Max -Planck -Institu t fUr Meteorologie
REPORT No. 88
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OPTIMAL FINGERPRINTS FOR THE DETECTION OF TIME DEPENDENT CLIMATE CHANGE
by
KLAUS HASSEL~ANN
HAt.4BURG, AUGUST 1992
Max-Planck-Institut fUr Meteorologie
REPORT No. 88
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-1,00 10 20 30 40 '50 60 70 80 90 100 I
OPTIMAL FINGERPRINTS FOR THE DETECTION OF TIME DEPENDENT CLIMATE CHANGE
by
KLAUS HASSElMANN
HAMBURG. AUGUST 1992
AUTHOR:
Kious Hasselmann
MAX-PLANCK-INSTITUT FOR METEOROLOGIE
BUNDESSTRASSE 55 D-2000 HAMBURG 13 F.R. GERMANY
Max -Planck-I nstitut fUr Meteorologie
Tel.: Telex:
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MPI.METEOROLOGY +49 (40) 4 11 73-298 REPb88
OPTIMAL FINGERPRINTS FOR THE DETEcrrON OF
TIME DEPENDENT CLIMATE CHANGE
Abstract
K.Hasseimann
lO.Aug. 1992
An optimal linear filter (fingerprint) is derived for the deteclion of a
given lime-dependent. multi-variate climatc-change signal in
of natural climmc variability noise. Applic.niol1 of the
the observed (or model-simulated) climate data yields a
the presence
fingcrprint to
climate-change
detection variable (detector) with maximal signal-to-noise ratio. The
optimal fingerprint is given by the product of the assumed signal pattem
and the inverse of the climate variability covariance matrix. The data can
consist of any, not necessarily dynamically eompletc climate data set for
which estimates of thc natural variability covariance matrix exist. The
single-pattern
climate-change
anal ysis readil y
signal lying
lOW-dimensional)
applied separately
pattern space.
signal-pattern
to each
Multi-pattern
generalizes to tile multi-pattcl11
in a prescribed (in practice
space: thc single-pattern result
case of a
rc!atively
is simply
individual base pattern spalming the signal
detection methods can bc applied either to
test the statistical significance of individual compollcnts of a predicted
muiti-componem
detection tests,
climate
or to
change response,
detennine the
using
statistical
separate single-pattern
significance of the
complcte signal, using a multi-variate test. Both detection modes make use
of the same set of detectors. The difference in direction of the assumed
signal pattem and computed optimal fingcrprint vector allows alternative
interpretations of the estimated
detectors. The present analysis
signal
yields
associated with the sel of optimal
an estimated signal lying in the
assumed signal space, whereas an
detection problem by Hasselmann
earlier analysis of the
yielded an estimated
intelpretations can
time-independent
signal in tbe
be explained computed fmgelprint
by different choices
space. The different
of the metric used to relate the signal space to the
fingerprint
metric,
space (inverse covariance matrix versus standard Euclidean
respectively). Two simple nmural
considered: a space-time separability model, and
POPs (pJincipal Oscillation Pattcl11s). For each
valiabilily models arc
an expansion in temlS of
model the application of
the optimal fingerprint method is illustrated by an example.
ISSN 0937-1060
I, Introduction
The general public coneem and ongoing scientific debatc on the
anticipated global warming due to increasing greenhouse-gas concentrations
and on the impact of other activities of man on lIle earth's climate has
gencrated a strong demand for the devclopment of improved tcchniques for
the early
idcntification
detection of thc predicted elimme change signal. A clear
of the anthropogenic signal in climate obselvations would
reduce the present scientilic ullcertainties regarding the magnitude and
foml of the anticipated climate change and would provide a more reliable
quantitative basis for the dcvelopmCI1l of rational political abatement and
adaptatioll strategies,
At the core of the detection problem is the development of a suitable
strategy for distinguishing between the anticipated externally
time dependent climate change signal and the natural intemal
of the climate system, The problem can be divided into three
identification of lhe climate change signal lhill one wishes
generated
variability
parts: (I)
to detect,
(ii) detelmination of the relevant statistical properties of the natural
climate variability background, and (iii) developmclll of an optimal
detection method, This paper addresses the third problem, However, the
question of tile representation (but not the estimation) of the second
moment statistics of the natural climate variability noise needed for
optimaal detection, which relates to tile second problem, will also be
considered briefly,
It will be assumed throughout that the first problem has been rcsolved and
the general structure of the space and time dependent climate cbange
signal that onc is sceking to detect has been detelmincd, for cxample from
model simulations, It is well known that ill attempting to detcct signals
in noisy Illulti-variate d<lW, the number of degrees of freedom of the
signal must be scvcrcly curtailed in order to cxtract statistically
Significant results, Spccific<llly, it will be assumed thai the signal is
defined to within thc unknown coefficients or a relatively Sill all set of
prescribed time-varying response pallCIl1S eeL Cubaseh et aL, 1992),
2
This should not
anticipated climate
gloss over the di ff1cultics, however,
chnnge signal. This is generally a
of defining the
110n-tIivial problem
reqUlnng an inlercompmisoll and detailed assessment of different climate
change simulations with differcnt models. The problem is compounded by the
fact that realistic time dependent climate change simulations can be
carried out only with coupled ocean-atmosphere general circulution models
(CGCMs), which generate Ulcir own natural climate variability (Washington
and Meehl, 1989, Stouffer et aI., ] 989, Mnnabc Cl aI., 1991. Cubasch et
aL, 1982, Santer et aI., I 992a; see also tile stochastically forced ocean
cxperiment of Mikolajewicz and Maier-Reimer, 1990). The signal detection
problem arises therefore already in the attempt to define the climate
response signal in model simulations, enDing into question the basic
premise of the separability of the signal definition and signal detection
problems. This will nevertheless he ussulllcd in the following as a conceptlJaI starting point.
In the same Spilit, thc estimation of
natural climate variability required for
the core of the second problem, will also
again a non-tliviul ussumpllon, but is
well-defined detection problem. In practice,
whether an anthropogenic climate signal
the statistical properties of the
optimal detection, which is at
he regarded as resolved. This is
revolves around uncertainties over the
natural variability of the real climate.
necessary in
much of the
cun al read y
stIUcturc and
Nevertheless, in
a consistent conceptual framework it will be assumed,
point, that estimates of tile rcquircd natural
statistics are available.
order to pose a
ongoing debate on
bc detected today
magnitude of the
order to develop
again as a starting
climate valiability
This is not such a severe restriction, however, as may at first sight
appear. The detection strategy will be developed in
'observed climate stnte vcctor', which refers \0
the following for an
any set of climate
variables or
The observed
for example,
consist of
indices for which adequate statistical data are available.
climate state need not be dyn<lm ically complete. as required,
for a climate model. Thus the 'observed climate state' can
time series of vnrying Icngth and for diffcrcnt climate
valiables, measured at inilomogeneollsl y distributed stations. Timc
3
dependent signal detection is concemed only with the kinematics. not the
dynamics of climate change. and a dynamically incomplete represenlntion of
Ule climate stale has no consequences for the method of detection (apart.
of course. from the unavoidable loss of detection power associated with a
loss of infonnalion).
After an optimal detection stmtegy lws becn developed under these two
working assumptions. the general
thrce elements of tile ovcrall
question
detection
of the
problem
possible
interdependence of
can be revisited
the
in
approaches to modeling fimhcr iterations. The discussion of some
the relevant second moments' properties
presentcd at the end of this paper
direction.
of the natural climate vaJiahility
The detcction
viewed as the
problem. in the
task of identifying
represents a first step in this
separable fon11 discussed here, is often
tile most sensitive climate index. from a
large set of potentially available indices. for which the anticipated
anthropogenic climate signal can he most readily distinguished [rom the
nal1n'al climate noise. Global or regional mean surface temperature.
veltical temperature differences, sea icc extent. sea level change and
integrated deep ocean temperatures are examples 01 indices which have been
discussed in this context (e.g. Wigley and Jones. 1981. Barnett. 1986,
Barnett and Schlesinger. 1987. Karoly. 1987.1989. Munk and Forbes. 1989.
Mikolajcwicz et aI., 1991). A more systematic approach. however, is the
tlngerprint method. Here all climate variables are regarded as containing
potentially useful information on climate change, and the task is to
extract from the filII set of availahle observed climate variables an
optimal
1980,
net climate change detection index (cf. Madden and Ramanathan,
MacCracken and Moses. 1982), This approach will he pursued in this
paper. While it is theoretically conceivable that a single climate
variable will turn out to be the most effective detector. in the general
case all variables will calTY some signal information. although wi th
varying levels of noise contamination. and the optimal detector. (lefined
as the variable which has a maximal signaHo-noise mIlo. wjlJ consist of
a weighted linear combination (the . fingcrpIinl ') of all variables.
4
Most investigations in the past have considered only partial aspects of
the full multi-vtlliate, space-time dependent problem, The detection of a
time independent equilibrium change of a multi-variate climate system in
response to a constant extelllal forcing has been investigated by
Hasselmann (1979, referred to in the following as H), Hannosch6ck and
Frankignoul (1985), Hense (1986), Hense ct al (1990), Bell (1982, 1986)
and others, The com plem elltary problem of detecting a time-dependent
climate
nature
change signal, while disregarding the spatial and multi-variable
(1992),
of (he climate signal,
These authors did
optimization of the
detection problem
(1988), Wigley and
detection
has been studied by Bloomfield and Nychka
not address, however, the question of the
variable, Various other facets of the signal
have been investigated or reviewed by PreisendOl'fcr
Bamell (1990), Bamett (1991), Barnell el aL (1991),
Solow (1991) and other authors, Recently, however, North et al (prelimary
pre print) have independently investigated the full space-timc depcndcnt
climate change detcction problem, also using pattern analysis tcchniques
rather similar to the approach pursued here,
In com paring
studies appl)'
the theoretical signal predictions with
some form of filtering or pallem
which in effect projects the observed data onto
pattern, This corresponds essentially to the usual
a signal pattern to dllta such that the variance
minimized, While this removes mueh of the irrelevant
observed data, many
correlation technique
the predictcd signal
least-square filling of
of the residual is
noise
does
should
not, however, represent the optimal signal detection
in the data, it
solution, This
maximize the signal-to-noise ratio rather than the explained
variance,
In this paper an optimal space-time dcpendent filter (fingerprint) is
derived which maximizcs the signal-to-noise ratio for the associated
detector for any prescribed multi-variable, spacc-time dependent climate
signal. TIle singlc-pattern solution is then generali7,ed to tile
mulli-pattel11 case to determine it set of p optimal fingerprillls and
associated detectors for any climate-change signal lying in a prescribed
space spanned by p given climatc-changc pattellls, To detenninc the
statistical signiflc,mce of the estimated climate signal it must be
5
assumed that the natural climale vm'iilblility is Gaussian (or is otherwise
known). However, the dct1nilioll or the optimal )'ngerprillls as such is not
dependent on the Gaussian hypothesis.
The detection problem is dolined in Sc~tion 2. Assuming the space-time
depcndelll stlUcturc of the externally generated climate signal is
prcsclibed to within an unknown amplitude factor, tile optimal space-time
integrated detector is dctived in Section 3 as a linear combination (the
optimal 'fingerprint') of the complete set of time dependent climate
vaIiables. The result is generalized in Seclion 4 10 the case of a signal
defined only as an unknown linear combination of a finltc set of
presClibed time dependent pallems. The analysis follows the basic ideas
of H, extended in a straightforward manner to include the time dimension,
but is simplified by the introduction of lile 'fingerprint' tenninology.
Tbe problem of modeling the complex space-time (or space-frequency)
dependent second moments of the naltmli climate valiability required for
optimal detection is discussed in Section 5. Two simplifications arc considered and illustmtcd by examples: the assumption of space-time
separability, and 1I1C il1lroductioll of upproximate POP (Principal
Oscillation Pallem) representations. The results are summarized and some
open questions mentioned in the concluding section (J.
Consider the evolution of ,m 'observed climate state' = in
response to some lime dependent external forcing over a time interval 0 $
t ~ T. The 'observed climate state' can be represented as u discretized
composite vector 2 = (<ili) whose indices i = (v,z) lUll through the climate
variables v (temperature, pressure, moisture, ... ) and the disrete spatial
coordinates Z' which can refer, for example, to a set of observing
stations or a model grid. The climate trajectory 2(1) can represent either
a set of vadables of the real climate system or
some numerical model experiment. As has been
tile simulated response in
pointed out, it is not
necessary that 2(t) provides a dynamically complete description of the
climate system. The 'observed state vector' can (and, indeed, must) be
limited to variables for which sufficient Observational information is
available to adequately define the statistics of the ensemble of
6
trajectories ~(t) characterizing the natural variability of the system,
The detection problem is then to decide whether the climate time history
9!(t) generated by the external forcing can he distinguished, at some given
level of significance, from the statistical ensemble of
natural-variability trajectories ~(t),
For the lime-independent problem, an optimal signal detection strategy has
been presented in H, In the following, the analysis of H is extended to
the gcneral multi-variate, time dependent t:<lse and recast in a simpler
'fingclprint' telminology,
Fonnally, the approach of H can Ile
dependent case by simply discretizing
Ule time index in a combined index
immediately generalized
the time variable and a ;;::: (i,t) nmning
then
from
to the timc
incorporating
1 to n, The
extended climate vector
summarizing tile complete
discretizcd time interval
is
a
set to unity), The
= (v,lS,t) not only
represents a constant vector
climate trajectory:
1,2, ... 1'
IJI a set of all <l>i(t) in the
(the time discretization interval
imroduction of a compact
simplifies the notation, but
variabk-space-time-index
also focusses on the
essentially very simple linear-algebra geometry of the detection problem.
An important requirement 101' a successful detection strategy is the
reduction of the number of degrees of freedom of the signal. This is
achieved in H by considering a signal which is defined II priori only in a
relatively low-dimensional sub-space of the full climate system. Attempts
to test whether the full climate response vector )I! can be distinguished
from an element )I! of the natural variability ensemble
in dimensionality of tilC signal will generally fail
reasons (cf.H, Bamctt and Hassclmann, 1979):
without a reduction
1'01' the following
The climate response te) given cxtcmal forcing can be represented
generally (ignoring nonlinear illlcraetions between the natural climate
variablility and the eXlcl'Ilally forced response) as a superposition
of the forced delenninistic climate signal )l!s and a particular realization
7
of the
denoting the
natural
direction
signal is statistically
variablility ensemble. Let v be
of the signal, ~s~ I ~ll y. significant in the sense that
with the noise component in the v I ~s I is large compared
a unit· length vector
Assume now that the
the signal ampliLUde
direction,
where the cornered parentheses denote
is used, a superscript T denoting the
noise is taken to be zero, <\jI,? priori, it is then possible to test
T 2 the !let square response (y~) in the
the inequality (2) would be positive.
(2)
cnscmble meaDS :md matrix notation
transpose. The mean of the eliuHlte
D. If the direction v is known a
for the statistical signilieancc of
v direction. and the outcome under
If, on the other hand, the signal paltem is unknown, one can lest only
the statistical significance of the complete n·dimensional response vector
~. This requires considering the magnitude and orientation of the response
vector in relation to the joint probability density of the ensemble of
vectors ~ in the full n dimensional climate trajectOlY space. If the
probability density decreases monotonically with distance from the origin,
as is nonnally the case. each (n-I )-dimensional hypersurface of constallt
probability density will divide the n-dimensional space into an intcmal
closed region of some probability measure P around the origin and an
extemal open region of probahility measure (l-P), in Which the
probability density is everywhere smaller than in the intemal region. The
response vector ~ is thcn normally termcd statistically significant at the
significance levcl P if the end.point of vector, drawn from the origin,
lies in the external region. Without entering here into the details of the
analysis, it is qualitatively apparent that the larger the number of
irrelevant noise dimensions, the smaller the relative contribution of the
signal to the total magnitude of the responsc vcctor, and the more
difficult it will be to detect the signal in the full n-dimensional space
- evcn when the signal component is significantly larger than the noise
component in tl1e one (unknown) direction v.
Fortunately, this second 'needle in a haystack' situation
not apply in practice: the direction of the hypothesized
8
will normally
elimnte Signal
can be assumed to be known from model simulations, or at least to lie
within a known sub-space of relatively small dimension_
The case that the climate change signal is known exactly applies for the
simplest yes-no question of climate change detection: one wislles to
detemline whether a specific time-dependent global climate signal which
has been predicted by a model can be detected in the data (or at what
future time it should become detectable in the data). TllC more general
case that the response signal is assumed only to lie within some
prescribed low-dimensional sub-space of the full climatc trajectory space
arises if the predicted climate change signnl is only imperfectly known
(for example. because diITercnt models have predicted different climate
change patterns) or if one wishcs to distinguish hetwccn different climate
Ci1atlgC signals produced by different anthropogenic or natural cxtcmal
forcing mechanisms.
A mulli-pattcm analysis is necessary also if one wishes to test the
statistical significance not only of tllC complete global climate change
signal, but also of particular sub-components of tile global signal. II can
be anticipated. for example. that the most effective single nct global
climate change detector will be based primaJily 011 the largeescale
features of the climate ficlds. But for policy-makers. the regional
climate changes (which at present arc not predicted very reliably by
climate models) will presumably be of grealer eoneem than globally
integrated quantities. They will therefore wish to know not only whelher
the global climate change predicted by models has been dClected, bUI also
whether thc model predictions of climate change on lhe regional scale can
be confirmed by observations.
The following analysis CHn be extcnded to
than detenninistically prescribed climate
lIppropriate, for example. if thc predicted
ensemble of different modcl simulalions
the case
signals.
signal
with
of statistically rather
This could be
is inferred from an
differcnt levels of
credibility. or from a mixture of model simulations and general
theoretical considerations (for cxamplc, regarding the expectation of a
land-sea-contrast signal). However. this Bayesian gcnerali/alian will not
be pursued fUJ1her in the prcsent paper_
9
If the pallern g of the space-lime signal
an unknown constant amplitude factDr c,
. s InueelDry ]jI is knDwn ID within
(3)
the signal deteclion problem reduces (if one limits oneself to linear
techniques) tD the tusk of tlcriving an optimal detection variable Dr
'detector'
d l' (! ]jI), (4)
computed from ]jI by applicatiDn of a linear ' filter function' or
'fingerprint' f co (I' ), for - il
which the sqUilrc signal-lo-noise ratio
(5)
is maximized. Here
s ([1' ]jIs) d . (6)
ilnd
0 (LT \ill (7)
represent, respeeti vel y, the signal and climate-variability noise
cDmponents of the net detector d ~ dS+ o.
The fingerprint vector r (5)) only to
defined only
will be
within a
to within
cstimated
is determined through the condition R2 = max (Eq.
factor. Similarly, the signal pattern g need be
an arbitrary factor, since the amplitude of the
by the detection procedure. Although .f and g signal
could therefore be nonnalizcd [0 unit vectors. it is notationally more
convenicnt to leave the vectors unnormalizcd at this point.
The maximization of R2 with respect to the arbitrarily normalized vector f is equivalent to the minimization of <02> under the side condition (ds)2 ~ COllSt. This yields as determining equation for the I1ngerprim !
10
where C z
variability
= C I~ = au and ), is
normalization chosen lor [.
·u' ( TC,-l )-1 WI I r, = - g "" g .
Tile optimal llllgcl'print
assumed signal direction,
(and is oftcn assumcd
o (8)
t:ovariance matrix of the natural climatc
L<lgrange multiplier
Thc solution is
r
direction is in
as may perhaps
in detcction studies).
whose
general
have
This
value depends on the
(9)
not parallel to the
been expected intuitively
is best understood by
transforming to statisticall y orthogonal coordinates (denoted in the
following by primes),
\jIa = I \jib Cba {I 0)
h
which arc defined with respect to an orthontllmaJ basc cha = ~a consisting
of the eigenvectors (empirical orthogonal functions, EOFs) £a of C;;;,
with
(12)
In EOF coordinates, the covariance matrix of the natural climate
variability lakes thc diagonal fnrl11
2 " = on vab (13)
2 where (Ja is the variance associillcd with the EOF £a' Equation (9) tilus
becomes
11
f' a (14)
The multiplication of lhe signal with the inverse of the covariance matrix
is seen to weight Ihe fingerprinl components f' in tile EOF frame relative a 2
to the signal componenls g:' by the inverse «( of the EOF variances, u a
thereby slewing the fingerprint vector away from the EOF directions with
high noise levels towards the low-noise directions. (In practice, the EOF
spectrum, if estimated rrom daw. should be truncated after a finite
num ber of terms, since tbe higher-index eigenvalues lend 10 he
underestimated, leading to a spurious mnpli Ikation of the higher-index
fingerprint eomponenls,
1988.)
For the special case
cr. v.Storch
of ,1 single
and H annoseh<lek, 1985, Prciscndorfer,
time-dependent variable, Ihe resuli (13)
is well known fl'Ol11 classical signal processing theory (ef. Wninstein and
Zubakov, 1(62). The EOFs for a statistically stationary Ii me sClics are
so simply the harmonic j'unctions of Ihe Fourier series representation,
thai Eq.(14) reproduces in this case the basic Iheorem that the optimal
signal detection IIlter for a station,lry time series is given by the
Fourier transform of the signal divided by thc noise variance spectrum.
Implicit in the definition
which maximizes square
statistical significance of
signal-ta-noise ratio. For
of the optimal detector as the linear variable
signal-to-noise ratio is the assllmption that the
the detector increases monolonically with the
most climate variability distributions, this
will be the case. It !la;; also been assumed thai the only source of
statistical noise in the detector is the natural climate variability. In
practice, data errors will also contribute to the detector noise. However,
these are generally slIIall compared with the climmc variabilily, and, to
avoid complicating the analysis, will be ignored.
The statistical si gni ficanee of the optimally detected signal d can be
computed from the probability distribution of the noise variable a for the
Null hypothesis that there is no signal. For this purpose it is normally
assumed that the natural climate variability is Gaussian, so that all
distributions can be derived from the covariance matrix C.
12
In this case, if !;; is known exactly (as opposed to being estimated from a
I1nitc data set), a is also Gaussian with variance
(15)
If the covDriance matrix is estimated from a finite data set, a has
statistics generally similar to a Student-t variable (Morrison, 1976).
Howeverl a differs from a Student distribution in that the direction for
which the variance of a is estimated is not prescribed a priori. but is
modified relative to the prescribed signal direction by mulliplicatioll
with the inverse of U,e estimated covariance matrix. The resultant
variable does not correspond to a standard tabulated statistical variable.
and its distribution must therefore be estimated by approximate analytical
techniques or Monte Carlo simulations.
In principle. Ule significance level can. of course, be computet! also for
an arbitrary non-Gaussian. but known statistical distribution. However. it
will normally be difficult in tbe gcneral timc-depcndent case to obtain
reliable dircct estimates or statistical distribution using, for example,
pcnnutation methods. This requires creating an ensemble of realizations.
for which time series are needed which arc significantly longer than the
analysis time interval T (the same problem "rises also in the estimation
of the covariance nHlIrix Q. The only recourse in this case may be to
augment the observational or modcl simulatcd data with still longer model
simulations of the llatural Climate vnriability.
The analysis so far has addressed only the problem of delecting a signal
with known direction g. How can the optimal detcctor d and associated
fingerprint ! be tnmslatcd now into an estimate of the signal? This
requires defining the direction amI mab~litude of tile estimated signal as a function of g. ! :md d. The answer is nOt unique and depends on how the
fact that f and g are not panlllcI is interpreted.
In H, the optimal dctection problem was l'OllllUlntcd as the task of finding
an optimal unit-length Nignul detection vcctor g, given the signal
direction 11, which maximizes the signal-to-noise ralio for the
cocft1cient c1 of the estimated signal
13
(16)
The coefficient c I itself was determined in the standard manner by
minimizing the mean square 01- the residual 1'r of the net response
(17)
which yielded
(18)
The optimal detection direction was round to lie in the direction of the
optimal fingerpJint, as given by Eq.(9),
b = .vIII,
so that the best signal cstimate was given by
Alternmively. one
signal is assumed
lie in this direction,
T 2 1'e = «I 1'> / I f I ) I
can adopt the view thm since lhe
to be g, the estimated signal should
(19)
(20)
direclion of the
also be taken to
(21)
The coefficicnt c2 should then be detellllined by the condition that in tile
absence of noise one should recover the true signal (or, equivalently,
that in the presence of noise tile mean-square deviation from the tlUe
signal should be minimized). Tilis yields
T -I -I c2 = d (g ~ g)
Thus tile estimated signal is given in this case by
",e -T Te-l-l :: = <L ~ (g ~ g) g
14
(22)
(23)
This intcrpretation will be adoptcd in the I-ollowing_
One can argue
of Ule signal
interpreted with
either vicwpoinL The suppression i 11 the fingerpri III pallem
components associated with high noise levels call be
change pattcm
existence of
lloise-COlllamillated
H to imply that one has actually
only in the low-noise fingerprint
eli milte-change signal
directions cannot be
components
supported by
tested tile climate
direction,
in the
the data_
and the
suppressed
On the
other hand, onc can adopt the prcscI1l vicw that the climate change signal,
if it exists, is specified a priori as a complete vector, and one is at
liberty to test any linear projection of the signal on to some chosen
direction as evidence of the existence of tire complete signal.
TI]~ _ n.ll!!!!:p~!!~~l] _ [lrgl?!~!!'
The single-pattern analysis can be readily
which the direction of the signal vector is
general ized
no longer
to the case
prescribed b\ll
in
is
postulated only to lie in a space spanned by p given gue.>s vcctors Il.y' y =
l, __ p ,
p s
~- I Cv £y (24)
y=l
The base signal space can be chosen used to represent the prescribed
arbitrarily. orthonormalized, however, in The guess vectors will not be
order to preserve their original physical meaning. The guess vectors may
represent, for example, differellt possible time-dependent climate change
pattcrns induced by a CO2 increase or enhanced acrosol concentrations, or
the climate change associated with variations in Ihe solar constant, or
some regional climillc change pallern. In general, these space-time
patterns will not be orthogonal.
In most applicmions, the coefficicnts cy arc cstimated from the net
response ~ , Eq.(1), by minimizing tile mean ~qllare residual ~, yielding
the standard least-squares Solulion
15
L H~:l T (25) c (gfl ;Jf) Y
II
where
fl vll = (g~ g~) (26)
However. in the present application the goal is not to maximize the
explained variance but lO maximize the signal-to-noise ralio for the
estimated signal. This yields a different solution. The [ollowing
deJivation is based on fl, but is simplified lhrough the application of the
single-pattern fingerprint cOllcept.
As before, the problem is solved in two steps. First, a set of p detectors
dy is derived for which the p-dimensional statistical significance
(assuming a Gaussian distribution) is maximized. It will be shown tl1m
thesc ilrc just the single-pattern detectors of the individual patterns &y'
(27)
where
(28)
In a second step, the coefficicllls dy arc then assigned to p base vectors
2y to construct an estimate of the signal.
From Eqs (27), (28), (4) and (9) it follows lirst that for a.ny signal t of the ronn (24) with given coefficients cY ' and therefore given signal
pallem g = t, the optimal fingerprint and associated optimal detector
arc given by, respectively,
[ (29)
and
(30)
Tbe SCI of
represent a
fingerprints r -v straighlforw;lJ'(1
Eg. (28) and detectors dv' Eq (27), therefore
generalization of the single-patlern solution
16
to the multi-pattern case ill the sense lila! they yield the optimal
fingerprint, Eq.(29), and max i m al-si gna l-lo-no i se detector, Eq.(30), for
any givcn signal, Eq.(24), in the space spmmed hy the sct of pallcrns gy
This result is relevant if one wishcs to test the sLatistical significance
of individual a priori defined components of a mulli-pallcm signal.
Although this is importanL for many llpplications, the more general
situaLion in multi-patLem signal detection is that the direction of the
signal is unknown, except thaL iL is nssullled
signal pattem space, The task is Lhen La
change signal found within Lhe space sp<llliled by
to lie
decide
in the
whether
p-dlmenslonal
the climate
guess pattems
direction of the gil
signal
is sl:itistleally significmH,
a priori,
the sci
withont
of p prescribed
specifying the
On needs in this case to find a set of p detectors whose associated
p-dimensional statistical significance mcasure is maximized for all)' signal
lying in the p-dimcnsional
solution to this prohlem
detectors dVI Eq, (27), defined
guess-pallerll space,
is again given by
by the fingerprints Lv'
H will be
the set
Eq. (28),
shown that the
of single-pattern
The standard measure of the stmistical signifieam;c of the p-dimcllsional
vector d ~ (d) in tilC prescncc of a Gaussian background noise neld is 2 - y
the p statistic
(31)
where
(32)
:jnd
(33)
represent the covariance matrix and individual components, respectively,
of the detector vector il associated with the illituml variability noise in
the absence of a signal (underlined symbols denotc vectors or matrices in
the p-dimensional space component indices ,t,v"" in contrast to the
vectors considered hitherto in [he n·dimensional observed
climate-trajectory space wilh component inciices a,ll",,).
17
If the
indeed
elimate variability and thus the probability distribution
Gaussian, the (p-l)-dimensional hypcrsurfaces p2(d) =
of a is v tonstant
represent
measure
surfaces of conslalll prob<ibility
of the statistical significance
density, Thus P2CQl of the coeff'icielll
provides a
veClor, as
discussed above. Jf the joint probability distribution is nOll~Gaussian. a
rigorous
fraught
computation
with still
one-dimensional case,
of the
greater
SO thal
statistical
sampling p2 will
significance will generally
uncertainties than in
remain a useful statistic also
be
the
in
this case, providing at least a lowest order (second moment) estimate of
the statistical significance,
For a Gaussian distribution
Q), p2@ is a 2 X variable
from data, p2@ is a
However, in analogy with
HatcHing, as
and known (:ovariancc matrix
with p degrees of freedom,
Hotelling-type variable (eL
singlc""pallem case, the
);' (and therefore
If C is estimated ~
Morrison, 1976),
p2 distribution is the
the set of estimated eoefl1cients is not strictly
modified by the
covariance Inalrix.
multiplicmion witl1
The p2 distribution
the
must
inverse );,-1 0 r the
lhcrefnre be estimated
estimated
alst) in
tllis c~se by approximate annlytieal or Monic Carlo techniques,
respect to any linear transrorm alion III a new signal pallclll
property will bc used now
forms an
begitmillg
the signal
similarly
orthonormal base,
of this section
bllSC gv' The
nOI1-orthononnal,
to require that the optimal set of
Similarly, the set of signal pallerns
that orthonormality would not be presumcd
result ani fingerprint solutions (28) wcre
Orthonorm,llity is invoked here only as
interim convenience and will be dropped again in the tinal result)
18
for
then
an
Finally, the climate state space will be tmnsformed,
(34)
to a new coordinate system in which the covariance matrix £ becomes the
unit matrix,
" T C =ACA =1 <:;:; z = = = (35)
The transformalion :;} can be obtained, for example, by first transfonning ,
to EOF coordinates '!! and then normalizing tlle EOF coordinates to unit " ,
variance) '1a \jIa lOa'
In order that tllC signal and detectors ey ' dy remain invaliant under this
transformation, the signal patterns must transform in thc samc way as the
climate state vector,
(36)
while the fingerprints tranSfOll1l as adjoint vectors,
(37)
After these transfol111utions, the statistic p2 reduces [0 the Euclidean
fonn
(38)
It follows immediatcly that the optimal set of fingerpri nt5 which
maximizes p2 for any signal ,!!S lying in " signal patterns gIl is given by
(or any equivalent rotated u
the base g) For this v .
orthonormal base
solution, p 2 =
the space spanned
which spans the
1 '!!" 12
, while
by the set of
(39)
same space as
for any other
fingerprint space, part of the sign;li will be lost in the projection onto
19
2 1 "'s 12. the fingerprint space, so lImt p S :t
Transfomling back to the original coordinates, Eq.(39) becomes, applying
Eqs. (36) and (37),
f -v
or, invoking Eq.(35),
T A A " "'v
(40)
(41) (=(28)
The 0l1hononnality conditions imposed 011 the guess pallcl'lIs J!,v and " fingerprints Iv in the transformed climate stale space transfonn into
similar condilions in the original
products defined
climatc state
now with respect
coordinate
to the
system, but
non-Euclidean with the scalar
metrics ~-l and ~, respectively. However, thcse orthonomlality conditions
can now be dropped, since only the signal and fingerprint spaces as such
arc of interest, the choice of base for either space being arbitrary. The
signal base vectors can therefore be identified with thc original guess
pMtems,
fingerprints
without normalization and orthogonality restrictions, and the
can be Similarly dcllncd, witllout ortllonormality
considerations, by the relation (41). Thus the optimal p-dimensional
detector vector d is identical to tile detcctor vector found prcviously for
the prescribed-pattern casco
It can be shown for the optimal fingerprint solution (using Eqs. (4)-(7)
and (27)-(33)) tilal the multivariate significance measure p2 for any signal ~s = g lying in the signal space gv is identical to the
single-pattern square signal-to-noise ratio R2:
p2(~sl = ~T !;?-l d T -1 (ds)2j < a 2> R2 (42) =g!;;g '" -
The result (42) holds only for the optimal fingerprints and tor the siglml
~s itself, not for the net response ~ consisting of lhe signal plus noise.
conlains (For the case that 1 <p2@> = p, while R = L)
no signal, l()r example, ~ = jij, onc finds
There remains now the
the detect ion vector d.
second step of attribliling an estimated signal ~c lO
Adopting again the view that the estimated signal
20
should reproduce the true signal when the noise eontHm ination of the
observed response ~ is negligible (or, equivalenlly, should exhibit the
smallest n1ean~squarc deviation rrom the true signal in the prescn~c of
noise) one finds, in analogy wil11 the singlc-pallern case,
ul=\d b ;r L V-y (43)
where the base VCCLOrs £y of the estimated signal arc given by
(44)
with
(45)
Thus (he estimaled signal associmcd with the set of maximally significant
detectors dv is gi ven by
)[Ie = L (1f~~~I)[I) (~r\y If" (46)
~L, Y
[n H, an altemative derivation of the maximally significant signal
estimate was given which yielded the same set of optimal dctcctors dy ' but A
a different set of base vectors Qv' and thus a different optimally
estimated signal pattern. Starting from the gcncml form (43) (with b A ~
replaced by ~v)' the detectors d" were first detcrmined by a least square A
fit to each realization )[I for a fixed base !!v' Subscqucntly, thc base was detcrmined by applying a maximal·significance condition to the set of
detectors. This yielded (as already menlionec\ fo!' the single~pattcrn case)
U,e solution ~y = Lv = C 1 gv' In contrast to the base ~v found in the
present analysis (Eq. (44», which defines the same space as the signal A
space Ji.v' the space different from the signal
spanned by the fingc'print base ~v is gene!'ally
The present result
solution using an
metric'
space.
(46) can be derived also as altcrnative definition or the square crror
C~ I rather than the usual Euclidean ~
leasl ~square~error
based on the
metric This 'significance
con'esponds to finding an estimated signal in the prescribed signal space
21
[or which the probability llllll the residual error reprcse11ls a realization
or tile natural variability ensemble is maximize';.
Representing the obscrved climalc response as the superposition
of an estim atcd sigmll
)IIc = A
\' c !' L v;;;:v V
(47)
(48)
in the signal space &1' and
error norm
I' a residual error )J!. minimization of the square
(49)
one obtains in this case UIC solution
1\ [(0-1) (,TC-I c = )II) v = V~l gft '"
(50)
~l
Equations (48) lind (50) yield the estimated
previously. but expressed now in lerms of Ihe
signal
original
(46) derived
set of base
functions g v instead of the transformed base it
I t should be emphasized
detectors dv = (L: )II) or
derivations, inde[lcndcl1t or
signal.
again A
C is v ' Ihe
thai the
maximized
definition
SLatisticil! significance
and is idclllicni for
o[ the associated
of the
all three
estimated
In practice. multi-pattern signal detection will nOlmally be carried out
in a hieruchical mode: first, a single pallem is tested; if this is
detected as signi ficalll, a second pallem is added. and so forth until the
p2 statistic is no longer accepted as statistically significant The
success of the method depends critically, as always in signal detection
problems, on the realistic choice or the prescribed signal [lallcrns &1"
22
The EOFs cab characterizing the 'pace-time covariance matrix s.;ub
represcnl familiar functions. For a fixed SUb-index (suppressed in lhe
following relations), the vector 'VCi,l) = 'Vl can first be decomposed into
EOFs eft with respect to the time index. As pointed oul in Section 3, for
a statistically stationary process, Ihe limc-(lomain EOFs arc simply the
hanllonie fUlletions of the Fourier representation (Illis foliows from thc
statistical orthogonality of the Fourier components, although these arc
nOI normally ordered with respect [0 variance, as in other applications).
Diagonalizatioll of Ihe covariance spectra with respect to the remaining
index characterizing the climate variables ami spatial coordinates
(rcfcl'cd to for brevity in tile following as U,C 'spalial' index) yields
then the standard complex EOF represelllation of the covariance spectrum of
a lIlulti-variate process (Wallace and Dickinson, 1972, Bamcll, 1983).
The complete
consists of a
space-time SCI of BOFs
different set or complex
Cab is generally
BOFs with rcspect
vcry large: it
to the spatial
index i for eacl1 frequency bund r of Ihe SpectlUnl. In practicc, it will be
difficult both to cstimale such a large set of functions from a finite
data set and to work with the complete representation in numcrical
computations. One will therefore need 10 resort to some form of
approximation or simplified model based on a reduced set of functions. Two
such models me considered in the l'olloll'ing. The first assumes statistical
separability of the lime and spHlial dimensions, while the second uses a reduced Principal Oscillation
1988, v.Slorch et aI, 1988).
Separability
defined ilere
of
as lhe space and
tile propel'll'
Pattern (POP)
time coordinmcs
thm lhe
represenla!ion (Hasselmann,
and t, respectively, is
call be factorized
inlo a spatial ei"cllvceto!' cS. and il temporal
o JI
eigenvector eab
eigenvector eh,
(51)
23
\vhcre a (i ,t), b (i,I) and the upper indices sand t will be used
gencrally
respectively.
to distinguish between spatial and temporal quantilies,
This propCl'ly holds if thc time dependent coefficients of the sWndllrd
spatial EOFs are complelel), uncorrelated, i.e. if the coefficients are
uncolTclatcd not onl)' for
definition of the spatial EOFs,
the same tillle arguments, as
but also for non-zero time lags.
impliCit in the
Cnder these condiliol1s the spatial eigenvectors satisfy the eigenvalue
equations
IC s s 2 s (52) (i,l)(i,u) ekj (011l ,k) e
ki j
in which the time indices t,u appear as parameters. Only the eigenvalues
(0~U,k)2 depend on these paramcters, not the eigenvectors themselves.
The temporal eigenvectors satisfy tile eigenvalue equmiolls
\' (s )2 et I.. 0 m,k fu (53)
u
which yields then for tile flill system tile cigcllv,llue equation
\' C e S c t I.. (i,t)(j,u) kj fu j, U
Tilc eigenvalue (0(k.O)2 rcprcscllIs the variance
of the Hutovanance spectrum of the k'th spatial
s),stem, the full space-time climate CDvariance
characterized by the spatial EOFs cki and 2
0(k,r)
(54)
of the !"th spectral band
EOF. Thus for a separable
matrix Cab is completely
their aUlOvariance spectra
In tenns v ,
of EOF coordinates (g(k,f)l
now as a superscript to relieve index
(writing the signal-pattern
congeslion) the expressions
index v for the
24
guess-signal patterns take the form
which yields for the fingerprints
The hasie advantage of
first be determined in
a separable system
tile standard manner
is thai
without
(55)
(56)
Ihe spatial EOFs can
filtering in the timc
domain. The analysis in the time-frequency dOlllllin can thell be carried out
for each EOF separately as a second step.
As illustrmion, consider singic-pattern signal consisting of a superposition of a number of spalial EOPs C~i'
(57)
where each of Ihe time dependent spatial-EOP coefficients Ykt can be
represented as a linear trend,
(58)
with mk = cons!.
Taking the Pourier trnnsform of Ykt one obtains for the coefl1cient of the
signal. in EOF coordinates.
,IT exp( - n:if/T) / 2sin(n:f/T) for f '" 0 (59)
for f = 0
The IIngerprint C,Hl then be obtained from Eq (56).
25
a)
-1
-4
-5
IT
0.5
E ·c e-j 0.0 r--
-0.5
-1.0 0 10 20 30 40 50 60 70 80 90 100 t
Fig. 1: Noise spectra 0 2f (panel a) and optimal fingerprints (panel b) for the
detection of a linear time-dependent signal in presence of power law noise. Definitions of the cases a - i and values of the associated signal-to-noise enhancement factors are given in Table I. The length of the time series is 100 (0 $ t $ 99; -50 $ IT $ 49; only the positive-frequency branehes of the specn'a are shown.)
26
__ M
Case (cf. Fig. 1) I a b c d e f g h i :
spectral power q
!
0 0.2 0.4 0.6 0.8 1.0 1.5 2 !
3
. --: :
: enhancement factor E : I 1.01 1.05 1.14 1.27 1.5 3.1 10.4 > 100 I i
i I I Table 1: Signal-to-noise enhancement factors E = [R2 (optimal fingerprint) IR2 (signal pattern)]
for linear signal and various power law spectra.
Fig. J shows the spcclrn nnd fingerprints f(i,l) computcd far red power-law
spectra, (J~k,f) = ~ (I' + T- i (1 1'01' various values of q> 0 (thc frequency
is offset by one disc!'Cle frequency unit T- l to avoid the singularily at f
0), Thc enhancement of lile square signal-la-noise !'alio R2 computed with
me optimal jingerpriIll relativc 10 the refcrence non-optimized case in
which the fingerprint is simply set equal to the signal pallclll is shown
in Table 1. The computations were carried OUi for T = 100 discrete time
sleps,
The curves detliOnSIl'ale Ihnt the optimal detection fingerprint generally
differs
by a
significantly
straight-line.
frolll
The
the nOil·optimized
non-optill1ii,ed solution
fingerprilll,
reduces
which is given
to the estim atioll of a lincarly increasing signal by
in the present case
standard method the
of constructing a regression line through the data, However, this
represents the optimal analysis method only for a whilc-lloise natural
vaJiability spectrum, For rcd the optimal weighting is
27
I
!
distributed more
spectra steeper
towards
than -I I' ,
the
the
end-points or the time series. For power-law
optimal detection strategy is to use the end
points of the lime series only. The enhancement of the statistical
significance through optimization of the fingerprilll can be quite large
for steep spectra.
An allel11ative method of reducing the complexity of the 1'1111 covaJiance
matrix Cab wi thout introducing the rather srlingenl assumption of
space-time separability is to represent the natural climate variability as
a superposition of a !inite number of Principal Oscillation Patterns
(POPs, d. Hasselmann, 1988, v.Storch et al., 1988). The basic idea of the
POP method is to combine an EOF-type pattern expansion in the spatial
domain with an ARlvlA-type dynamical modeling approach in the time domain.
In the original papers of Hassclmann and v.Storch et al. (and in a number
of subsequent applications, cf. Xu, 1992a, 1992b, Latil et aI., 1992a,
1992b), the POP method was regarded primarily as a technique for
constlUcting simple dynamical models, usually for forecasting or
diagnostic purposes. Howcver, the POP method is equally useful as an
approximate muili-variate spectral compression technique.
Tile POP method approximates the natural
here to the usual decomposed notation) as
damped oscillations 0:,
I 0: . I \Y lcrc Pi IS a constant comp ex pattern,
0: 0:( I) + i po:(2) Pi = Pi I
and the complex amplitude
satisfies the damped oscillator equation
28
variablility ~ = ~.(t) (retul11ing a 1
a supelPosition of anum ber of
(60)
(61 )
(62)
(63)
with eigcnrrcqucncy [.fl. ( > 0) nnd damping factor )/J. ( > 0); IP(t) is a
complex white-noise rorcing fUllction,
(64)
Decomposed into spectral comlxments <\J Y'(f) (to avoid notational
proliferation the Slune symbols will be used for lime and frequency-domain
functions) the solution of (63) is given by
(65)
where
(66)
is the POP transfer function, yielding for the POP oscillalion (60)
<jJ~(t) I TO:(r; lP'(f) P~ e,p (2rcirt) + complex conjugate
f
I ['1'0:(1) n''-(O py + T('-(-I)' n('-(-I/ prj exp (2rcifl) (67)
f
(nOie IIlat Ta(O '" TO:(_ f) *, n(I.(1) '" n a(t/ since both the POPs and the
noise forcing are complex).
'vl/hile
rotating
seiluence
consists
opposite
the free
clockwise po:(l) _>
I
POP solution consists of
in the _ p a(2)_>
I
complex (J.( I) >
-p i -
plane, po:(2)
I
generally
directions.
of a superposition or These arise from
frequencies ill the rcpreSCIll1l1 ion of
a single damped oscillation
patlel11 by the characterized _> P <?(l), the forced solution
two
the
n(I),
I
oscillations rotating in
positive
which
and
force
negativc
both the
basic POP pair and the complex conjugate pattern pair.
Assuming that the forcing components nU, nP for differcnt POPs a and pare
ullcorrelaled, thc complex cross SPCCtlUIll of the process
by
29
iii. is thcn given 1
F .. (JJ = <[$.(f)}* $.(/» IJ I J
O.
where
NO. = <,,rl(l) * net(lJ> = const
Met = <n(l.(O n(l.(.I» = consl
(69) (70)
The general expression (68) can be simplified by assuming that the
excitation of the conjugate POP pair, which occurs at negalive frequencies
rar removed rrom the resonant POP eigcn·rrequencies, is neglible. In this
case the second and third tcons in (68) are small for positive f (and
similarly thc first and third tenns I()r negative fl, so tilal for f > 0,
(68) reduces to
The corresponding
relation F.(!) IJ
example considered
exprcsssion * F.(I) .
IJ below, it
only the third telll1 in (68),
(71 )
for f < 0 follows from (71) and the symmeli),
(In applications, such as Ihe numcrical
will generally be more convenient 10 drop
as Ihis avoids a discoIHinuous change in thc
expressiDn for lile covariance spectrul11 which otherwise occurs at zero
frequency.)
The simplified POPs rcprcsc11lalion oj' thc Cl'Oss·specl'ul11 is seen 10 have
the same form as tile standard eom plcx EOF rcprcsc11lation
(72)
of thc cross·spcctrum in tcn1lS of complex EOFs e". The variance spectllJl11
FV (1) = < leV (f) 12> of the cocrricicnls c v (I) of the I complex EOF expansion
correspond to the spectrum N°'1 T(I.(O 12 of the POP repl'CSClllHlion while the
30
complex EOF CY itsclf corresponds to the Prillcipal Oscillation Pattcm p~, However. in contrast to the standard rcpresel1lalion, the cross-spectrum is
now no longer decomposed into a different SCI of EOFs for each spccll1Il
hand. but into a single set of complex POP pallerns applicable for the
cntire spcctrum,
The COllllibutioll of individual POPs to dillercnt spectral bands is
detem1ined by the weighting factor I T<\f) i 2 in (71). As pointed out,
individual POPs will contribute mainly to spectral bands in the
neighborhood of the POP eigcnfrcqucndcs {( the effective spectral
handwldth being proportional to the damping factor ,,0..
v In eontrasl to the twe EOF, C i' the complex POPs do nOl represent tile
eigenvectors of an Henllitian matrix ilnd will therelorc generally not be
(spatially) orthogonaL They can be readily orthogona!ized, however,
through a suitable (frequenc)' depcndenl) complex rOlmion p~' ->
f1.' I so.p p~ where So:p is unitary matrix: I (so:y{ sPy oo:P, The p, cO , a cO
I I
P y transformation preserves Ihe essenlial sl atistical orthogonality of the
POP coeflicients,
Thc detcl1llinmion of the approximate form (72) is normally carried out in
the time domain by filling n first order vcctor MlIl'kov process to the data
lime series (ef. v,Storch et ai, 1988), Howcvcr. a direct Ilt of the model
covariance spectrum to the observed covari,mcc spectnllll in the frequency
domain. using methods applied by Frankignoul and Hasselmann (J 972), Lemke
et aL (1980). HCl1crieh and Hassclmann (1982) and Dobrovolsky (1992) in
similar problems of stochastic modet fitting, should be feasible also in
lie POP-model case. Model filling in the spectral domain generally has the
advantage of providing better quantitative estimates of error hounds.
To illustrate tile impact of optimal filtering for a POP
consider the case of a signal lying in the pallcrn space
POPs, Since the di I'fcrent POP pairs are assumed 10
orthogonal, one can consider each POP signa! component
31
noise spectrum,
spanned by the
bc statislically
g.(t) = y(t) p. + complex conjugate 1 1
(73)
separately (y(t) denotes a complex time .. dependent coeJ'licicnt, and the
POP index a. has becn dropped).
Tile optimal fingcrprint
fiCt) = <\l(L) Pi + complex conjugatc (74)
can be rcpre~ented in this case in closed [01111. In the Fourier domain, the
complex fingerprint coernci~nt is givcll hy (Eqs.(14),(7J»
(75)
which yields in the time domain
The fingerprint is scen to havc the right structurc to reduce the POP
noisc conllibutions. For a modulation factor Y(1l cxp (2rcifot ),t)
eOlTcspondig to 11 pure POP oscillalion, Eq. (76) yields <\let) = 0, i.e.
pure POP oscillmions are rejected by tile fingerprint. (This result
appears paradoxical, since the fingerprint is defined in the Fourier
domain as the product of thc signal and the inverse noise SpeCtl1l111, which
cannot vanish identically. The explanation is that a pure POP oscillation
is not a permitted signal form, since it becomes infinitc lor t -> - ~, so
that its FOUlicr transform docs nnt exist. A signal which is zero [or t $;
o and represents a POP oscillation only for > 0 is not completely
removed by the fillgclprint.)
Fig.2
'\let) sholYs
for the
the POP spcctr;\
casc of a linear signal
100 lime incrcments, for taking again T
and 20 and the ), = 20
fingerplint modulation factors
modulnlion factor yet) thc frequency values r , 0
(measured in rrequency
(Eq.(58»,
(J, 10
increment . T- 1) unHS . For
damping
a linear
factor
signal (which appears as a periodic saw-tooth in
32
a)
1.0
0.8
E 2 t3 0.6 '" 0.
'" tI> 0.. 0 D..
0.4
0.2
-30 -10
-0.5
10 30 50 IT
. . . , . , . I 60 70 80 90 100
c)
oF=================~
-5
-15
-25 0 10 20 30 40 50 60 70 80 90 100 t
Fig. 2: Noise spectra (panel a) and real and imaginary components (panels, b, c, respectively) of optimal fingelprints for a POPs spectrum with damping factor 'J- = 20 and frequencies foT = 0, 10,20 (the imaginary fingerprint component vanishes for foT=O and is the same for fOT=10 as for fOT=20 to within a factor). Signal-to-noise enhancement factors for various values of I., and foT are given in Table 2. The length of the time series is 100 (05 t :s; 99; -50:S; IT :s; 49).
33
~ 0 10 20
I
10 3.5 11.21 1.16
I
20 1.63 1.12 1.08
40 1.14 1.06 1.04
i i
Table 2: Signal-to-noise enhancement factors E = lR2 (optimal fingerplint) I R2 (signal pattern)] for linear signal and POP variance spectra for variolls eigcnfrcqucncies fO and damping factors A.
the Fourier sum representation) the lirst derivative in Eq.(76) consists
of the sum of a constant tenn and a negative o-function (negative spike)
al each end of the lime illlcrval. The second derivative is given by the
derivative of a o-function, which in the discetc representation takes the
form of a posilivc and negative and spike at the hegirming and end,
respectively, of the time intclval.
The real pan of the I1ngerplill( modulation factor consists (hen of the
linear signal modulation racLOr
difference in the response at
latter term gaining more weight
unifonn white spectrum. This IS
for red power-law spectra in
separability model.
itseIr plus a term representing the
the endpoints of the tilne interval, the
the lllorc the spectrum differs from a
qualitatively similar to the result found
the previous example of a space-time
34
The imaginary component of thc footprint modulation factor consists of two
equal negative contri butions from thc endpoints of thc lime interval and
an equally weightcd positive contribution averaged over the full time
interval.
The enhanccmcnt of thc square signal-to-noise ratio R2 achieved
optim al fingerprint solution relative to thc reference case
using the
without
optimization is shown in Table 2. As in lire previous example, the largest
enhancement is achieved for noise spectra containing large variance
contributions at low frcqueneies.
The signal pallclll detection method developed by Ii for the time
independent problem can be rcadily extcndcd to the timc depcndent case.
The introduction of thc fingerprint concept leads to a significant
simplification of the thcory both conceptmilly and analytically. The
optimal fingerprint (Iilter) for the deteclion of an anticipated
space-time dependent signal pattern g in thc presence of natural climate
variability noise, characterized by the space-time dependent covariance
matrix ~, is given by I = ~-! g. Application of the optimal fingerprint to
an observed climate trajcctory ~ yields a detector d = (IT~) with maximal
signal-to-noise ratio. This result generalizes immediately to the
mulli-pattcm case: the set of fingerprints Iv= ~-I gv associated with a
set of p guess pallerns gv yields a set of detectors dv = (I~ ~s) for which
the relevant p-dimensional statistical-significance statistic p2
= L dvD~~d~l is maximized for any signal ~s lying in the space spann cd by
v, ~l the set of guess pallems gv' Here
covariance matrix of the dctector noise
variability ili.
The direction of the Jingerprint
direction of the associated
vector
signal.
interpretations of the CS(imaled signal
g~l) represcnts the
induced by the clim ate
normally di ffers from the
This permits alternative
inferred from the set of
dcteetors. In H, lire detcction and cslimaLion problems were regardcd as coupled parts of a single problem, and the estimatcd signal was defined to
35
lie in the space spanned by the sct of fingerprint veclors. In the present
analysis, the detection problem was solved first and the attribution of an
estimated signal iO the set of detectors was addressed subsequently as an
independent problem, From this viewpoint it appears more consistent to
regard the estimated signal as lying in the space spanned by the
prescdbed signal pallerns, given for eHller Arguments ean be
interpretation, The statistical eSlimated signal is significance of the
detcmlined in both cases dctectors and is thus by identical sets of
independent of thc interprctation.
Thc detection technique call be applied to any set of observed or
model-simulated data for which the second momcms can be adequately
estimated, independent of the completeness of the data set with rcgard to
the dynamical dcscliption of thc climate system,
Two practical difllculties are encountered in applying the technique,
First, a complete description of the space-time dependent covariance
matrix of the natural climate variability noise involves large quantities
of infollnation which cannot nonnally be effectively handled and also
cannot be inferred from a finite amount of observed or simulated data,
Thus some form of simpli ficd
expansion in POPs (Ptincipal
slatistical model must imroduced. An
Oscillation Pallcms) provides an
gencral reduction techniquc. In some cases a still simpler
separability model may be applicable. Examples given for both
effective
space-timc
types of
model dcmonstrMe that the optimal fingerprints can deviate significantly
from the
signal-to-noise
original
ratios
signal
compared
pattclll
with
and yield
straightforward
considerabl y
projection
enhanced
onto the
signal pallem, The strongest enhancement is obtained for red speetra with
high variance conlributions :11 vcry low frequencies,
Secondly, the optimally estinwtcd detectors dv have a known Gaussian
distribution only if the naturnl climate variability is Gaussian with
known (rather than estimated) covarinnee m:ltl1x. In the normal ease that
tile climate variability is nOll-Gaussian or the covariance matrix is
estimated from a limited data scI, the stntisticai significance of tlte
computed detectors must be cstimnlcd by Monte Carlo simulations or other
approximate techniques,
36
Not addressed in the pl'cscm paper were problems or data (un~Crlaintics were associated solely witil the natural
errors
climatc
patlCl1l,
allow for a priori probabilities of
variability) or qucstions related [0 tile
including extensions of the theory to
the anticpated signal distribution within
intended to pursue thcse qucstions
Oleo!'y.
Critlcal reviews of a first draft or
delinition of the signal
a prescribed signal space. It is
latcr in appliCiltions of the
this
helpful commelllS by Hans Vall Storch,
manuscript and a number of
Bcnjamin SanteI' and Wolfgang
Brilggcmann are gratefully acknowledged.
37
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