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Key Questions and a Case Study

Huybers (H07)

δ18O Record

Ashwin and Ditlevson (AD15)

Multiple Climate Model Comparison of the Mid-Pleistocene Transition

using Ensemble Empirical Mode DecompositionAuthors: Andrew Gallatin, Ryan Smith, Charles D. Camp

Mathematics Department, California Polytechnic State University, San Luis Obispo

Motivation

• How can modern time series analysis techniques be used to

improve the characterization of empirical records and model

outputs?

• How can we clearly distinguish the information in the empirical

records that is used for hypothesis development / model tuning

from information used for model validation?

• Can EEMD better identify the timing and character of the MPT?

Saltzman and Maasch (SM91)

Methodology

• X is continental ice mass

• Y is atmospheric CO2

• Z is deep ocean temperature

• f(t) is normalized insolation

forcing at 65°N in July

• Wi is stochastic forcing

• Vt is ice volume

• Tt is the deglaciation threshold

• vt is stochastic forcing

• θt is obliquity forcing

When Vt ≥ Tt , ice volume

linearly resets back to 0 in

10 time steps.

Empirical Mode Decomposition (EMD)• EMD assumes that the data may have coexisting oscillatory modes on

varying timescales present at any point. EMD achieves temporal

decomposition to create a set of time series on distinct time scales, called

intrinsic mode functions (IMFs).

1. Create an IMF from the data

a. Locate local extrema and create an upper and lower bound by

splining the maxima together and the minima together

b. Subtract the mean of the bounds from the series

c. Repeat steps (a) and (b) until the mean of the extrema of the

resulting time series is zero. This leaves an IMF

2. Subtract the IMF from the time series and repeat step 1 to find the

next IMF

Ensemble Empirical Mode Decomposition (EEMD)

• Improves EMD analysis by adding white noise to reduce mode-mixing.

1. Create an EMD with noise

a. Add white noise to the data

b. Decompose the modified data with EMD

2. Repeat step 1 with different white noise

3. Average the IMFs of the entire ensemble of decompositions

• Proxy measure for global ice volume, based on a depth-derived

age model (Huybers, 2006).

• IMF 5 shows the start of the MPT at approximately 1.2 Mya.

• IMFs 4 and 5 show strong evidence of persistent 41 kyr cycles

throughout the Pleistocene and the emergence of 100 kyr cycles

at the transition.

• Late Pleistocene cycles show a sharp triangular characteristic

between glacial build up and deglaciation.

• IMF 5 cycles exhibit growth in the late Pleistocene; consistent

of a super-critical Hopf bifurcation.

• Milankovitch forcing is the external forcing for the system in

the form of variations in Earth’s orbital mechanics. Obliquity

and eccentricity forcing shown in red on IMFs 4 and 5,

respectively.

SM91 Parameter Estimation

Ashwin P, Ditlevsen P. 2015. The middle Pleistocene transition as a generic bifurcation on a slow manifold, Climate Dynamics, DOI:10.1007/s00382-015-2501-9

Huang, N. E., & Wu, Z. 2008. A review on Hilbert‐Huang transform: Method and its applications to geophysical studies. Reviews of Geophysics, 46(2).

Huybers, P., 2007. Glacial variability over the last two million years: an extended depth- derived age model, continuous obliquity pacing, and the Pleistocene

progression. Quat. Sci. Rev. 26, 37–55.

Huybers, P. 2006. Pleistocene Depth-derived Age Model and Composite d18O Record. IGBP PAGES/World Data Center for Paleoclimatology Data Contribution

Series # 2006-075.NOAA/NCDC Paleoclimatology Program, Boulder CO, USA.

Huybers, P. 2006. Early Pleistocene Glacial Cycles and the Integrated Summer Insolation Forcing. Science, 313(5786), 508–511.

Saltzman, B., & Maasch, K. A. 1991 A first-order global model of late Cenozoic climate. II. Further analysis based on a simplification of the CO2 dynamics. Clim.

Dyn. 5, 201–210. (doi:10.1007/BF00210005)

ConclusionPhysical Conclusions

• Qualitatively, SM91 seems more consistent with the data than H07 or

AD15; particularly in capturing the persistence of the 41 kyr cycles

during the late Pleistocene.

• The comparisons of both SM91 and AD15 suggest that a bifurcation of

internal dynamics is more consistent with data than the change in

response to forcing as shown in H07.

• We see the possibility of phase locking in SM91 through the correlation

of eccentricity. This may provide insight into how many models can

match the emerging 100 kyr glacial cycle.

• In SM91 and AD15, the periodicity of the 100kyr cycle varies as the

insolation forcing amplitude is changed; we can use EEMD for better

parameter estimation

EEMD Analysis Conclusions

• EEMD allows the comparison of subtler details of the model outputs

and data records by comparing corresponding IMFs from each since the

IMFs contain the variations acting on distinct time scales.

• More analysis can be done in the frequency domain; comparing models

using cross-spectral analysis on the EEMD analysis may lead to an even

subtler comparison.

• We can use EEMD to better tune models and distinguish between

information used for tuning and validation.

• H is the function for

the manifold

• v is ice volume

• y is a state variable

• I is insolation forcing

• A reduction in the CO2 to a low enough value allows ice sheets

to form and excites the internal main free oscillator which

phase-locks with the orbital forcing.

• Continuing research in implementing more realistic forcing

such as an integrated insolation to reduce effects of precession.

• Model seen with u = 1.55 tuned for ratio resulting in

significantly higher value than original paper.

• Each deglaciation is triggered by high obliquity forcing, but the

deglaciation threshold becomes more resistant to triggers

causing skipped cycles and the emergence of a the longer and

deeper glaciations of the late Pleistocene.

• IMFs 3 and 4 exhibit artifacts of EEMD from abrupt transitions.

• Relaxation oscillator with multiple states for interglacial, mild

glacial, and deep glacial periods.

• Dominant cycles caused by the topology of the slow manifold

H=0; drifting lambda triggers the bifurcation.

Mid-Pleistocene Transition• The past few million years in proxy climate data has shown the growth and

decay of land ice in glacial-interglacial cycles; Milankovitch theory

hypothesizes that the timing of these cycles is linked to cycles in the Earth's

orbital dynamics.

• Approximately 0.8 to 1.2 myr ago climate records show a transition from

dominant 41 kyr glacial cycles to 100 kyr cycles.

Qualitative ComparisonPositive Negative

SM91 •Entire SM91 output has good

qualitative match with the proxy

data through the MPT.

•Persistent 41 kyr cycle in IMF 4.

•No sharp triangular

characteristic.

• IMF 5 does not continue to

grow after the MPT.

• IMF 3 has large amplitude.

AD15 •Sharp triangular characteristic.

• IMF 5 continues to grow

slightly

•More regular in early

Pleistocene.

• IMF 4 dampens after the MPT

and is not regularly persistent.

H07 •Sharp triangular characteristic.

• IMF 5 continues to grow.

• IMF 4 is not persistent through

MPT.

The long standing questions about changes in glacial (ice age)

cycles in paleoclimate provide a useful test case for developing

such techniques; e.g., the Mid-Pleistocene Transition (MPT).

• For many physical systems, multiple models exist with

• distinct underlying physics

• Distinct mathematical structures

Yet all can match empirical results to some degree.

• The case for better analysis tools

• Improve parameter estimation for any given model.

• Identify and compare subtler features of both empirical and

model outputs

• We can identify characteristics in the EEMD to better tune models; we

distinguish between information for tuning and validation.

• Using Huyber’s integrated summer insolation, IMF 3 can be reduced to

more accurately match empirical records.

• Remaining parameters are tuned using periodicity of late IMF 5 and ratio

of early IMF 4 to late IMF 5 as benchmarks.