1. A robust seasonality detector for geophysical time series: application to satellite SO2 observations over China
Taylor, M.1*, Koukouli, M.E.1, Theys, N.2, Bai, J. 3, Zempila, M.M.4, Balis, D.S.1, van Roozendael, M.2 , van der A, R.5
1 Laboratory of Atmospheric Physics, Aristotle University of Thessaloniki, Greece.
2 Belgian Institute for Space Aeronomy, BIRA-IASB, Brussels, Belgium.
3 Institute of Atmospheric Physics (IAP-CAS), Beijing, China.
4 Natural Resource Ecology Laboratory, Colorado State University, Fort Collins, USA.
5 Koninklijk Nederlands Meteorologisch Instituut (KNMI), De Bilt, The Netherlands.
* URL:
COMECAP 2016, Thessaloniki ,19th of September: Remote Sensing of Atmospheric Composition Session 14:30-16:00

2. Motivation:
Geophysical time series often exhibit broad peaks on top of a continuous background (trend + periodicity + noise) BUT
Trends are often nonlinear
Peak frequencies which “fingerprint” limit cycles (of the underlying dynamical system) are often not sinusoidal
Noise blurs spectra and makes it difficult to provide confidence intervals
 Need for a general cycle detection tool for real data
Radiative forcing estimates in 2011 relative to 1750 and aggregated uncertainties for the
main drivers of climate change. Values are global average radiative forcing (RF15) partitioned according to
the emitted compounds or processes that result in a combination of drivers. The best estimates of the net
radiative forcing are shown as black diamonds with corresponding uncertainty intervals; the numerical values
are provided on the right of the figure, together with the confidence level in the net forcing (VH – very high, H
– high, M – medium, L – low, VL – very low). Albedo forcing due to black carbon on snow and ice is included
in the black carbon aerosol bar. Small forcings due to contrails (0.05 W m–2, including contrail induced
cirrus), and HFCs, PFCs and SF6 (total 0.03 W m–2) are not shown. Concentration-based RFs for gases can
be obtained by summing the like-coloured bars. Volcanic forcing is not included as its episodic nature makes
is difficult to compare to other forcing mechanisms. Total anthropogenic radiative forcing is provided for three
different years relative to For further technical details, including uncertainty ranges associated with
individual components and processes, see the Technical Summary Supplementary Material. {8.5; Figures
8.14–8.18; Figures TS.6 and TS.7}
Ghil (2002) Encyclopedia of global environmental change 1:
COMECAP 2016, Thessaloniki, 19th of September

3. Signature of an annual cycle on a nonlinear trend
Etheridge et al (1998)
data.okfn.org/data/core/co2-ppm
Change in level due to trend
Radiative forcing estimates in 2011 relative to 1750 and aggregated uncertainties for the
main drivers of climate change. Values are global average radiative forcing (RF15) partitioned according to
the emitted compounds or processes that result in a combination of drivers. The best estimates of the net
radiative forcing are shown as black diamonds with corresponding uncertainty intervals; the numerical values
are provided on the right of the figure, together with the confidence level in the net forcing (VH – very high, H
– high, M – medium, L – low, VL – very low). Albedo forcing due to black carbon on snow and ice is included
in the black carbon aerosol bar. Small forcings due to contrails (0.05 W m–2, including contrail induced
cirrus), and HFCs, PFCs and SF6 (total 0.03 W m–2) are not shown. Concentration-based RFs for gases can
be obtained by summing the like-coloured bars. Volcanic forcing is not included as its episodic nature makes
is difficult to compare to other forcing mechanisms. Total anthropogenic radiative forcing is provided for three
different years relative to For further technical details, including uncertainty ranges associated with
individual components and processes, see the Technical Summary Supplementary Material. {8.5; Figures
8.14–8.18; Figures TS.6 and TS.7}
Clear annual cycles (single peak and trough)
These cycles need to be understood in
the context of the overall (nonlinear) trend
Noise is apparent in the small differences
between annual cycles
COMECAP 2016, Thessaloniki, 19th of September

4. Satellite SO2 over China (apriori gap-free data)
Radiative forcing estimates in 2011 relative to 1750 and aggregated uncertainties for the
main drivers of climate change. Values are global average radiative forcing (RF15) partitioned according to
the emitted compounds or processes that result in a combination of drivers. The best estimates of the net
radiative forcing are shown as black diamonds with corresponding uncertainty intervals; the numerical values
are provided on the right of the figure, together with the confidence level in the net forcing (VH – very high, H
– high, M – medium, L – low, VL – very low). Albedo forcing due to black carbon on snow and ice is included
in the black carbon aerosol bar. Small forcings due to contrails (0.05 W m–2, including contrail induced
cirrus), and HFCs, PFCs and SF6 (total 0.03 W m–2) are not shown. Concentration-based RFs for gases can
be obtained by summing the like-coloured bars. Volcanic forcing is not included as its episodic nature makes
is difficult to compare to other forcing mechanisms. Total anthropogenic radiative forcing is provided for three
different years relative to For further technical details, including uncertainty ranges associated with
individual components and processes, see the Technical Summary Supplementary Material. {8.5; Figures
8.14–8.18; Figures TS.6 and TS.7}
Point source megacities (purple triangles) & power plants: orange diamonds (<2200 MW), blue squares ( MW) & green circles (>3500 MW) overlaid on the smoothed mean SO2 load over China (2011).
Koukouli, Balis, van der A, Theys, Hedelt, Richter, Krotkov, Li & Taylor (2016) Atmos Env 145: 45-59
Trend is “bumpy” and nonlinear
Annual cycle is apparent (summer peak and
winter trough)
COMECAP 2016, Thessaloniki, 19th of September

5. What does Fourier analysis tell us?
Subjective threshold 3 modes
3 modes produces a decent fit (R=0.891)
BUT: it is not clear that this is better than
fitting with a single mode (R=0.858)
Radiative forcing estimates in 2011 relative to 1750 and aggregated uncertainties for the
main drivers of climate change. Values are global average radiative forcing (RF15) partitioned according to
the emitted compounds or processes that result in a combination of drivers. The best estimates of the net
radiative forcing are shown as black diamonds with corresponding uncertainty intervals; the numerical values
are provided on the right of the figure, together with the confidence level in the net forcing (VH – very high, H
– high, M – medium, L – low, VL – very low). Albedo forcing due to black carbon on snow and ice is included
in the black carbon aerosol bar. Small forcings due to contrails (0.05 W m–2, including contrail induced
cirrus), and HFCs, PFCs and SF6 (total 0.03 W m–2) are not shown. Concentration-based RFs for gases can
be obtained by summing the like-coloured bars. Volcanic forcing is not included as its episodic nature makes
is difficult to compare to other forcing mechanisms. Total anthropogenic radiative forcing is provided for three
different years relative to For further technical details, including uncertainty ranges associated with
individual components and processes, see the Technical Summary Supplementary Material. {8.5; Figures
8.14–8.18; Figures TS.6 and TS.7}
COMECAP 2016, Thessaloniki, 19th of September

6. Required (usually linear or constant)
A nonlinear approach: Singular Spectrum Analysis (SSA)
Step 1: Choose window size (“embedding dimension” or “number of time lags”) M ≤ N/2
Step 2: Construct the “trajectory” matrix X using M-lagged copies of the time series (i.e. “Hankelize”)
Step 3: Calculate the lag-covariance matrix C of X
Step 4: Diagonalize C  eigenvalues λ which are the artial variance in the direction of each of the eigenvectors E (“empirical orthogonal functions or EOFs”)
Step 5: Calculate break-point of eigenspectrum separating high-variance quasi-periodicity from low variance (trend + noise)
Step 6: Calculate time-varying principal components (PCs) by projecting the time series onto each E
Step 7: Reconstruct groups of PCs by de-Hankelizing
Seitola et al (2015) Tellus A, 67
Ghil et al (2002) Rev Geophys 40(1:3):1-41.
Radiative forcing estimates in 2011 relative to 1750 and aggregated uncertainties for the
main drivers of climate change. Values are global average radiative forcing (RF15) partitioned according to
the emitted compounds or processes that result in a combination of drivers. The best estimates of the net
radiative forcing are shown as black diamonds with corresponding uncertainty intervals; the numerical values
are provided on the right of the figure, together with the confidence level in the net forcing (VH – very high, H
– high, M – medium, L – low, VL – very low). Albedo forcing due to black carbon on snow and ice is included
in the black carbon aerosol bar. Small forcings due to contrails (0.05 W m–2, including contrail induced
cirrus), and HFCs, PFCs and SF6 (total 0.03 W m–2) are not shown. Concentration-based RFs for gases can
be obtained by summing the like-coloured bars. Volcanic forcing is not included as its episodic nature makes
is difficult to compare to other forcing mechanisms. Total anthropogenic radiative forcing is provided for three
different years relative to For further technical details, including uncertainty ranges associated with
individual components and processes, see the Technical Summary Supplementary Material. {8.5; Figures
8.14–8.18; Figures TS.6 and TS.7}
Fourier Analysis
SSA
Basis Functions
Sines & Cosines
Limit Cycles
Variance Measure
Coefficients
Eigenvalues
Periodicity
Sinusoidal
Quasi-Periodic
Stationarity
Required (usually linear or constant)
Nonlinear trend
Noise
Implicit
Explicit
COMECAP 2016, Thessaloniki, 19th of September

7. SSA components (optimal M=18)
Break in slope of SSA spectrum distinguishes
“signal” from “noise” and can be identified by
the minimum distance length (MDL) of Wax &
Kailath (1985) IEEE 33(2):
Neighbouring pairs of equal variance are suggestive of a single quasii-periodic cycle
Radiative forcing estimates in 2011 relative to 1750 and aggregated uncertainties for the
main drivers of climate change. Values are global average radiative forcing (RF15) partitioned according to
the emitted compounds or processes that result in a combination of drivers. The best estimates of the net
radiative forcing are shown as black diamonds with corresponding uncertainty intervals; the numerical values
are provided on the right of the figure, together with the confidence level in the net forcing (VH – very high, H
– high, M – medium, L – low, VL – very low). Albedo forcing due to black carbon on snow and ice is included
in the black carbon aerosol bar. Small forcings due to contrails (0.05 W m–2, including contrail induced
cirrus), and HFCs, PFCs and SF6 (total 0.03 W m–2) are not shown. Concentration-based RFs for gases can
be obtained by summing the like-coloured bars. Volcanic forcing is not included as its episodic nature makes
is difficult to compare to other forcing mechanisms. Total anthropogenic radiative forcing is provided for three
different years relative to For further technical details, including uncertainty ranges associated with
individual components and processes, see the Technical Summary Supplementary Material. {8.5; Figures
8.14–8.18; Figures TS.6 and TS.7}
(First 6 components only are shown)
COMECAP 2016, Thessaloniki, 19th of September

8. Sensitivity analysis on M
CONDITION:
Maximum R for M > MA(13)
Radiative forcing estimates in 2011 relative to 1750 and aggregated uncertainties for the
main drivers of climate change. Values are global average radiative forcing (RF15) partitioned according to
the emitted compounds or processes that result in a combination of drivers. The best estimates of the net
radiative forcing are shown as black diamonds with corresponding uncertainty intervals; the numerical values
are provided on the right of the figure, together with the confidence level in the net forcing (VH – very high, H
– high, M – medium, L – low, VL – very low). Albedo forcing due to black carbon on snow and ice is included
in the black carbon aerosol bar. Small forcings due to contrails (0.05 W m–2, including contrail induced
cirrus), and HFCs, PFCs and SF6 (total 0.03 W m–2) are not shown. Concentration-based RFs for gases can
be obtained by summing the like-coloured bars. Volcanic forcing is not included as its episodic nature makes
is difficult to compare to other forcing mechanisms. Total anthropogenic radiative forcing is provided for three
different years relative to For further technical details, including uncertainty ranges associated with
individual components and processes, see the Technical Summary Supplementary Material. {8.5; Figures
8.14–8.18; Figures TS.6 and TS.7}
COMECAP 2016, Thessaloniki, 19th of September

9. SSA trend (cross-validation)
The statistically significant trends in D.U. per decade for 137 OMI_BIRA point sources having N>30 monthly averages.
Radiative forcing estimates in 2011 relative to 1750 and aggregated uncertainties for the
main drivers of climate change. Values are global average radiative forcing (RF15) partitioned according to
the emitted compounds or processes that result in a combination of drivers. The best estimates of the net
radiative forcing are shown as black diamonds with corresponding uncertainty intervals; the numerical values
are provided on the right of the figure, together with the confidence level in the net forcing (VH – very high, H
– high, M – medium, L – low, VL – very low). Albedo forcing due to black carbon on snow and ice is included
in the black carbon aerosol bar. Small forcings due to contrails (0.05 W m–2, including contrail induced
cirrus), and HFCs, PFCs and SF6 (total 0.03 W m–2) are not shown. Concentration-based RFs for gases can
be obtained by summing the like-coloured bars. Volcanic forcing is not included as its episodic nature makes
is difficult to compare to other forcing mechanisms. Total anthropogenic radiative forcing is provided for three
different years relative to For further technical details, including uncertainty ranges associated with
individual components and processes, see the Technical Summary Supplementary Material. {8.5; Figures
8.14–8.18; Figures TS.6 and TS.7}
Koukouli, Balis, van der A, Theys, Hedelt, Richter, Krotkov, Li & Taylor (2016) Atmos Env 145: 45-59
COMECAP 2016, Thessaloniki, 19th of September

10. So, how many cycles? (ANSWER = 1  an annual cycle only)
SUBJECTIVE
Radiative forcing estimates in 2011 relative to 1750 and aggregated uncertainties for the
main drivers of climate change. Values are global average radiative forcing (RF15) partitioned according to
the emitted compounds or processes that result in a combination of drivers. The best estimates of the net
radiative forcing are shown as black diamonds with corresponding uncertainty intervals; the numerical values
are provided on the right of the figure, together with the confidence level in the net forcing (VH – very high, H
– high, M – medium, L – low, VL – very low). Albedo forcing due to black carbon on snow and ice is included
in the black carbon aerosol bar. Small forcings due to contrails (0.05 W m–2, including contrail induced
cirrus), and HFCs, PFCs and SF6 (total 0.03 W m–2) are not shown. Concentration-based RFs for gases can
be obtained by summing the like-coloured bars. Volcanic forcing is not included as its episodic nature makes
is difficult to compare to other forcing mechanisms. Total anthropogenic radiative forcing is provided for three
different years relative to For further technical details, including uncertainty ranges associated with
individual components and processes, see the Technical Summary Supplementary Material. {8.5; Figures
8.14–8.18; Figures TS.6 and TS.7}
OBJECTIVE
(at the 95% level)
COMECAP 2016, Thessaloniki, 19th of September

11. How does Fourier analysis really compare with SSA?
Radiative forcing estimates in 2011 relative to 1750 and aggregated uncertainties for the
main drivers of climate change. Values are global average radiative forcing (RF15) partitioned according to
the emitted compounds or processes that result in a combination of drivers. The best estimates of the net
radiative forcing are shown as black diamonds with corresponding uncertainty intervals; the numerical values
are provided on the right of the figure, together with the confidence level in the net forcing (VH – very high, H
– high, M – medium, L – low, VL – very low). Albedo forcing due to black carbon on snow and ice is included
in the black carbon aerosol bar. Small forcings due to contrails (0.05 W m–2, including contrail induced
cirrus), and HFCs, PFCs and SF6 (total 0.03 W m–2) are not shown. Concentration-based RFs for gases can
be obtained by summing the like-coloured bars. Volcanic forcing is not included as its episodic nature makes
is difficult to compare to other forcing mechanisms. Total anthropogenic radiative forcing is provided for three
different years relative to For further technical details, including uncertainty ranges associated with
individual components and processes, see the Technical Summary Supplementary Material. {8.5; Figures
8.14–8.18; Figures TS.6 and TS.7}
COMECAP 2016, Thessaloniki, 19th of September

12. Seasonality Detection Mean R (linear vs SSA trend)
Seasonality detection for time series with gaps
Historical records usually contain gaps
 need for a robust imputation approach
Seasonality Detection
N (sites) %
Mean gaps
Mean R (linear vs SSA trend)
month cycle 5
3.5 14
0.993
month cycle 44
31.2 16
0.979
month cycle 28
19.9 17
0.984
a 12 month cycle 99
70.2 19
a 6 month cycle 60
45.4 20
0.962
a 3 month cycle 39
27.7
0.971
non-12, 6 or 3 month cycles
135
95.7 22
0.951
Radiative forcing estimates in 2011 relative to 1750 and aggregated uncertainties for the
main drivers of climate change. Values are global average radiative forcing (RF15) partitioned according to
the emitted compounds or processes that result in a combination of drivers. The best estimates of the net
radiative forcing are shown as black diamonds with corresponding uncertainty intervals; the numerical values
are provided on the right of the figure, together with the confidence level in the net forcing (VH – very high, H
– high, M – medium, L – low, VL – very low). Albedo forcing due to black carbon on snow and ice is included
in the black carbon aerosol bar. Small forcings due to contrails (0.05 W m–2, including contrail induced
cirrus), and HFCs, PFCs and SF6 (total 0.03 W m–2) are not shown. Concentration-based RFs for gases can
be obtained by summing the like-coloured bars. Volcanic forcing is not included as its episodic nature makes
is difficult to compare to other forcing mechanisms. Total anthropogenic radiative forcing is provided for three
different years relative to For further technical details, including uncertainty ranges associated with
individual components and processes, see the Technical Summary Supplementary Material. {8.5; Figures
8.14–8.18; Figures TS.6 and TS.7}
COMECAP 2016, Thessaloniki, 19th of September

13. Conclusions
SSA and the smoothed periodogram with a χ2 noise test is able to automatically detect statistically-significant cycles
SSA nonlinear trends fit annual sub-series well and are consistent to other fitting approaches
For satellite SO2 over China, nonlinear trends extracted by SSA appear to close to linear (0.951 ≤ R ≤ 0.993)
70% of time series exhibit at least an annual cycle
45% show sub-annual variability (semi-annual cycle)
A simple hybrid gap-filling approach with LOESS and splines appears to be robust to porous time series and noise
Radiative forcing estimates in 2011 relative to 1750 and aggregated uncertainties for the
main drivers of climate change. Values are global average radiative forcing (RF15) partitioned according to
the emitted compounds or processes that result in a combination of drivers. The best estimates of the net
radiative forcing are shown as black diamonds with corresponding uncertainty intervals; the numerical values
are provided on the right of the figure, together with the confidence level in the net forcing (VH – very high, H
– high, M – medium, L – low, VL – very low). Albedo forcing due to black carbon on snow and ice is included
in the black carbon aerosol bar. Small forcings due to contrails (0.05 W m–2, including contrail induced
cirrus), and HFCs, PFCs and SF6 (total 0.03 W m–2) are not shown. Concentration-based RFs for gases can
be obtained by summing the like-coloured bars. Volcanic forcing is not included as its episodic nature makes
is difficult to compare to other forcing mechanisms. Total anthropogenic radiative forcing is provided for three
different years relative to For further technical details, including uncertainty ranges associated with
individual components and processes, see the Technical Summary Supplementary Material. {8.5; Figures
8.14–8.18; Figures TS.6 and TS.7}
COMECAP 2016, Thessaloniki, 19th of September

14. THANK YOU
Taylor, M.1*, Koukouli, M.E.1, Theys, N.2, Bai, J. 3, Zempila, M.M.4, Balis, D.S.1, van Roozendael, M.2 , van der A, R.5
1 Laboratory of Atmospheric Physics, Aristotle University of Thessaloniki, Greece.
2 Belgian Institute for Space Aeronomy, BIRA-IASB, Brussels, Belgium.
3 Institute of Atmospheric Physics (IAP-CAS), Beijing, China.
4 Natural Resource Ecology Laboratory, Colorado State University, Fort Collins, USA.
5 Koninklijk Nederlands Meteorologisch Instituut (KNMI), De Bilt, The Netherlands.
* URL:
Taylor, Koukouli, Theys, Bai, Zempila, Balis, van Roozendael, van der A et al (2017) Nonlinear Processes in Geophysics (in preparation)
COMECAP 2016, Thessaloniki ,19th of September: Remote Sensing of Atmospheric Composition Session 14:30-16:00