak synoptic forcing, while Purr et al. (2019) reported an underestimation of mean precipitation intensity in long-living, extreme convective objects and a general overestimation of the lifetime of convective objects. The 120 results presented below are all based on the 0.025 • CPM ensemble.

Tracking algorithms
We make use of two tracking algorithms. In the first, convective objects are tracked based on advection by the steering flow; we refer to this algorithm as ADV. In the second, convective objects are tracked based on the overlap method; we refer to this algorithm as OVER. These algorithms are chosen (i) because they are representative of two standard approaches to tracking 125 convective objects (i.e. advection-and overlap-based tracking), and (ii) for their low levels of complexity, facilitating generalizability of the results.
The ADV algorithm is based on the method of Brendel et al. (2014), which was developed for tracking convective objects in radar data and was adapted for convection-permitting models by Brisson et al. (2018). The OVER algorithm, on the other hand, is a simple temporal overlap procedure. The algorithms have been summarised in a schematic (Fig. 2). For both algorithms, 130 non-convective precipitation is first masked out using the method of Poujol et al. (2020b). All precipitation below a chosen threshold (P min ) is also masked out. Objects are then identified as contiguous precipitation areas exceeding a minimum chosen area (A min ), based on the number of grid boxes within the object. Objects whose lifetime is shorter than a chosen threshold (T min ) are discarded, as are objects which are not fully in the domain.
In ADV (Fig. 2a), once an object has been identified, its position at the next timestep is estimated based on the steering 135 flow, here the wind velocity averaged across the 500, 700, and 850 hPa levels. From the expected location at the next timestep, convective objects are searched within a defined search radius whose length is proportional to the wind speed (see Brendel et al. (2014)). For the object nearest to the expected location, the procedure is further iterated until no object is found. For object splits, the object nearest to the expected location is chosen, while the remaining object(s) is (are) considered as a new object(s). For object mergers, the largest of the original objects is continued, while the other track is ended. In OVER (Fig. 2b), the spatial footprint of an identified object is first determined and an overlap between this footprint and any footprints at the next timestep is sought. The process is further iterated until no overlap is found. For both object splits and mergers, the object with the largest overlap (by precipitation volume) is continued, while the other object is considered new (splits) or to have ended (mergers).
Both algorithms compute the following lifetime diagnostics for each object: mean and maximum areal precipitation intensity 145 (P avg , P max ), mean and maximum object area (A avg , A max ), mean and maximum integrated precipitation volume (V ol avg , V ol max ), lifetime (T ), total distance travelled (D) and average speed (S). We use 5 min precipitation totals in our study.

Analysis
The ensemble setup of 14-day CPM simulations provides an ideal platform to test a wide range of options for defining a convective object and comparing two tracking algorithms. The aim is to see whether, in the presence of climate warming, the 150 tracking algorithm or how a convective object is defined may impact the magnitude of any detected changes in the characteristics of convective objects. As mentioned in Section 3.2, we analyse the object characteristics P avg , P max , A avg , A max , V ol avg , V ol max , T , S and D over the lifetime of each object. For each ensemble member, we consider the median of these metrics; the change of the associated ensemble mean is then computed. In addition to the characteristics of convective objects, we also consider the total number of convective objects (N obj ) and the total volume of convective precipitation (P tot ). Our and T min =15 min. These reference settings are chosen based on a balance between previous studies in our region (Brisson et al., 2018;Purr et al., 2019Purr et al., , 2021 and finding a mid range between the tested options. Results for the reference settings are 165 presented in Table 1, where the absolute values can also be found.

Uncertainty and significance
To test and conveniently display the statistical significance of any differences in the detected change signals, we employ bootstrap resampling in conjunction with the confidence intervals (CIs) proposed by Goldstein and Healy (1995). All ensemble members are first resampled 10,000 times with replacement and the change signal is re-computed each time, giving a dis-170 tribution of 10,000 changes. Under the normal approximation, the bootstrap CIs for the statistic t i can be constructed as Davison and Hinkley, 1997), where α is the two-tailed probability, z α the corresponding positive gaussian quantile, and σ the standard deviation. In the case of two change statistics t i and t j , their differences will be statistically significant at level α if the condition |t i − t j |/ σ 2 i + σ 2 j > z α is satisfied. Their CIs, meanwhile, will be non-overlapping if |t i − t j |/(σ i + σ j ) > z α . Rewriting the left-hand side of the latter in terms of the former, it can be shown that differences 175 significant at level α will have non-overlapping CIs constructed as where This can be repeated across multiple categories to compute a single z β , which is the average taken across all pairs i, j; each 180 category i ∈ Z + then has CIs t i ± z β σ i (Goldstein and Healy, 1995). Statistically significant differences between the different change signals can hence easily be discerned from an absence of overlap between the Goldstein-Healy CIs. In our study, we take α = 0.95.
4 Results I: Sensitivity of climate-change signal 4.1 Reference setup 185 We begin with a reference setup for both algorithms (ADV and OVER): a minimum area A min = 8 grid boxes, a minimum precipitation threshold P min = 8.5 mm h −1 (0.7 mm/5 min) and a minimum lifetime T min = 15 min. This setup serves as a threshold "base-state" at which in the following sections at least one threshold (A min , P min , T min ) is held constant while the remaining threshold(s) vary singularly or jointly. Under this setup (Table 1), we find ensemble medians of about 4,500 objects per member, which are concentrated in the western half of the analysis region (Fig. 3). Median lifetimes and distances travelled 190 for the objects are roughly 35 min and 12 km, respectively, for each algorithm. For the lifetime object mean precipitation rates and areas, an equivalent hourly rate of 18 mm h −1 and an area of 96 km 2 are found. In the PGW ensemble, the total numbers of objects increases by over 45%. Changes in the object characteristics in response to the PGW signal range from -6 % to +38 % (Table 1), depending on the object characteristic. The greatest increase is seen in distance travelled, with minimal change in object lifetime. Object areas and volumes increase, with areal mean precipitation intensity decreasing. The net effect 195 of the aforementioned changes on total convective precipitation is an increase of roughly 87 (ADV) to 98 (OVER) %, which is the most noticeable difference between the two tracking methods. Amongst all change signals, no statistically significant differences between ADV and OVER are evident.

Minimum size of object (A min )
In this subsection, we hold T min and P min constant at their reference values. A min (the minimum area threshold) is varied, 200 with values of A min = 2 i grid boxes, where i = 1 . . . 6 (Fig. 4). For P avg and P max (the object lifetime mean and maximum precipitation intensity), the minimum object size has no significant impact on the response to warming; this is mostly true for the object lifetime too. For the remaining metrics, however, the A min threshold has a significant impact on the resulting climate- much less of an impact on the magnitude of the climate-change signal than varying A min . Across the sampled range of P min thresholds, clear statistically significant differences (Fig. 5) are most evident for diagnostics which characterize the object's precipitation intensity: P avg and P max show a monotonic upward trend in their climate-change signal with increasing P min ; 225 this is in contrast to varying A min , which was shown to have no effect on the climate-change signals of P min and P max . Some smaller but statistically significant differences are also seen in the object area's response to warming (A avg , A max ; Fig. S2) and in the total number of objects. For the remaining object characteristics, the range of tested P min thresholds produces very few significant differences in the response to warming. The speed of the objects does, however, show a clear monotonically  decreasing trend (Fig. S2), suggesting that over a wider range of P min thresholds, significant differences may emerge. As with 230 the A min threshold, no statistically significant differences between the tracking methods are evident.

Minimum lifetime of object (T min )
Here we vary the minimum-lifetime threshold T min of the objects, while keeping P min and A min at their reference values

Total convective precipitation
Changes in the characteristics of convective objects do not necessarily inform us about changes in total convective precipitation.
An additional metric of interest in object-oriented precipitation analysis may thus be the total amount of convective precipitation attributable to the identified objects (P tot ), and how this responds to warming. By jointly varying (i) T min and A min , and (ii) T min and P min , a large range of P tot responses is found across 132 setups, with a strong P tot increase in all cases (Fig. 7), 250 ranging from about +70 % to +120 %, depending on the combination of the three thresholds. As with the reference setup (Section 4.1), considerable differences are often evident between the two algorithms, with those for OVER typically stronger.
However, due to the large range of uncertainty in the magnitude of these increases, no statistically significant differences between the tracking methods are found.
A general, though not uniform, pattern of a stronger warming response with higher T min thresholds and lower P min thresh-255 olds can be discerned, while no clear influence of the A min threshold on the P tot climate-change signal is evident. The higher increases in total precipitation with higher T min thresholds mirror the changes seen for the number of objects as T min increases ( Fig. 6), suggesting that the latter explains differences in the climate-change signal of P tot as T min is varied. Higher increases in P tot as P min decreases, meanwhile, appear to be explained by differences in the A avg signal as P min is varied (Fig. S2f).

Results II: Future projections 260
In this second results section, projected future changes to cell characteristics based on our PGW simulations are evaluated.
Here, we consider the magnitude of the projected changes, as opposed to differences in the magnitude under the different thresholds presented in previous sections. We consider how Lagrangian projections might best be presented based on the lessons learned in Sections 4.2 to 4.5 and compare the projections with those found in the literature for our region. As it has been shown in previous sections that the choice of tracking method has no impact on our results, we will for clarity show 265 results for just the ADV algorithm. Purr et al. (2021) performed Lagrangian analysis of convective cells over Germany under present and future (RCP8.5) conditions using continuous 30-year simulations. Their model and resolution were the same as in this study (COSMO-CLM, 0.025 • ) and thresholds of A min = 4 grid boxes, T min = 15 min and P min = 8.5 mm h -1 were used. Under these thresholds and accounting for uncertainties, our results for T are in agreement, whereas our projected changes for A max and S are higher 270 and for P avg and P max are lower (our other metrics weren't considered in the study). This confirms that our 14-day study period of high convective activity is not representative of climatological conditions; our projections should thus be considered as indicative for the synoptic conditions present during our study period.
In Sections 4.2 to 4.5 it was shown that the choice of thresholds for defining a convective object can significantly impact the magnitude of the climate-change signal. We therefore propose analysing the output of the tracking algorithm by first 275 partitioning the data into bins delineated by different values of A avg , P avg and T , the metrics on which the object thresholds are based. To maximize the range covered by all bins, the trackings with each of the three lowest thresholds -A min = 2 grid boxes, P min = 4.5 mm h −1 and T min = 15 min -alongside their counterpart reference thresholds are used (Fig. 8).
Considering all metrics, it is found that, in the RCP8.5 scenario, the greatest projected increase is for the number of convective objects (N obj ), which has maxima for (i) low-and high-intensity objects (close to a doubling), and objects with (ii) large 280 area and (iii) long lifetimes. Partitioning objects based on their intensity reveals a strong increase in most metrics across the spectrum (Fig. 8a), which reaches a maximum for objects with mid-range intensities (∼ 9 -15 mm h -1 ): future increases of 10 -20 % for object area and volume, or as high as 60 % for the distance an object travels. Using object area as a basis for partitioning ( Fig. 8b), meanwhile, a general negative change for most object characteristics is found: decreases of 5 -20 % in volume and intensity for the smallest objects. Exceptions to this are for distance travelled and speed, where increases of over 285 50 % are projected in some bins. One thing that all object characteristics have in common is that as object area increases, their projected change tends asymptotically to a common value. This mirrors many of the results shown in Figs. 4 -6 and suggests that the spatial homogeneity of the precipitation field is an important factor in the sensitivity of Lagrangian projections to object thresholds, i.e. larger area thresholds (A min ) give projections whose magnitude is less sensitive to further increases in the area threshold. Similar to the T min thresholds presented in Fig. 6b, the object lifetime only has a clear impact on future 290 changes in objects' distance travelled and speed (Fig. 8c), ranging from 50 % for the shortest-lived objects (15 min) to 10 % for the longest-lived objects (> 3 h). Meanwhile, the other object characteristics show future changes which vary little across the spectrum of object lifetimes, with projected changes mostly within ± 10 %.
The results following from the above partitioning (Fig. 8) may appear somewhat contradictory. For example, a generally positive change in characteristics when partitioning by object intensity (a) and a generally negative change in most character-295 istics when partitioning based on object area (b). Or, additionally, a negative change in volume when partitioning by object area (b) and a positive change in volume when partitioning by object lifetime (c). In Fig. 7 it has already been shown that total precipitation increases in all cases. The increase in P tot can be explained -without having to test a myriad of object thresholds -by the insights revealed from the partitioning applied in Fig. 8. From here, the following picture emerges for our study period: (1) the total number of objects increases in (almost) all cases, (2) future objects have larger areas and volumes, regardless of 300 how long they live or how intense they are, (3) despite this, objects of all areas and lifetimes have lower mean intensities, (4) it can thus be concluded that the increase in P tot is driven by the combined effect of more objects and an increase in the area of these objects, (5) the increase in object volumes despite a decrease in intensity shows that the effects of more objects and higher areas are dominant over the reduction in mean object intensity, which acts in the opposing direction. A final note would be that the area of the most intense objects is actually found to decrease (-6 %) and their maximum local precipitation intensity Aided by the growing use of kilometre-scale climate models (Lucas-Picher et al., 2021), Lagrangian methods for analysing the response of convective precipitation to climate change have become increasingly popular (e.g. Prein et al., 2017;Poujol et al., 2020a;Purr et al., 2021). This object-oriented approach is particularly useful for studying changes in the characteristics 310 of convective cells. In our study, we have tested the sensitivity of Lagrangian projections to the choice of (i) tracking algorithm and (ii) how a convective object is defined. Two simple tracking algorithms, each representative of a common approach to Lagrangian analysis, were employed to track convective objects in convection-permitting PGW ensemble simulations, allowing their respective climate-change signals to be compared. Furthermore, for each algorithm, the sensitivity of the climate-change signal to how a convective object is defined was examined by systematically varying the threshold criteria for identifying 315 a convective object, namely: minimum size (A min ), intensity (P min ) and lifetime (T min ). In total, 132 configurations were tested. Our PGW simulations encompassed a 14-day period with elevated levels of both strongly-and weakly-forced convection (Section 2), offering a diverse representation of convective objects against which the different algorithms and configurations could be tested.
Our first main result is that the tracking method appears to have no significant impact on how the properties of convective 320 objects, or the total number of convective objects, respond to climate change. The representative advection-and overlap-based algorithms which we implemented produce very similar climate-change signals for all object properties, with no statistically significant differences found. Additional tests of this conclusion using a set of climate-length simulations, those used in Meredith et al. (2019), show that the insensitivity of the climate-change signal to the tracking method remains consistent ( Fig. S6 and accompanying discussion).

325
Our second main result is that, unlike the tracking algorithm, the definition of what constitutes a convective object has a potentially large impact on the climate-change signal for all object properties, as well as for changes in the total number of objects. The minimum precipitation intensity (P min ), minimum size (A min ) and minimum lifetime (T min ) thresholds for defining a convective object were all found to be relevant. How the climate-change signal responds to varying these thresholds was found to depend on the object property under investigation. For example, the minimum object size had no significant 330 impact on changes in the object's precipitation intensity, but did lead to different climate-change signals for changes in the total number of objects, as well as changes in object properties like the integrated precipitation volume, distance travelled and more. Similarly, the minimum intensity threshold affected the climate-change signal of object intensity, but was not relevant for, e.g., changes in the object volume. Changes in total convective precipitation were also sensitive to how an object is defined.
As discussed in the introduction, the definition of what constitutes a convective object shows considerable variance in the 335 literature. An open question in climate-change research is whether the spatial extent of convective storms will increase or decrease with warming (Fowler et al., 2021). Our results suggest that, at least in some regions, the answer may be dependent on how a convective storm is defined. For our case study, the most intense convective objects showed a decrease in spatial extent (-6 %), while less intense objects showed an increase (up to +15 %).
is for the climate-change signal of precipitation intensity as P min increases, which sees a levelling-off at higher thresholds.
Otherwise, the main difference is that, in many cases, the uncertainty in the climate-change signal grows, so that the number of statistically significant differences based on different object definitions reduces ( Figures S7-S9 and accompanying discussion).
Uncertainty due to the higher quantiles would be expected to decrease with a larger sample of convective objects, e.g. from longer, climate-length, simulations.

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To reduce the sensitivity of Lagrangian-based projections to how an object is defined, we suggest performing spectrum-based analysis by first, e.g., binning the data based on object area, intensity and lifetime before computing the desired statistic within each range of interest (Section 5). Using this approach, it was revealed for our study period that under an RCP8.5 scenario convective cells will (in general) increase in number, area and volume, but decrease in mean intensity. The latter will, however, be of secondary importance and total convective precipitation is projected to increase. For the convective cells with the most 350 extreme mean intensities, however, the total precipitation volume and area will decrease, while the maximum local intensities will increase.
Our results hint that the sensitivity of the climate-change signal to how an object is defined may, for certain (not all) object properties, decline as object size increases (Figs. 4,6,8b,S4). Were this the case, then studies focused on larger precipitation systems (e.g. Nissen and Ulbrich, 2017;Prein et al., 2017) could be expected to lead to higher certainty. This finding, however, 355 cannot automatically be extrapolated to other weather situations or studies at climate timescales and, thus, requires further investigation. It is similarly true that the sensitivities found for our test period would not necessarily be the same sensitivities found in other studies, as our experiment encompasses a specific period, region and climate-change profile. What we have demonstrated, is the principle that in Lagrangian analyses of convective cells, the climate-change signal of different object properties can be dependent on how an object is defined. This dependency also has consequences for identifying the physical 360 mechanisms underlying future changes in total convective precipitation. The relative importance of specific object properties in interpreting changes in total convective precipitation will not remain constant if these properties' climate-change signals respond differently to changes in how an object is defined. As such, analysing Lagrangian projections by first partitioning the data based on specific object properties (e.g. intensity, area, lifetime) can clarify the underlying mechanisms by which future precipitation changes.

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For researchers studying future changes in convective precipitation using Lagrangian methods, the first message is that, amongst the standard approaches, the choice of tracking algorithm will likely have little impact on the results. The second message is that the minimum thresholds for what constitutes a convective object should be carefully chosen based on what is most appropriate for (1) the study region and (2) the aims of the study. Alongside this, the change signal across a range of object intensities, areas and lifetimes should be explored (see Fig. 8). While computational resources may limit how low the 370 object thresholds can be set, the lower the object thresholds then the wider the range of responses that can be investigated. To conclude, Lagrangian analysis is an important technique for studying future changes in precipitation. To make best use of this approach, the uncertainties in the climate-change signal associated with how a convective object is defined should be examined wherever possible. cess) at https://doi.org/10.5281/zenodo.6977074 (Meredith et al., 2022b). The 0.025 • simulation data have been archived under an open access license at the DKRZ Long Term Archive with the permanent link <https://www.wdc-climate.de/ui/entry?acronym=DKRZ_LTA_ 1152_ds00302> (Meredith et al., 2022a). The ERA-Interim and MERRA2 reanalyses used as lateral boundary forcing are publicly available via https://www.ecmwf.int/en/forecasts/datasets/reanalysis-datasets/era-interim and https://gmao.gsfc.nasa.gov/reanalysis/MERRA-2/, respectively (last accessed 01.08.2022).