Introduction
Continuous recordings of meteorological data are available since the late
18th century. During the 20th century, observational networks have been
refined intensively, even at remote sites. However, these observations are
generally not distributed equally in space and their temporal resolutions
range from some hours (e.g. three measurements of temperature for each day)
to 1 day (e.g. rain gauges). Later, in the late 20th century, the
instrumentation of meteorological stations has been supplemented by the
installation of automatic weather stations (AWS), which are capable of
collecting meteorological data continuously with a frequency ranging from 1 h to 1 min or even shorter periods of time .
Figure depicts the global temporal evolution of data
availability for daily and hourly meteorological time series during the 20th
century and beyond. This diagram has been compiled using two freely available
data sets through querying the temporal coverage of available data of each
data set: daily data are collected continuously in the Global Historical Climatology Network (GHCN) daily database
, whereas the Integrated Surface Database (ISD) provides hourly time series of stations worldwide
. This comparison reveals that the availability of
hourly observations as provided by AWS is restricted to a few decades only.
When observing Fig. , it becomes obvious that a large
number of AWS have only been mounted in the last 2 or 3 decades.
In contrast, hourly meteorological time series are required for numerous
applications in geoscientific modelling. Typical applications in hydrology
include both derived flood frequency analyses e.g.
and water balance simulations e.g.. In ecological
modelling, sub-daily meteorological data are required for, e.g., the
estimation of epidemic dynamics of plant fungal pathogens
.
Consequently, the question arises how to generate hourly time series of
meteorological variables, e.g., by using available daily observations in
order to benefit from their longer temporal coverage and higher spatial
network density. In general, three completely different approaches exist
(listed in descending order regarding their potential to reconstruct the
originally measured hourly values that are representative for a given
location and time):
Temporal disaggregation of daily meteorological observation e.g.:
this method is the simplest approach among the methods listed here even though more complex methodologies are also available,
especially for precipitation e.g.. Simplicity holds, however, mostly true for computational needs
as well as for the complexity of the methods itself. Deterministic equations or simple statistical models are applied to daily
time series in order to derive hourly values. For each variable, the disaggregation is generally applied independently. Including
statistical evaluations might improve results at a specific site compared to simple methods that are independent from station
recordings . For instance, the rainfall disaggregation package “HyetosR”
provides an extensive parameter estimation methodology, which is based on observed time series. Despite their simplicity,
disaggregation methods have great potential to reconstruct the originally measured hourly values for a given day as they
are forced by actual daily values valid for that specific day.
Dynamical downscaling using limited area models (LAM) of the atmosphere and atmospheric (re-)analysis data e.g..
As globally available data are used (e.g. re-analysis data), this approach is mostly independent of local observations although
these local recordings might have contributed to the global data sets. It is a physically based approach that preserves physical consistency
among all meteorological variables, which holds not necessarily true for the first and third methodology. However, due to its physical base,
it is more complex and computationally expensive. Since atmospheric (re-)analysis data represent the actual weather for a given time, dynamical
downscaling of this kind of data is a sophisticated way to derive hourly values for that time and arbitrary locations in a realistic manner.
However, small-scale precipitation might not be covered as accurately by the LAM in some cases due to the very complex micro-physical nature of precipitation and its variability e.g..
Using weather generators to derive new synthetic time series that match the statistics of available hourly data: weather generators
calculate statistics of observed time series and apply these statistics using a random number generator to obtain new time series with equal
statistical characteristics . For hourly time steps, resampling techniques are applied in most cases
e.g.. Time series derived by weather generators only match the observations statistically. The sequence of events
is different due to its random nature, which is why sub-daily time series do not provide the originally measured values. Weather generators are powerful
tools that supplement deterministic modelling by stochastic methods, and thus add a probabilistic component to the otherwise pure mechanistic methodology
mixed deterministic-stochastic models; see, e.g.,. Combinations with disaggregation techniques are also possible .
Time series of the worldwide station data availability in the 20th
and 21st century according to the global ISD and the GHCN data sets.
In this study, we focus on the simplest method among the listed approaches,
the disaggregation of daily meteorological data (no. 1). For instance, in
hydrological modelling, simple methods are usually sufficient in order to
force conceptual, process-based models . To
the authors' knowledge there is neither any “best” way of disaggregating
meteorological data to hourly values nor any easy, ready to use and flexible
software package that enables this task for different meteorological
variables including precipitation, temperature, humidity, solar radiation,
and wind speed. Therefore, we propose a robust and fully documented
methodology including alternative approaches for all these variables in order
to make the best use of available data. Although there are more complex and
sophisticated methods available for obtaining hourly values,
MEteoroLOgical observation time series DISaggregation Tool (MELODIST) can be viewed as good balance among several aspects such as
data availability, user's prior knowledge, robustness, and computational
costs. Therefore, MELODIST addresses practitioners, who need to run
their model for long periods of time at 1 h time steps. Here, emphasis
is put on single stations rather than considering interdependencies among
different stations. However, the manuscript includes some specific remarks
with respect to this restriction.
The paper is organised as follows: first, the study sites investigated herein
are briefly presented in Sect. 2. The next section gives an overview of the
disaggregation methods. In the fourth section, the methods are statistically
evaluated with respect to their accuracy to reconstruct sub-daily features.
Finally, Sect. 5 includes concluding remarks and an outlook for possible
future work.
Study sites
The accuracy of disaggregation methodologies strongly depends on diurnal
characteristics of meteorological variables. In turn, these diurnal
characteristics might vary among different climates and environments. To test
the robustness of the methods described in the next section, a small number
of sites in different climates has been chosen (see Fig.
and Table ).
Map of stations investigated in this study. Dot size represents the
mean annual precipitation, whereas the colour of each station indicates the
mean annual temperature.
List of AWS investigated in this study. The elevation of each
station z is given in metres above sea level. Data availability refers to
the available station recordings (hourly data). The first period of time
refers to the calibration period, whereas the second period is preserved for
validation purposes. P is precipitation, T is air temperature,
H is humidity, W is wind speed, R is solar radiation, and
S is sunshine duration. The location of each station is shown on the map
in Fig. .
Station
z
Data availability
Source
1
De Bilt
2
1961–1990, 1991–2014, P, T, W, H, R, S
2
Ny-Ålesund
11
1993–2004, 2006–2011, T, W, H, R, S
3
Obergurgl
1938
2000–2007, 2008–2014, P, T, W, H, R
project data
4
Rio de Janeiro – São Cristovão
5
2003–2008, 2009–2014, P, T, W, H
5
Tucson International Airport
779
1973–1993, 1994–2014, P, T, W, H, R
,
Except for Obergurgl, all station data are available for free. For each
station, all relevant meteorological variables have been recorded for at
least 1 decade. Only shortwave radiation and precipitation are not
available for Rio de Janeiro and Ny-Ålesund respectively
(Table ).
The available data sets have been subdivided into two independent periods of
time, one for calibration purposes, if required, and the other for an
independent validation of the disaggregation results. This subdivision has
been defined in order to enable a split-sample test , which
requires an independent validation period for testing models. In this study,
the split-sample test is applied for the disaggregation methods described in
the next section.
Disaggregation of daily to hourly meteorological values
Overview
Overview of disaggregation methods included in MELODIST.
The fist letter indicates the parameter that is considered by each method
(P is precipitation, T is air temperature, H is humidity,
W is wind speed, R is solar radiation, and X is all variables). For each
method, key references are given.
Method
Type
Calib.
T1
Standard sine redistribution with different options
deterministic
no
H1
Tdew=Tmin
deterministic
no
H2
Tdew=aTmin+b
deterministic
yes
H3
Linear variation of Tdew overlaid by sine function
deterministic
yes
H4
Hmin, Hmax
deterministic
no
W1
Equal distribution
deterministic
no
W2
Cosine function
deterministic
yes
W3
Random distribution
stochastic
no
R1
Scaling of potential shortwave radiation
deterministic
no
R2
model for sunshine duration S, then R1
deterministic
yes
R3
model, then R1
deterministic
yes
P1
Equal distribution “(124)”
deterministic
no
P2
Cascade model
stochastic
yes
P3
Redistribution according to another station
deterministic
no
X1
Linear interpolation
deterministic
no
In this section, all disaggregation methods employed in the framework of this
paper are described in brief. For each meteorological variable different
options are available (Table ). Deterministic methods
generally provide the same output if input remains unchanged. In contrast,
stochastic methods are based on random numbers. This means that the output
differs in consecutive runs even if the input data set remains the same. Thus,
stochastic methods require multiple runs prior to a sound statistical
evaluation of these runs in order to draw conclusions. Some models require
the calibration of model parameters that need to be adjusted for each site.
Split-sample tests are applied to test the methods more
rigorously.
The subsequent sections provide details for each of the methods listed in
Table . For each variable an example figure is provided,
which gives an idea of how each of the methods works. The times and locations
of these figures have been randomly selected.
Temperature (T1)
Temperature on day i is disaggregated to hourly values j
using a cosine function, whose
amplitude is defined by the observed minimum Tmin,i and maximum
temperature Tmax,i on day i e.g.:
Ti,j=Tmin,i+Tmin,i+Tmax,i2⋅1+cosπ⋅(tj+a)12.
The parameter a is determined either through providing an a priori guess
of the temporal difference between the solar noon and the occurrence of the
maximum temperature or through calibration. Three options are provided by
MELODIST: minimum and maximum temperatures occur at 7:00 and
14:00 LT,
respectively (T1a). The second option (T1b) relies on radiation geometry in
order to calculate sunset as point in local time for minimum temperatures and
sun noon + 2 h as point in time for maximum temperatures (see,
Fig. ). As the temporal shift of 2 h might not be
viewed acceptable as a general rule of thumb, temporal shifts for each month
can be evaluated through statistical evaluation of observed hourly time
series (T1c).
Example application of the temperature disaggregation model T1
performed with different settings for Obergurgl. Observed and disaggregated
time series are shown for temperature. T1a: fixed abscissa values for minimum
and maximum temperature; T1b: minimum and maximum temperature are related to
sunrise and sun noon + 2 h, respectively; T1c: similar approach as T1b
with an additional empirical shift of the maximum temperature; T1d: option
T1a with modified nighttime option. The numbers in parentheses indicate the
root mean square error (RMSE) computed for each method.
In principle, the methodology is based upon the assumption that the diurnal
course of temperature simply tracks the diurnal course of the incoming
shortwave radiative flux with a shift in time. This assumption does not hold
true during polar nights, which is why another method is applied for
Ny-Ålesund. For this station, a linear interpolation between minimum and
maximum temperature is applied (T1d nighttime option). If temperature
increases compared to the previous day, minimum temperature is assumed to be
representative for the first 12 h of the current day and the maximum
temperature is likewise attributed to the second half of that day. If
temperature decreases from one day to the next, the opposite assignment is
applied. Even though this method is rather simple, it preserves minimum and
maximum temperatures while disaggregating.
Humidity
Humidity disaggregation based on dew point temperature (H1 to H3)
Relative humidity H [%] is defined as the ratio of actual vapour
pressure ea [hPa] to saturated vapour pressure es
[hPa]:
H=100⋅eaes.
It generally follows a diurnal course with the maximum around sunrise and the
minimum in the early afternoon .
All humidity disaggregation methods require already disaggregated temperature
recordings. Methods H1 to H3 generate hourly values of dew point temperature
Tdew [K], as the actual vapour pressure is assumed equal to the
saturated vapour pressure at dew point temperature. Hourly H values can
thus be calculated using hourly values of T and Tdew
as
H=100⋅es(Tdew)es(T).
Saturation vapour pressure for a given temperature T [∘C] is
calculated using the Magnus formula :
es=6.1078exp17.08085T234.175+TT≥0∘C6.1071exp22.4429T272.44+TT<0∘C,
while actual vapour pressure for a given temperature T and relative
humidity H [%] is calculated as
ea=es(T)⋅H100.
Methods H1 and H2 use a model in the form of Tdew, day=aTmin+b to calculate daily dew point temperature (i.e. no diurnal
dew point temperature variation is assumed). For H1, a=1 and b=0;
i.e. Tdew, day is assumed to be equal to the daily minimum
temperature. H2 uses hourly observations of temperature and humidity to
calculate the best fit for a and b for a given site. Tdew is
thereby calculated from T and H by inverting Eq. ():
Tdew=234.175lnea(T,H)6.107817.08085-lnea(T,H)6.1078T≥0∘C272.44lnea(T,H)6.107122.4429-lnea(T,H)6.1071T<0∘C.
H3 assumes a diurnal dew point temperature variation based on the assumptions
that dew point temperature varies linearly between consecutive days, and that
mean daily dew point temperature occurs around sunrise .
Dew point temperature for a given day (d) and hour (h) is thereby calculated as
Tdewd,h=Tdew, dayd+h24Tdew, dayd+1-Tdew, dayd+Tdew,Δh,
where
Tdew,Δh=12sinh+1πkr-3π4.
kr should be set to 6 for sites with average monthly radiation
higher than 100 W m-2, and to 12 otherwise . An
example application of these methods is shown in Fig. .
Example application of different humidity disaggregation models for
Obergurgl. Observed and disaggregated time series are shown for relative
humidity. The numbers in parentheses indicate the root mean square error
(RMSE) computed for each method.
Minimum and maximum humidity disaggregation (H4)
Method H4 uses records of daily minimum and maximum temperature and daily minimum
and maximum relative humidity as well as the disaggregated hourly temperature
values to generate hourly humidity values:
H=Hmax+T-TminTmax-TminHmin-Hmax.
If Hmin and Hmax are available for each day, this method is the best available option among all available disaggregation methods .
Wind speed
Wind speed is a meteorological variable subjected to high variability at
small temporal scales. This small-scale variability can be observed, e.g.,
from eddy-covariance measurements . The methods compiled in
this study focus on suitable wind speed time series for hourly time steps
without taking into account these sub-hourly considerations. This idea best
corresponds to averages of wind speed for a given increment of time (e.g.
1 h) rather than instantaneous measurements.
Equal distribution (W1)
As for precipitation, this method applies one unique value for each hour of
the considered day. The daily mean value is assumed to be valid for hourly
values as is (W1). For many applications, this assumption might be
sufficient.
Cosine function (W2)
Due to local and microclimatic conditions, wind speed is subjected to diurnal
variations on days with calm weather in absence of synoptic-scale weather
patterns that obliterate local and microclimatic forcings .
Typical diurnal patterns in wind speed (and wind direction as well) are
related to mountain-valley or land-sea wind systems. Besides these local
climatic wind systems, wind speed typically increases during daytime and
almost always diminishes after sunset. This phenomenon is related to
increased radiation-induced momentum flux on fair weather days. Again,
synoptic-scale weather patterns such as low pressure systems might obliterate
local-scale effects. These patterns of diurnal wind speed variations can be
simply represented by a cosine function (W2), which requires calibration
using data observed at the considered site. This model is similar to the
temperature disaggregation method T1 see Eq. in
vi,t=aw⋅vi⋅cosπ⋅(t-Δtw)12+bw⋅vi.
The wind speed representative for day i is disaggregated to vi,t for
hour t (Fig. ). aw, bw, and
Δtw are parameters that need to be calibrated for each site
prior to the application of this method.
Example application of the wind disaggregation models
W1, W2 and W3 for Obergurgl. Observed
and disaggregated time series are shown for wind speed. For option W3, 10
realisations are shown. The numbers in parentheses indicate the root mean
square error (RMSE) computed for each method.
Random wind speed disaggregation (W3)
According to a random disaggregation of wind speed (W3)
might also perform reasonably:
vi,t=vi⋅[-ln(rnd[0,1))]0.3.
The function rnd is a random number generator, which draws random numbers
between 0 and 1 from a uniform distribution. Figure
includes 10 runs (realisations) for this option. The daily average is not
necessarily preserved by this method.
Shortwave radiation
Radiation model and disaggregation of daily mean shortwave radiation (R1)
Shortwave radiation R0 in W m-2 is computed for hourly time steps
using the methodology described by , which predicts
potential shortwave radiation R0 for each time step. A simplified formula
is provided that assumes a flat surface :
R0=1370Wm-2⋅cosZ⋅(Ψdir+Ψdif).
The solar constant (1370 W m-2) is scaled according to the solar
zenith angle Z, which depends on time (day of year and hour measured from
local solar noon) and latitude . Details on these
calculations as well as on the direct and diffuse radiation scaling values
Ψdir and Ψdif are given by .
This methodology is applied for all three options. R1 assumes daily averages
of shortwave radiation. This type of data is generally only available if
hourly recordings of shortwave radiation have been aggregated prior to the
data dissemination. In contrast, options R2 and R3 do not require shortwave
radiation data as input.
Disaggregation of sunshine duration (R2)
The method R2 builds upon the same methodology as R1 but runs the
model prior to the disaggregation computations. This
model relates sunshine duration to mean shortwave radiation for daily time
steps:
RR0=a+b⋅SS0.
Relative sunshine duration S/S0 is transformed to relative global
radiation R/R0 and then the radiation model is applied
using this data.
Example application of the radiation disaggregation model R2 and R3
for De Bilt. Observed and disaggregated time series are shown for shortwave
radiation. Option R2 is based on sunshine duration, whereas option R3
requires minimum and maximum temperature as input. The numbers in parentheses
indicate the root mean square error (RMSE) computed for each method.
The parameters a and b are by default set to 0.25 and 0.75, respectively
, but can also be determined by optimisation using
observations of daily mean solar radiation, if available.
Figure shows an example based on method R2 for
summertime radiation in De Bilt (Fig. ). The constants a
and b have been obtained through linear regression of R and S time
series covered by the calibration period. If shortwave radiation and sunshine
duration recordings are available, it is recommended to calculate these
values for the site of interest.
The Bristow–Campbell model (R3)
If radiation is not available, option R3 might provide reliable radiation
estimates based on minimum and maximum temperature. It is assumed that small
differences between maximum and minimum temperatures typically occur on
cloudy days. However, larger differences are common on sunny days with
radiative cooling during nighttime and surface heating caused by shortwave
radiative flux during daytime. The corresponding method is named after its
inventors, :
RR0=A⋅1-exp(-B⋅ΔTC).
Here, relative global radiation R/R0 is related to the diurnal temperature
range ΔT, which is estimated using maximum and minimum temperatures
on specific day i and the subsequent day i+1:
ΔTi=Tmax,i-(Tmin,i+Tmin,i+1)2.
Besides the parameters A=0.75 and C=2.4, which might be viewed as
constants in a first step, B is a site-specific parameter:
B=0.0036⋅exp(-0.154⋅ΔT‾).
In contrast to ΔT, which refers to a certain day, ΔT‾ is the long-term average of differences between maximum and minimum
temperature for the month of the current day. Based on these computations,
radiation estimates are used as input to the radiation model R1 (see
Fig. ). A site-specific adjustment of the
parameters A and C is possible by optimisation using observations of
shortwave radiation, daily minimum and maximum temperature.
Example for precipitation disaggregation using the cascade model:
1 h rainfall observed at Rio de Janeiro – São Cristovão on
5 December 2010 (blue). Based on statistical evaluations of long-term hourly
precipitation series and their aggregation to coarser temporal resolutions,
all relevant steps of the cascade disaggregation applied to daily totals are
presented (green). The time series of each cascade level are shown for three
realisations of the model n = 1 (left), n = 2 (centre), and
n = 3 (right).
Precipitation
Equal redistribution (P1)
Reconstructing sub-daily precipitation intensities from daily values is
challenging as precipitation intensities strongly vary in time and space. In
the framework of this study, three methods are presented. The first method is
the simplest way of disaggregating daily precipitation to hourly intensities
by dividing the daily value by 24.
Cascade model (P2)
In order to provide a more sophisticated model that preserves sub-daily
precipitation characteristics and is still less complex than typical weather
generators, a simple statistical precipitation disaggregation approach has
been set up: the microcanonical, multiplicative cascade model by
. Some enhancements proposed in the literature
, such as weighting, have been taken into account as
well. This method is a probabilistic approach providing different
disaggregation results for each run (realisation). However, the statistical
characteristics of each realisation are equal by definition.
The disaggregation is carried out assuming a doubling of temporal resolution
for each step. Due to this stepwise doubling of resolution, the model is
referred to as cascade model see Fig. 1 in. The time
series of cascade level i with time step Δti is disaggregated to
level i+1 with time step Δti+1=12⋅Δti.
The procedure is applied successively until the desired temporal resolution
is reached. The doubling of elements of each subsequently derived time series
implies that each box of the higher level's time
series has to be split in the next cascade level. Thus, the question arises
how the separation of the precipitation volume Pi into two temporally
equidistant time steps Pi+1,1=W1⋅Pi and Pi+1,2=(1-W1)⋅Pi=W2⋅Pi (branching) is done, whereby W1 is the relative
weight of branching for the first box of the subsequent level with respect to
the total precipitation volume to be branched (W2 is the weight assigned
to the second box). Three cases are foreseen in the so-called branching
generator :
W1,W2=0 and 1with probability P(0/1)1 and 0with probability P(1/0)x and 1-xwith probability P(x/(1-x));0<x<1.
The first case indicates a branching that fills the second box of the
subsequent level only, whereas the second case indicates the opposite. In
contrast, the third case accounts for a weighted branching into both boxes of
the subsequent level. For these cases, probabilities are provided for four
different types of wet boxes with Pi>0:
Starting box: this type of box indicates a dry box in the previous and a wet box in the next time step.
Ending box: an ending box follows a wet box and is followed by a dry box.
Isolated box: in this case, the adjacent boxes of the previous and the next time step are dry.
Enclosed box: the adjacent boxes of the previous and next time step are wet.
Histograms of observed and disaggregated time series for each
variable (columns) and each station (rows). Observed (disaggregated) time
series are displayed in grey (white).
These probabilities for the three different branching possibilities
(Eq. ) can be achieved by a reverse scaling procedure.
Highly resolved precipitation time series are aggregated by applying the
cascade level branching assumption backwards. In each case boxes are summed up pairwise representing the respective total volume of the antecedent higher level. Statistics are calculated for the branching types mentioned above
(probabilities are derived through dividing counts of each case by the total
number of elements of the time series). Separate evaluations are prepared for
precipitation intensities below and above the mean precipitation value.
Additional statistics need to be computed for the case P(x/(1-x)) for which
the relative weight x is evaluated as well. For all box types and both
intensity classes, the relative weight ranging from 0–1 is simply
divided into 7 bins see histograms in
and counted according to the previously mentioned criteria (four box types,
two
intensity classes, seven classes of x). This procedure is applied for the
aggregation steps 1→2 h (21 h), 2→4 h
(22 h), 4→8 h (23 h), 8→16 h (24 h),
and 16→32 h (25 h). According to , a
count-related weight is assigned to the probabilities P(0/1), P(1/0), and
P(x/(1-x)) in each aggregation step prior to averaging the probabilities of
all steps. The same procedure is applied to the weights. Finally, as a
result, matrices of probabilities and weights are derived that represent the
station's precipitation scaling. The parameterisation is done by applying the
empirical distributions of P(0/1), P(1/0), P(x/(1-x)), and x to a
random number generator (without fitting analytical distributions).
In turn, these matrices of probabilities and weights are used to disaggregate
daily time series. The type of branching is determined by drawing random
numbers for each branching step incorporating the probabilities P(0/1),
P(1/0), and P(x/(1-x)), which are evaluated cumulatively. If the random
number is within the range of P(x/(1-x)), a similar procedure is applied to
determine the weight x using another random number. In contrast to the
aggregation procedure, disaggregation is applied including the following
steps (see Fig. ): 24→12 h
→6 h →3 h →1.5 h →0.75 h
. The time series with a 45 min time step are equally
distributed to time series with a 15 min time step. These, in turn, are
transformed uniformly to obtain time series with 1 h time step.
For all disaggregation steps described above, the cascade model preserves mass, which means that the precipitation total of the disaggregated time series
is equal to the respective value of the original time series (microcanonical cascade model). Despite its simplicity with respect to model complexity and
parameter estimation , cascade models have been already used successfully in different climates . In contrast to more
sophisticated models, the autocorrelation structure might not necessarily be preserved .
Remarks on spatial representativeness: if this procedure is applied to more than one station, the sub-daily temporal distribution
of precipitation is randomly derived for each station. These spatial patterns do not represent the actual spatial structure of the events at
sub-daily timescales. For practical applications at the mesoscale, it is
therefore suggested to redistribute the sub-daily intensities for each
station according to the cumulative relative sum of the station that is
subjected to the highest daily precipitation depth , which
can be performed using the method described in the next paragraph. Areal peak
intensities at sub-daily time steps might be overestimated due to this
assumption, which limits the universal applicability of this approach.
However, this overestimation might be acceptable for some applications like,
e.g., derived flood frequency analyses for hydrologic design purposes
. A more sophisticated but much more complex approach
that has been developed recently takes
spatial consistency explicitly into consideration.
Redistribution according to another station (P3)
Finally, a third method is supplied that addresses the generally higher
network density of precipitation gauges compared to other meteorological
variables. If a mixed network including hourly and daily observational sites
is considered and if the distance among these stations is small, the relative
mass curve of the station recordings at 1 h time step can be transferred to
the other sites for which only daily recordings are available. The values for
the target sites are obtained through multiplying the relative mass of the
highly resolved station's curve with the daily precipitation depth observed
at the target site. This methodology is also applied in the tool IDWP
(Inverse Distance Weighting for Precipitation), which is part of the
hydrological modelling system WaSiM (Water balance Simulation Model)
. The applicability is limited to the period of time
covered by recordings at a 1 h time step.
Results and discussion
Overview
This section follows the same structure as the methodology section. For each
variable long-term averages of disaggregated and observed time series are
presented and evaluated in order to assess the model skill of the
disaggregation methods. The time series used for disaggregation represent
hourly observations aggregated to daily averages and totals. Emphasis is put
on prediction of diurnal features since most methods described herein are
founded upon assumptions that imply a certain diurnal course for a given
variable. This holds especially true for temperature, humidity, wind speed,
and radiation. For precipitation, results are compiled and discussed for the
cascade model. Due to the involvement of a random number generator in this
method, evaluations with respect to model skill require the analysis of
multiple runs (realisations).
Not all methods provided by
MELODIST are evaluated. We focus on a subset of methods, which might be
relevant to a broad range of users with respect to typical data availability
settings and typical applications. For each variable, the same methodology is
applied to all stations listed in Sect. .
Model performance measures for temperature disaggregation
(T1b model). x¯o and x¯s are the mean
values of observed and disaggregated temperature, respectively. The standard
deviation of the observed (σ̃o) and disaggregated
(σ̃s) time series are also specified. The root mean square error (RMSE), the correlation coefficient r, and the Nash–Sutcliffe
model efficiency (NSE) are calculated using the observed and disaggregated
time series for each station.
Station
x¯o
x¯s
σ̃o
σ̃s
RMSE
r
NSE
(unit of temperature [K])
[–]
[–]
De Bilt
283.57
283.45
6.88
6.94
1.74
0.97
0.94
Ny-Ålesund
269.55
269.65
7.24
7.27
1.63
0.97
0.95
Obergurgl
275.90
276.36
7.87
8.03
2.00
0.97
0.94
Rio de Janeiro
298.25
298.74
4.03
4.34
1.66
0.93
0.83
Tucson
294.35
294.45
9.53
9.49
2.69
0.96
0.92
In order to put light on the model skill in a more quantitative way,
statistical parameters have been derived for both the observed and the
disaggregated time series (see, e.g., Table ). All
statistical parameters refer to the validation period listed for each station
in Table and have been calculated for hourly time steps.
The mean value as well as the standard deviation have been computed for both
time series for each station and each variable. The comparison of mean values
gives an idea about possible biases, whereas the comparison of standard
deviations is relevant to assess the comparability of the variability
inherent in both time series. Moreover, the root mean square error (RMSE),
the correlation coefficient r, and the Nash–Sutcliffe model Efficiency
(NSE) have been calculated based on observed and disaggregated time series.
Model performance measures for humidity disaggregation
(H3 model). x¯o and x¯s are the mean values
of observed and disaggregated relative humidity, respectively. The standard
deviation of the observed (σ̃o) and disaggregated
(σ̃s) time series are also specified. The root mean square error (RMSE), the correlation coefficient r, and the Nash–Sutcliffe
model efficiency (NSE) are calculated using the observed and disaggregated
time series for each station.
Station
x¯o
x¯s
σ̃o
σ̃s
RMSE
r
NSE
(unit of relative humidity [%])
[–]
[–]
De Bilt
81.82
81.63
15.12
15.48
12.67
0.66
0.30
Ny-Ålesund
74.96
74.82
12.61
12.30
17.41
0.02
-0.91
Obergurgl
70.83
66.42
17.67
13.40
16.43
0.51
0.14
Rio de Janeiro
70.95
67.45
14.13
10.76
10.39
0.72
0.46
Tucson
35.31
33.31
21.96
10.51
18.52
0.55
0.29
RMSE is a measure of deviations between observed and disaggregated time
series on an hour-to-hour basis. Smaller values are generally better than
larger values. The correlation coefficient is ideally close to one and
describes the coincidence of phase for two series without considering biases.
In contrast, NSE can be viewed as a combined measure addressing deviations in
terms of biases and shifts in phase. It ranges from negative infinity
indicating a low skill to one indicating a perfect fit. A value of zero means
that the model is as good as applying the average value.
In order to gain some insight on how well the distributions of disaggregated
time series match the observed ones, histograms for each variable and each
site are displayed for both disaggregated and observed values in
Fig. .
Temperature
Despite the fact that only one option is available for temperature (T1), the
standard-sine method enables different options to define the boundary
conditions of the sine function (see Fig. ). This
method uses minimum and maximum temperature as input data. Here, results
using the day-length-dependent option are presented, where maximum
temperature is assumed to occur 2 h after the solar noon. For Ny-Ålesund, the modified nighttime option was activated as well in order to
reliably disaggregate nighttime temperatures during polar nights, when the
assumption of a distinct diurnal course does not hold true.
Long-term averages of diurnal courses of observed (dashed) and
disaggregated (solid) temperature. Option T1b has been chosen and the
sine curve was modelled based on sunset and sun noon computations for minimum
and maximum temperature, respectively. The period of time involved in this
analysis is listed in Table for each station.
Long-term averages of hourly temperature derived for all sites are compiled
in Fig. alongside with the corresponding observations.
The disaggregated diurnal course of temperature coincides well with
observations for each station. Diurnal features are reliably preserved in the
disaggregated time series. However, the amplitude is slightly overestimated
for each site, attributable to the fixed assignment of minimum and maximum
temperature for a given day of year. This assumption is mostly valid on fair
weather days with surface heating but in some cases, e.g. when fronts cross
the site of interest, minimum and maximum temperatures might occur at
different times. Thus, minimum and maximum temperatures are more spread
throughout the day in the observed data sets, resulting in a slightly smaller
amplitude on average.
Besides this visual comparison, Table summarises the
model skill of temperature disaggregations for each station. Mean temperature
values are well represented in the data set given that the mean temperature
was assumed to be unknown and only minimum and maximum temperatures have been
involved in the analyses. The differences are smaller than 0.5 K. Due to the
prescribed difference between minimum and maximum temperature, the standard
deviations of observed and disaggregated time series are very similar.
However, the magnitude of RMSE values shows that differences on an
hour-to-hour basis exceed the average bias.
However, given that only two values per day are used as input data, the RMSE
values can be viewed as good model performance. This holds also true for r
and NSE, indicating a high model skill.
Disaggregated time series of each station are of similar model performance.
Only Rio de Janeiro has a slightly lower model skill, which can still be
viewed as good model representation. Observations derived at Ny-Ålesund
indicate that even an application of average values might be sufficient as
disaggregation procedure, which can be explained by the lower impact of
radiation on diurnal features of meteorological variables for that site. To
conclude, temperature disaggregation based on minimum and maximum temperature
should provide reliable estimates. This finding is also supported by the good
agreement of the histograms constructed for both disaggregated and observed
time series (Fig. , 1st column).
Humidity
Long-term averages of diurnal courses of observed (dashed) and
disaggregated (solid) relative humidity (H3 model). The period of time
involved in this analysis is listed in Table for each
station.
Long-term averages of diurnal courses of (a) observed and
(b) disaggregated “normative” wind speed (W2 model). The
normative wind speed indicates the ratio of the long-term mean of the
wind speed observed or modelled at a specified hour to the respective value
averaged for the entire day. The period of time involved in this analysis is
listed in Table for each station.
As for temperature, Fig. depicts the long-term mean of
the diurnal course of relative humidity for all stations (H3 model). The
diurnal patterns of relative humidity are reasonably disaggregated through
simulating a drop in humidity in the afternoon, which is observed at most
stations. However, the accordance is less pronounced than for temperature. It
is worth noting that the disaggregation of relative humidity depends on
hourly temperature values. For these analyses, the results described for
temperature in the previous sections have been applied for the disaggregation
of relative humidity. Hence, uncertainties involved in the prior step also
contribute to deviations between observation and disaggregation.
A closer look at the statistical evaluations derived for humidity
disaggregation as compiled in Table shows that the
model performance is lower than the corresponding values obtained for
temperature. The mean values are reproduced within a range of ±5 %.
Even though no information about daily minimum and maximum values of humidity
have been involved in the disaggregation procedure, the standard deviations
computed for observed and disaggregated time series are of similar magnitude.
The RMSE amounts to 20 % indicating comparably large differences between
observed and disaggregated values even though the mean bias is substantially
lower. For all but one station, the correlation coefficient is higher than
0.5. In Ny-Ålesund a correlation close to zero could be interpreted as
inadequate model skill, which is underlined when considering the negative NSE
value. It may be assumed that the generally lower impact of radiation on
other meteorological variables would suggest to use an equal redistribution
of humidity values for that station.
However, the model performance achieved for the other stations is better
given that the RMSE is lower and r and NSE are higher, respectively. In
contrast to temperature, the humidity disaggregation performs best for Rio de
Janeiro. To summarise, the disaggregation of humidity is reliable considering
the fact that disaggregated temperature time series and only one humidity
value per day have been used as input. Hence, minimum and maximum humidity
are not preserved by this approach. This finding becomes apparent when
considering the mismatch of minimum and maximum humidity reconstructions for
some sites (e.g. Tucson, see Fig. , 2nd
column for further details). These findings prove previous work that also
discussed the accuracy of humidity disaggregation techniques
. If daily minimum and maximum values of
relative humidity are available, the redistribution of these values should be
pursued see Fig. and.
Model performance measures for wind speed disaggregation (W2 model).
x¯o and x¯s are the mean values of observed
and disaggregated wind speed, respectively. The standard deviation of the
observed (σ̃o) and disaggregated
(σ̃s) time series are also specified. The root mean square error (RMSE), the correlation coefficient r, and the Nash–Sutcliffe
model efficiency (NSE) are calculated using the observed and disaggregated
time series for each station.
Station
x¯o
x¯s
σ̃o
σ̃s
RMSE
r
NSE
(unit of wind speed [m s-1])
[–]
[–]
De Bilt
3.49
3.49
1.89
1.59
1.05
0.83
0.69
Ny-Ålesund
4.03
4.03
3.23
2.57
1.95
0.80
0.64
Obergurgl
1.38
1.38
1.51
1.06
1.08
0.70
0.49
Rio de Janeiro
1.41
1.41
1.21
0.75
0.85
0.72
0.50
Tucson
3.27
3.27
2.07
1.17
1.70
0.57
0.32
Model performance measures for radiation disaggregation
(R2 model). x¯o and x¯s denote the mean
values of observed and disaggregated shortwave radiation, respectively. The
standard deviation of the observed (σ̃o) and
disaggregated (σ̃s) time series are also specified.
The root mean square error (RMSE), the correlation coefficient r, and the
Nash–Sutcliffe model efficiency (NSE) are calculated using the observed and
disaggregated time series for each station.
Station
x¯o
x¯s
σ̃o
σ̃s
RMSE
r
NSE
(unit of radiative flux [W m-2])
[–]
[–]
De Bilt
113.65
123.05
188.11
182.17
61.30
0.95
0.89
Ny-Ålesund
59.73
63.21
92.42
89.38
31.90
0.94
0.88
Wind speed
Wind speed disaggregation has been accomplished using the modified sine curve
(W2). In Fig. the long-term averages of the diurnal
course of wind speed is plotted separately for observed and disaggregated
wind speed, respectively. In this figure, wind speed is scaled as
“normative” wind speed; i.e. the value for each hour is divided by the mean
value. Maximum wind speed, which is typically observed during the afternoon
hours, is well represented in the disaggregated time series. Small-scale
variability, as discussed in the methodology section, is not reproducible by
this approach.
As the mean value is simply redistributed according to a sine function, mean
values are exactly reproduced by the disaggregation approach. As already
mentioned, variability (i.e. fluctuations) is neglected resulting in lower
predicted standard deviations when compared to the corresponding standard
deviations derived for the observed time series (Table 5). This also becomes evident when observing the falling limb
of the histograms of disaggregated values shown in
Fig. (third column). If these fluctuations
are not relevant for further evaluation, this disaggregation methodology for
wind speed has an acceptable model skill, which can be observed from the
correlation coefficients and NSE values. Although these values are lower than
those derived for temperature, they indicate a good model performance for all
sites. The best model skill is achieved for De Bilt, whereas the lowest
performance is achieved for Tucson, where a secondary wind speed maximum is
observed in the morning. This diurnal pattern might be related to a local
wind system that is subject to a change in wind direction and, hence, to a
change in wind speed. Such phenomena are not addressed by this method.
Long-term averages of diurnal courses of observed (dashed) and
disaggregated (solid) shortwave radiation. Disaggregation is based on daily
recordings of sunshine duration (R2 model).
Radiation
Even though radiation observations are available to most of the sites
investigated in this study, the availability of daily mean shortwave
radiation in absence of sub-daily time series is not so common. One exception
is climate model output, which is typically aggregated to daily values. A
typical real-world-case is, however, a long data set of sunshine duration
recordings. Therefore, method R2 is applied even though it is only applicable
to De Bilt and Ny-Ålesund. The diurnal course of mean hourly values
derived through averaging the observed and disaggregated data sets is
displayed in Fig. .
Model performance of the precipitation cascade model (P2) evaluated
for each station. For the validation period, mean values are given for some
relevant characteristics of the precipitation time series. An event is
defined through consecutive hours with precipitation intensity greater than
0 mm h-1. Numbers in parentheses refer to the respective
observed time series of each station.
De Bilt
Obergurgl
Rio de Janeiro
Tucson
Duration of events [h]
3.91
4.72
3.41
2.90
(2.99)
(3.73)
(2.70)
(2.20)
Rainfall of events [mm]
2.52
2.78
4.87
3.46
(2.45)
(2.78)
(4.63)
(3.81)
Duration of dry spells [h]
21.76
23.05
39.24
118.44
(22.02)
(24.00)
(37.87)
(131.47)
Number of events per year
342
316
206
72
(351)
(316)
(216)
(66)
Given that the disaggregation is based on sunshine duration, the model skill
can be viewed as very good for both sites. The timing of solar noon radiative
fluxes as well as the phase of the disaggregated time series track
observations very well, which is also underlined by the performance measures
presented in Table . Deviations between the mean values
can be related to uncertainties involved in the model,
which has been fitted prior to disaggregation for both stations using the
data from the calibration period. However, the disaggregated time series are
subjected to similar variabilities as the observed time series, which is
expressed by the very similar standard deviations and the coincidence of
histograms computed for disaggregated and observed time series as displayed
in Fig. (fourth column). As expected, the RMSE is comparably high when compared to
the mean value of the time series since shortwave radiation is subjected to
fluctuations due to the presence and absence of clouds causing rapid changes
in shortwave radiation even for small increments in time. Notwithstanding
these restrictions, the model skill expressed through the correlation
coefficient and the NSE can be viewed as very good.
Precipitation
In contrast to the meteorological variables previously described,
precipitation has been disaggregated using the cascade model (P2), which is a
probabilistic model. As already explained, this change from deterministic to
probabilistic methods requires a modified evaluation of model performance.
Even though the precipitation total is preserved for each day throughout the
disaggregation procedure, the occurrence and sequence of precipitation
intensities differ from run to run. For rigorous testing and validation of
the method, multiple runs are needed and their results have to be
statistically evaluated. Figure (fifth column)
shows histograms for both disaggregated and observed time series for each
station. The comparison of histograms derived from disaggregated and observed
values reveals that the empirical distributions are similar. The falling limb
of the histograms is also reliably reconstructed by the cascade model for
which 100 runs have been considered to compute the histograms.
In addition to this visual comparison, the evaluation has been carried out
according to the validation approaches described by and
. Following their ideas, Quantile–Quantile plots (Q–Q
plots) of precipitation intensities are shown in
Fig. , with close attention paid to the highest
1 % of precipitation intensities. Since autocorrelation structure is not
explicitly warranted by the cascade model, this feature is also tested (see,
Fig. ). As the common performance measures
cannot be applied appropriately for random distributions of daily
disaggregations, other performance criteria have to be considered. An
approach similar to that described by was chosen for that
reason (see Table ).
First, the simulation of peak intensities is studied through comparing
observed and disaggregated intensities in a Q–Q plot
(Fig. ). For each station for which precipitation
is available the highest 1 % of disaggregated intensity values is plotted
against the corresponding sorted time series of observed values. The cascade
model was run 100 times, which is why 100 realisations are similarly
evaluated. The areas shaded in light blue represent the range of values
achieved through involving all realisations in the analyses. In contrast, the
area shaded in dark blue corresponds to the standard deviation of the
considered quantile. Moreover, the mean of all realisations is drawn as
blue line for each station.
Even though intensity peaks are only represented implicitly through branching
probabilities, precipitation peaks are well captured from a statistical point
of view. For Rio de Janeiro, Tucson, and De Bilt, precipitation intensities
are slightly underestimated. In contrast, an overestimation can be observed
in the results of Obergurgl. The range of values indicate that some of the
highest values in the observed data sets are even exceeded in some
realisations, which might underline the need for multiple runs.
Other characteristics that are also relevant for evaluations of sub-daily
precipitation characteristics are summarised in Table .
The mean duration of events ranges from 3 to 5 h and is overestimated for
all stations, which was also found by and
. In contrast, the mean precipitation total of events
derived through disaggregation is on average similar to the respective
observed value. This finding holds for all stations. It is evident that this
value is higher in the subtropics than in the mid-latitudes. Although the
total annual rainfall in Tucson is comparably small and the number of events
per year is low, the average rainfall of events is also higher than in the
mid-latitudes. This feature is correctly predicted by the cascade model. The
duration of dry periods is also in good agreement compared to observations.
Even though the length of events is over-predicted, the characteristics of
the observed precipitation time series are captured very well for each site
by the cascade model.
To conclude, the cascade model preserves major characteristics of the
observed hourly time series. However, these sub-daily characteristics can
only be statistically evaluated due to the probabilistic nature of the
approach. The model skill achieved for the stations listed in
Table can be viewed as reasonable reconstruction.
Modelled (blue) vs. observed (black) precipitation intensities for
the 1 % highest intensities derived using a split-sample test for the
cascade model (P2). For all panels the shaded areas refer to the standard
deviation (dark blue) and range of values (light blue) computed for 100
realisations, respectively.
As for the intensity plot, shaded areas are added to the diagrams
in Fig. 14 to show the variability in terms of
total range and standard deviation of values. The autocorrelation derived for
the disaggregated time series match observed values very well for Rio de
Janeiro, Tucson, and De Bilt. For Obergurgl, higher rk values are
observed, which are not covered by the model results. The results derived
using the cascade model for these sites can be viewed as good reconstruction
of hourly precipitation features given that intensities, major
characteristics of precipitation events, and the autocorrelation structure of
the disaggregated time series are in good agreement with observation.
Conclusions and outlook
The application of a simple and easy-to-use toolbox of disaggregation methods
has been presented. Most of the methods included in MELODIST are
parsimonious with respect to theory and computational costs (disaggregating
5 years of daily precipitation recordings using the cascade model takes less
than 4 s on a notebook with a 2 GHz i7 CPU). The basic levels of complexity
have been chosen keeping practitioners in mind, who need a package that is
capable of disaggregating all relevant meteorological variables needed for
environmental modelling. Available studies on disaggregation often focus on
single variables such as precipitation rather than providing a unified
framework for disaggregation. However, the presented package can be easily
extended by more complex methods available in the literature as it provides
basic functionalities for handling of time series with different temporal
resolutions.
Autocorrelation rk as function of time lag k (in hours) plotted
for modelled (blue) and observed (black) precipitation time series. For all
panels the shaded areas refer to the standard deviation (dark blue) and range
of values (light blue) computed for 100 realisations, respectively.
A set of methods relevant for real-world cases has been presented based on a
split-sample test and statistical evaluations performed for the validation
period. The presented methods perform well for different stations situated in
different climates, which underlines the robustness of the methods applied in
the framework of this study. The highest model skill is achieved for
temperature. Humidity disaggregation is, however, less reliable given that
only one value per day is provided. The availability of minimum and maximum
relative humidity improves the model skill. Wind speed disaggregation based
on diurnal variations also works well if fluctuations are not required for
further analyses. In contrast, the random wind speed function might be an
alternative as it provides higher variabilities. Hourly radiation time series
can be obtained with good agreement compared with observations, even if daily
recordings of sunshine duration are used as input. Although precipitation was
disaggregated using a stochastic approach, which matches observations only in
terms of long-term statistical evaluations, major characteristics of hourly
precipitation features coincide well with observations. Based on this
validation and the fact that different meteorological variables and stations
have been involved in the validation analyses, MELODIST can be
viewed as a reliable and robust tool.
Some of the methods provided by MELODIST are based upon analyses of
time series for parameter estimation, which requires a certain quality of
data to derive sound parameters for performing the disaggregation runs. In
general, it is important to note that data homogeneity might not always apply
to long time series as changes in the instrumentation, microclimate, and
processing of data might have caused discontinuities in the time series
see, e.g.,. For instance,
describe trends in the Ny-Ålesund data sets, which are also tested herein.
This is especially important if statistical disaggregation methods are
applied that have been tuned for small periods of time only. Moreover, the
limited availability of hourly observations involved in the statistics
achieved in this study has to be carefully reviewed with respect to
representativeness from a climatological point of view. In this study,
different stations have been considered to investigate the robustness of
methods rather than drawing conclusions in terms of climatic differences.
Homogeneity might be also relevant for disaggregation of time series that are
subject to changes in climate. Ideas to cope with changing climatic
conditions for disaggregation approaches are currently investigated. Two
examples relevant in this context for the statistics-based cascade model are
a circulation-based parameterisation in order to better predict changing
weather patterns related to changing climate and an
intensity-based categorisation . Current research also
focuses on the incorporation of the Clausius–Clapeyron relation to better
predict rainfall intensities in future climates . These
studies only address single stations or a limited study area without the
consideration of different climates. Hence, the applicability of new methods
should also be critically reviewed with respect to transferability.
In contrast to weather generators and dynamical downscaling approaches,
physical consistency among the meteorological variables considered in this
framework is not inherent in the methodology. This limitation might restrict
the methodology to derive input data only for conceptual models that are not
pure physics-based approaches as the latter are more demanding with respect
to this consistency. However, for most conceptual “grey box” models
see, e.g., the quality of data provided by this
disaggregation methods should be sufficient as tested in the framework of
other model experiments . A better representation of the
dependencies among the most relevant meteorological variables should be
addressed explicitly in the future. Moreover, further emphasis should be on
spatial consistency in disaggregation as already pursued by some authors
see, e.g.,. The ongoing research on
disaggregation methods underlines the need for sound and robust tools for
disaggregating meteorological variables.
Even though MELODIST provides robust methods that do not include
those very recent developments, it might serve as tool for both practitioners
and scientists. For the latter group, MELODIST could be viewed as
framework for performing future research on disaggregation since new
disaggregations methods can be easily plugged in.