HESSHydrology and Earth System SciencesHESSHydrol. Earth Syst. Sci.1607-7938Copernicus PublicationsGöttingen, Germany10.5194/hess-22-3983-2018 Technical note: Assessment of observation quality for data assimilation in flood modelsAnalysis of observation uncertainty for flood assimilation and forecastingWallerJoanne A.j.a.waller@reading.ac.ukhttps://orcid.org/0000-0002-7783-6434García-PintadoJavierMasonDavid C.DanceSarah L.https://orcid.org/0000-0003-1690-3338NicholsNancy K.School of Mathematical, Physical and Computational Sciences, University of Reading, Reading, UKMARUM – Center for Marine Environmental Sciences and Department of Geosciences, University of Bremen, Bremen, GermanySchool of Archaeology, Geography and Environmental Science, University of Reading, Reading, UKJoanne A. Waller (j.a.waller@reading.ac.uk)23July20182273983399230January20181February201817May201822May2018This work is licensed under the Creative Commons Attribution 4.0 International License. To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/This article is available from https://hess.copernicus.org/articles/22/3983/2018/hess-22-3983-2018.htmlThe full text article is available as a PDF file from https://hess.copernicus.org/articles/22/3983/2018/hess-22-3983-2018.pdf
The assimilation of satellite-based water level observations (WLOs) into 2-D
hydrodynamic models can keep flood forecasts on track or be used for
reanalysis to obtain improved assessments of previous flood footprints. In
either case, satellites provide spatially dense observation fields, but with
spatially correlated errors. To date, assimilation methods in flood
forecasting either incorrectly neglect the spatial correlation in the
observation errors or, in the best of cases, deal with it by thinning
methods. These thinning methods result in a sparse set of observations whose
error correlations are assumed to be negligible. Here, with a case study, we
show that the assimilation diagnostics that make use of statistical averages
of observation-minus-background and observation-minus-analysis residuals are
useful to estimate error correlations in WLOs. The average estimated
correlation length scale of 7 km is longer than the expected value of 250 m.
Furthermore, the correlations do not decrease monotonically; this unexpected
behaviour is shown to be the result of assimilating some anomalous
observations. Accurate estimates of the observation error statistics can be
used to support quality control protocols and provide insight into which
observations it is most beneficial to assimilate. Therefore, the
understanding gained in this paper will contribute towards the correct
assimilation of denser datasets.
Introduction
In data assimilation (DA), observations are combined with numerical model
output, known as the background, to provide an accurate description of the
current state, known as the analysis. In DA the contributions from the
background and observations are weighted according to their relative
uncertainty. The observation error statistics are the sum of the instrument
error and representation error . The error of representation
arises due to the mismatch in the observation and its model equivalent, and it
is often correlated and state dependent . In DA,
observation error statistics are typically assumed to be uncorrelated. The
data density is reduced in order to satisfy this assumption
. Yet having adequate estimates of these uncertainties is
crucial in order to obtain an accurate analysis. Since the true state of the
system is not known, the exact observation errors and their statistics can
not be calculated. Instead observation uncertainties must be estimated
statistically e.g..
provide a diagnostic to estimate observation uncertainties using the
statistical average of observation-minus-background and
observation-minus-analysis residuals. The diagnostic has been applied to
operational numerical weather prediction (NWP) settings to estimate
observation uncertainties
. The use of
these estimated statistics in NWP results in a more accurate analysis and
improvements in objectively measured forecast skill .
The development of DA systems has largely been driven by its use in NWP, but
the methodologies are applicable to any system that can be modelled and
observed. There have been recent advances in real-time 2-D hydrodynamic
modelling and the acquisition and processing of relevant remote sensing
observations (earth observations, EOs) .
Consequently, several studies have shown the benefit of applying DA to
operational flood forecasting
.
review the potential of EOs for inundation mapping
and water level estimation and their use for calibration, validation and
constraint of real-time hydraulic flood forecasting models.
A predominant EO technique to obtain water level observations (WLOs) is
synthetic-aperture radar (SAR). SAR provides high-resolution observations of
radar backscatter which, after processing, serve to delineate the flood
extent. Then, the intersection of the flood extent with a high-resolution
lidar digital terrain model is used to obtain the WLOs. The resulting WLOs
are discontinuous but locally dense in space; consequently, the errors in
the observations may be highly correlated. However, the current practice when
assimilating WLOs is to neglect the error correlations. To make the
assumption of uncorrelated errors valid the current approach is to thin the
data. Hence, in hydrology, one scenario that would benefit from improved
understanding of the observation uncertainties is the assimilation of the
satellite-derived water level observations (WLOs) for either operational
flood forecast or hindcast analyses .
A more detailed understanding of the observation
uncertainties would be highly useful as understanding the error statistics
may permit more observations to be included in the assimilation, which should
allow the information from dense observation sets to be fully exploited.
Additionally, accurate estimates of observation uncertainties can inform the
thinning strategy and suggest which observations may benefit the assimilation
most . There is a clear potential to improve the flood
forecast if all the SAR WLOs could be assimilated in an appropriate way.
In this article we use the diagnostic of , described in
Sect. , to estimate the observation error statistics for SAR
WLOs that are assimilated using a local ensemble transform Kalman filter (LETKF)
into the LISFLOOD-FP 2-D hydrodynamic model. For this study, we use a
sequence of real SAR overpasses in a flood event that occurred in November 2012
in SW England. A description of the SAR WLOs and experimental design are
given in Sect. . Results are discussed in Sect. .
First, we estimate average WLO error statistics across the
entire domain for the duration of the flood event. It will be seen later that
these globally estimated error statistics show an anomalous pattern. To
determine the cause of these anomalous results we consider if observations in
different sub-domains have different error characteristics. We also consider
if the error statistics differ for different phases of the flood event. From
the results we infer that the anomalous pattern is not related to the
distribution of observations over the domain but to observations during the
later stages of the flood. To the best of our knowledge this is the first
time that the diagnostics have been applied to estimate error statistics for
hydrological data assimilation. Importantly, we show that the diagnostic of
can be used to identify anomalous observation datasets
that are not suitable for assimilation.
The diagnostic of Desroziers et al. (2005)
Data assimilation is a technique used to provide the best estimate, the
analysis, of the current state of a dynamical system. The analysis is denoted
xa∈RNm. The analysis is determined by combining
the background xb∈RNm, a model prediction, with
observations, y∈RNp, weighted by their respective
error statistics. Here the dimensions of the observation and model state
vectors are denoted by Np and Nm, respectively. To compare observations
and background it is necessary to project the background into observation
space using the observation operator, H:RNm→RNp,
which may be non-linear. The analysis can be used to initialize a forecast
which in turn provides a background for the next assimilation.
In the analysis is calculated using
xa=xb+BHTHBHT+R-1y-Hxb,=xb+Kdbo,
where R∈RNp×Np and
B∈RNm×Nm are the observation and background error
covariance matrices, K is the Kalman gain matrix and
H is defined as the observation operator linearized about the background state.
The observation-minus-background residuals dbo=y-H(xb),
also known as the innovations, are assumed to be unbiased. Hence any bias should
be removed before assimilation .
The observation error covariance matrix can be estimated using the
observation-minus-background, dbo=y-H(xb),
and observation-minus-analysis, dao=y-H(xa), residuals .
Assuming that the observation and background errors are mutually
uncorrelated, the statistical expectation of the product of the analysis and
background residuals is
EdaodboT≈R.
As the resulting matrix is estimated statistically it will not be symmetric.
Therefore, it must be symmetrized before it can be used in a data assimilation scheme.
The form of the diagnostic in Eq. () is not suitable to calculate
observation error statistics when each assimilation cycle uses different
observations. Instead components of the background and analysis residuals must
be paired and binned, with the binning dependent on the type of correlation
being estimated. For example, when calculating spatial correlations the bins
may depend on the distance between observations, whereas for temporal correlations
the bins would depend on the time between observations. For each bin, β,
the covariance, cov(β), is then computed individually using
cov(β)=1Nβ∑k=1Nβdioadjobk-1Nβ∑k=1Nβdioak1Nβ∑k=1Nβdjobk,
where (dioadjob)k is the kth pair
of elements of dao and dbo in bin β, and
Nβ is the number of residual pairs in bin β. It is assumed that
the observation-minus-background and observation-minus-analysis residuals are
unbiased, but this is not guaranteed. Hence the second term of
Eq. () ensures that the computation of the observation error
statistics is not affected by bias . To calculate the
spatial correlation, the covariance in each bin, cov(β), is divided by
the estimated variance (the covariance at zero distance, cov(0)).
The diagnostic in Eqs. () and () only gives a
correct estimate of the observation error uncertainties if the error
statistics used in the assimilation are exact. Even if the assumed statistics
are not exact the diagnostic can still provide useful information about the
true observation error statistics . Further
limitations include the use of an ergodic assumption in order to obtain
sufficient samples and the assumption that the observation
operator is linear .
One further issue is that the standard diagnostic is derived assuming that
the analysis is calculated using minimum variance linear statistical
estimation. If local ensemble DA is used to determine the analysis, the
diagnostic does not result in a correct estimate of the observation
uncertainties. However, by using a modified version of the diagnostic some of
the observation error statistics may be estimated. It is possible to estimate
the error correlations between two observations if the observation operator
that determines the model equivalent of observation yi acts only
on states that have been updated using the observation yj. Since we use a LETKF assimilation scheme in this study, we
must take this into account when estimating observation error statistics for the WLOs.
Methodology
In this article we estimate the observation error statistics for SAR WLOs
that are assimilated using a LETKF into the LISFLOOD-FP 2-D hydrodynamic
model. This study makes use of the observation, model and assimilation system
described in . We direct the reader to this
reference, and references therein (particularly ),
for a thorough description of the derivation of WLOs and
the assimilation design. Here we summarize the methodology and provide a
description of the data used specifically in this study.
Derivation of WLOs
The original observations used in the deviation of WLOs are obtained using
SAR which observes the surface backscatter. In a SAR image flood water
appears dark so long as the surface water turbulence is insignificant.
Therefore, to obtain flood extent, the pixels in a SAR image are grouped into
homogeneous regions. A mean backscatter value is calculated for each region
and if this value is below a given threshold, the region is classified as
flooded. The threshold is determined by using training data from “flood” and
“non-flood” regions. This initial estimate of flood extent is then refined
by, for example, (1) correcting for any high backscatter that is a result of vegetation
either within the flooded region or at the flood edge; (2) correcting for high
backscatter near flooded areas that is a result of water with a rough
surface; (3) performing a “nearest neighbour” check, where any local flood height
that is significantly larger than those nearby is reclassified as non-flooded.
To provide the WLOs the refined flood extent is intersected with high-resolution digital elevation model (DEM). In order to improve the accuracy
of the WLOs, they are only calculated if the slope in the DEM is sufficiently
shallow. A further refinement takes into account, for example, the emergent
vegetation at the flood edge.
The WLO derivation process results in a large number of WLOs that exist in
clusters. It is expected that many of the observations in a cluster will be
highly correlated and hence not contribute independent information. At this
stage in the processing, thin the WLOs to reduce spatial
correlation. However, we postpone this step until after the quality control
procedures for the data assimilation have been performed.
Model and data assimilation
The observations are assimilated into a 75 m resolution LISFLOOD-FP flood
simulation model using a LETKF . Due to the
formulation with the diagnostic described in Sect. , the
localization in the LETKF is set in standard 2-D Euclidean space rather than
the physically based distance along the river channel described in
, which would require a further adaptation of
the diagnostic calculation. The localization radius is set using a compactly
supported fifth-order piecewise rational function Eq. 4.10
with length scale 20 km.
To compare the modelled field with the observed quantity it is necessary to
define an observation operator that maps from model to observation space. In
this study we use the “nearest wet pixel” approach described in
. The mapping in the nearest wet pixel approach
is dependent on the inundation status at the model location. If at an
observation location the model is flooded, the model equivalent of the
observation is simply the water level predicted by the model. However, if the
model is dry at the observation location the model equivalent of the
observation is taken to be the model water level at the wet pixel nearest to
the observation location.
Quality control and data thinning
Data assimilation techniques can lose accuracy if presented with an
observation that is grossly inconsistent with the model state
. Thus, before being assimilated, the WLOs are
subjected to several quality control (QC) protocols according to the physical
characteristics of the terrain and land cover. An additional background check
is performed where observations that result in anomalous
observation-minus-background residuals are discarded. The QC procedures
result in dense cluster of discontinuous observations in which both the
observations and their errors may be highly correlated. A direct assimilation
of this dense dataset would lead to an analysis biased towards the
observations and, for covariance-evolving methods (e.g. ensemble Kalman
filters), an over-reduced posterior covariance and unstable long-term
forecast/assimilation cycles. Thus, to reduce the number of correlated
observations and to avoid dealing with the spatial correlation in the
assimilation, the current approach is to further thin the data (as is
standard in other assimilation applications such as NWP and oceanography;
). The applied thinning, as described in ,
uses a top down clustering approach in which principal component analysis is
used to select observations that have the highest information content. The
spatial autocorrelation of the resulting observations is calculated, and if
any significant correlation exists the thinning procedure is applied
iteratively until no significant correlation remains. Typically the thinned
dataset contains approximately 1 % of the pre-thinned observations. The
measured standard deviation for the thinned dataset can be calculated by
fitting a plane by linear regression to the WLOs. The variance of the
difference between the WLO and planar surface can be used as an estimate of
the observation error variance. This approach is considered adequate for this
case study as the floodplain in the downstream observed areas is reasonably flat.
Potential observation error sources
In data assimilation the observation uncertainty has contributions from both
measurement errors and representation errors. The representation error arises
due to the difference between an actual observation and the modelled
representation of an observation; this difference can be a result of the following:
Pre-processing/QC errors are errors introduced during the observation
pre-processing or quality control procedures.
Observation operator errors are errors that arise due to approximations
in the mapping between model and observation space.
Errors due to unresolved scales and processes are errors that result
from the mismatch between the scales represented in the model field and the observations.
For the WLOs it is clear that a pre-processing error will exist as there is
potential for errors to be introduced in the derivation of the WLOs. For
example if the water surface is rough it may be assumed that the pixel is
dry; as a result the flood extent would be incorrect and hence an error would
be introduced in the WLO. For nearby pixels it is possible that there will be
similar errors in the derivation process, thereby introducing correlated
observation errors. The procedures in provide an estimated
standard deviation for the WLO pre-processing error and thin the data to
ensure that the pre-processing error is uncorrelated. However, we note that
in this study we use a denser dataset than is typically produced. Therefore,
there is potential for some correlated pre-processing error to remain.
A potential source of correlated error for WLOs is the observation operator
error. As described in Sect. the observation operator
uses the “nearest wet pixel” approach. For observations in locations where
the model is flooded it is expected that there is minimal error in the
observation operator (since the corresponding water level is predicated
directly by the model). However, if the observation location does not
coincide with a flooded model pixel it is necessary to find the nearest wet
pixel in the model. It is possible that in locating the nearest wet pixel and
extrapolating information we introduce correlated error.
The error due to unresolved scales and processes is also a possible source of
observation error correlations. Although in this case the model is of
relatively high resolution compared to the observation resolution, there are
still scales that are unresolved. Previous studies that have considered these
scale mismatch errors have found that they are typically correlated .
Calculation of WLO error statistics
(a) Flood model domain where the colour bar denotes the
height in metres and (b) position of SAR WLOs on OSGB 1936 British
National Grid projection; coordinates in metres. For (b) the line
denotes the west/east domain split discussed in Sect. ,
crosses: 27–29 November, circles: 30 November and 1 December, squares: 2 and
4 December.
We estimate observation uncertainties for observations from a real flood
event that occurred in West England on an area of the lower
Severn and Avon rivers in November 2012 (Fig. a). The WLOs
were extracted from a sequence of seven satellite SAR observations (acquired
by the COSMO-SkyMed constellation) using the method described in
. During the flood event the WLOs are available daily for the
period 27 November to 4 December 2012 (with the exception
of 3 December). Observations on the first day illustrate the flood
levels just before the flood peak in the Severn. On 30 November the
river went back in bank; however, a substantial amount of water remained on
the floodplain see Fig. 2 in.
Before being assimilated, the WLOs are subject to the QC and thinning
procedures described in Sect. . When used in previous studies
such as the dataset has been thinned to a
separation distance of 250 m, at which the observation errors are assumed
uncorrelated. However, in this article a denser observation set (although
still sparse) with thinning distance of 125 m is used, in which some spatial
correlation should remain. The location of the observations is plotted in Fig. b.
We apply the diagnostic of to the
observation-minus-background and observation-minus-analysis residuals
resulting from the flood assimilation. We first use all available data to
calculate the average horizontal error variance and correlations. We then
consider if the observations of the flood on the Severn are similar to the
error statistics for the Avon. Finally we consider if the error statistics
vary for different periods of the flood. For all cases the observation error
correlations are calculated at a 1 km bin spacing. As we use an LETKF we must
use a modified form of the diagnostic (see Sect. 2). As a result we are not
able to calculate observation error correlations for observation pairs with a
separation distance greater than 19 km. When evaluating the correlations we
assume that they become insignificant when they drop below 0.2 .
For this assimilation system we assume that the ensemble background error
covariance matrix gives a reasonable estimate of the true background error
statistics. The assumed standard deviation for the WLOs is 59 cm; this is
calculated as described in Sect. . The value accounts only for
the preprocessing error, and not for any error introduced by the
approximations in the observation operator or scale mismatch errors and,
therefore, may be an underestimate of the true error standard deviation.
As is typical for most DA systems, the observation errors are assumed
uncorrelated. With these assumed error statistics the theoretical work of
suggests that the observation error statistics estimated
using the diagnostic will have the following:
an underestimated standard deviation
an underestimated correlation length scale.
Therefore, we would expect the true standard deviations and length scales to
be larger than those we estimate using the diagnostic.
ResultsAverage observation error statistics
We first estimate average horizontal error covariances across the entire
domain for the duration of the flood event. We plot in Fig.
the estimated correlation, along with the number of samples used, for the WLOs.
Estimated SAR WLO error correlations (black line) and number of
samples (bars) used for the calculation. Estimated error standard deviation
is 54 cm.
The estimated statistics give a standard deviation of 54 cm. This is slightly
lower that the assumed error standard deviation of 59 cm. Following the theory
of we expect the estimated standard deviation to be an
underestimate of the true observation error standard deviation, and hence the
results suggest that the assumed standard deviation is likely set at the correct level.
Our results show that the correlations become insignificant (< 0.2) at
approximately 8 km, but there is some unexpected behaviour before 8 km. The
correlations drop smoothly between 0 and 4 km then increase again up to 6 km before
dropping off. This behaviour is seen for a variety of different binning
widths (not shown). We investigate the cause of this “local maximum” in the
estimated correlations in Sects. and . In
general we find that the correlation distance is much longer than the
thinning distance of 125 m, which was chosen to try to ensure that the
observation errors are uncorrelated. Furthermore, theoretical results of
suggest that, with this design of assimilation experiment,
the correlation length scales will be underestimated.
Correlations in different parts of the domain
It is possible that the local maximum in the correlations is a result of
observations on different tributaries of the river. To test this hypothesis
we split the domain in two (as shown in Fig. ): the western
domain covering the river Severn and eastern domain covering the river Avon.
We plot the estimated correlations, along with the number of samples used for
the SAR WLOs, for the western part of the domain in Fig.
and for the eastern part of the domain in Fig. . We note
that there are fewer observations in the eastern domain. This results in
fewer available samples for the calculation in Eq. ()
and hence the results are subject to greater sampling error.
Estimated SAR WLO error correlations (black line) and number of
samples (bars) used (bin width = 1 km), west domain. Estimated error
standard deviation is 58 cm.
Estimated SAR WLO error correlations (black line) and number of
samples (bars) used (bin width = 1 km), east domain. Estimated error
standard deviation is 43 cm.
From Figs. and we see that the “local
maximum” in the correlations is still present in both parts of the domain. In
the eastern domain it is very pronounced. This suggests that the cause of the
increase in correlations between 4 and 6 km is not observations on different
tributaries of the river.
Correlations at different times
We next consider if the correlation structure changes over time. We plot in
Figs. , and the
correlations calculated for the first three days, the second two days and the
final two days respectively. At the beginning of the flood period, the
observations have similar standard deviations to those estimated for the
entire flood event; however, the correlation length scale is short, approximately 2 km.
Estimated SAR WLO error correlations (black line) and number of
samples (bars) used (bin width = 1 km), 27–29 November. Estimated error
standard deviation is 53 cm.
Estimated SAR WLO error correlations (black line) and number of
samples (bars) used (bin width = 1 km), 30 November and 1 December.
Estimated error standard deviation is 43 cm.
During the middle of the flood event the observation error standard deviation
decreases and the correlation length scale increases slightly. For the final
two days the river is back in bank; for this period the standard deviation is
largest, as is the correlation length scale, which is approximately 8 km. It
is also in this final period where the “local maximum” appears in the correlations.
Figure shows the estimated error statistics for the
recession stages for the flood. During this period a high proportion of the
observations were in areas which remained flooded but were disconnected from
the main river flow. For this same sequence of SAR overpasses
showed that the assimilation of the last three
overpasses was still able to exploit the background ensemble covariances to
pass some of the information from these WLOs to the main flow. However, two
effects became evident: (a) the assimilation increments were of a smaller
magnitude in these last stages, and (b) the corrections to the flow in these
last stages were gradually more short-lived. This was a result of the reduced
information content in these WLOs regarding the inflow errors upstream, which
in the end control the flood and flow evolution. Here the
diagnostic has been able to identify a corresponding anomalous structure in
the WLO errors at these last stages. The correlation structure shown in
Fig. indicates that apart from the longer correlation errors,
which can be expected from the smoother flood dynamics at the end of the
flood, an increase in the correlation appears at ∼ 6 km. The increasing
disconnection of the WLOs in the flood plain from the main flow appears to be
the cause for the local maximum in the estimated correlation structure.
However, further work is required to determine why the “local maximum” in the
estimated correlation function appears at 6 km.
Estimated SAR WLO error correlations (black line) and number of
samples (bars) used (bin width = 1 km), 2 and 4 December. Estimated error
standard deviation is 57 cm.
Conclusions
We have shown that the diagnostic is a useful tool to
identify the error covariance in WLOs from satellite SAR. Further, the
diagnostic has been able, in the case study, to isolate an unexpected anomaly
in the correlation structure, pointing to the applicability limits of the
satellite WLOs in the flood plain in the recession stages of the flood. The
diagnostic has been useful in this study for highlighting anomalous data.
Given its low-cost calculation, we propose it be customarily calculated in
flood forecasts and hindcast analyses to support the understanding of the
observation errors and to support QC protocols for selection of adequate
observations. However, due to the dependence of the observation error on the
choice of observation operator and model resolution, results will differ for
each individual user. Therefore, further study may be required to understand
how the diagnostic results can best support QC protocols.
The data used in this study are available in .
JW, JG-P and DM prepared the data and ran the experiments.
JW and JG-P analysed the results and drafted the manuscript. DM, SD and
NN contributed to the discussion and manuscript editing.
The authors declare that they have no conflict of interest.
Acknowledgements
Joanne A. Waller, Nancy K. Nichols and Sarah L. Dance were supported in part by
UK NERC grants NE/K008900/1 (FRANC), NE/N006682/1 (OSCA). Joanne A. Waller and
Sarah L. Dance received additional support from UK EPSRC grant EP/P002331/1 (DARE).
Nancy K. Nichols was also supported by the UK NERC National Centre for Earth
Observation (NCEO). Javier García-Pintado, David C. Mason and Sarah L. Dance were
supported by UK NERC grants NE/1005242/1 (DEMON) and NE/K00896X/1 (SINATRA).
The data used in this study may be obtained on request, subject to licensing
conditions, by contacting the corresponding author.
Edited by: Florian Pappenberger
Reviewed by: two anonymous referees
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