Phase Space Reconstruction and Predictability of
Spectral Solar Irradiance from SOLSTICE
Guoyong Wen1,2, and Robert F. Cahalan1
1NASA/Goddard Space Flight Center
2Goddard Earth Science and Technology Center, University of Maryland, Baltimore Co
Introduction
Example of Prediction
y λ (n) = [Iλ (n),I λ (n + T),Iλ (n + 2T),...,Iλ (n + (d −1)T)]
The Method of Analogues
λ
λ
Quasi-Annual Cycle
The average mutual information is a m easure of
depende ncy bet ween t wo rando m variables
I( X ,Y ) = ∑ ∑ r( x i , y j ) log
i
y(i+3)
Figu re 2. In the met hod of ana logues
one f inds the nea rest ne ighbo r of the
cu rrent state at t im e ste p L in the
vecto r space. The ev olut ion of phys ica l
process afte r tim e step L i s ana logue
to that fo r its nea rest ne ighbo r at t im e
step i.
Figure 4. The normalized average
AMI(i)
) as a
mutual information ( NAMI(i) = AMI(1)
function of time lags for different
wavelengths. A p eak near 360-day lag
is clear for wavelength > 300 nm.
Average Mutual Information
A simple way to p redict a time series is the
method of an alogues. In th is method one looks at
the past ( y (1),y (2),...,y (L)) to f ind the vecto r y λ (i) that
is nea rest to the latest y λ (L) in the d-dimensional
space. The sca lar values afte r i’th obse rvation
(i.e., Iλ (i + 1),Iλ (i + 2),Iλ (i + 3),Iλ (i + 4),...) are the p redictions
after time step L (i.e., Iλ (L + 1),Iλ (L + 2), Iλ (L + 3),Iλ (L + 4),....).
λ
r
y (i)
Figure 3. (Upper panel): To predict the
SSI after time step L, we find the
vector
y(i) = [I(i), I(i − 1),..., I(i − N + 1)] (dashed) is the
closest to the cu rrent vector y(L) (solid),
where L = 2199 (UARS days), i=1623
(UARS days). (Lower panel): The
observed SSI aft er i’th step is used as
prediction after L’th time step.
y(i+2)
y(L+2)
y(i+1)
y(i)
y(L+1)
I(T) = ∑ r(s(n), s(n + T )) log
y(L)
Autoco rrelation
C(T) =
∑ (x
i
i
− x)(x i +T − x)
∑( x
i
− x) 2
• A quasi-annual cycle in those wavelengths
is evident.
r(s(n),s(n + T ))
p(s(n)) p((s(n + T))
i
Summary
• The predictability of SSI is associated with
a distinct peak in relative mutual
information at 360 day time lag.
is often used in time
series analysis. The autocorrelation measures
the linear dependency of a time series and the
delayed time series. The average mutual
information is a m ore general measure of
depende ncy bet ween t wo rando m variables.
Figure 5. The enhancement parameter
is defined to characterize if there is a
peak near 360 day lag in mutual
information. Similar to core to wing
ratio, the enhancement parameter
2AMI(360)
E=
is large when a peak is
AMI(340) + AMI(380)
evident. Two spectral regions are
distinctive.
• The method of analogues is used to predict
SOLSTICE time series in the time delayed
phase space. The prediction is reasonably
well in some spectral regions (e.g., >
300nm) in a time scale of year.
r( x i , y j )
p(x i )q( y j )
where r(x, y) is the joint probability distribution of
x and y , and p(x) and q(y) are marginal
probability distributions of x and y . When x
y are indepen dent to each other
and
r(x, y) = p(x)q(y) , the ave rage mutual information is
zero.
The averaged information between obse rvations
at t ime step n and a time lag T later, namely, the
average information about s(n+T) we have when
we make the obse rvation of s(n), is
n
y(L-1)
j
Scaled Irradiance
Figu re 1. Tim e se ries of SOLS TICE
spect ral irr ad iances at (top) 121.6 n m
(Ly man - α ), (mi dd le) 280 n m (M g-II),
(botto m) 393 n m (C a II ).
where Iλ (n) is observed spectral irradiance at
wavelength λ and time step n. The time delay T is
determined such that the ave rage mutual information
between a time series and a de layed time series
reaches minimum. The dimension d may be
determined by th e false nearest neighbor analysis, or
simply allowing y λ (n) to span 8~1 0 27-day rotation
cycles. y λ (n) is a data po int in d dimensions vector
space at time step n. The method of analogue is
used to predict the time series.
In a sepa rate study we demonst rated the low
dimension for wave length < 280 nm and h igh
dimension for wavelength > 280 n m. In this study
we found that SSI may be predictab le in spect ral
region > 300 nm using the m ethod of ana logues.
Furthermore, we found the average mutua l
information has a peak at ~ 360 -day time lag for
those wavelengths. The time series of SSI at
those w ave lengths appea r to have a 360 -day
cycle (Fig. 6 ).
Normalized Mutual Information
Observations of TSI and SSI have been no w made
for a relatively long time, allowing us to pe rform
statistical analysis. This work focuses on the
predictability of spect ral solar irradiance by ana lyzing
SOLSTICE data. Inst ead of scalars we analyze the
time series in the vecto r space. A time delay T ddimension vector is constructed
Predictability vs Mutual Information
UARS DAY
Figure 6. Scaled SOLSTICE solar
irradiance at 363 nm as a function of
UARS days. A 360-day cycle is
evident.