Generally, the algorithm used in WaterSip works based on trajectories of atmospheric air parcels, described by a time-dependent coordinate vector x(t) of positions given by longitude λ, latitude ϕ, and height z: 1x(t)=[λ(t),ϕ(t),z(t)].

Air parcel trajectories on the way to a target area A (Fig. a, black frame) may pass over the underlying surface of either ocean (blue area) or land regions (green area), thereby acquiring moisture at the moisture sources (hatched area). Acquisition of moisture by an air parcel is termed a moisture uptake, and the ensemble of all moisture uptakes to an air parcel then comprises the moisture source. Identifying the relevant set of air parcels arriving over a target domain and their corresponding trajectories is the first step of the algorithm (Fig. c, step 1).

Next, in order to determine the moisture sources of an air parcel, specific humidity q and air temperature T, or relative humidity RH and T, must be available along the same trajectory. These are commonly obtained by interpolation from four-dimensional NWP model output fields. For example, the 4-D specific humidity field Q is used to obtain q(t) along the transport pathway of air parcels towards a target area at every time interval i with time step Δt: 2qi=Q[x(t=i⋅Δt)].

The conceptual starting point of the algorithm is then the Lagrangian water budget, describing the change in specific humidity along an air parcel trajectory resulting from evaporation (e) and condensation (c) : 3DqDt=e-p

The discretized Lagrangian water budget, giving the change in q along a trajectory with time step Δt of ∼ 1–6 h at time step i is then 4ΔqiΔt=ei-pi.

The arrival time at target area A is given by t=0 at step 0, and i increases backward in time along the trajectory (Fig. d, horizontal axis). The change in specific humidity from the previous interval, i+1, to interval i is then obtained from 5Δqi=qi-qi+1.

Importantly, Δqi can either be positive, zero, or negative (Fig. d). A positive Δq (i.e., Δq>0) is interpreted as a so-called moisture uptake, while a negative Δq (i.e., Δq<0) is interpreted as a moisture loss along the trajectory. Thereby, Δqi is always a net effect of both ei and pi that may have affected the water budget of the air parcel during time step i. The algorithm necessarily assumes that either evaporation or precipitation exclusively act on an air parcel during discretisation time Δt, which introduces uncertainties depending on the discretisation time Δt (see Sect. ). In practice, a critical specific humidity threshold Δqc is used to filter out smaller variations in Δq that are due to numerical noise (see Sect. , Fig. d).

Conceptually, each moisture uptake Δqn along the trajectory path contributes a share Δqn to the arriving moisture q0: 6q0=∑n=1UΔqn+qr.

Thereby, qr is a residual amount of moisture that already was present in the air parcel at the earliest available time step of the trajectory with unidentified sources (see Sect. ). Due to rainout and mixing effects along the further transport, a part of the uptake moisture can be lost, and Δqn will only contribute with a fraction fn to the moisture at the arrival location: 7q0=∑n=1Ufn⋅Δqn+qr.

The objective of the accounting step of the algorithm is now to obtain an estimate of the contributing fraction fn for each of the U identified moisture uptakes (Fig. c, step 2). The moisture accounting provides all fn, which then allow to diagnose where, when, and under what conditions the air parcel water budget changed, and to what extent this is reflected in the precipitation falling from the air parcel (Fig. c, step 3).

At the time of a moisture uptake, the fractional contribution fn for each uptake n is directly obtained from 8fn=Δqnqn.

If at a later time another moisture uptake contributes to the specific humidity in the same air parcel, qn will change, and fn needs to be recalculated to reflect its decreased share. This sequential updating of the fractional contributions is referred to as the accounting of specific humidity along the trajectory. In Fig. d, a moisture increase between time intervals 4 and 5, for example, that is larger than the threshold value Δqc, gains a smaller relative share of the arriving moisture q0 after a large uptake occurs between time steps 1 and 2. A more detailed, quantitative example is given in .

In the case of another moisture uptake at a later time step i, the fractions fm of all previous uptakes m>i simply need to be recalculated using Eq.  as 9fm′=Δqmqiform<i.

In the case of a moisture loss (due to precipitation) at a later time i, some of the moisture contributed from previous uptakes m<i will be lost. Since the algorithm assumes that the water vapour is well mixed within a given air parcel, precipitation events do not change the fractional contributions at that time. However, a later moisture loss Δqi does remove moisture from previous uptakes Δqm according to their fractional contribution fm at time m: 10Δqm′=Δqm-fm⋅Δqi.

The accounting is complete when the above steps in Eqs. (7)–(9) have been applied from the earliest time of the trajectory up to the arrival time t=0. compactly expressed the final result of this accounting method in two update equations. While the algorithm is presented here in a detailed representation of each of the computational steps, the equations in are almost entirely equivalent, and may facilitate a numerical implementation of the algorithm (for one exception see below). WaterSip currently implements only this stepwise accounting procedure.

The vertical location of the air parcel as moisture uptakes take place can be used to infer the reliability of the assumption that water vapour stems from evaporation at the current location. If air parcels are within or close the the height of the (well-mixed) boundary layer, this assumption is considered more likely to be valid than if the uptake is detected well into the free troposphere (Fig. b, dashed line). The boundary-layer height scale factor sh is used to separate between uptakes considered to be within the boundary layer or within the free troposphere (see Sect. ).

Up to this point, the algorithm has provided quantitative information about the contributions of uptakes to the moisture within an air parcel at x(t=0) at some height z. While it is possible to directly use the information about the sources of the water vapour, more commonly one would be interested in the moisture sources of surface precipitation. In order to transfer the water vapour information to surface precipitation at the arrival point of trajectories, moisture loss from a precipitating air parcel at the last time step before arrival (Δq0<0) is assumed to directly contribute to surface precipitation. To obtain a reliable estimate of the total precipitation in an area, a sufficiently large set of particle trajectories is required to faithfully represent the atmospheric column above a target area (Fig. b, crosses). Then, the Lagrangian precipitation estimate P̃ (in units of kg m-2(Δt)-1) can be computed as the column integral of all Δq0k for all K air parcels with mass mk residing over the target area A at time t=0: 11P̃=1A∑k=1K|Δq0k|⋅mk,

To obtain a spatially coherent view, the contribution of each individual air parcel is gridded onto the arrival grid using a gridding kernel (see Sect. ). Thereby only air parcels fulfilling certain requirements are considered in the analysis of precipitation moisture sources. For example, to further support the presence of clouds leading to precipitation, a critical relative humidity threshold RH_c is applied, where clouds are assumed when the grid-scale relative humidity exceeds 80 % (see Sect. ).

The sum of all fractional contributions fik from identified moisture uptakes U along trajectory k, the total accounted fraction ftotk, provides information about the known sources of q0k and thus Δq0k: 12ftotk=∑i=1Ufik.

Hereby, U is the number of all moisture uptake events identified along trajectory k. Based on the vertical position (particle elevation) z where the uptake is identified, the fractional contributions are subdivided into the two categories: the accounted fraction in the boundary layer, fblk, and the accounted fraction in the free troposphere, fftk. The sum of both fractions is the total explained fraction ftotk, which should obey the relation 13ftotk=(fblk+fftk)≤1.

Then, ftotk provides information about the consistency of the algorithm. In general, and on average, this relation is found to be indeed fulfilled. However, if the Δqc threshold is set to values outside a meaningful range, the total explained fraction can exceed 1.0. Typically, this will happen for large values of the precipThreshold parameter, which can lead to undetected rainout events (see Sect. ). Users are thus advised to inspect ftotk in their results and to make adjustments if needed. Notably, this is a difference to the algorithm of where ftot≤1 by definition.

The explained fractions fblk, fftk can be used further to subdivide the Lagrangian precipitation estimate P̃ into its respective explained fractions. The total explained precipitation P̃tot is obtained from 14P̃tot=∑k=1Kftot⋅P̃k=1A∑k=1Kftotk⋅|Δq0k|⋅mk.

Corresponding quantities P̃bl and P̃ft can be computed for fbl and fft. A choice to consider in this respect is which precipitation should be used for the weighted average of diagnosed quantities, P̃ or only the explained fractions of (P̃tot, P̃bl, P̃ft). Given that the Lagrangian precipitation estimate P̃ is typically in error by about 20 %–30 % or more , this choice might be secondary for the interpretation of the final results. In the algorithm as implemented in WaterSip, P̃ is used, rather than P̃tot.

The core parameters of the algorithm are a set of threshold parameters that determine the identification of moisture uptakes and precipitation events from the Lagrangian moisture budget along trajectories.

When used to diagnose precipitation origin, air parcels are retained for analysis if (i) they reside within the target region, (ii) during the preceding time step had a specific humidity decrease larger than the precipitation threshold value (see Sect. ), and (iii) had a relative humidity within chosen bounds (see Sect. ). The geographic location of the target region and the considered time period are therefore fundamental choices for an analysis with WaterSip. Other important elements regarding how the air mass is discretised are the format and length of trajectories, and the time interval between particle locations.

Another fundamental setting is the start- and end-time of the analysis, regulated by parameters startDate and endDate in setting group Case. Since the analysis proceeds backward from the time of arrival, the start date is larger than the end date. Importantly, in order to obtain results for, e.g., 20 d accounting along a trajectory, the time difference between start date and end date must be at least 20 d plus one time step (to enable calculating the required number of specific humidity changes). The length of the analysis period is defined by parameter trajPoints in settings group Case, and the duration of the entire accounting period for each trajectory is then given by the parameter timeStep times trajPoints (see Sect. ).

WaterSip uses the mass represented by each air parcel to transfer the results of the diagnostic onto a grid which will then be written to an output file for further use. There are two output grids in WaterSip. The coarser source and transport grid covers the entire potential source regions of the traced moisture, and is typically of hemispheric or global dimensions. The typically smaller arrival grid covers the target region of the analysis, and usually has a finer grid spacing to reveal spatial details within the arrival domain.

Both grids are specified by a respective set of parameters in the setting group Grids (Table ). The source grid is defined by the parameters sourceGridMinLon, sourceGridMaxLon, sourceGridMinLat, sourceGridMaxLat, which provides the boundaries in latitude-longitude coordinates. The parameters sourceGridDx, sourceGridDy set the grid spacing in longitude and latitude direction, respectively. A corresponding set of grid parameters exists for the arrival grid (arrivalGridMinLon, arrivalGridMaxLon, arrivalGridMinLat, arrivalGridMaxLat, arrivalGridDx, arrivalGridDy).

Quantities are gridded onto both grids using weighting factors. The weight mwk for gridding a quantity ξ (for example the air temperature at the time of arrival) is obtained from the water mass Δq0k lost from an arriving air parcel k as 17mwk=mk⋅Δq0k.

With K trajectories arriving as a vertical stack (Fig. b), the weighted average of all ξk at a location λ,ϕ is thus obtained as 18ξ̃(λ,ϕ)=∑k=1Kξk⋅mwk∑mwk.

This arrival point information needs to be transferred onto the arrival grid. Instead of simply gridding the information to the grid cell closest to the air parcel location, WaterSip allows to specify a spatial representativeness of this information, which is then used to distribute it to surrounding grid cells. Representativeness is provided by a gridding radius rg, specifying a circular distance from the arrival point. The areal overlap between the circular weight and neighbouring grid cells are then used as weights to distribute air parcel quantities ξ onto the arrival grid. For example, with trajectories computed at a horizontal distance of 50×50 km, quantities can be assumed to be representative for a circular area with radius 30 km. If the arrival grid for this setup is then chosen to be arrivalGridDx = arrivalGridDy =0.25°, and the gridding radius is set to rg=30 km, one can expect to reveal finer interpolated structures within the arrival domain. The gridding radius is set as parameters arrivalGridRadius in settings group Grids. Corresponding reasoning applies to the source grid radius sourceGridRadius. Here, a common combination for a hemispheric output grid is sourceGridDx = sourceGridDy =1.0°, and sourceGridRadius =110 km.

The gridding proceeds by finding the area overlaps between the circular gridding area and each grid box. A corresponding fraction of the total quantity to be gridded is then added to each grid box . It is important to be aware of the impact user choices have on the results. A too small gridding radius for a given grid will lead to gaps between grid points, while a too large gridding radius will cause smoothing and potentially loss of detail in the gridded fields.

A final choice regarding the gridding is whether uptake events are gridded with the time stamp of the arrival in the target region, or to the time when uptakes take place. The choice of the assigned time is important for later analysis of the results. In the first case, moisture sources are valid for all precipitation arriving within a chosen time period in the target area, and can be compared to other quantities regarding the arrival time. In the second case, the moisture sources can be compared to properties at the source regions during valid time, for example total evaporation fields. The assignment time is set as parameter assignToUptake in settings group Grids, where 0 denotes that the uptakes are assigned to arrival time, and 1 denotes assignment to uptake time (see Sect.  for an example).

There are some general considerations for the input data to be made such that the WaterSip diagnostic output becomes representative for the chosen target area. The setup for creating trajectory input data to WaterSip varies depending on whether a particle dispersion model (FLEXPART) or an air mass trajectory model (LAGRANTO) is used.

When using the particle dispersion model FLEXPART, commonly, hemispheric or global domain-filling forward simulation are set up. In domain-filling mode, FLEXPART automatically discretises the air mass over the chosen target area into a specified number of particles of equal mass, and distributed according to pressure. With a larger number of particles, WaterSip will provide statistically more robust results, in particular for more long-range and long-duration water vapour transport. Within FLEXPART, particles are advected forward continuously for the simulation time. WaterSip then extracts a sub-set of these trajectories according to the run parameters. A global domain-filling setup has also been applied in the example case presented here (Sect. ). So-called particle dump needs to be activated in FLEXPART to obtain the input data needed by WaterSip. Writing out additional variables with each particle position requires modification of the routine partoutput_pos.f90 or partoutput_short.f90 of the FLEXPART code. Example code for this modification for FLEXPART versions 8.2 through 10.4 is available in the online supplement .The most recent version 11 of FLEXPART writes particles in netCDF output. Until WaterSip has been enabled to ingest this input directly, users will have to convert the FLEXPART version 11 particle dump from netCDF to the binary format. A corresponding helper tool is planned to be made available on the WaterSip respository. Running FLEXPART in backward mode can be a more efficient alternative to a global domain-filling run, but requires a new release interval to be defined for every time step. General instructions for setting up and running FLEXPART are available in the respective literature .

When used together with FLEXPART input, the WaterSip configuration file needs to contain a setting group Flexpart, where several parameters need to be specified (Table ). The parameter particleMass contains the mass per particle as either specified in the FLEXPART releases file, or as written out by FLEXPART in domain filling mode in the output file part_mass. The input files need to be contained in a folder specified by parameter inputDir in setting group Case in the configuration file. As individual particle dump files are read in from FLEXPART output for each time step, the pathname is specified with a wild card pattern (%s) representing the date string, e.g. inputDir = /path/to/directory/partoutput_pos_%s.

When using output from a trajectory model, such as LAGRANTO, the air mass over the target area needs to be discretised manually. This is achieved by defining the start locations of the trajectory calculations in LAGRANTO's startf file. One common way to discretise a target region is to define a regular distance spacing in projected coordinates (e.g., one starting point every 25 or 50 km in both directions). In the vertical, it is recommended to space the starting points at equal pressure intervals. This way, each air parcel trajectory represents an equal amount of mass, which can be entered in the configuration file for a LAGRANTO setup as described earlier (Eq.  and Sect. ).

It is recommended to run the LAGRANTO program caltra with the flag “-j” to prevent trajectories from intersecting with topography. WaterSip currently only works with the text-based output format of LAGRANTO. Since trajectory variables such as air temperature and specific humidity can be at any column in these files, the column number for each variable needs to be specified explicitly in the configuration file as parameters (Table ). The input path for LAGRANTO trajectory files is again specified using parameter inputDir in setting group Case of the configuration file, using a file name pattern with wildcard %s to match trajectory files to the desired date range.

The WaterSip code is written in the programming language C++. A sample makefile is included with the code repository (https://git.app.uib.no/GFI-public/watersip, last access: 20 November 2025) to compile the program code, and that users may adapt to their computational environment. Installed netCDF, netCDF-C++, and netCDF-C++4 libraries are in addition required to compile the program code. Lightweight parallelisation has been included by means of OpenMP directives in the code, but compiling with OpenMP is optional. Results of some performance tests are described in Sect. .

After compiling WaterSip, the input files need to be made available from an external source, such as running FLEXPART or LAGRANTO. The input data and configuration file for the test case in Scandinavia presented throughout this manuscript is available as a supplement to this manuscript . A script is included with the test dataset that sets up a recommended directory structure to run WaterSip. Users are invited to experiment with the test case before creating their own setup to get familiar with the usage and output from WaterSip. When running an own case, the directory names for input and output data need to be specified in the configuration file. Then, all other configuration file settings, such as the time period, the target region, diagnostic parameters, need to be set by the user. Importantly, the sections in the configuration file specific for the used input data need to be included and edited to match the input data (Tables  and ).

To start a diagnostic run, the compiled WaterSip program is then executed from the command line. The name of the configuration file is the only argument that needs to be provided when WaterSip is started. WaterSip then starts executing, and provides either output to the terminal about the progress of the analysis, including a time estimate until completion, or an error message that will give an indication how the setup and configuration file need to be adjusted.

When a diagnostic run completes, a set of output files is written to the file location specified as parameter outputDir in the configuration file setting group Case. The main output files, containing gridded fields and time series for different averaging times (time step, daily, monthly, annually, all), are created in netCDF format (Table ). In particular the grid files for days and time steps can become rather large for longer runs. The creation of gridded output files for selected averaging times can therefore be deactivated by setting the respective parameters in parameter set Output from 1 (true) to 0 (false, see Table ). The series files for different intervals contain time averages of the gridded fields, characterised by the precipitation-weighted mean of each quantity, the unweighted standard deviation, and the minimum and maximum values in the grid area. A sectorisation of the results into user-specified latitude/longitude regions are also contained in the series files (Sect. ). Further output files comprise histogram files (Sect. ) and trajectory files (Sect. ).

Which variables are contained in the output files can be controlled by the parameters in setting group Variables in the configuration file (Table ). Output variables are included in the output by setting the corresponding parameter values from 0 to 1. Users can also choose to include static fields, such as topography or the land-sea mask using parameter staticFields. The choice in the setting group Variables affects both grid and series files. If output is activated for variables that are not contained in the input file, WaterSip calculates these from the available information if possible, and otherwise fills in missing values during output. Finally, WaterSip can optionally write out air parcel trajectories in a compact binary format to inspect the result of the moisture diagnostics in detail for individual air parcels (Sect. ).

To exemplify the use of the WaterSip diagnostic, a case study for a 10 d Northern Hemisphere summer period (5 to 15 August 2022) over Scandinavia is now presented. The weather situation during the period was characterised by westerly flow, which was steered to higher latitudes by a high pressure system over Eastern Europe. This flow configuration brought several low-pressure systems from the North Atlantic from southwesterly directions towards the Norwegian coast and land areas downstream, which then shed precipitation over Scandinavia.

In order to obtain the required input data for the WaterSip diagnostic, the FLEXPART particle dispersion model in Version 10.4 was run with input fields from the ERA5 reanalysis for a Northern Hemisphere domain with 10 000 000 particles in domain filling mode. The simulation was initialised on 00:00 UTC on 15 July 2022 and run forward for 31 d until 00:00 UTC on 15 August 2022. FLEXPART particle output files (see Data availability section) contain latitude, longitude, height, air temperature, specific humidity, boundary-layer height, and air density at each particle's location, as well as skin temperature, 2 m specific humidity and topography height below the particle. WaterSip was then configured to identify the moisture source and transport properties for a time period of 10 days (5 to 15 August 2022) in the region 54–71° N and 4–32° E. A 6 h time step, 20 d backward tracing, and threshold parameter values RHc=80 and Δqc=0.2 gkg-16h-1 were used for identifying moisture uptakes and losses along the trajectories. Output was activated for all available variables and time steps. The WaterSip setup and output files for this example run are available with the sample data .

The most fundamental result of the moisture source diagnostic are the moisture sources, that is the spatial distribution of the moisture uptakes that contribute to precipitation in the target region. Figure  shows an example for the total moisture sources of precipitation in a arrival domain over Scandinavia (yellow frame). This moisture source area (shading) may be interpreted as the subset of all evaporation at the moisture sources that ends up as precipitation in the target region. The units of kgm-2d-1, also corresponding to mmd-1, are obtained from gridding the contribution of moisture uptakes to precipitation in the target region (fi⋅Δq0). When integrated spatially, taking the area of each source grid box into account, one obtains the total precipitation mass in the target area during this time interval. If divided by the target area, the precipitation rate for the analysis period then corresponds exactly to the value of the Lagrangian precipitation estimate (see Sect. ).

In the example case, the largest contributions to precipitating water come from southern Scandinavia and the western North Atlantic, but the moisture sources also stretch into Central and Eastern Europe (Fig. a, grey to red areas). In addition, a large area is covered by moisture uptakes with 0.25 mm d⁻¹ or less (blue shading), reaching deep into continental North America. In the example here, the gridding radius on the global maps with a 1.0 × 1.0° spacing was set to 120 km (see Sect. ). A coarser grid spacing and a larger gridding radius would give more smoothed output fields. It is important to consider that a large area with relatively small contributions can still be an overall important moisture source. To quantify the overall contributions for different areas, the source contributions can be averaged within predefined geographic sectors (Sect. ).

The interpretation of moisture source maps can be greatly facilitated in a percentile representation . This can be obtained by determining, for example, the 50th and 80th percentiles of the accumulated moisture source contributions, sorted from largest to smallest grid point values. Such a percentile representation is particularly helpful to compare the shape of a moisture source footprint across different events or seasons. In this example case, the 50th and 80th percentile contours for the total moisture source footprint (Fig. a, solid and dotted red lines) highlight that almost half of the precipitation is contributed by areas bounded by the orange shading (contour value 0.06 mm d⁻¹), and the large majority is bound by the light blue area (contour value 0.24 mm d⁻¹).

The moisture source detection algorithm distinguishes between uptakes that take place within or close to the boundary layer top, and those that are in the free troposphere, above the boundary layer. For the example case, the moisture sources in the boundary layer dominate (Fig. b), while the free-troposphere moisture sources supplement the boundary-layer uptakes over the North Atlantic and Central Europe (Fig. c). The collocation of free-troposphere and boundary-layer sources for this event suggests that boundary-layer air may have been vented, moistening the free troposphere. The interpretation could therefore for this case focus on the total moisture uptakes .

The moisture source footprint is here compared to the result of the method for the same case (Fig. d). The vertical integral of e-p of all the particle trajectories over 10 d (the quantity (e-p)20d introduced by shows patches where evaporation dominates (red shading) and where precipitation dominates (blue shading). While evaporation regions dominate further away, precipitation overcompensates north of Iceland and over the arrival domain, masking potential evaporation from these regions. The presence of such compensating effects, as well as the strong dependency on a selected time scale (not illustrated here), are two of the most important differences compared to the results from the algorithm. The red contours, indicating the 50th and 80th percentile of total moisture sources from WaterSip, further underline the differences between the results from the two methods.

In addition to providing the moisture source footprint for an entire analysis period, the WaterSip diagnostic provides gridded moisture sources for each time step in the steps grid file (Sect.  and Table ). At 06:00 UTC on 10 August 2022, the precipitation in the target area was sourced from a rather large banded region stretching across the North Atlantic, and reaching into Eastern North America (Fig. a). This map shows the moisture uptakes corresponding to precipitation on that day in the arrival region. If one wants to compare the moisture uptakes directly with, for example, gridded evaporation fields, one can use the parameter assignToUptake (Sect. ). The corresponding plot for uptakes on 06:00 UTC at 10 August 2022 shows all evaporating moisture at the synoptically valid time, irrespective of when it will precipitate over the target area. In the given example, relevant uptakes at that time are detected south of Iceland, over Scandinavia, and over western Eurasia (Fig. b). The moisture from uptakes at that time will end up as precipitation over the target area at a later time.

A number of additional quantities that are indicative of the moisture transport conditions can be obtained on the same grid as the moisture source information. These so-called transport quantities can support the interpretation of the obtained moisture sources in various ways.

The transport pathways of air parcels towards the target region are visible from the particle mass density (Fig. a). The particle mass density is an integral of the pathways of all particles' mass bound to the target region in a given time interval. It is obtained by gridding mk/L for all time steps of a trajectory. The example case shows the relatively large overall area from where air parcels originate in a 20 d time frame, and the important role of Greenland's topography for guiding the air parcel transport. The quantity “moisture transport” shows the water vapour movement in these air parcels towards the target area. The moisture transport field is obtained by gridding the water mass in tracked air parcels on the way to the target area using qi⋅mk/L for each time step. For the example case, the moisture transport map highlights a dense area of moist air west and south of the target area, and where the moist air crosses the North Atlantic and advances towards the target area (Fig. b). This map can be interpreted as a share of total column water bound to the Scandinavia domain. In combination, these two transport quantities allow one to assess the main pathways of air mass transport and the differences to moist air advection to the target region.

Decreases of specific humidity in air parcels during transport can be due to either precipitation events or mixing with drier air (Sect. ). The moisture rainout field contains the amount of humidity lost from trajectories that are identified when RHi≥RHc from the diagnostic, gridded using Δqi⋅mk along each trajectory. In the example case, moisture rainout events are concentrated over the arrival domain and narrow transport pathways over the North Atlantic and east of Greenland (Fig. d), probably related to precipitation formation in air masses that ascend as they approach the target area. In contrast, mixing events are identified as moisture decreases if RHi<RHc, and also gridded using Δqi⋅mk. Both quantities are therefore easy to relate to the (e-p)20 d field (Fig. d). For the present example case, air mass mixing is mostly identified over continental North America and over Central Europe (Fig. c). It is interesting to note that for this case only a small part of the events are classified as mixing, while the large majority are rainout events. Note that it does not make a difference to the source identification and accounting whether a specific humidity decrease is classified as a mixing or as a rainout event.

For larger target areas, there can be spatial differences in the atmospheric conditions during air parcel arrival within the domain. It can be interesting to investigate different quantities that are connected to the arrival of air parcel trajectories. These quantities are termed here arrival quantities.

One of the most important arrival quantities is the Lagrangian precipitation estimate P̃ (Sect. ). In the example case, there are clear distinctions between the coastal regions of western Scandinavia with an estimated precipitation of above 6 mm d⁻¹, and large parts of the Baltic Sea, southern Sweden and Finland with below 2 mm d⁻¹ during the case study period of 5–15 August 2022 (Fig. a, shading). The diagnosed precipitation pattern corresponds qualitatively to ERA5 simulated precipitation during the period (white contours), even though the domain average during the period is about 25 % larger in ERA5 (2.43 mm d⁻¹ compared to 2.07 mm d⁻¹). A direct comparison of the two fields shows that P̃ is more smoothed out compared to the sharper pattern in ERA5 (Fig. ). Regions with high P̃ approximately correspond to locations where more air parcels arrive that precipitate according to the selection criteria, as reflected by the arrival count (Fig. d). In regions where only few air parcels contribute to arrival quantities, gridded results may show large spatial variations that are not statistically robust. The arrival count can be used for masking such grid points.

Another set of important arrival quantities are the accounted fractions (Sect. ). The total accounted fraction in the example case is uniformly above 95 % (Fig. f). Subdividing this quantity into the fractions accounted for by boundary layer uptakes (Fig. d) and free troposphere uptakes (Fig. e) shows a uniform split into approximately 60 %–70 % boundary layer and 30 %–40 % free troposphere contributions. These shares appear consistent with the corresponding moisture source maps (Fig. b, c).

In addition, some thermodynamic conditions of the air parcels at the time of arrival over the target domain are gridded onto the arrival grid. For example, the arrival temperature is the temperature of each air parcel at arrival, gridded using weights corresponding to P̃totk at arrival grid location (λ,ϕ) for the total accounted fraction: 19Tarr(λ,ϕ)=∑k=1KT0k⋅P̃totkP̃tot

In the case of water vapour diagnostics, weighting is instead done using the air parcel specific humidity (Sect. ). In the present example, arrival temperatures vary between about 0 and 8 °C (Fig. c). Thereby, regions where most precipitation fell have arrival temperatures that are lowest, with 0–2 °C. The air pressure at arrival is also available as gridded quantity (not shown, Table ).

As WaterSip identifies the location of a moisture source, different properties of the source region can be computed, such as the average sea surface temperature, or its latitude and longitude. In particular for larger arrival regions, and for those that contain geographic or climatic gradients, the source region location and its properties may vary substantially within the arrival domain. In order to make such spatial differences in the source region properties visible, WaterSip provides Lagrangian forward projections (LFPs) of the source region properties on a spatial grid covering the arrival domain.

LFPs are obtained as the weighted average of a given moisture source property for each trajectory. The weighted average is then gridded on the arrival grid, weighted by fk. Thus, for each trajectory k, a moisture source or transport quantity ζ is transferred from the geographic uptake location (λu,ϕu) to the geographic location at the arrival region (λa,ϕa) using the fractional contributions fik obtained from the accounting step as a weight, and resulting in LFP quantity ζ̃ for air parcel k: 20ζ̃k(λa,ϕa)=∑i=1Uζk(λu,ϕu)⋅fik∑i=1Ufik

Hereby, U is again the number of all moisture uptake events identified along trajectory k. The different ζ̃k are then again gridded using Eq. (). In creating LFPs, an important choice is whether the total accounted fraction of the precipitation estimate is used in the gridding, or only the boundary layer or free troposphere fraction. Parameter forwardProjectionMode in parameter section Diagnostics determines this choice (1: boundary layer, 2: free troposphere, 3: total fraction).

For the case study over Scandinavia, the mass-weighted moisture source latitude (centroid latitude) ranges from 50–60° N, with a clear latitudinal gradient (Fig. a). Corresponding moisture source longitudes range from -60 to 40° E (Fig. b). The source distance map for the example case shows a similar pattern as the source longitude, with more nearby sources (< 1500 km) over Denmark and southern Sweden in the southwest, and patches of more long-distant transport (> 3500 km) in the north of the domain (Fig. c). The distance of the moisture sources is computed as a spherical distance for each uptake location and then gridded according to Eq. (). These examples already illustrate how forward projections of source region properties provide insight into the spatial variability within the arrival domain. When used on large (global) domains, Lagrangian forward projections can provide interesting hydro-climatic insights .

The source land fraction (Fig. d) reveals a northwest-southeast oriented gradient in land uptakes. While land sources in the northwest of the domain are < 30 %, their share reaches above 80 % over the Baltic states. The fraction of land sources is obtained from evaluating a land-sea mask at the locations below the moisture uptakes. In the case where the mask file contains fractional land cover information, values above a threshold of >0.5 will classify the source location as land.

The moisture source temperature is obtained here from the skin temperature (SST over ocean regions, SKT over land) using Eq. () (Fig. e). For the present case, moisture source temperatures are warmest in regions where land sources dominate with up to 22 °C. Additionally, the LFP of moisture source relative humidity, air pressure at the sources, and specific humidity at 2 m (q2 m) at the sources may be available, depending on what has been written out to the FLEXPART and LAGRANTO input files (Table ).

Stable water isotopes are an important diagnostic in moisture source analyses e.g.,. Measurements of the secondary water isotope parameter d-excess can reflect the relative humidity environment during evaporation. In the case that SKT and q2 m or RH and T2 m are available on the input files, an approximation for the stable water isotope parameter d-excess at the moisture sources dsrc can be computed from the relation of : 21dsrc=-0.54‰%-1⋅RHSST+48.2‰

Thereby, the relative humidity with respect to SST (RH_SST) is obtained using saturation vapour pressure according to . For the example case, the computed dsrc shows a gradient that mirrors land properties, with higher values (8 ‰–10 ‰) where land sources dominate, while ocean source dominated regions show with ∼ 4 ‰ values below the global average in precipitation of 10 ‰ . The d-excess parameter is both an interesting diagnostic for paleoclimatic studies, and can be used to relate moisture transport to water isotope measurements on meteorological time scales e.g.,.

In a similar way as moisture source quantities (Eq. ), it is possible to project transport quantities onto the arrival region. The only difference is that all trajectory points, and not just the uptake points are included, using the current accounted fraction. For example, ξ̃k, the forward projected moisture transport temperature valid during arrival of air parcel k is obtained by averaging the air parcel quantity ξk along the entire transport path of N time steps, using the explained fraction at step fik as a weight: 22ξ̃k(λa,ϕa)=∑i=1Nξik⋅fik∑i=1Nfik

The LFP of the moisture transport temperature obtained this way shows relatively warm transport for the majority of air masses between 6–12 °C (Fig. a). In comparison, the condensation temperature, computed as transport temperature conditioned on RH >80°C, shows that saturation is reached at colder and more variable temperatures of between 2–8 °C throughout the domain (Fig. b). Comparison to the transport pressure shows transport at higher pressures in the regions of warmer condensation temperatures, with air pressure of above 990 hPa over the Norwegian Sea, compared to ∼ 950 hPa over the Baltic Sea (Fig. c).

A quantity that allows to inspect the time scale of moisture transport is the transport time, obtained as the difference between the time of each uptake event and the corresponding arrival time. For the example case, the transport time shows the largest values with 5–7 d in northern Scandinavia, whereas Denmark and southern Sweden have transport times of 3 d and below (Fig. d). The transport time in this case thus bears similarities with the transport distance (Fig. e) and the direct source distance map (Fig. c). In contrast to the moisture source distance, the transport distance is obtained as the total distance along an air parcel trajectory. The difference plot between both distance quantities underlines that the transport distance is always larger than the source distance, and more so for regions where transport takes more time, with a differences of up to 2000 km in Northern Scandinavia (Fig. f). In combination, these transport quantities enable one to interpret the transported moisture in terms of processes that occurred underway from source to sink. Such information can for example be relevant for understanding changes of isotope composition and the mixing with other air masses. For more detailed discussion of the transport time from WaterSip, see and .

The temporal evolution of different diagnostic quantities in the arrival domain can provide further insight into moisture transport dynamics. WaterSip provides time series output of all arrival and forward-projected source (ζ) and transport quantities (ξ) described above (Sect. –). Thereby, the mean, median, and the maximum and minimum value are reported for different integration times, from the trajectory time step to daily, monthly, yearly, and the entire analysis time period (Table ). Longitude is thereby converted to a Cartesian grid for the averaging operation. Averaging over time steps is done by weighting the contribution at each time step i of the analysis period with the corresponding precipitation estimate P̃ at grid location (λa,ϕa), and dividing by the total of the precipitation estimate for a given averaging period T: 23ζ‾(λa,ϕa)=∑i=0Tζi(λa,ϕa)⋅P̃i(λa,ϕa)∑i=0TP̃i(λa,ϕa)

Time series of P̃‾ of the case study obtained this way show a pronounced precipitation maximum in the first half of the study period on 5 to 7 August 2022 with 1.0 mm (6 h)⁻¹ (Fig. a, black solid line). A second maximum occurred with about 0.8 mm (6 h)⁻¹ on 10 August 2022. The ERA5 precipitation shows that WaterSip underestimates domain average precipitation during these periods, but not during weaker rainfall events (Fig. a, blue line). The standard deviation of the precipitation estimate in the arrival domain reflects some of the peak values present in ERA5 (dashed line).

The period with most intense precipitation was associated with gradually shorter transport distances of less than 3000 km (Fig. b). Land sources were dominating in the first period, with a fraction up to 60 % on 6 August 2022, compared to below 30 % on 12 August 2022 (Fig. c). Source latitudes were more northerly during the first half of the case study, and dropped to below 50° N on 10 August 2022 (Fig. d). Both characteristics match well with the moisture source snapshot for that day (Fig. a).

During the forward projection, some grid points have only a very small number of trajectories arriving, which can then lead to relatively large grid maximum and minimum values compared to the overall spread. For some applications, a more refined statistical quantification of the gridded fields may be required, for example such as excluding grid locations below a minimal precipitation estimate. In such a case, users may choose to output the gridded fields for every time step, and then follow up with their own post-processing steps.

Another post-processing available within WaterSip is the option to categorise the moisture source maps by geographic domains, the so-called sectorisation . During sectorisation, the coordinates of each uptake in the boundary layer and free troposphere are checked to be within a set of J latitude and longitude bounds [λj,ϕj], which define the source sectors Sj. The geographic bounds of each sector are implemented in the program code as a function that returns the sector number corresponding to a given geographic coordinate. Currently, 11 different sectorisations are available in WaterSip (Table ). Implementing additional sectors requires users to modify the WaterSip code. Further details are given in the Appendix .

The output from the sectorisation is included in the time series files (Table ). All quantities available on the source and transport grids contain an additional dimension for the sector number. Furthermore, quantities are given for land regions only, sea only, and for the total. This allows to define relatively simple geographic sector shapes that still make the important distinction between land and ocean regions, based on the land-surface information provided to WaterSip. The sector areas are also available to convert the fractional output to mass units.

For the case study over Scandinavia, the sectorisation “Norway” has been chosen, which includes 8 latitude sectors (Fig. c). The total arriving precipitation is dominated by sources at 50–60° N (Fig. a, light green) in the first half of the study period, before being dominated by more southerly sources in the second half (40–50° N, dark green). Splitting up the moisture originating from land areas only further emphasises that the shift to more northerly sources is driven by land contributions from 30–40° N (Fig. c). In a tabular summary, it becomes evident that ocean sources still dominate for the entire event period, and that less than 10 % of the precipitation originate from south of 40° N (Table ).

For some applications, it may be valuable to have the source and transport information from WaterSip available from every single uptake event. During the gridding to source or arrival grids, this detail of information is lost. Since the number of individual uptakes can be very large, histograms are a convenient way to aggregate such data. For example, used histograms of transport time to identify the age spectrum of precipitation. Other examples for the use of such output are the distribution of SSTs at the moisture sources . WaterSip creates an output file in comma separated value (csv) format containing histograms of a fixed set of variables as output if the parameter saveHistogram in setting group Output is set to 1 (true). These variables include convection count and depth, uptakes, rainout, air mass mixing, source SST or T2m, source distance, source d-excess, and RH_SST at the sources (Table ). As the histogram classification is implemented directly within WaterSip, changes to the classification parameters, or adding more variables requires changes to the WaterSip program code in file Water.cpp.

For the example case of Scandinavia during August 2022, the probability density function of the uptake contributions shows a maximum at 2 d before arrival, and a rapidly decreasing tail thereafter to 10 d before arrival (Fig. a). The distribution of rainout events (not weighted by f) also peaks at 2 d before arrival time, but has otherwise an even distribution throughout the 20 d analysis period (Fig. b). The PDF of mixing events (also not weighted by f) peaks at 4 and 8 d before arrival (Fig. c). Finally, the PDF of source distances shows a rapidly flattening tail with the majority of sources within about 5000 km (Fig. d). These examples illustrate the additional insight that one can gain from the histogram data, beyond the mass-weighted averages provided by the gridded and time series output.

WaterSip processes air parcel trajectory data obtained from the Lagrangian models LAGRANTO and FLEXPART. During the processing of these data, trajectories are filtered based on criteria such as being located within the arrival domain, precipitating during arrival, and so on. Essentially, thus, the WaterSip tool creates a subset of the original trajectory data, enhanced by additional information, such as the identified uptakes, and the accounted fraction. Setting parameter saveTraj of settings group Output to either 1 (output every time step) or 2 (output in monthly files) instructs WaterSip to write out these trajectory data in a compact binary format (*.traj). Since the total output size can become rather large, it is possible to use parameter skipTraj with a value n>1 to only write out trajectories for every nth particle. A python script for reading the binary format is provided in the WaterSip code repository.

The potential value of the trajectory output is illustrated here for a subset of the air masses arriving over the target area in Scandinavia at 06:00 UTC on 10 August 2022. During this time, 20 d backward trajectories tap into moist air masses in the Gulf of Mexico and the subtropical North Atlantic, and to a lesser extent into the Pacific basin (Fig. a). The colour shading by moisture uptakes and losses emphasises regions of moisture gain over the eastern and western North Pacific (blue), over North America, and the North Atlantic, as well as moisture loss as air parcels approach Scandinavia (red). The time vs. height diagram illustrates that in this case much of the moisture gained more than 10 d before arrival has been lost due to the vertical lifting and drying of air masses at about -10 d (Fig. b). Another branch of air descended from the upper troposphere between 6–10 km (dark brown colours) before taking up moisture in the lower troposphere. In some cases, the remaining moisture may also have been been overwritten by uptake events during the last 10 d before arrival, as exemplified by the trajectory highlighted in red that rises and then sinks again towards the surface multiple times.

Using the Scandinavia case study discussed above, a range of sensitivity experiments was conducted to illustrate the sensitivity of the diagnostic to several core parameters. The domain averages of the precipitation estimate P̃, total accounted fraction ftot, and the source distance have been chosen as bulk properties of the moisture sources during the event. When the critical relative humidity RH_c was varied from the reference value 80 % to 70 % and 90 %, P̃ increased or decreased about 25% compared to the reference value of 2.074 ± 1.162 kg m⁻² (Table ). This can be expected due to more (less) precipitation events being detected at arrival for a lower (higher) RH_c. In comparison to the domain-average ERA5-simulated precipitation of 2.43 kg m⁻² d⁻¹, the underestimation amounts to 17 % for the reference case. The moisture source distance increases slightly with a higher RH_c, while ftot remains nearly unchanged.

Sensitivity with regard to the value of Δqc was tested with 0.1 and 0.4 g kg⁻¹ 6 h⁻¹. The precipitation estimate remained unaffected by this change, since the humidity change during the last time step is set by the separate parameter arrivalPrecipThreshold. The total accounted fraction decreases with increasing Δqc, since less events are identified as moisture uptakes. A higher Δqc also favours slightly more distant moisture sources, as less discounting of remote moisture takes place for the case study.

The trajectory length has a clear imprint on ftot, with 13.4 % less accounted sources for 10 d (L=41) compared to 20 d (L=81) trajectories. In comparison, the difference between 15 and 20 d trajectories is only 4.3 %. The moisture source distance is also sensitive to the trajectory length, with on average 327 km closer moisture sources for the 10 d trajectories. Still, the average value is less than the 1σ standard deviation within the domain of 527 km, and remains within 15 % of the reference value for all of the above parameter variations.

Finally, the sensitivity to the time step Δt has been tested using a 3 h time interval. To enable a fair comparison to the reference, this run also required adjustment of Δqc to 0.1 g kg⁻¹ 3 h⁻¹, and increase of the trajectory length to 161 time steps to maintain 20 d trajectories. The results for this sensitivity test deviate more markedly from the reference than the previous sensitivity experiments, with a 50 % higher precipitation estimate, a 3.1 % higher ftot, and a 431 km closer source distance than the reference. This kind of response in the results of the diagnostic can be explained by the increasingly ambiguous distinction between variations in specific humidity from moisture uptakes that the diagnostic aims to identify, and variations from interpolation errors and other random factors. Users should carefully evaluate which time step and specific humidity thresholds are suitable for their study.

As other Lagrangian moisture source diagnostics, WaterSip calculates the Lagrangian water budget of air parcels that are moving with the atmospheric flow. At initial time t0, these air parcels are defined by their location, an assumed spatial extent, and represent an atmospheric mass mk. As air parcels move horizontally and vertically, their shape is assumed to be maintained, as well as the total mass. However, exchange processes across the air parcel's imaginary walls can modify the amount of water vapour within the air parcel, as quantified by the specific humidity q (g kg⁻¹). Interpolation errors can then, among others, cause spurious variations in the specific humidity. Another assumption that is made in the algorithm is that during one time step, all specific humidity changes are assumed to either be due to uptakes or losses, while in fact it is a net change that may include both uptakes and losses in the time interval. While this assumption is more valid the shorter the time interval between trajectory points, shorter time intervals also lead to using smaller Δqc between time steps, which then becomes more difficult to separate from interpolation noise. Another assumption in WaterSip is that diagnosed evaporation took place at the geographic location where the moisture uptake has been detected. However, this does not necessarily need to be the case, for example when water vapour that had evaporated earlier is at a later time mixed into an air parcel as a result of mixing from turbulence or convection.

While threshold parameters, such as Δqc are used to suppress some of the numerical noise introduced during the trajectory calculation and interpolation, there may either be too many uptakes diagnosed, or some uptake events may be missed. The total explained fraction ftot can then provide important guidance whether the accounting works consistently (i.e., ftot≤1) with the chosen threshold parameters. Spurious variations that are larger than the threshold value will lead to more remote moisture being discounted in the favour of later uptakes, while the assumption that either evaporation or precipitation take place during one time step acts to overestimate remote sources. Such potential biases are important to keep in mind during interpretation.

Other important sources of uncertainty are the number of particles chosen to represent the investigated air mass, as well as the time step of the trajectory output. In general, a larger number of trajectories will provide better statistical representation of the diagnosed quantities, in particular when used with a particle dispersion model such as FLEXPART that parameterises turbulent and convective motions. The number of particles should also be chosen in line with the resolution of the source and arrival grids, providing reasonably smooth and continuous gridded output. Other considerations of offline trajectory diagnostics are the effects of data assimilation, which provides humidity increments during the process , and from parameterisation schemes in the parent numerical model. Here in particular the role of convection schemes is important to consider, providing both a pathway for rapid specific humidity changes and vertical particle motion that are hard to capture in offline calculations, even when using parameterisation schemes for convection in the trajectory model . Here, the use of online trajectory calculation methods could provide a pathway towards quantifying such uncertainty.

Future development of the WaterSip software and similar approaches will benefit from quantifying uncertainty, and from identifying the contribution of different factors to uncertainty. In the absence of a direct observable quantity for the water vapour origin, model inter-comparison efforts are important. First steps in this direction have been made by who employed a regional tagged water tracer simulation as the reference standard for a Eulerian moisture source identification algorithm. Similarly, and compared the WaterSip diagnostic to tagged tracer simulations from a regional model. While tagged tracers do have limitations with regard to the representation of different processes in the atmospheric water cycle at small scales, it can be expected that using such online simulations as input data for offline diagnostics can help to quantify uncertainty, essentially serving as an internal “gold standard” for such inter-comparison efforts . Thanks to the availability of an increasing number of approaches to provide (Lagrangian and Eulerian) offline moisture source diagnostics as open access software , the feasibility of such an inter-comparison effort has greatly increased in recent years. The present description of the WaterSip diagnostic is another piece in the puzzle that may help to advance ongoing debates. On the longer run, the availability of suitable water vapour isotope measurement data sets, in particular such that connect moisture source signals with measurements along the transport pathway and the arrival region, have the potential to indeed constrain diagnostic methods by measurements .

The test case here is run only for a short time period and over a spatially limited domain. Running on a single core on a 36-core Intel(R) Xeon(R) Gold 6140 CPU at 2.30 GHz, the diagnostic completes after about 21 min, with a peak memory use of 4.6 GB (Table ). Important aspects of the duration of the computational time are the amount of input/output (I/O) operations, and the number of particle trajectories to be processed. The test case has at most about 3000 new particles that are qualified for tracking within each time step of the test case period. This means that here I/O dominates over the amount of time spent for evaluating the moisture source changes and gridding the results. When the amount of particles considered for analysis is reduced by a factor of 10 (Test “Stride A”) or 100 (Test “Stride B”), the computation time only decreases by a factor of 0.4 or 0.6, respectively. Nonetheless, a stride larger than one provides results substantially faster, which is useful for example when testing an analysis setup. Note that results will however contain more gaps with fewer particles due to the stride setting. The memory footprint scales less than the computational time, since this demand is dominated by the output grids and thus the length of the analysis period. Finally, the parallelisation performance is illustrated with a run over the same time period, but with a larger domain covering half the Northern Hemisphere. The use of 4 cores (Test “Large B”) shows a 2.75× speedup compared to the single-core run (Test “Large A”). However, performance gains quickly become lower with additional cores due to the need for processes to wait for one another (not shown). It may be possible to gain faster computation times and better scaling if the code were further optimised for parallelisation.