Gauge-based radar rainfall adjustment techniques have been widely used
to improve the applicability of radar rainfall estimates to
large-scale hydrological modelling. However, their use for urban
hydrological applications is limited as they were mostly developed
based upon Gaussian approximations and therefore tend to smooth off
so-called “singularities” (features of a non-Gaussian field) that
can be observed in the fine-scale rainfall structure. Overlooking the
singularities could be critical, given that their distribution is
highly consistent with that of local extreme magnitudes. This
deficiency may cause large errors in the subsequent urban hydrological
modelling. To address this limitation and improve the applicability of
adjustment techniques at urban scales, a method is proposed herein
which incorporates a local singularity analysis into existing
adjustment techniques and allows the preservation of the singularity
structures throughout the adjustment process. In this paper the
proposed singularity analysis is incorporated into the Bayesian
merging technique and the performance of the resulting
singularity-sensitive method is compared with that of the original
Bayesian (non singularity-sensitive) technique and the commonly used
mean field bias adjustment. This test is conducted using as case study
four storm events observed in the Portobello catchment
(53 km
Traditionally, urban hydrological applications have relied mainly upon
rain gauge data as input. While rain gauges generally provide accurate
point rainfall estimates near the ground surface, they cannot properly
capture the spatial variability of rainfall, which has a significant
impact on the urban hydrological system and thus on the modelling of
urban runoff
First, errors in radar reflectivity measurements may arise from
blockage of the radar beam, attenuation, ground clutter, anomalous
propagation of the signal, among other sources
In order to overcome these drawbacks of radar rainfall estimates while
preserving their spatial description of rainfall fields, it is
possible to dynamically adjust them using rain gauge
measurements. Many studies on this subject have been carried out over
the last few years, though most of them focus on hydrological
applications at large scales
To address this limitation and improve the applicability of adjustment
techniques at urban scales, a method is proposed herein which
incorporates a local singularity (identification) analysis
The proposed singularity-sensitive method was initially developed and
preliminarily tested in the reconstruction of a storm event which led
to reported flooding in the Maida Vale area, Central London, in
June 2009
Schematics of
In the present paper the formulation of the proposed
singularity-sensitive method is explained in detail and new numerical
strategies aimed at improving the use of singularity information are
introduced. Moreover, the method is further tested using as case study
four storm events observed in the Portobello catchment
(53 km
The paper is organised as follows. In Sect.
Firstly, a description is provided of the two key techniques used in this paper: the Bayesian data merging method and the local singularity analysis. Afterwards, the proposed method for integrating these two techniques is explained. Intermediate results of each of the steps described in this section, which help illustrate the main features of the proposed methodology, can be found in the Supplement.
The Bayesian data merging method (BAY) is a dynamic adjustment method
(applied independently at each time step) intended for real-time
applications For each time step The interpolated rain gauge rainfall field is
compared against the radar field ( Using a Kalman filter It can be seen that the Kalman gain is a function of the co-variances
of radar and rain gauge estimation errors. When
It is in steps b and c where the problems associated
with the Bayesian merging technique, and geostatistical techniques in
general, arise. The (second-order) co-variance function that these
techniques employ to characterise radar–rain gauge errors cannot well
capture local singularity structures. Instead, in second-order models
singularities may be mistakenly regarded as errors in the radar data,
thus leading to higher estimated radar uncertainty,
Schematic of the local singularity analysis (adapted from
Various types of hazardous geo-processes, including precipitation,
often result in anomalous amounts of energy release or mass
accumulation confined to narrow intervals in time and/or space. The
property of anomalous amounts of energy release or mass accumulation
is termed
A schematic of the estimation of the constant value
Going back to the definition of singularity, Eq. (
In practice however, there is a drawback to this local singularity
analysis. Because it carries out a “local” analysis, the singularity
exponents are usually obtained from a small number of data
samples. This increases the uncertainty of the estimation of
The second numerical strategy is to decompose the rainfall field using
an iterative procedure
Moreover, in this work a spatial-scale range of 1–9 km,
which results in a total of five rainfall intensity samples (at scales 1,
3, 5, 7 and 9 km), was used in the singularity analysis. This range
was selected for two main reasons. Firstly, our analyses revealed that
a good linear behaviour was generally observed within this scale range,
while small-scale structures were still preserved in the resulting
rainfall product. As such, the selected spatial scale range was deemed
to represent a good balance between estimation uncertainty (which depends
upon the number of samples employed in the calculations) and local feature
preservation. Secondly, a scaling break at approximately 8–16 km has been
reported in studies in which 1 km radar rainfall data were analysed
Lastly, a 10-iteration singularity analysis was applied in order to ensure that most of the singularity exponents could be extracted. The downside of conducting many iterations is the longer computational time, which may be an issue for real-time applications. Nonetheless, in practice, approximately 4–6 iterations are sufficient for effectively removing most of the singularity.
Portobello catchment
The underlying idea of the proposed method is to use the local
singularity analysis to decompose each radar image into a non-singular
image and a singularity map before applying the Bayesian merging
(step (i) in Fig.
It is worth noting that the proposed singularity-sensitive merging method does not always increase the reliability of RD estimates. Such increase only happens when the RD estimates exhibit high singularity and thus cannot be well handled using Gaussian approximations.
A particular phenomenon which may cause problems in the application of the proposed methodology and is therefore worth highlighting is the eventual presence of singularity structures in the interpolated rain gauge field (i.e. BK field) and in the resulting supposedly non-singular Bayesian (NS-BAY) merged field. While BK fields are generally highly smooth, singularity structures may appear in the special case in which a rain gauge is located within a convective cell or a local depletion. Singularity structures in the BK field may be preserved in the NS-BAY field. When this is the case, the application of the singularity map back and proportionally to the NS-BAY field may result in double-counting of singularities. This can ultimately result in a merged (SIN) product with more singularities than those originally observed in the radar image. In order avoid this, a “moving window” smoothing has been applied to the BK field before it is merged with the NS-RD field. That is, each pixel value of the BK field is replaced by the mean of the original value and neighbouring pixel values within a 9 km diameter (which is equal to the coarsest scale considered in the local singularity analysis). In this way singularity structures potentially present in the BK field are smoothed-off.
The proposed SIN merging method is tested using as case study four
storm events observed in the Portobello catchment (Edinburgh, UK)
during 2011 and for which radar estimates, dense rain gauge and flow
records, as well as a recently calibrated urban drainage model were
available. Portobello is a coastal town located 5 km to the east of
the city centre of Edinburgh, along the coast of the Firth of Forth,
in Scotland (Fig.
Selected rainfall events over the Portobello catchment.
Note: the accumulation and peak intensity values shown in this table correspond
to areal mean values for the entire domain under consideration (as shown in Fig.
A semi-distributed model of the storm-water drainage system of the
Portobello catchment, including its sewer system
(Fig.
The Portobello model contains a total of 1116 sub-catchments, with
areas ranging between 0.02 and 24.42 ha and a mean area of
2.3 ha. Sub-catchment slopes range from 0.0 to
0.63 m m
Following UK standards
Local rainfall and flow data were collected in the Portobello
catchment through a medium-term flow survey carried out between April
and June 2011. The survey comprised 12 tipping bucket rain gauges and
28 flow monitoring stations (each comprising a depth and a velocity
sensor, based upon which flow rates were estimated). Both rain gauge
and flow records were available at a temporal resolution of
2 min. However, rain gauge records were linearly
interpolated to 5 min, in order to ensure agreement with the temporal
resolution at which radar estimates were available (see Sect.
The Portobello catchment is within the coverage of C-band radars
operated by the UK Met Office (Fig.
During the monitoring period (April–June 2011), four relevant storm
events were captured which comply with UK standards for calibration
and verification of urban drainage models (i.e. these events have
instantaneous rainfall rates
As can be seen in Table
The performance of the proposed singularity-sensitive Bayesian method
(SIN hereafter) is assessed by inter-comparison against radar (RD),
rain gauge (RG) and block-kriged (BK) interpolated RG estimates, as
well against adjusted estimates resulting from the original Bayesian
(non singularity-sensitive) technique (BAY) and the commonly used mean
field bias (MFB) adjustment method. It is important to note that, in
this work, the MFB was implemented in a relatively dynamic way by
computing a sample cumulative bias (
Two evaluation strategies were applied:
Through analysis of the different rainfall estimates,
using as main reference local rain gauge records, while
also inter-comparing the behaviour of other estimates. Through analysis of the hydraulic outputs obtained by feeding
the different rainfall estimates as input to the hydraulic model of
the Portobello catchment and comparison of these with available flow
records. Note that the RG estimates were applied to the model using
Thiessen polygons.
Both evaluation strategies have inherent limitations which are next
described. However, they provide useful and complementary insights
into the performance of the proposed merging method.
The first strategy is a natural and widespread way of assessing the
performance of rainfall products. However, the fact that all
precipitation estimates entail errors and that the true rainfall field
is unknown, in addition to the differences in the spatial and temporal
resolutions of RG and RD estimates (and the resulting merged rainfall
products), renders any direct comparison of rainfall estimates
imperfect
The second strategy (i.e. hydraulic evaluation) allows some of the
limitations of the rainfall evaluation strategy to be overcome,
and is particularly useful when dense flow records are available, as is
the case in the Portobello catchment. However, it has two main
deficiencies: the fact that flow records (obtained based upon
depth and velocity measurements) used in the evaluation contain errors,
and the fact that the hydraulic modelling results encompass
uncertainties from different sources in addition to rainfall
input uncertainty
The performance of the SIN rainfall products in relation to other rainfall estimates (including RD, RG, BAY and MFB) is evaluated in terms of accumulations and rainfall rates at the areal level (i.e. at a scale corresponding to the area over which the Portobello catchment stretches) and at individual point gauge locations. In addition, a qualitative assessment of the spatial structure of the different (gridded) rainfall products is carried out based upon visual inspection of images of the rainfall fields at the time of areal average peak intensity.
In view of the high density and coverage of the RG network over the
Portobello catchment, the areal average RG estimates in the areal
level analysis are assumed to be a good approximation of the “true”
areal (average) rainfall over the experimental catchment (i.e. the
areal reduction effect is expected to be minor –
In the analysis of rainfall estimates at rain gauge point locations
a cross-validation strategy was adopted and three
performance statistics are used. The cross-validation
strategy, also referred to as “leave-one-out”, is an iterative method
in which, at each iteration, data from one RG site is omitted from
the calculations and the value at the “hidden” (i.e. omitted) location
is estimated using the remaining data. Performance statistics are then
computed from the comparison between the estimated and the known
(but not used) values
A qualitative analysis of the hydraulic outputs is carried out based
upon visual inspection of recorded vs. simulated flow hydrographs (for
the different rainfall inputs) at different points of the
catchment. Furthermore, similar to the rainfall analysis, a simple
linear regression analysis is applied to each pair of recorded and
simulated flow time series (at each flow gauging location). The
performance of the associated hydraulic simulations is evaluated using
the
In order to minimise the influence of the errors in the flow
measurements, the available flow records were quality-controlled (QC)
before carrying out the statistical analysis of hydraulic outputs. The
QC was carried out following UK guidelines
Areal average rainfall accumulations and peak intensities for the different rainfall products.
Table
As would be expected, the BK estimates exhibit areal average
accumulations and peak intensities similar to those of the RG. Small
differences are observed (in general BK values are slightly lower than
RG ones) which can be generally attributed to the area-point rainfall
differences
Histogram of singularity exponents (
When looking at the adjusted rainfall products (i.e. MFB, BAY and SIN), it can be seen that all of them can improve the original RD estimates, but the degree of improvement is different for each method. As expected, the MFB successfully reduces the difference in event areal average accumulations (i.e. bias), leading to areal average accumulations close to those recorded by RG. In terms of peak intensities, the MFB method leads to some improvement, but the resulting peak intensities are still significantly lower than the RG ones. Although the MFB was applied dynamically with an hourly frequency of bias correction, these results suggest that more dynamic and spatially varying (higher order) methods than the MFB are required in order to successfully adjust radar rainfall estimates for urban hydrological applications.
Scatterplots of instantaneous areal average RG vs. RD/BK/MFB/BAY/SIN rainfall rates over the Portobello catchment for the four selected events, where SIN1–SIN5 represent the SIN estimates with different “truncated” singularity ranges (from widest to narrowest).
The BAY estimates show the least improvement in terms of event bias,
with a general tendency to underestimate RG areal accumulations, which
is even more marked than for BK estimates. This is particularly the
case in Storms 3 and 4, in which strong singularity structures, as
represented by the high frequency of
Figure
The aforementioned features of the different rainfall estimates are
further highlighted through analysis at each rain gauge location; the
associated statistics, including sample bias (
Boxplots displaying the distribution of sample bias ratio (
Snapshot images of the different spatial rainfall products at the time of peak areal intensity for Storms 1 (top panels) to 4 (bottom panels) over the Portobello catchment. From left to right panels: RD, BK, MFB, BAY and SIN3 (with singularity range [1, 3]) estimates. The black polygon indicates the boundary of the Portobello catchment, and the black and white markers respectively represent the location of flow and rain gauges.
As expected, the RD estimates (before adjustment) display the largest
differences from point RG estimates: in general, they possess the
largest cumulative bias (
Similarly to the results of the areal (average) analysis, the
individual-site BK estimates display the closest behaviour to the RG
ones. This is of course expected given that the BK estimates are
obtained by simple interpolation of point RG data. It can be seen that
the BK estimates are nearly unbiased (
With regard to the adjusted rainfall estimates, the MFB method is
found to bring original radar estimates slightly closer to RG ones,
but the improvement seems insufficient. As expected, the main
improvement of MFB estimates is found in the bias (
When looking at the statistics of the BAY estimates, it can be noticed
that these behave similarly to the BK ones: their bias is also
small, the
With regard to the SIN estimates, it can be seen that their bias is
small (close to 1) and that the distribution of their
Snapshot images of the different gridded rainfall products at the time
of peak areal intensity for the four storm events under consideration are
shown in Fig.
Observed flows vs. simulated flows with RG, RD, BK, BAY and SIN3
rainfall inputs at selected flow gauging sites of Portobello catchment during
Storm 3. Selected gauging sites: FM3: upstream end of the catchment (top
panels); FM10: mid-stream area (bottom panels). The location of the selected
monitoring sites is shown in Fig.
Observed flows vs. simulated flows with SIN1–SIN5 rainfall inputs at
selected flow gauging sites of Portobello catchment during Storm 3. Selected
gauging sites: FM14: upstream end of a small branch of the sewer system (top
panels); FM19: downstream end of the catchment (bottom panels). The location
of the selected monitoring sites is shown in Fig.
It can be seen that the spatial structure of the BK rainfall field
(fully based upon rain gauge data) is highly symmetric and smooth, and
is rather unrealistic. With regard to the adjusted rainfall products
(MFB, BAY and SIN), it can be noticed that the proportion of radar (RD)
and BK interpolated rain gauge features that are preserved varies
according to the method. The MFB fields fully inherit the spatial
structure of the RD fields; the only change is that the actual
intensity values are scaled up or down by an areal ratio derived from
the sample bias between mean rain gauge and radar rainfall
estimates. In agreement with the quantitative results presented above,
it can be seen that the structure of the BAY peak rainfall fields is
often similar to that of the BK ones and is smoother than
the original RD image. Singularity structures are often present in
rainfall fields during peak intensity periods (such as the ones shown
in Fig.
Boxplots displaying the distribution of regression
coefficient (
Figures
From the hydrographs in Figs.
The preliminary conclusions drawn from the visual inspection of the
selected hydrographs are corroborated by the statistics in
Fig.
Regarding the difference in hydraulic performance between the
events used for model calibration (i.e. Storms 1–3) and the
independent event (i.e. Storm 4), it can be seen that the statistics
of the hydraulic outputs during Storm 4 are generally lower than for
the other three storm events (Fig.
In this paper, a new gauge-based radar rainfall adjustment method was
proposed, which aims at better merging rainfall estimates obtained
from rain gauges and radars, at the small spatial and temporal scales
characteristic of urban catchments. The proposed method incorporates
a local singularity analysis into the Bayesian merging technique
Using as case study four storm events observed in the Portobello
catchment (53 km
In this study the sensitivity of the SIN results to the “degree” of singularity that is removed from the radar image and preserved throughout the merging process was also tested. While the impact of it was found to be generally small, the results suggest that partially removing singularities could have a negative impact on the results. Therefore, removing most singularity exponents from the original radar image is advisable.
While the proposed singularity method has shown great potential
to improve the merging of radar and rain gauge data for urban
hydrological applications, further testing including more
storm events and pilot catchments is still required in order to ensure
that the results are not case specific and to draw more robust
conclusions about the applicability of the proposed method. Other
aspects on which further work is recommended are the following:
The current version of the singularity-sensitive method shows
a slight tendency to overestimate rainfall rates and accumulations.
This is likely to be due to one of two aspects, or a combination of them:
In the eventual case in which a rain gauge is located
within the core of a convective cell, the resulting interpolated
(block-kriged; BK) field may end up having singularity structures and, as
explained in Sect. The asymmetric distribution of singularity
exponents and the numerical stability of singularity extraction
from a small set of data samples. This drawback could be improved
by forcing the mean of non-singular components to remain equal
to the original radar estimates Given that the proposed singularity-sensitive merging method is
particularly intended to improve rainfall estimates for (small-scale) urban areas,
it would be interesting to test it using higher spatial-temporal resolution data (e.g. from
X-band radars).
Lastly, a suggestion often made to us and therefore worth briefly discussing is to use a transformation in order to bring the distribution of the radar field closer to normality before the merging (be it with the Bayesian or other geo-statistical method) is conducted. However, doing this would somehow miss the point of the proposed method. The key point here is that for a non-Gaussian structure, moments beyond the second order are important, as each brings new information worth preserving. To create a more “normal” field is not the purpose of the singularity extraction; instead, it is the consequence after removing singularities from the rainfall fields, which can be physically associated with abnormal energy concentration, such as “convective” cells, and which in the proposed method are set aside to ensure their preservation throughout the merging process.
The authors would like to acknowledge the support of the Interreg IVB NWE RainGain project, the Research Foundation-Flanders (FWO) and the PLURISK project for the Belgian Science Policy Office of which this research is part. Special thanks go to Alex Grist and Richard Allitt, from Richard Allitt Associates, for providing the rain gauge data and the hydraulic model, and for their constant support with the hydraulic simulations. Thanks are also due to the UK Met Office and the BADC (British Atmospheric Data Centre) for providing Nimrod (radar) data, to Innovyze for providing the InfoWorks CS software, and to Cinzia Mazzetti and Ezio Todini for making freely available to us the RAINMUSIC software package for meteorological data processing. Lastly, the authors would like to thank the reviewers, Scott Sinclair and Cinzia Mazzetti, and the Editor, Uwe Ehret, for their insightful and constructive comments which helped improved the manuscript significantly. Edited by: U. Ehret