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Document NWPSAF-KN-TR-021

Version 1.0

06-07-2012

Triple collocation

Jur Vogelzang and Ad Stoffelen

KNMI, de Bilt, the Netherlands

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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

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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

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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.,

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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.

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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

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,

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.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

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)

(

()

.

><+><+=

><+><+=

><+>=<

zzzz

yyyy

xx

tabM

tabM

tM

ε

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,

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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

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.

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

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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.

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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

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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

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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

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measurements and YM and ZM satellite observations. In this case 0>=

<

yx

ε

ε

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.

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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.

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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.

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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.

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,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.

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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.

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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.