Technical ReportPDF Available

Abstract and Figures

Triple collocation is a method that is now widely used to characterize systematic biases and random errors in in-situ measurements, satellite observations and model fields. It attempts to segregate the measurement uncertainties, geophysical, spatial and temporal representation and sampling differences in the different data sets by an objective method.
Content may be subject to copyright.
Document NWPSAF-KN-TR-021
Version 1.0
06-07-2012
Triple collocation
Jur Vogelzang and Ad Stoffelen
KNMI, de Bilt, the Netherlands
NWP SAF Triple collocation Doc ID : NWPSAF-KN-TR-021
Version : 1.0
Date : 06-07-2012
2
Triple collocation
Jur Vogelzang and Ad Stoffelen
KNMI, De Bilt, The Netherlands
This documentation was developed within the context of the EUMETSAT Satellite Application
Facility on Numerical Weather Prediction (NWP SAF), under the Cooperation Agreement dated
16 December, 2003, between EUMETSAT and the Met Office, UK, by one or more partners
within the NWP SAF. The partners in the NWP SAF are the Met Office, ECMWF, KNMI and
Météo France.
Copyright 2012, EUMETSAT, All Rights Reserved.
Change record
Version Date Author Approved Remarks
0.1 July 2011 Jur Vogelzang First draft
0.2 Dec 2011 Ad Stoffelen Second draft
0.3 Jan 2012 Jur Vogelzang Third draft
1.0 July 2012 Jur Vogelzang Ad Stoffelen First public release
NWP SAF Triple collocation Doc ID : NWPSAF-KN-TR-021
Version : 1.0
Date : 06-07-2012
1
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Error model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.3 Calibration coefficients and error variances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.4 Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.5 Error variances for calibrated data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.6 Resume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3 Representation errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.1 Resolution and representation errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.2 Calculation methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
4 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
NWP SAF Triple collocation Doc ID : NWPSAF-KN-TR-021
Version : 1.0
Date : 06-07-2012
2
NWP SAF Triple collocation Doc ID : NWPSAF-KN-TR-021
Version : 1.0
Date : 06-07-2012
3
1 Introduction
Triple collocation is a method that is now widely used to characterize systematic biases and random errors
in in-situ measurements, satellite observations and model fields. It attempts to segregate the measurement
uncertainties, geophysical, spatial and temporal representation and sampling differences in the different
data sets by an objective method.
In scientific literature often dual comparisons are provided for validation, verification and calibration.
However, implicit assumptions are made that limit the accuracy of dual comparisons. A frequent and
often biased assumption is that all errors are due to the system that is being tested against a reference
system, that is in turn assumed perfect, but Stoffelen [1998] also refers to biases associated with
regression and with error distributions. Problems with dual comparison, such as satellite data verification
against in-situ measurements, may be caused by differences in:
temporal and spatial representation (instantaneous versus hourly mean and for example local, average
over a satellite footprint or NWP grid volume);
geophysical representation (real winds versus equivalent neutral winds, bulk versus skin SST, etc.);
spatial and temporal sampling (over a whole basin or only at in-situ stations, twice daily or sampled
over full diurnal cycles);
error distributions, including aspects of error amplitude and skewness.
These issues cannot be clearly resolved in dual comparisons, as scatter will be caused simultaneously by
all issues above for both observing systems and there is no clear objective way to assign errors to one or
the other. In other words, dual comparisons are really difficult to comprehend. Stoffelen [1998] formalizes
some of these problems in his section 3 and Appendix A. This basically has driven the discovery of triple
collocation.
In triple collocation, three (ideally) independent data sets are brought together, so three scatter plots can
be made. The plot with the least scatter obviously denotes the two systems that agree most, while the
worst scatter plot, indicates that the excluded and third measurement system is the best performing.
Moreover, triple collocation provides the relative linear calibration (scaling) of the three systems. Also,
assuming known (normal) error distributions and after mutual linear calibration (rescaling) of the
distributions with the errors, matching of the cumulative PDF leads to higher order calibration as well
(known as CDF matching). This is described in the original paper (Stoffelen [1998]) dealing with the
triple collocation method and elaborating it for buoy, scatterometer and NWP wind data sets.
Following Stoffelen [1998], it has been and is being used in wind and stress comparisons [Portabella and
Stoffelen, 2009; Vogelzang et al. , 2011] wave height comparison [Caires and Sterl, 2003; Janssen et al.,
NWP SAF Triple collocation Doc ID : NWPSAF-KN-TR-021
Version : 1.0
Date : 06-07-2012
4
2007], sea surface temperature (SST) [O’Carroll et al., 2008], soil moisture [Scipal et al., 2010];
Delphine et al., 2011], ice drift [Hwang and Lavergne, 2010], precipitation analyses [Roebeling et al.,
2011], etc. Some of these authors consider the measurements as already properly calibrated and perform
only random error estimation.
A limitation of the triple collocation method may be the fact that three independent measurement systems
are needed that must deliver simultaneous collocated measurements. Collocation in space and time may
be hard to achieve and it takes typically one year to gather enough data for successful application of the
method. The method can moreover only be applied at those locations and times where triple collocation
data is available. Usually this is limited by the availability of in-situ observation locations, i.e., similar to
dual in-situ comparisons, and the satellite overpass times, i.e., twice a day for polar satellites in sun-
synchronous orbits.
Another hurdle in application may be that the triple collocation method is felt difficult to comprehend.
Where in dual comparisons often implicit assumptions are made on the error distributions (see above), in
triple collocation explicit assumptions are needed on the random and systematic error distributions, i.e., a
realistic error model needs to be defined and tested. The method has been described in the scientific
literature [Stoffelen, 1998], but such a presentation must necessarily be very concise. Implementation of
the method is tedious, since a more detailed description on text book level is missing. This report is
intended to fill that gap.
This report gives a full description of the triple collocation error model, the derivation of the calibration
coefficients and the measurement error variances, the assumptions needed in the triple collocation
method, and its numerical implementation. Also attention is paid to the role of representation errors and
the scale dependency of measurement errors.
NWP SAF Triple collocation Doc ID : NWPSAF-KN-TR-021
Version : 1.0
Date : 06-07-2012
5
2 Derivation
2.1 Definitions
In this chapter the following definitions hold:
Assume measurements of a quantity
N
α
, denoted as N,...,,i,
i21
=
α
. The first moment of
α
, or its
average, is denoted as and satisfies
α
M
,
N
MN
ii
=
>==<
1
1
αα
α
(2.1)
where <> denotes statistical averaging. If there are also measurements of a quantity
N
β
, the mixed
second moment satisfies
αβ
M
=
>==<
N
iii .
N
M
1
1
βααβ
αβ
(2.2)
In case
α
β
= one obtains the ordinary second moment .
αα
M
The covariance is defined as
αβ
C
(2.3)
.MMMC
βααβαβ
=
In case
α
β
= equation (2.3) yields the variance of
α
, .
2
ααα
σ
=C
2.2 Error model
2.2.1 Calibration and measurement errors
Suppose three measurement systems X, Y, and Z, giving collocated measurements of the same
quantity
),,( zyx
t
. Supposing that system X is the reference system with respect to which systems Y and Z are to
be calibrated. Suppose also that linear calibration is sufficient for the whole range of values under
consideration, and that the reference system is free of bias (i.e., there are no systematic errors or these are
corrected for). The measurements then satisfy
()
()
,
zzz
yyy
x
tabz
taby
tx
ε
ε
++=
++=
+=
(2.4)
where is the common part of the signal (sometimes referred to as “truth”) and
tzyx ,,, =
α
ε
α
the true
random error in each measurement. These random measurement error components are assumed unbiased,
NWP SAF Triple collocation Doc ID : NWPSAF-KN-TR-021
Version : 1.0
Date : 06-07-2012
6
.z,y,x, =>=<
α
ε
α
0 (2.5)
It is assumed that the error variances are constant over the whole range of values under consideration, so
. (2.6) z,y,x, =>=<
ασε
αα
22
Further, it is assumed that the errors are independent of the common signal t, so
,z,y,x,tt =>><>=<<
α
ε
ε
αα
(2.7)
which yields zero due to (2.5).
2.2.2 Representation error
The true or calibrated measurement errors are also assumed uncorrelated,
,z,y,x,, =>=<
β
α
ε
ε
βα
0 (2.8)
unless common representation errors play a role. Suppose that system Z has a much coarser resolution
than system Y and that system Y has coarser resolution than system X. High resolution signal that is
common to X and Y will not be detectable for system Z and therefore be regarded as error. It can be
represented as a correlated error between X and Y, so
, (2.9)
r
yx 2
>=<
εε
with 2
r
the variance of the representation error, i.e., the signal in X and Y that is not detected by Z. The
other error correlations are zero according to (2.8). It is noted that coarse resolution may refer to spatial or
temporal resolution, but there may also be geophysical representation issues that make systems X and Y
look more alike while system Z lacks certain geophysical sensitivity. For example, for SST two systems
X and Y could measure skin temperatures and a third system Z bulk temperature. X and Y then measure
signal that is lacking in Z, and therefore <
ε
x
ε
y > 0.
Note that with this procedure, the error model (2.4) gives the measurement error variances at the scale of
the coarse observations made by system Z. It will be shown in section 2.4 how the measurement error
variances at the scale of the intermediate resolution system Y can be obtained as well. In chapter 3 the
concept of scale-dependent errors will be elaborated further, together with procedures to obtain the
representation error.
2.3 Calibration coefficients and error variances
Forming the first statistical moments of (2.4) yields
NWP SAF Triple collocation Doc ID : NWPSAF-KN-TR-021
Version : 1.0
Date : 06-07-2012
7
)
(
()
.
><+><+=
><+><+=
><+>=<
zzzz
yyyy
xx
tabM
tabM
tM
ε
ε
(2.10)
The first moments of the random measurement errors all equal zero by (2.5). The first equation of (2.10)
therefore yields and can be used to eliminate
>=< tMx>
<
t from the others. This results in
(2.11)
.
MaMb
MaMb
xzzz
xyyy
=
=
The ordinary second-order moments of (2.4) are
(2.12)
.
222
222
2
222222
222222
22
><+><+><+><++><=
><+><+><+><++><=
><+><+>=<
zzzzzzzzzzzzz
yyyyyyyyyyyyy
xxxx
abtatbaabtaM
abtatbaabtaM
ttM
εεε
εεε
εε
This can be simplified by application of (2.4)-(2.9) and by using >
=
<tM x to
(2.13)
,
2
2
22222
22222
22
xzzzzzzzz
xyyyyyyyy
xxx
MbaabtaM
MbaabtaM
tM
+++><=
+++><=
+>=<
σ
σ
σ
Now and can be eliminated from (2.13) using (2.11). A little algebra and introduction of the
covariances (2.3) results in
y
bz
b
(2.14)
(
()
.
2222
2222
222
zxzzz
yxyyy
xxxx
MtaC
MtaC
MtC
σ
σ
σ
+><=
+><=
+>=<
)
In the same way the mixed second order moments of (2.4) read
(2.15)
.
2
2
2
><+><+><+><+><+><=
><+><+><+
+><++><+><+><+><=
><+><+><+><+><+><=
xzzxzxzzzzzzx
zyzyyyzyyz
zzyzyyzzzyzyzyyz
yxyxyxyyyyyxy
abtatatbtaM aaabtaa
abbbtbataatbataaM abtatatbtaM
εεεεε
εεεε
εε
εεεεε
This can be simplified using (2.4)-(2.9) and >
=
<tM x to
(2.16)
,
2
2
22
xzzzx
zyxyzxzyzyyz
yxyyxy
MbtaM
bbMbaMbataaM
raMbtaM
+><=
+++><=
++><=
where, according to (2.9), we assumed that system Z has coarser resolution than systems X and Y. Again,
NWP SAF Triple collocation Doc ID : NWPSAF-KN-TR-021
Version : 1.0
Date : 06-07-2012
8
y
b and can be eliminated from (2.16) using (2.11). A little algebra and introduction of the covariances
(2.3) results in
z
b
(
)
()
()
.
22
22
222
xzzx
xzyyz
xyxy
MtaC
MtaaC
rMtaC
><=
><=
+><=
(2.17)
Eliminating and from the second equation in (2.17) using the first and third, respectively, yields
y
az
a
.
222
yz
xyzx
xC
CC
rMt =+>< (2.18)
Substituting (2.18) back into (2.17) yields
., 2
raC
C
a
C
C
a
yxy
yz
z
zx
yz
y
== (2.19)
Substituting (2.18) and (2.19) into (2.14) and solving for the error variances yields
()
(
)
()
.,, 2
2
2
2
2
2
raC
CC
C
C
CraC
C
C
raCC
C
yxy
zxyz
zzz
zx
yzyxy
yyy
yz
yxyzx
xxx
=
=
=
σσσ
(2.20)
This completes the derivation of the calibration coefficients and measurement error variances of the
calibrated data with the triple collocation method. However, the actual implementation of the method
makes use of a slightly different formulation. Moreover, there are two other issues that receive further
attention:
1. If the representation error plays a role, the error variances are with respect to the system with coarsest
resolution;
2. The representation error variance is valid for the calibrated data, not for the raw data. However, note
that ay r 2 is a scaled representation error that may be determined from the raw Y data.
These issues will be addressed in the next sections.
2.5 Analysis of the calibrated data
The measurement error variances for the calibrated data are obtained from error model (2.4). Suppose that
one has an estimate of the calibration coefficients and
α
a yxb ,,
=
α
α. The calibrated measurements x,
y, and z are given by
NWP SAF Triple collocation Doc ID : NWPSAF-KN-TR-021
Version : 1.0
Date : 06-07-2012
9
.
z
z
y
y
abz
z
a
by
y
xx
=
=
=
(2.21)
Note that no errors are involved in (2.21), because it just gives the inverse of the linear calibration as it is
performed on the raw data. Now the triple collocation error model (2.4) can be applied to the calibrated
data. The model is solved the same way as in section 2.3 under the same assumptions. The results are
,, xzzzxyyy MaMbMaMb ==
,, 2
raC
C
a
C
C
a
yxy
yz
z
zx
yz
y
== (2.22)
()
(
)
()
,,, 2
2
2
2
2
2
raC
CC
C
C
CraC
C
C
raCC
C
yxy
zxyz
zzz
zx
yzyxy
yyy
yz
yxyzx
xxx
=
=
=
σσσ
where the bar indicates that all quantities are for calibrated values. Note that the representation error has
been defined with respect to the calibrated data.
If the values of the calibration coefficients and
α
a yxb ,,
=
α
α are correct, then the measurements x, y,
and z are properly calibrated and we must obtain
.
1
0
==
==
zy
zy aa
bb (2.23)
Using (2.22) the relation 1== zy aa leads to
(
)
2
rCCC xyyzzx == , and the error variances for
calibrated data reduce to
.,, 222222 rCCrCCrCC xyzzzxyyyyxyxxx +=+=+=
σσσ
(2.24)
Equation (2.24) is valid with respect to the system with coarsest resolution, say system Z. The equations
in this section can be solved iteratively on a computer. More details are given in chapter 4.
2.4 Resolution
If all three measurement systems have roughly the same resolution then the representation error
and the equations for calibration coefficients and measurement error variances further simplify. When the
representation error plays a role, then the results of the previous section apply to the system with coarsest
0
2=r
NWP SAF Triple collocation Doc ID : NWPSAF-KN-TR-021
Version : 1.0
Date : 06-07-2012
10
resolution, because the representation error has been taken into account as an error correlation between
the two systems with higher resolution.
Suppose that system Z has coarsest resolution. It is easy to obtain the error variances with respect to the
system with intermediate resolution: just subtract the representation error variance r 2 from and ,
since it represents part of the common resolved signal in X and Y. System Z does not resolve this part of
the signal and error variance r
2
x
σ2
y
σ
2 should be added to . Now the signal detected by systems X and Y but
not by Z is counted as measurement error of system Z (lack of resolution) and as signal of X and Y.
Denoting the error standard deviation at intermediate resolution by
2
z
σ
zyx ,,,
ˆ
=
α
σ
α, one has
.
ˆ
ˆ
ˆ
22
2
2
rCC
CC
CC
xz
zzz
xyyy
y
xyxx
x
+=
=
=
σ
σ
σ
(2.25)
Suppose now that system X has much finer resolution than system Y. Equations (2.25) then gives the
measurement error variances with respect to the resolution of Y. It is not possible to say something on the
measurement error variances at the finest resolution of system X, unless additional assumptions are made
on the measurement error distributions. For example, if X is a calibrated local in-situ measurement
system, then the measurement error may be known. This could be used to compute a temporal or spatial
representation error for Y and Z, such that the errors of X, Y and Z in representing a local in-situ
measurement may be estimated.
For our scatterometer application, system X corresponds to moored buoys, system Y to the scatterometer,
and system Z to the NWP background. The triple collocation method enables us to calculate error
variances at the scales resolved by the NWP background and at the scale of the scatterometer, which
scales have our main interest.
2.6 Resume
The triple collocation method requires the following assumptions:
1. Linear calibration is sufficient over the whole range of measurement values;
2. The reference measurement values are unbiased and calibrated;
3. The random measurement errors have constant variance over the whole range of calibrated
measurement values;
4. The measurement errors are uncorrelated with each other (except for representation errors);
5. The random measurement errors are uncorrelated with the geophysical signal.
NWP SAF Triple collocation Doc ID : NWPSAF-KN-TR-021
Version : 1.0
Date : 06-07-2012
11
Under these assumptions the calibration coefficients and measurement error variances are given in terms
of the first and second order statistical moments as
,MaMb,MaMb xzzzxyyy
==
,, 2
raC
C
a
C
C
a
yxy
yz
z
zx
yz
y
==
()
(
)
()
,,, 2
2
2
2
2
2
raC
CC
C
C
CraC
C
C
raCC
C
yxy
zxyz
zzz
zx
yzyxy
yyy
yz
yxyzx
xxx
=
=
=
σσσ
where system Z has coarser resolution than the other systems, resulting in a representation error of 2
r
.
These results hold with respect to the resolution of system Z and are valid for the uncalibrated data.
The error variances with respect to the system with medium resolution, which may be either X or Y, is
obtained by subtracting the representation error from and , and adding it to . Within the triple
collocation method it is not possible to retrieve the measurement error variances with respect to the
system with finest resolution (if any), unless additional assumptions on the error distributions are made.
2
x
σ2
y
σ2
z
σ
The error variances with respect to the calibrated data are found by implicitly solving the triple
collocation error model. When the calibration coefficients and
α
a yxb ,,
=
α
α are correctly estimated,
the error variances are
,,, 222222 rCCrCCrCC xyzzzxyyyyxyxxx +=+=+=
σσσ
with the bars indicating that the corresponding quantity is for calibrated data. Note that the error variances
with respect to the system of medium resolution (underlined) are independent of the representation error
for the two systems with finer resolution (X and Y)
.,, 2
222 rCCCCCC xzzzzxyyyyxyxxx +===
σσσ
NWP SAF Triple collocation Doc ID : NWPSAF-KN-TR-021
Version : 1.0
Date : 06-07-2012
12
NWP SAF Triple collocation Doc ID : NWPSAF-KN-TR-021
Version : 1.0
Date : 06-07-2012
13
3 Representation errors
3.1 Resolution and representation errors
Geophysical data originate from a variety of sources: in-situ measurements, satellite observations, and
model predictions. Each type has its own geophysical, spatial and temporal characteristics: in-situ data are
representative of a point whereas satellite and model data give a value that is representative for some area.
Geophysical in-situ data are often presented as time averaged (for instance the wind speed and direction
which is commonly given as average over 10 minutes or 1 hour), while satellite data are almost
instantaneous. NWP models compute a new time step every 10 or 20 minutes and will represent temporal
scales at about 5-7 times this time step [Skamarock, 2004].
For wind calibration, some authors use the assumption of frozen turbulence, called Taylor’s hypothesis
[Taylor, 1921; Richardson, 1926], to manipulate (in-situ) data sets in order to reduce spatial resolution to
match a comparison data set. We note that this is not a very accurate method of obtaining a certain spatial
resolution. At a wind speed of 8 m/s, one would thus average 3600 s or one hour to match a satellite
footprint of ~30 km. When averaging over one hour one would however obtain an effective resolution of
~15 km at 4 m/s, but only ~60 km at 16 m/s. Extreme high winds thus would be severely smoothed, while
low winds would not be smoothed much. Spatial and temporal smoothing of a field change the PDF,
because extreme high values will become less frequent. Therefore, calibration or regression with respect
to a temporally smoothed data set will tend to follow the extreme values and be representative for a
coarser resolution than anticipated. Temporal smoothing of transient fields in order to reduce spatial
resolution is thus prone to rather complex error characteristics. Following Kolmogorov’s hypothesis
[Kolmogorov, 1941], it appears more attractive to model small-scale variability through a
representativeness error [Stoffelen, 1998].
Spatial satellite and model data are produced at different resolutions. With both temporal and spatial
resolution the true resolution of a measurement system is meant, i.e., the size of the smallest detail
discernible with that system in resp. time and space. In most cases this resolution is coarser than the grid
size on which the product is presented.
As noted earlier, coarse resolution may refer to spatial or temporal resolution as described above, but
there may also be geophysical representation issues that make systems X and Y look more alike while
where system Z lacks certain geophysical sensitivity. For example, for SST two systems X and Y could
measure skin temperatures and a third system Z the bulk temperature. X and Y then measure signal that is
lacking in Z, and therefore in triple collocation <
ε
x
ε
y > 0. Another example may exist in soil wetness
NWP SAF Triple collocation Doc ID : NWPSAF-KN-TR-021
Version : 1.0
Date : 06-07-2012
14
from a NWP model, where the geophysical representation in the NWP model lacks local soil
characteristics.
The difference in resolution between various data types becomes important when intercomparing data or
when assimilating observed data into numerical prediction models. It gives rise to representation errors,
also referred to as error of representativeness. The term error may be a bit misleading, because the
representation error is not a real error but originates from merging data at different resolution. Moreover,
its role depends on the point of view that one takes.
Suppose, again, three measurement systems X, Y, and Z, and three resolution classes: coarse or low
resolution (L), medium resolution (M), and fine or high resolution (H). Suppose further that the resolution
class is given as subscript to the system, and that we have the situation {XH, YM, ZL}. For the case of
wind scatterometry this corresponds to XH being the buoy measurements, YM the scatterometer
observations, and ZL the NWP model background. The highest resolution scale seen by system XH will, of
course, be missed by systems YM and ZL. Medium scales will be seen by systems XH and YM, but also be
missed by system ZL. From the point of view of XH the other two systems have a hopelessly blurred view
of reality. In case of comparison to and verification of system YM the high resolution details seen by XH is
unwanted variance and treated as error, while system ZL misses medium resolution details that are
important for the verification of YM. In case of verification of system ZL all medium and high resolution
details is unwanted variance and treated as error. In particular, the medium resolution part resolved by XH
and YM will appear as a correlated error in case of ZL verification or calibration and will add to the error
correlation >< yx
ε
ε
.
The triple collocation method uses the common signal of {XH, YM, ZL} and thus the measurement error
variances are specified at the lowest resolution. It is important to include the representation error variance
r 2 into the error model. Not only does it affect the results (the representation error is of the same order of
magnitude as the measurement errors in the case of wind scatterometry), but it also offers the possibility
to obtain the measurement errors at the medium resolution by subtracting r 2
as common XH and YM
signal variance from and , as discussed in section 2.5. The representation variance r
2
x
σ
2
y
σ
2
should in
this case be added to error since system Z
2
z
σ
L cannot resolve this signal. The system XH measurement
error at medium resolution, x
σ
ˆas specified in (2.25), contains geophysical signal that is resolved by
system XH, but not by the other systems, denoted rx 2
. While error x
σ
ˆ is retrieved from the triple
collocation method, we would need further information on the local measurement errors, xM ,
σ
, to obtain
an estimate of the additional geophysical variance resolved by XH, this is .
22
,
2
ˆxxMx r+=
σσ
The argument can be repeated for other resolutions of the three measurement systems. For instance,
O’Connor et al. [2008] discuss the case {XH, YM, ZM} with XH in-situ sea surface temperature
NWP SAF Triple collocation Doc ID : NWPSAF-KN-TR-021
Version : 1.0
Date : 06-07-2012
15
measurements and YM and ZM satellite observations. In this case 0>=
<
yx
as there is no common
temperature signal of YM and XH not resolved by ZM. Table 3.1 summarizes a number of cases. All other
cases follow from interchanging the roles of X, Y, and Z, and the fact that representation errors do not
play a role when all three systems have the same resolution.
Resolution Representation error
X Y Z >
<
yx
ε
ε
>
<
zy
ε
ε
>
<
xz
ε
ε
H H M/L 2
r
0 0
H M L 2
r
0 0
H M M 0 0 0
H/M L L 0 0 0
Table 3.1 Contribution of the representation error 2
r
to the error covariances for various resolution cases
3.2 Calculation methods
In the literature several methods are described for calculating representation errors. Which method can be
used depends, of course, on data availability.
Stoffelen [1998] and Portabella and Stoffelen [2009] obtain the representation error by considering the
variance spectra of NWP background and scatterometer observations. They assume a scatterometer wind
spectrum of [Lindborg, 1999]. The constant is determined by requiring that the scatterometer
spectrum equals the NWP spectrum at a certain spatial frequency , corresponding to a spatial scale of
the order of 1000 km. Then the representation error equals the difference between the two spectra
integrated from to the highest spatial frequency , corresponding to the smallest scales in the
spectrum.
35 /
ck c
sep
k
sep
kmax
k
Vogelzang et al. [2011] refine this approach by using observed scatterometer wind spectra rather than a
theoretical form. This is applied to various operational scatterometer wind products, and consistent
results are obtained for corresponding to a spatial scale of 800 km.
35 /
k
sep
k
Janssen et al. [2007] do not consider representation errors, but extend the number of datasets to five by
using various NWP model results (first guess, analysis, and hindcast) that contain different levels of
assimilation. The additional sets give rise to additional equations, allowing one to solve for six error
correlations. However, error correlations between satellite observations (from altimeter) and in-situ
(buoy) measurements are neglected, so basically the representation error is assumed to be zero.
NWP SAF Triple collocation Doc ID : NWPSAF-KN-TR-021
Version : 1.0
Date : 06-07-2012
16
O’Connor et al. [2008] discuss the effect of representation errors on the error correlations, but give no
estimate for the magnitude of the representation errors.
NWP SAF Triple collocation Doc ID : NWPSAF-KN-TR-021
Version : 1.0
Date : 06-07-2012
17
4 Implementation
Calculation of the first and second moments needed for the triple collocation model must be done with
some care, since notably the second moments are sensitive for outliers in the data set. Outliers may be
caused by anomalous geophysical conditions. For example, for scatterometers one may think of excessive
sub-WVC wind variability, e.g., at the passage of a front or caused by a large convective downburst, or,
particularly for Ku-band scatterometers, blurring of the ocean wind signature by rain. Most of these
conditions are removed in a quality control step, but some outlier may remain due to a non-perfect
probability of detection. Outliers also exist in in-situ data for various reasons, e.g., due to salting or icing
conditions. To avoid problems caused by outliers it is better to filter them out, for instance by rejecting all
data that lie away more than a specified times the standard deviation from the expected value. However,
whether or not some measurement must be classified as outlier depends not only on the selection
criterion, but also on the calibration, since the calibration determines the expected value of some
measurement relative to the reference measurement value.
It is clear that this problem is best solved in an iterative scheme as shown in figure 4.1. Adopting the
formalism of section 2.5 one starts with an initial estimate of the calibration coefficients, , ,
, and , as well as for the expected distances from the calibration line, , and . For
scatterometer applications good results are obtained with the values
(0)
y
a(0)
z
a
)0(
y
b)0(
z
b)0(
xy
d)0(
yz
d)0(
zx
d
,bb zy 0
(0))0( ==
,aa zy 1
(0)(0) ==
m
9
(0))0((0) === zxyzxy ddd 2s-2
Now the iterative scheme starts to solve the triple collocation model. The next step is to go through all
collocated triplets, calculate the calibrated triplets )( jjj z,y,x , with
j
the triplet number. The triplet is
rejected if any of the conditions
,16 (0)
xy
xy
jd>
δ
,16 (0)
yz
yz
jd>
δ
,16 (0)
zx
zx
jd>
δ
holds (this is called the
σ
4 test, but is it written here in quadratic form). If the triplet is accepted, the f irst
and second moments of the calibrated wind components are updated.
NWP SAF Triple collocation Doc ID : NWPSAF-KN-TR-021
Version : 1.0
Date : 06-07-2012
18
If all triplets have been processed, the values for the error variances and calibration coefficients are
calculated from (2.23). The calibration coefficients are updated according to
Iteration loop
Initialization
Data loop
Calibration
4σ test
Update moments
Update calibration distances
Calculate error variances
Update calibration coefficients
Calculate calibration distances
Calculate re
p
resentation errors
Convergence?
Completed
Figure 4.1 Flow chart for triple collocation algorithm.
NWP SAF Triple collocation Doc ID : NWPSAF-KN-TR-021
Version : 1.0
Date : 06-07-2012
19
,z,y,
a
a
ai
i==
+
α
α
α
α
)(
)1(
,z,y,bbb ii =+=
+
α
ααα
)()1(
with the iteration number. The expected distances to the calibration line are set equal to the sum of the
estimated error covariances
i
22)1(22)1(22)1( ,, xz
i
zxzy
i
yzyx
i
xy ddd
σσσσσσ
+=+=+= +++
For normal unbiased random error distributions, only one in 15,787 data points would fall outside the 4-
sigma range for each of the three differences in the triple collocation data set. Since these data points
would be far away from the diagonal, one might want to check their contribution to the estimated errors
and calibration. In this respect we note that one distance of 4 x
σ
in 15,787 points adds 0.1% to the
variance and thus 0.2% to the estimated standard error, i.e., is likely to be negligible in the context of the
assumptions in the error model used for triple collocation.
Now the next iteration can be started, unless the calibration coefficients have converged.
NWP SAF Triple collocation Doc ID : NWPSAF-KN-TR-021
Version : 1.0
Date : 06-07-2012
20
NWP SAF Triple collocation Doc ID : NWPSAF-KN-TR-021
Version : 1.0
Date : 06-07-2012
21
References
Byongjun, H., and T. Lavergne, 2010,
Triple comparison of OSI SAF low resolution sea ice drift products,
Ocean & Sea Ice SAF Associated & Visiting Scientist Activity Report, CDOP-SG06-VS02
Version v1.3, 28/09/2010, http://saf.met.no/docs/OSISAF_TripleCollocationIceDrift_V1p3.pdf
Caires, S., A. Sterl, 2003,
Validation of ocean wind and wave data using triple collocation,
J. Geophys. Res. 108, 3098-3113, doi:10.1029/2002JC001491
Janssen, P.A.E.M., S. Abdalla, H. Hersbach, and J.-R.Bidlot, 2007,
Error estimates of buoy, satellite, and model wave height data.
J. Atm. Ocean. Tech. 24, 1665-1677.
Kolmogorov, A. N., 1941,
Dissipation of Energy in a Locally Isotropic Turbulence.
Dokl. Akad. Nauk SSSR 32, 141. (English translation in Proc. R. Soc. London A 434: 15, 1991).
Leroux, D., Y. Kerr, and P. Richaume, 2011,
Estimating SMOS error structure using triple collocation,
IGARSS 2011, http://www.slideshare.net/grssieee/estimating-smos-error-structure-using-triple-
collocation.ppt .
Lindborg, E., 1999,
Can the atmospheric kinetic energy spectrum be explained by two-dimensional turbulence?,
J. Fluid Mech. 388, 259–288.
O’Connor, A.G., J.R. Eyre, and R.W. Saunders, 2008,
Three-way error analysis between AATSR, AMSR-E, and in-situ sea surface temperature
observations.
J. Atm. Ocean. Tech. 25, 1197-1207.
Portabella, M. and A. Stoffelen, 2009,
On scatterometer ocean stress.
J. Atm. Ocean. Tech. 26, 368-382.
Richardson, L. F., 1926,
Atmospheric Diffusion Shown on a Distance-Neighbour Graph.
Proc. R. Soc. London A 110, 709.
Roebeling, R., E. Wolters, and H. Leijnse, 2011,
Triple collocation of precipitation retrievals from SEVIRI with gridded rain gauge data and
weather radar observations,
http://www.knmi.nl/research/climate_observations/meetings/presentations/Roebeling_20110526.
pdf.
NWP SAF Triple collocation Doc ID : NWPSAF-KN-TR-021
Version : 1.0
Date : 06-07-2012
22
Scipal, K., W. Dorigo, R. deJeu, 2010,
Triple collocation – a new tool to determine the error structure of global soil moisture products,
Proc. IGARSS 2010, page(s): 4426 - 4429, Doi: 10.1109/IGARSS.2010.5652128 ,
http://publik.tuwien.ac.at/files/PubDat_189396.pdf .
Skamarock, W.C., 2004,
Evaluating mesoscale NWP models using kinetic energy spectra.
Monthly Weather Review 132, 3019-3032.
Stoffelen, A., 1998,
Towards the true near-surface wind speed: Error modeling and calibration using triple
collocation.
J. Geophys. Res. 103C3, 7755-7766.
Taylor, G. I., 1921,
Diffusion by Continuous Movements.
Proc. London Math. Soc. 20, 196.
Vogelzang, J., A. Stoffelen, A. Verhoef, and J. Figa-Saldaña, 2011,
On the quality of high-resolution scatterometer winds.
J. Geophys. Res. 116, C10033, doi:10.1029/2010JC006640.
... For surface soil wetness, verification McColl et al. (2014) have developed the Extended Triple Collocation (ETC) method to estimate the temporal correlation between a model or observation system and the unknown truth. The ETC method makes the same assumptions as the more widely known Triple Collocation (TC; Scipal et al. 2008;Vogelzang and Stoffelen 2012;Zwieback et al. 2012;Dorigo et al. 2010;Yilmaz and Crow 2014;Gruber et al. 2016;Draper et al. 2013) method. ETC and TC require three mutually independent time-series estimates of the same quantity. ...
Technical Report
Full-text available
Accurate soil dryness information is essential for the calculation of accurate fire danger ratings, fire behavior prediction, flood forecasting and landslip warnings. Soil dryness also strongly influences temperatures and heatwave development by controlling the partitioning of net surface radiation into sensible, latent and ground heat fluxes. Rainfall forecasts are crucial for many applications and many studies suggest that soil dryness can significantly influence rainfall. Currently, soil dryness for fire danger prediction in Australia is estimated using very simple water balance models developed in the 1960s that ignore many important factors such as incident solar radiation, soil types, vegetation height and root depth. This work presents a prototype high resolution soil moisture analysis system based around the Joint UK Land Environment System (JULES) land surface model. This prototype system is called the JULES based Australian Soil Moisture INformation (JASMIN) system. The JASMIN system can include data from many sources; such as surface observations of rainfall, temperature, dew-point temperature, wind speed, surface pressure as well as satellite derived measurements of rainfall, surface soil moisture, downward surface short-wave radiation, skin temperature, leaf area index and tree heights. The JASMIN system estimates soil moisture on four soil layers over the top 3 meters of soil, the surface layer has a thickness of 10 cm. The system takes into account the effect of different vegetation types, root depth, stomatal resistance and spatially varying soil texture. The analysis system has a one hour time-step with daily updating. For the surface soil layer, verification against ground based soil moisture observations from the OzNet, CosmOz and OzFlux networks shows that the JASMIN system is significantly more accurate than other soil moisture analysis system used at the Bureau of Meteorology. For the root-zone, the JASMIN system has similar skill to other commonly used soil moisture analysis systems. The Extended Triple Collocation (ETC) verification method also confirms the high skill of the JASMIN system.
... It can help to identify individual relative error structure of in situ, remote sensing, and reanalysis datasets. For the detailed description of TC method, readers are referred to Vogelzang and Stoffelen [66] and Yilmaz et al. [63]. When there are only two SM products, the variances of the two available products will be used to derive, deterministically, the weight coefficients according to Equation (3). ...
Article
Full-text available
The inter-comparison of different soil moisture (SM) products over the Tibetan Plateau (TP) reveals the inconsistency among different SM products, when compared to in situ measurement. It highlights the need to constrain the model simulated SM with the in situ measured data climatology. In this study, the in situ soil moisture networks, combined with the classification of climate zones over the TP, were used to produce the in situ measured SM climatology at the plateau scale. The generated TP scale in situ SM climatology was then used to scale the model-simulated SM data, which was subsequently used to scale the SM satellite observations. The climatology-scaled satellite and model-simulated SM were then blended objectively, by applying the triple collocation and least squares method. The final blended SM can replicate the SM dynamics across different climatic zones, from sub-humid regions to semi-arid and arid regions over the TP. This demonstrates the need to constrain the model-simulated SM estimates with the in situ measurements before their further applications in scaling climatology of SM satellite products.
... Here we will demonstrate the impact of differing spatial representativeness of the data sets by means of differing underlying soil moisture signal components using the covariance notation. Note that this approach is not different to other approaches, where the correlated signal components are considered as cross-correlated random errors in the data sets (Stoffelen, 1998;Vogelzang and Stoffelen, 2012). Two different cases will be distinguished: (i) one point-scale in situ measurement together with two coarse-scale data sets that have a comparable spatial representativeness, and (ii) three data sets with significantly different spatial representativeness. ...
Article
Full-text available
To date, triple collocation (TC) analysis is one of the most important methods for the global-scale evaluation of remotely sensed soil moisture data sets. In this study we review existing implementations of soil moisture TC analysis as well as investigations of the assumptions underlying the method. Different notations that are used to formulate the TC problem are shown to be mathematically identical. While many studies have investigated issues related to possible violations of the underlying assumptions, only few TC modifications have been proposed to mitigate the impact of these violations. Moreover, assumptions, which are often understood as a limitation that is unique to IC analysis are shown to be common also to other conventional performance metrics. Noteworthy advances in TC analysis have been made in the way error estimates are being presented by moving from the investigation of absolute error variance estimates to the investigation of signal-to-noise ratio (SNR) metrics. Here we review existing error presentations and propose the combined investigation of the SNR (expressed in logarithmic units), the unscaled error variances, and the soil moisture sensitivities of the data sets as an optimal strategy for the evaluation of remotely-sensed soil moisture data sets.
... Naively rescaling the measurement systems (e.g., by matching their temporal variances) and applying this simplified estimation equation will deliver biased RMSE estimates, since error estimation and calibration are fundamentally intertwined (Stoffelen, 1998). In practice, consistent estimates of calibration parameters and error estimates can be obtained by solving the equations iteratively (see Vogelzang and Stoffelen (2012) for more details), since triple collocation achieves the optimal rescaling (Yilmaz and Crow, 2013). In this study we calculate RMSEs using equation 5 rather than rescaling and using equation 7. ...
Article
Full-text available
Calibration and validation of geophysical measurement systems typically requires knowledge of the “true” value of the target variable. However, the data considered to represent the “true” values often include their own measurement errors, biasing calibration and validation results. Triple collocation (TC) can be used to estimate the root-mean-square-error (RMSE), using observations from three mutually-independent, error-prone measurement systems. Here, we introduce Extended Triple Collocation (ETC): using exactly the same assumptions as TC, we derive an additional performance metric, the correlation coefficient of the measurement system with respect to the unknown target, ρ_(t,X_i ). We demonstrate that ρ_(t,X_i)^2 is the scaled, unbiased signal-to-noise ratio, and provides a complementary perspective compared to the RMSE. We apply it to three collocated wind datasets. Since ETC is as easy to implement as TC, requires no additional assumptions, and provides an extra performance metric, it may be of interest in a wide range of geophysical disciplines.
Article
Full-text available
Buoys provide key observations of wind speed over the ocean and are routinely used as a source of validation data for satellite wind products. However, the movement of buoys in high seas and the airflow over waves might cause inaccurate readings, raising concern when buoys are used as a source of wind speed comparison data. The relative accuracy of buoy winds is quantified through a triple collocation (TC) exercise comparing buoy winds to winds from ASCAT and ERA5. Differences between calibrated buoy winds and ASCAT are analyzed through separating the residuals by anemometer height and testing under high wind-wave and swell conditions. First, we converted buoy winds measured near 3, 4, and 5 m to stress-equivalent winds at 10 m (U10s). Buoy U10s from anemometers near 3 m compared notably lower than buoy U10s from anemometers near 4 and 5 m, illustrating the importance of buoy choice in comparisons with remote sensing data. Using TC calibration of buoy U10s to ASCAT in pure wind-wave conditions, we found that there was a small, but statistically significant difference between height adjusted buoy winds from buoys with 4 and 5 m anemometers compared to the same ASCAT wind speed ranges in high seas. However, this result does not follow conventional arguments for wave sheltering of buoy winds, whereby the lower anemometer height winds are distorted more than the higher anemometer height winds in high winds and high seas. We concluded that wave sheltering is not significantly affecting the winds from buoys between 4 and 5 m with high confidence for winds under 18 ms−1. Further differences between buoy U10s and ASCAT winds are observed in high swell conditions, motivating the need to consider the possible effects of sea state on ASCAT winds.
Article
Full-text available
In this paper, the advantages of shaping a non-conventional triple collocation-based calibration of a wave propagation model is pointed out. Illustrated through a case study in the Bagnoli-Coroglio Bay (central Tyrrhenian Sea, Italy), a multi-comparison between numerical data and direct measurements have been carried out. The nearshore wave propagation model output has been compared with measurements from an acoustic Doppler current profiler (ADCP) and an innovative low-cost drifter-derived GPS-based wave buoy located outside the bay. The triple collocation—buoy, ADCP and virtual numerical point—make possible an implicit validation between instrumentations and between instrumentation and numerical model. The procedure presented here advocates for an alternative “two-step” strategy. Indeed, the triple collocation technique has been used solely to provide a first “rough” calibration of one numerical domain in which the input open boundary has been placed, so that the main wave direction is orthogonally aligned. The need for a fast and sufficiently accurate estimation of wave model parameters (first step) and then an ensemble of five different offshore boundary orientations have been considered, referencing for a more detailed calibration to a short time series of a GPS-buoy installed in the study area (second step). Such a stage involves the introduction of an enhancement factor for the European Centre for Medium-Range Weather Forecasts (ECMWF) dataset, used as input for the model. Finally, validation of the final model’s predictions has been carried out by comparing ADCP measurements in the bay. Despite some limitations, the results reveal that the approach is promising and an excellent correlation can be found, especially in terms of significant wave height.
Article
Full-text available
Quantitative information on the spatial and temporal error structures in large-scale (regional or global) precipitation datasets is essential for hydrologic and climatic studies. A powerful tool to quantify error structures in large-scale datasets is triple collocation. In this paper, triple collocation is used to determine the spatial and temporal error characteristics of three precipitation datasets over Europe-that is, the precipitationproperties visible/near infrared (PP-VNIR) retrievals from the Spinning Enhanced Visible and Infrared Imager (SEVIRI) instrument on board Meteosat Second Generation (MSG), weather radar observations from the European integrated weather radar system, and gridded rain gauge observations from the datasets of the Global Precipitation Climatology Centre (GPCC) and the European Climate Assessment and Dataset (ECA&D) project. For these datasets the spatial and temporal error characteristics are evaluated and their performance is discussed. Finally, weather radar and PP-VNIR retrievals are used to evaluate the diurnal cycles of precipitation occurrence and intensity during daylight hours for different European climate regions. The results suggest that the triple collocation method provides realistic error estimates. The spatial and temporal error structures agree with the findings of earlier studies and reveal the strengths and weaknesses of the datasets, such as inhomogeneity of weather radar practices across Europe, the effect of sampling density in the gridded rain gauge dataset, and the sensitivity to retrieval assumptions in the PP-VNIR dataset. This study can help us in developing satisfactory strategies for combining various precipitation datasets-for example, for improved monitoring of diurnal variations or for detecting temporal trends in precipitation.
Article
Full-text available
Triple collocation is a powerful method to estimate the rms error in each of three collocated datasets, provided the errors are not correlated. Wave height analyses from the operational European Centre for Medium-Range Weather Forecasts (ECMWF) wave forecasting system over a 4-yr period are compared with independent buoy data and dependent European Remote Sensing Satellite-2 (ERS-2) altimeter wave height data, which have been used in the wave analysis. To apply the triple-collocation method, a fourth, independent dataset is obtained from a wave model hindcast without assimilation of altimeter wave observations. The seasonal dependence of the respective errors is discussed and, while in agreement with the properties of the analysis scheme, the wave height analysis is found to have the smallest error. In this comparison the altimeter wave height data have been obtained from an average over N individual observations. By comparing model wave height with the altimeter superobservations for different values of N, alternative estimates of altimeter and model error are obtained. There is only agreement with the estimates from the triple collocation when the correlation between individual altimeter observations is taken into account. The collocation method is also applied to estimate the error in Environmental Satellite (ENVISAT), ERS-2 altimeter, buoy, model first-guess, and analyzed wave heights. It is shown that there is a high correlation between ENVISAT and ERS-2 wave height error, while the quality of ENVISAT altimeter wave height is high.
Article
Full-text available
Scatterometers estimate the relative atmosphere-ocean motion at spatially high resolution and provide accurate inertial-scale ocean wind forcing information, which is crucial for many ocean, atmosphere and climate applications. An emprirical scatterometer ocean stress (SOS) product is estimated and validated using available statistical information. A triple collocation dataset of scatterometer, moored buoy and numerical weather prediction (NWP) observations together with two commonly used surface layer (SL) models are used to characterize the SOS. First, a comparison between the two SL models is performed. Although their roughness length and the stability parameterizations differ somewhat, the two models show little differences in terms of stress estimation. A triple collocation exercise is then conducted to assess the true and error variances explained by the observations and the SL models. The results show that the uncertainty in the NWP dataset is generally larger than in the buoy and scatterometer wind/stress datasets, but depending on the spatial scales of interest. The triple collocation analysis also shows that scatterometer winds are as close to real winds as to equivalent neutral winds, provided that we use the appropriate scaling. An explanation for this duality is that the small stability effects found in the analysis are masked by the uncertainty in SL models and their inputs. The triple collocation analysis shows that scatterometer winds can be straightforwardly and reliably transformed to wind stress. This opens the door for the development of wind stress swath (level 2) and gridded (level 3) products for the Advanced Scatterometer (ASCAT) onboard MetOp and for further geophysical development.
Article
Using co-locations of three different observation types of sea surface temperatures (SSTs) gives enough information to enable the standard deviation of error on each observation type to be derived. SSTs derived from the Advanced Along-Track Scanning Radiometer (AATSR) and Advanced Microwave Scanning Radiometer (AMSR-E) instruments are used, along with SST observations from buoys. Various assumptions are made within the error theory including that the errors are not correlated, which should be the case for three independent data sources. An attempt is made to show that this assumption is valid and also that the covariances between the observations due to representativity error are negligible. Overall, the AATSR observations are shown to have a very small standard deviation of error of 0.16K, whilst the buoy SSTs have an error of 0.23K and the AMSR-E SST observations have an error of 0.42K.
In my note (Kolmogorov 1941 a ) I defined the notion of local isotropy and introduced the quantities B d d ( r ) = [ u d ( M ′ ) − u d ( M ) ] 2 , ¯ [ u n ( M ′ ) − u n ( M ) ¯ ] 2 , where r denotes the distance between the points M and M' , u d (M) and u d (M') are the velocity components in the direction MM' ¯¯ at the points M and M' , and u n (M) and u n (M') are the velocity components at the points M and M' in some direction, perpendicular to MM' .
Article
High-resolution wind products based on space-borne scatterometer measurements by ASCAT and SeaWinds are used widely for various purposes. In this paper the quality of such products is assessed in terms of accuracy and resolution, using spectral analysis and triple collocation with buoy measurements and NWP model forecasts. An experimental ASCAT coastal product is shown to have a spectral behavior close to k-5/3 for scales around 100 km, as expected from theory and airborne measurements. The NWP spectra fall off more rapidly than the scatterometer wind spectra starting at scales of about 1000 km. Triple collocation is performed for four collocated data sets, each with a different scatterometer wind product: ASCAT at 12.5 km and 25 km, and SeaWinds at 25 km processed in two different ways. The spectral difference between scatterometer wind and model forecast is integrated to obtain the representation error which originates from the fact that global weather models miss small-scale details observed by the scatterometers and the buoys. The estimated errors in buoy winds and model winds are consistent over the data sets for the meriodional wind component; for the zonal wind component consistency is less, but still acceptable. Generally, enhanced detail in the scatterometer winds, as indicated at high spatial frequencies by a spectral tail close to k-5/3, results in better agreement with buoys and worse agreement with NWP predictions. The accuracy of the scatterometer winds is about 1 ms-1 or better. The calibration coefficients from triple collocation indicate that on average the ASCAT winds are slightly underestimated.
Article
Kinetic energy spectra derived from observations in the free atmosphere possess a wavenumber dependence of k -3 for large scales, characteristic of 2D turbulence, and transition to a k -5/3 dependence in the mesoscale. Kinetic energy spectra computed using mesoscale and experimental near-cloud-scale NWP forecasts from the Weather Research and Forecast (WRF) model are examined, and it is found that the model-derived spectra match the observational spectra well, including the transition. The model spectra decay at the highest resolved wave-numbers compared with observations, indicating energy removal by the model's dissipation mechanisms. This departure from the observed spectra is used to define the model's effective resolution. Various dissipation mechanisms used in NWP models are tested in WRF model simulations to examine the mechanisms' impact on a model's effective resolution. The spinup of the spectra in forecasts is also explored, along with spectra variability in the free atmosphere and in forecasts under different synoptic regimes.
Article
The statistical features of turbulence can be studied either through spectral quantities, such as the kinetic energy spectrum, or through structure functions, which are statistical moments of the difference between velocities at two points separated by a variable distance. In this paper structure function relations for two-dimensional turbulence are derived and compared with calculations based on wind data from 5754 airplane flights, reported in the MOZAIC data set. For the third-order structure function two relations are derived, showing that this function is generally positive in the two-dimensional case, contrary to the three-dimensional case. In the energy inertial range the third-order structure function grows linearly with separation distance and in the enstrophy inertial range it grows cubically with separation distance. A Fourier analysis shows that the linear growth is a reflection of a constant negative spectral energy flux, and the cubic growth is a reflection of a constant positive spectral enstrophy flux. Various relations between second-order structure functions and spectral quantities are also derived. The measured second-order structure functions can be divided into two different types of terms, one of the form r2/3, giving a k3-range in the energy spectrum. The structure functions agree better with the two-dimensional isotropic relation for larger separations than for smaller separations. The flatness factor is found to grow very fast for separations of the order of some kilometres. The third-order structure function is accurately measured in the interval [30, 300] km and is found to be positive. The average enstrophy flux is measured as [approximate]1.8×103 and the constant in the k3-range can be explained by two-dimensional turbulence and can be interpreted as an enstrophy inertial range, while the k−5/3-range can probably not be explained by two-dimensional turbulence and should not be interpreted as a two-dimensional energy inertial range.