Aliasing

The normalization issues discussed in Appendix B are benign in that they are easily corrected. A more insidious problem arises in that unresolved variability in a climate record will masquerade as lower frequency variability — a phenomenon Tukey called aliasing. First, a simple example of aliasing is given, and afterward the requirements for correcting for its affects are discussed.

Origins

The origins of aliasing can be understood by considering a discretely sampled signal, x_n, where the spectrum exists for frequencies 1/(N4t),2/(N4t), ....1/(24t). Now consider what happens to the spectrum whenxn is sampled at half the rate,vn =x2n. To compute the spectrum of the subsampled series, define another vector that has every other index ofx set to zero,

un = The utility of this second vector is thatu2nequalsx2n. The final relationship suggests the effects of subsampling is to combine x(n) with a higher frequency process, which becomes more clear after taking the Fourier transform,

ˆ of the above equality. It is straight-forward to relate vn andu2nbecause un is zero at odd indices. Substituting into the above equation yields,

ˆ

where the total number of observation, N, is now cut in half. The expected peri-odogram becomes,

assuming the frequencies s ands+ 1/24tare uncorrelated. Decreasing the sampling resolution by a factor of two results in the power which is no longer resolved, fre-quencies ₂₄¹_t to ₄¹_t, being folded into the resolved band, _N¹_4t to ₂₄¹_t. That is, all the energy in Ψx(s) is present in Ψv(s) but in half the number of estimates. This result can be generalized to any degree of undersampling, [see e.g. Priestley, 1984] where the estimated spectrum is related to the true spectrum, Ψ_t(s),

Ψv(s) =

X∞

r=0

Ψt(s+ r

24t). (1.4)

All spectral energy above the Nyquist frequency will alias into the resolved lower frequencies, biasing those estimates towards too large of values. See Wunsch and Gunn [2003a] for a general discussion of the effects of aliasing on the interpretation of paleoclimate records.

Effect on power-law processes

Aliasing can have serious effects on power-law estimates made from records of varying lengths and sampling intervals. Assume Ψt represents a band-limited white noise process with a uniform spectral distribution between zero frequency and a cutoff frequency, s_c, and is zero at all higher frequencies. If the sampling interval gives a Nyquist frequency greater than or equal to the cut-off frequency, 1/24t ≥ sc, no aliasing will occur. Building from the previous example, assume a sampling interval, 4t = ^2(h_s⁻_c¹⁾. When h= 0 the spectrum of Ψt is fully resolved, but for h >0 aliasing will occur; from Eq 1.4,

Ψv =

2^(h−1)

X

r=0

Ψt(s+ r 24t).

In the case of white noise this reduces to,

Ψv = 2^(h−1)Ψt(s).

The Nyquist frequency of each spectral estimate is sc2^−h, and the mean frequency is sc2^−(h+1). Writing the ratio of power density to the mean frequency gives,

2^(h−1)Ψt

2^−(h+1)sc

= 2^hΨt

2^−hsc

,

showing that the power density of the composite spectra with h = 0,1,2...will scale as, 2^h/2^−h. In logarithm-power and logarithm-frequency space this will be,hlog(2)/− hlog(2), giving rise to a spurious red spectrum with a power law ofq = 1 (the power-law is defined as s^−q so that positive q’s indicate that energy decreases with higher frequency). Note that the aliased version of each individual spectra will itself be white, but because greater amounts of aliasing occurs for the lower-frequency spectral estimate, the overall power-law estimated from the composite spectra will be red.

If one seeks to correct for the effects of aliasing, assumptions about the unresolved higher-frequency variability are required. Wunsch [1972] gives an example of account-ing for the bulge in the high-frequency spectral estimates of a tidal-gauge record by assuming a constant power-law withq = 2 and applying Eq 1.4. The solution involves an infinite sum, which only converges when q is greater than one. This implies that if one measures a finite spectrum with a power-law of one or less, in order to keep the total energy bounded, at higher frequencies the power-law must become more red. Therefor a plausible assumption regarding unresolved high-frequency tempera-ture variability would be a steady power-law up to a cut-off frequency, above which there exists negligible energy.

So far the discussion has focused on stochastic processes. A further consideration is that when the annual cycle is not resolved we can expect its energy to be aliased to lower-frequencies. Were paleo-proxies sampled at a uniform sampling interval, it would be straight-forward to calculate where the annual variability would appear. In practice, this is hardly ever the case, and thejitter in the sampling interval is expected to distribute the unresolved annual variability over a broad range of frequencies [see Moore and Thomson, 1991]. As a final consideration, the manner in which each proxy averages the climate variability will also influence the degree of aliasing. For example, if a tree-ring record represented a uniform average over the year, an annually resolved record would have no aliasing. But if the tree has a differing sensitivity to temperature at different times of the year, and different points in its growth cycle, some aliasing is inevitable. Then, in addition to a model of high-frequency temperature behavior, one needs to model the proxy sampling characteristics when assessing the effects of aliasing. Such a detailed analysis is not further pursued here.

The power-law found for aliased white noise, when successive records are sub-sampled by a factor of two, is the same as for tropical sea surface temperatures [see Figure 1-1]. The scope of the bias in the proxy records is unclear, but it appears that

for white, or nearly white, signals there will be a general trend towards increasingly over-estimated spectra as the sampling interval grows. This will tend to bias power-laws towards being too red. Some further observations can be made that argue against the power-law estimates being wholly an artifact of aliasing. First, multiple different proxies of tropical sea surface temperature variability are each consistent with a power-law of one. Second, the high-frequency estimates which resolve the annual variability are consistent with the low-frequency estimates where they overlap.

Further study of the impact of aliasing on power-law processes is required.

Chapter 2

On Insolation Forcing

The previous chapter presented a spectral description of the spatial and temporal variations in long-term climate variability. Much of climate variability is characterized by simple spectral power-laws. While there remain important questions regarding the origins of the climate continuum, the following Chapters focus on the Milankovitch bands of variability. Such a focus can be justified by the wide-spread interpretation of long-term climate variability as being strongly influenced by orbital forcing which can be traced back to the identification of Milankovitch bands of variability in deep-sea cores by Hays et al. [1976]. In seeking to better assess the role of orbital forcing in causing climate variability, this chapter investigates the orbital variations, the attendant changes in insolation forcing, and the manner in which these signals are likely to appear in the climate recorded.

The spatial and temporal variations in insolation are examined from a signal pro-cessing point of view, with attention paid to the frequency and amplitude modulation occurring at both the annual and longer period timescales. Two aspects of precession variability are considered in detail. First discussed is how precession period signals in the climate record should be interpreted; there are a number of potential sources, and there are certain requirements the climate system must meet in order to produce this variability. Second, the frequency modulation of the precession variability is dis-cussed in relation to orbitally derived age estimates. It is shown that the amplitude modulation of precession period signals cannot be used to test the accuracy of these orbital age-estimates.

The concentrations of insolation variability at both the annual and secular¹timescales

1In astronomy, secular changes refer to the long-term variations in a planet’s orbit. There are many short-term variations due to, for example, the gravitational influence of other planets, but

makes insolation difficult to represent and interpret. Often, the representation of insolation variability is simplified to a single one-dimensional low-frequency signal, inevitably altering or ignoring important features of the variability. These simpli-fications are briefly reviewed and compared with one another, after which a more complete representation of the insolation forcing is presented. The new representa-tion is developed in terms of spatial modes of variability, and retains a full descriprepresenta-tion of the seasonal and secular variability. The description is accurate and compact, and provides insight into how the spatial modes of insolation forcing vary seasonally and at long time periods.

2.1 Earth’s orbital parameters

The insolation for any time and point on the globe can be represented as [e.g.

Vernekar, 1972],

I =Io

1 +ecos(λs−$) (1−e²)

!2

(sinφsinθsinλs+cosφcosθcosλscos (λ−η)), I ≥0, (2.1) where I_o is the solar constant (about 1368W/m²), φ latitude, λ longitude, e orbital eccentricity, θ obliquity, $ the argument of perigee, λs solar longitude, and η the hour angle. The frequency of variation for the last five variables is e ∼ (400Kyr)⁻¹ and (100Kyr)⁻¹, θ ∼ (41Kyr)⁻¹, $ ∼ (21Kyr)⁻¹, λs = 1yr, and η = 1 day; thus the frequency range extends over more than seven orders of magnitude. The ’∼’ symbol indicates a band of variability centered on the specified frequency. All of the orbital parameters are frequency modulated [Hinnov, 2002] and thus, in fact, can only be partially described using a single frequency. A diagram of earth’s present orbital configuration is shown in Figure 2-1. The form of Eq. 2.1 indicates the secular variations in insolation are controlled by eccentricity, obliquity, and the argument of perigee — each orbital parameters is discussed in turn.

which over relatively short periods average to zero.

P A

do

d1

d2

d3 ϖ

λ

Figure 2-1: Earth’s orbit around the sun. The argument of perigee ($), measured from the vernal equinox (d_o) to perihelion (P), is shown with its current configura-tions, but for visual purposes, the eccentricity of the orbit is shown with ten times the current value. Relative to the fixed stars, do has a fixed period of 25.8 KY while P moves with periods varying from 100 to 400 KY. Relative to one another the motions of do and P give a climatic precession period ranging from 23 to 18 ka. The vernal equinox currently occurs on March 20th, and up to the small variations caused by the non-integer number of days in the year (hence the use of leap-years), has a fixed solar-longitude. Also shown are Northern Hemisphere summer solstice (λ = 90^◦), autumnal equinox (λ = 180^◦), and winter solstice (λ = 270^◦), which currently oc-cur on June 21st (d₁), September 22nd (d₂), and December 21st (d₃) respectively.

These latter set of dates are associated with varying solar-longitudes, depending on the degree of eccentricity and on the argument of perigee. The maximum variations in solar-longitude associated with the dates of each solstice and spring equinox over the last 1000KY are indicated on the above figure by the red arc segments.