The framework of universal multifractals (UM) characterizes the spatio-temporal variability in geophysical data over a wide range of scales with only a limited number of scale-invariant parameters. This work aims to clarify the link between multifractals (MFs) and more conventional weather descriptors and to show how they can be used to perform a multi-scale evaluation of model data.

The first part of this work focuses on a MF analysis of the climatology of precipitation intensities simulated by the COSMO numerical weather prediction model. Analysis of the spatial structure of the MF parameters, and their correlations with external meteorological and topographical descriptors, reveals that simulated precipitation tends to be smoother at higher altitudes, and that the mean intermittency is mostly influenced by the latitude. A hierarchical clustering was performed on the external descriptors, yielding three different clusters, which correspond roughly to Alpine/continental, Mediterranean and temperate regions. Distributions of MF parameters within these three clusters are shown to be statistically significantly different, indicating that the MF signature of rain is indeed geographically dependent.

The second part of this work is event-based and focuses on the smaller scales. The MF parameters of precipitation intensities at the ground are compared with those obtained from the Swiss radar composite during three events corresponding to typical synoptic conditions over Switzerland. The results of this analysis show that the COSMO simulations exhibit spatial scaling breaks that are not present in the radar data, indicating that the model is not able to simulate the observed variability at all scales. A comparison of the operational one-moment microphysical parameterization scheme of COSMO with a more advanced two-moment scheme reveals that, while no scheme systematically outperforms the other, the two-moment scheme tends to produce larger extreme values and more discontinuous precipitation fields, which agree better with the radar composite.

Validation of precipitation fields simulated by a numerical weather
prediction model is a delicate task as reference data (rain gauges, radar
scans) are typically available at a different spatial and temporal resolution
than the model. Traditional point-based verification scores are generally
unable to provide sufficient information about the forecast quality, as they
do not take into consideration the spatial structure and are affected by the
so-called “double penalty”

Multifractals (MFs) offer a convenient way to analyze the variability in
complex geophysical systems globally over a wide range of scales. In the
context of MFs, the statistical properties of a field are related
to the resolution by a power law

This article is structured as follows: in Sect. 2, the COSMO model and the Swiss radar composite are briefly described. The studied events, radar datasets and model variables are then described in detail. Section 3 provides a short summary of the UM framework. In Sect. 4, a climatological analysis of precipitation intensities simulated by COSMO is performed with the UM framework in relation to external geographical and meteorological descriptors. In Sect. 5, a spatial and temporal analysis of precipitation intensities on the ground simulated by COSMO is performed during three characteristic events. The results are then compared with the UM analysis of the radar composite. Finally, Sect. 6 gives a summary of the main results and concludes this work.

The COSMO model is a mesoscale limited-area numerical weather prediction
model initially developed as the Lokal Modell (LM) at the Deutscher
Wetterdienst (DWD). It is now operated and developed by various weather
services in Europe, including Switzerland. Besides its operational
applications, it is also used for scientific purposes in weather prediction
and for regional climate simulations. The COSMO model is a non-hydrostatic
model based on the fully compressible primitive equations integrated using a
split-explicit third-order Runge–Kutta scheme

In the operational one-moment scheme, the only free parameter of the PSDs is
the slope parameter

A more advanced two-moment scheme, which adds hail as a sixth hydrometeor
category, was developed for COSMO by

In COSMO, the interaction of various microphysical processes and their
feedback on the simulated flow fields are represented by a system of budget
equations for

Numerically, this system of differential equations is treated with a time
splitting method, in which the advection terms

In the operational setup, the COSMO model uses a prognostic turbulent
kinetic energy (TKE) closure at level 2.5 for the parameterization of
atmospheric turbulence. This scheme is similar to

List and description of all meteorological and topographical descriptors used in the MF characterization of the climatology of precipitation intensities.

In the first part of this work, a MF characterization of all
precipitation intensities simulated by COSMO during 5 years is performed.
The computed MF parameters are then compared with various
descriptors. A total of 43 115 hourly time steps of COSMO simulations in
analysis mode covering a period of 5 years (2011 to 2016) were retrieved from
the MeteoSwiss archives. These data are available at a 2 km resolution
(0.02

Short description of the three precipitation events considered.

Besides the MF parameters, which will be described later on, 11 local
descriptors of the geography and meteorology were computed from the
COSMO data within every square (Table

In the second part of this work, three precipitation events were simulated
with COSMO and compared to the radar QPE in terms of the MF properties of
their rainfall intensity fields. These events correspond to three typical
meteorological situations observed over Switzerland. A brief description of
the events is given in Table

500 hPa geopotential and pressure at mean sea level for the three considered events.

Situation map showing the theoretical maximum extent of available QPE (light blue), the location of the Swiss operational radars (blue dots), the region used for the climatological study of COSMO precipitation intensities (domain 1), and the subregions centered over the precipitation events used in the QPE analysis (domains 2 and 3).

In Sect.

The Weissfluhgipfel radar was not yet installed at the time of the considered events

operational polarimetric C-band radars. The QPE product of MeteoSwiss is computed in the following way: the linear equivalent radar reflectivity measurements at up to six 1Let

The fractal dimension

where

It is possible to interpret this result in a probabilistic way. Indeed, consider a line or cube of size

It is clear that the values of

It can be shown that Eq. (

The quality of the scaling can be studied with the trace moment (TM) method
which consists of a log-log plot of the up-scaled fields as a function of the
resolution

In the UM framework

The size of the sample limits the insight one can get of a statistical
process. For a MF processes, if

An example of the use of

In this work, the UM parameters are estimated with the double TM method

In the case of a non-conservative field

One way to consider non-conservative fields within the UM framework it to
assume that they can be expressed as the following:

The moment scaling function of the non-conservative field

Overview of all MF parameters.

The MF analysis of time series of two-dimensional fields, such as the ones considered in this study, can be performed both in space, by considering an ensemble of two-dimensional fields (one sample for every time step), or in time, by considering an ensemble of one-dimensional time series (one sample for every coordinate in the two-dimensional field).

A simple spatio-temporal scaling model

Within all 209 selected squares (Sect.

In order to test the effect of zeros on the overall analysis, the
MF parameters were also estimated in space

In time, it would not be possible to filter out non rainy time steps, as it would break the continuity of the times series.

by using only the fields where there is precipitation over at least 50 % of the surface. This did, however, not impact the main conclusions in terms of correlations and spatial structure of the MF parameters. Hence the subsequent study was performed on the raw precipitation fields without any kind of filtering.In the first step, the relationships between MF parameters evaluated in time
and space and the descriptors detailed in Table

The following conclusions can be drawn from the correlation plots.

Correlation plots showing the Spearman (rank) correlation between MF
parameters and descriptors.

The correlation values are roughly consistent in time and in space, although
the ones in time are generally higher. In time the correlation between

A hierarchical clustering of all 209 areas was performed based on the value
of their descriptors (Table

As could be expected, the clustering of the meteorological and topographical
descriptors results in a meaningful spatial distribution. Indeed, all
clusters are spatially very coherent, with Cluster 1 corresponding mostly to the Alpine
regions, from the Mediterranean Sea to Austria, Cluster 2 corresponding mostly
to the cooler temperate regions in the east of France and south of Germany,
and Cluster 3 corresponding to the warmer Mediterranean regions in the south of
France, in Italy and the Balkans. Note that over land areas, a somewhat
similar classification can be obtained by aggregating clusters of the famous

The distributions of MF parameters within these clusters, as illustrated in
Fig.

The statistical significance of these discrepancies was confirmed both with
the MANOVA (multivariate analysis of variance) and the non-parametric
Kruskal–Wallis statistical tests. All tests were performed with a
significance level of 2.5 %. The MANOVA reveals that the multivariate means
of MF parameters (both in time and space) are significantly different between
the three clusters, as well as between all three pairs of clusters taken
separately (1 vs. 2, 2 vs. 3 and 1 vs. 3). The non-parametric Kruskal–Wallis
test, performed separately for all MF parameters, reveals that distributions
of all MF parameters are significantly different between the three clusters.
A pairwise comparison (1 vs. 2, 2 vs. 3 and 1 vs. 3) reveals that, in time, all
MF parameters are significantly different between all pairs of clusters,
except for

Hierarchical clustering of the 209 regions into three
clusters based on the meteorological and topographical descriptors listed in
Table

Boxplots showing the distributions of MF parameters within the
clusters. MF parameters estimated in time are shown with a hashed pattern.
Note that the left

Spatial representation of the MF parameters estimated in space for
all areas. The special colormap for

To summarize, the statistical analysis shows that the MF parameters of precipitation intensities are significantly different within the three climatological clusters.

Figure

In time, similar conclusions can be drawn for

To summarize, the MF signature of precipitation is related to the topography, the climate and the typical meteorological conditions. As such, MFs can be used as a way to characterize precipitation fields and assess the realism of simulated atmospheric variables.

The previous section presented the MF approach in a climatological context, which helps to link meteorology/geography and MF parameters. In the present section, we evaluate the quality of the precipitation simulated by COSMO with two different microphysical schemes, by comparing it with quantitative precipitation estimations from the Swiss radar network using the UM framework. This comparison requires the COSMO model to be run at the radar temporal resolution (5 min) in a very expensive setup (two-moment scheme). As such, a climatological comparison over several years is not feasible from a computational point of view. Hence, the comparison is now conducted on the event scale, which also makes the intercomparison of the microphysical parameterizations easier.

The power law

For the two-moment scheme, things are more complicated as no one-to-one
relationship exists between rain rate and reflectivity. However, a rough
estimation of the error in

Overall, correcting precipitation fields for discrepancies in the

A MF comparison of the precipitation fields simulated by COSMO in its one-moment and two-moment schemes with the QPE product from the Swiss radar composite was performed. As a first step, a spectral analysis was performed both in time (ensemble of one-dimensional time series of precipitation intensities) and in space (ensemble of two-dimensional maps of precipitation intensities).

Figure

For the 26 March 2010, we observe a single scaling regime for the radar QPE, with a good scaling both on large (16–64 km) and small scales (2–16 km), as the spread around the line is relatively small. For the model intensities, we observe strong discrepancies with the radar QPE in terms of spectral slope at smaller scales (2–8 km), which are not well represented. A possible explanation for this break in scaling properties of the model is the fact that large scales are dominated by the dynamics of the model (primitive equations of the atmosphere), whereas smaller scales are dominated by the parameterizations of subgrid phenomena (turbulence, convection). However, even at larger scales (8–64 km), the agreement between radar QPE and model simulations is still quite poor in terms of spectral slope. Obviously, for this rainfall event, COSMO is not able to recreate the spatial structure of precipitation observed by the radar.

Spectral analysis in space of the QPE products during the three
events. The displayed lines are best-fit lines that take into account a
possible scaling break. The associated value of

For the 8 April 2014, the scaling is similar between radar and model precipitation intensities, possibly indicating that for this stratiform rain event, parameterizations and dynamics match better. Both radar and simulations show a weak scaling break at around 8 km.

Values of the non-conservation parameter

For the last event, we observe again a good scaling for the radar QPE and a
much worse scaling for the model precipitation intensities. However, in
contrast with the first event, the larger scales (

The spectral analysis in time (not displayed) generally shows similar
results, but with larger values of

Table

Scaling (TM) analysis of the QPE product during the three events
obtained with

Overall, it is worth noticing that the two-moment scheme almost always has
smaller

In order to account for the fact that the fields are mostly non-conservative
(

Figure

Values of

For the first event, both COSMO microphysical schemes give very similar
MF parameters and the discrepancy with the radar QPE is quite
important. In space, it can be observed that

For the stratiform rain event, the MF parameters of the COSMO
simulations are in better agreement with the radar QPE. In time, the
two-moment COSMO scheme gives values that are in relatively close agreement
with the radar QPE and, in this regard, outperforms the one-moment scheme.
COSMO simulations show generally smaller values of

For the last convective event, two scaling regimes are considered in space:
larger scales (16–128 km) and smaller scales (2–16 km). As already observed
in the spectral analysis, there is a better agreement between the radar
observations and the simulations with the one-moment scheme at smaller
spatial scales. In time, however, the temporal intermittency of COSMO is
smaller than for the radar QPE, which can be explained by the fact that COSMO
generally overestimates the extent of the convective systems. Compared with
the one-moment scheme and the radar QPE, the two-moment scheme has a smaller

In summary, the observations of the spatio-temporal analysis are consistent with the spectral and scaling analysis where (1) a strong discrepancy in scaling behavior was observed between COSMO and the radar QPE at small scales for the first event, (2) a better scaling of the model precipitation intensities was observed for the second event, (3) a discrepancy in scaling at large scales was observed between COSMO (especially for the two-moment scheme) and the radar QPE for the third event.

Overall, it can be observed that except for the first event where both
schemes give similar values, the two-moment scheme is usually characterized
by a larger

The upper panel displays the

To compare time series of UM parameters

Figure

During the convective event, four different phases can be identified. In the
first short phase (12:00–14:00 UTC), observations and simulations agree
relatively well in

For the two other events, the conclusions are similar: discrepancies in
MF parameters between simulated and observed precipitation
intensities are caused primarily by temporal and spatial shifts in the
simulated precipitation patterns. The effect of such shifts on the
MF analysis hints at the possibility of a further analysis based
not on a fixed study domain but on a study domain which follows the
precipitation system, in a way similar to

In this work a spatial and temporal analysis of precipitation intensities simulated by the COSMO NWP model was performed using the UM framework which allows for the representation of the variability across scales with a limited number of parameters.

Fraction of wet area during the event of the 13 August 2015.

The first part of this work focused on a MF analysis of the
climatology of precipitation intensities simulated by COSMO in its
operational analysis mode. Analysis of the correlations between MF
parameters and external meteorological and topographical descriptors revealed
that the fractal dimension (

The second part of this work focused on three different events, one cold
front associated with heavy snowfall, one stationary front associated with
stratiform rain and a stable atmosphere, and one summer convection event with
heavy rain. All events were simulated at a 2 km resolution with both the
standard operational one-moment microphysical parameterization of COSMO and a
more advanced two-moment microphysical scheme. A comparison of the
precipitation intensities at the ground simulated by COSMO and the Swiss
radar composite was performed in terms of their MF signature.
Although the radar data show a single scaling regime over the studied
spatial-scale ranges (1–128 km), the COSMO simulations display scaling breaks
for the first and the last event. It can be observed that during the
snowstorm event, COSMO is unable to properly reproduce radar observations at
small scales, which might be caused by the intrinsic difficulty of simulating
solid precipitation. During the last convective event, the opposite can be
observed and COSMO is struggling to reproduce the larger scales, due to its
difficulty in locating accurately the convective system in time and space
during this event. In the temporal scales, a scaling break is observed both
for the radar data and the COSMO simulations at around 3 h. Comparisons
of the one-moment and two-moment COSMO microphysical parameterizations show
that the fields simulated by the two-moment scheme tend to display a larger
intermittency and variability than the one-moment scheme. This does not
generally translate into a better agreement of MF parameters

Ultimately, the MF framework can be used to identify the scale ranges in which the model is able to simulate realistic fields of water contents, and as such this technique can be used as a diagnostic tool for model evaluation.

The data used in this work are property of MeteoSwiss and cannot be shared without their explicit authorization. The codes used in this work can be shared on request to the first author.

Spatial representation of the location of all areas used in the
climatological study of MF parameters and the corresponding local
descriptors. Note that the squares are mere indicators of the location of the
center of all areas and their size is not to scale with their real sizes. The
colors drawn below the squares correspond to the classification obtained in
Sect.

Figure

Figure

Illustration of the effect of

Illustration of the effect of

Figure

Distribution of MF parameters within Köppen areas (not displayed) show only
minor deviations from those obtained with the meteorological classification
(Fig.

Aggregated Köppen climate classification within all 209 subsquares
used in Sect.

The authors declare that they have no conflict of interest.

The authors would like to thank MeteoSwiss for access to the Swiss operational radar composite as well as the initial and boundary conditions used in the COSMO simulations.

The authors thank the Partenariat Hubert Curien – Germaine de Staël (Projet 32709UK) for financial support that made this collaboration possible.

The authors also thank Tim Raupach for proofreading of the final version of the manuscript. Edited by: Graham Feingold Reviewed by: two anonymous referees