Geology 591 – Fractals and Spectral Analysis
Filtering the Amplitude Spectrum
Amplitude spectra are shown in Figure 1 for the combined effects of the predicted
variations of orbital eccentricity, axial tilt and precession. This kind of plot should look
familiar to you and you should be comfortable translating frequencies into periods. Note
that there are several peaks in the spectrum, more than those commonly note for
precession as ~ 21,000 yr period, axial tilt as ~ 41,000 year period, and eccentricity as
~100,000 yr period. In this lab, you will learn how to use filtering methods to extract the
temporal behavior associated with a specified range of frequencies in the amplitude
spectra. Filtering can be used to isolate one or more of the peaks and exclude the
remainder. Another potential use of filtering is to exclude the high frequency noise that
often clutters our data and makes it difficult to see the underlying “signal.”
In the comparison you made of the two data sets, the next step will be check to see
if variations in the region of the spectrum corresponding to the predicted astronomical
influences are similar in both data sets. To do this you will need to isolate those
frequencies in the spectrum containing each orbital component and compare the
equivalent temporal response to variations observed over the same frequency range in
your second data set (see problem set at the end of this handout).
The “filtering” process is relatively easy to perform. Successful filtering, however,
resides in the design stage and not in the actual computation. Computation of the
spectrum on the other hand is a relatively mechanical process that involves little personal
interaction. Meaningful interpretation of the spectrum and realistic filter design are the
critical issues.
The filter option is executed from Psi-Plot’s MATH drop-down window. We’ll
pick the oxygen isotope data from the Caribbean or Mediterranean and go through this.
The starting point is to compute the amplitude spectrum and determine the regions of the
spectrum you’d like to isolate and examine. Leave the spectrum up for later reference.
First we will design a low-pass filter.
Click on
Import DeloCar.dat or DeloMed.dat (depending on which one you have been
assigned) into Psi-Plot. You’ve already computed the spectrum for O18data but let’s
take a few minutes to redo this and think about it in the context of filter design.
LOW PASS FILTER
First, design a low-pass filter to extract hypothetical low frequency eccentricity
variations. We will use a cutoff frequency of _______ (see figures 2 and 3). To undertake
the filter processClick on - MATH Filtering
A list appears to right >
Low Pass
High Pass
Band Pass
Notch
Select
Low-Pass Filter
Column List
time
delo
Sampling Intvel __0.002___
Cut-off Freq. ___?_____
The actual appearance of this menu will vary depending on what you have in your file.
SELECTING THE DATA SET TO BE FILTERED
A critical issue: make sure you highlight the column in the column list that you
want to filter. In this example, you would highlight delo or delocar (depending on how
your file is labeled)!
SAMPLING INTERVAL
The sample interval in the data sets you are analyzing will be 0.002 million years
for these climate data sets. For your project - make sure you pay attention to this since it
is likely your sample rate will be different from 0.002. (0.002 is a good number for
seismic since seismic data sets are often sampled at 2 milliseconds).
The low pass filtered output is compared to the raw O data in Figure 4.
BANDPASS FILTERS: LOW AND HIGH CUTOFF FREQUENCIES
To illustrate the application of the band-pass filter, we will continue in this guide
with the analysis of the DeloCar dataset. The design of a filter to extract the axial tilt
component is illustrated (see also figures 5 and 6). You can follow this same general
procedure to extract other astronomical components of interest to you.
The spectrum from the theoretical data presented in Figure 1 suggests that the
axial tilt influences are contained in a fairly narrow region of spectrum corresponding to
periods of about 40,650 years. The objective of filtering is to isolate or filter out the
region of the spectrum containing potential axial tilt influences using a band-pass. The
spectrum of the Caribbean O data (Figure 5) reveals that the low and high cutoff
frequencies are defined so that they straddle the region of the spectrum containing the
axial tilt component. We select cut-off frequencies by looking at the region on either side
of the axial tilt peaks and then place the high and low cut-off points to include the axial
tilt components but isolate other the potential influence of other components.
While we are looking at this plot, consider what cut-off frequencies you would
use to isolate or extract the precessional influences.
BACK TO THE BANDPASS FILTER WINDOW
You will have a menu similar to that appearing under the Low Pass option, but in
this case you must specify the low-cut and high-cut frequencies. Don’t forget to specify
the sample interval, and the correct data column to filter.
Complete your selections and Click OK.
DON’T USE FFT
Since it is unlikely that you have a power of 2 (4, 8, 16, 32, 64, 128, … etc.) number of
data points you will generally not want to use the “FFT”. Click on NO and let the
calculations proceed.
The filtered data appear in a column labeled FILTER(#) (or Filter (some column
number)). Plot and compare to other data sets (see Figure 6 for example).
The above design and discussion of filtering of the Mediterranean O data can be
referenced to Figures 7 through 9.
Composite
Normalized Equal-Weighted Effect
3
2
1
0
-1
-2
0
1000
2000
3000
4000
5000
Time (kiloyears past)
Spectrum
Amplitude
0.15
0.10
0.05
0.00
0.0
0.1
0.2
0.3
0.4
0.5
Frequency (cycles/1000yrs)
Amplitude
0.15
0.10
0.05
0.00
Periods (left-to-right)
370,000 years
123,456 years
92,593 years
40,650 years
23,585 years
22,222 years
0.01 0.02 0.03 0.04 0.05 0.06 0.07
18,904 years
Frequency (cycles/1000yrs)
Figure 1: The spectra shown in the lower two plots are derived from the theoretical behavior
predicted for the combined effects of orbital eccentricity, axial tilt and precession. Note that a
variety of peaks appear in the spectrum and not just three we might expect the generalized
discussions of these influences.
Amplitude Spectrum for O
0.20
9.4 cpMy
variations in the Caribbean Sea
21 cpMy
=0.048 My/cycle
0.15
Amplitude
18
0.10
38.7 cpMy
=0.026 My/cycle
0.05
53 cpMy
0.00
0
20
40
60
80
100
Frequency (cycles/million years)
In the above amplitude spectrum of the dO18 concentrations in the caribbean
we see bands of relatively high amplitude centered at (from low to high frequency)
approximately 9.4, 21, 38.7, and 53 cycles per million years (cpMy). The
correspondance between frequency and period is tabulated below.
Frequency
cpMy
9.4
21
38.7
53
Period
My/c years/cycle
0.11
110,000
0.48
21,000
0.0258 25,800
0.0189 18,900
Figure 2: Interpreted amplitude spectrum for the oxygen isotope variations observed in the
Caribbean Sea.
Amplitude Spectrum for O
0.20
9.4 cpMy
variations in the Caribbean Sea
Cutoff frequency
of 15 cpMy
0.15
Amplitude
18
0.10
0.05
0.00
0
20
40
60
80
100
Frequency (cycles/million years)
If our interest is to "tune" into the eccentricity variations, we have
to somehow eliminate or reduce the overlapping influences in time
of the other components present in the data.
The easiest way to do this is to design a low-pass filter that leaves
everything on the low frequency end of the spectrum where the
eccentricity variations occur and eliminate information having higher
frequency of frequency above some "cutoff" frequency.
In this example we could set our cutoff frequency at 15 cpMy.
Figure 3: The cutoff frequency used to extract the eccentricity variations is noted in the above
spectrum for the oxygen isotope variations from the Caribbean Sea.
Amplitude Spectrum for O
0.20
9.4 cpMy
variations in the Caribbean Sea
Cutoff frequency
of 15 cpMy
0.15
Amplitude
18
0.10
0.05
0.00
0
20
40
60
80
100
Frequency (cycles/million years)
A.
18
The unfiltered O data (black) are compared to
the lowpass filtered output (red)
1.0
0.5
O
18
0.0
-0.5
-1.0
-1.5
0.0
B.
0.1
0.2
0.3
0.4
0.5
0.6
0.7
time (in millions of years)
Figure 4: A) Amplitude spectrum of oxygen isotope variations observed in the Caribbean
Sea with highlighted 15cpMy cutoff frequency. B) The raw O18 temporal response is shown
in black and the extracted low frequency variations are highlighted in red for comparison.
Amplitude
18
Amplitude Spectrum for O variations in the Caribbean Sea
0.20
Low cut frequency
of 13 cpMy
0.15
High cut f frequency
of 33 cpMy
0.10
0.05
0.00
0
20
40
60
80
100
Frequency (cycles/million years)
Likewise, we can use filtering to extract temporal variations
18
of O occuring at periods that might be associated
with variations the earth's axial tile. In the above spectrum
we pick "low cut" and "high cut" frequencies that are used to
define a "bandpass" filter - a filter that passes frequencies in
a specified band or range of the spectrum.
In the case of the Caribbean O spectrum shown above,
we can isolate the region that would, theoretically, contain axial tilt
influences by extracting the band of data extending from 13 to 33 cpMy.
Figure 5: Amplitude spectrum of O18 variations observed in the Caribbean Sea. Vertical lines
specify the low cut and high cut frequencies we will use to extract frequencies in the range of
those hypothetically associated with axial tilt astronomical influences on climate.
Amplitude Spectrum for O
18
variations in the Caribbean Sea
0.20
Range of output frequencies
extracted by the bandpass filter
Amplitude
0.15
0.10
0.05
0.00
0
A.
20
40
60
80
100
Frequency (cycles/million years)
Variations in the range of periods associated with axial tilt variations
(in red) are compared to the unfiltered O
18
data (black)
1.0
0.5
O
18
0.0
-0.5
-1.0
-1.5
B.
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
time (in millions of years)
Figure 6: A) Amplitude spectrum of O variations in the Caribbean. High and low cutoff
frequencies define the region extracted by the bandpass filter. B) Filtered temporal variations
having frequencies in the passband are shown in red for comparison to the raw unfiltered O
variations (black).
18
Amplitude
Amplitude Spectrum for O variations in the Mediterranean Sea
0.5
2.5 cpMy
9.1 cpMy
0.4
23 cpMy
0.3
32.25 cpMy
46 cpMy
0.2
53 cpMy
0.1
0.0
0
20
40
60
80
100
Frequency (cycles/million years)
In the above amplitude spectrum of the dO18 concentrations in the Mediterranean
we see bands of relatively high amplitude centered at (from low to high frequency)
approximately 2.5, 9.1, 23,32.3, 46, and 53 cycles per million years (cpMy). The
correspondance between frequency and period is tabulated below.
Frequency
cpMy
2.5
9.1
23
32.25
46
53
Period
My/c years/cycle
0.04
400,000
0.1099 109,900
0.044
44,000
0.031
31,000
0.0217 21,700
0.0189 18,900
Figure 7: Amplitude spectrum of O variations in the Mediterranean Sea.
18
Amplitude
Amplitude Spectrum for O variations in the Mediterranean Sea
0.5
2.5 cpMy
9.1 cpMy
0.4
15 cpMy cutoff
frequency
0.3
0.2
0.1
0.0
0
20
40
60
80
100
Frequency (cycles/million years)
18
While the amplitude spectrum of the O concentrations in the Mediterranean
looks considerably different from those in the Caribbean, we can use the same cutoff
18
frequency to isolate the variations in O concentration that might be
associated with eccentricity effects. In this case we have a much higher amplitude
400,000 year cycle in the data, but we can extract the entire range of frequencies
including the 400,000 and 110,000 component using a lowpass filter with cutoff
frequency of 15 cpMy or 66,666 years. The output from this lowpas filter will contain
features in the data with periods greater than approximately 67,000 years.
Figure 8: The cutoff frequency used earlier to extract eccentricity related variations
using the lowpass filter is highlighted in the above spectrum of the O variations from
the Mediterranean Sea.
Eccentricity variations (red) compared to unfiltered
O
18
data from the Mediterranean (black)
3
2
1
O
18
0
-1
-2
-3
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
time (in millions of years)
Axial variations (red) compared to unfiltered
O
18
data from the Mediterranean (black)
3
2
1
O
18
0
-1
-2
-3
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
time (in millions of years)
Figure 9: Comparisons of lowpass filtered (top, in red) and bandpass filtered (bottom,
in red) O data to the unfiltered data from the Mediterranean Sea.
Comparison of the low pass outputs computed for O variations
in the Caribbean (blue) and Mediterranean (green) seas.
"Eccentricity" Components
2.5
Mediterranean
2.0
1.5
1.0
0.5
0.0
-0.5
-1.0
Caribbean
-1.5
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Low Pass Output from the Mediterranean
time (in millions of years)
Cross plot of the low pass filtered O variations
2.5
2.0
r = 0.48
1.5
1.0
0.5
0.0
-0.5
-1.0
-1.2
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
Low Pass Output from the Caribbean
Figure 10: The low pass filtered outputs from the Caribbean and Mediterranean seas are
compared in terms of their variation through time (top) and as a cross plot. The linear
regression derived correlation coefficient for this interrelationship is 0.48.
Comparison of the bandpass filtered outputs for dO variations
in the Caribbean (blue dotted) and Mediterranean (green) seas
"Axial Tilt " Components
1.5
1.0
0.5
0.0
-0.5
-1.0
-1.5
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Bandpass output from the Mediterranean
time (in millions of years)
Cross plot of the low pass filtered O variations
1.5
r = 0.13
1.0
0.5
0.0
-0.5
-1.0
-1.5
-0.60
-0.40
-0.20
0.00
0.20
0.40
0.60
Bandpass output form the Caribbean
Figure 11: The low pass filtered outputs from the Caribbean and Mediterranean seas are
compared in terms of their variation through time (top) and as a cross plot. The linear
regression derived correlation coefficient for this interrelationship is 0.13.
Geology 591 – Fractals and Spectral Analysis
Problem Set - Filtering the Climate Data
You’ve probably noticed that there is considerable difference in the frequency
distributions of oxygen isotope variations from different parts of the world. You will also
notice differences in the temperature and sea-level data. As the devil’s advocate (opposed
to the Milankovich theory), you hypothesize that if the Milankovich theory is correct then
the variations associated with this type of climate forcing mechanism must be felt
worldwide. You hypothesize that if there are astronomically induced variations occurring
in the Caribbean then there must be similar variations occurring in the Mediterranean. To
evaluate whether the O18 variations or the variations in other climate parameters are
similar in different areas you design filters to extract temporal variations occurring in
similar regions of the spectrum from both areas. If similar variations are observed, then
the glove fits, and you have to side with Milankovich. The best way to determine this is to
extract the region of the spectrum where that influence should be (even if you don’t see a
peak there) and compare those variations - in time - between the different areas. What do
the comparisons suggest? Are these influences shared in common? Do the variations
observed in one region of the spectrum from one area of the world follow those observed
in another? In the problem set associated with this technique you will have to be judge.
With the simulated data we know what the answer should be. But even in this
case, the extracted component can differ significantly from the actual component. These
differences can be produced by “edge effects” from improperly designed filters (to
narrow) and also to noise in the data. In general distortions associated with the filtering
process can be minimized by avoiding the use of excessively narrow filters.
It may happen that you will see peaks in the spectrum that do not coincide with
the frequency of precession (40ky), axial tilt (18-24ky), or eccentricity (>100ky)
variation. Perhaps something else contributes to the oxygen isotope variations that is not
associated with these phenomena? Remember also, that when we examined the
theoretical predictions for eccentricity, obliquity and precession we ended up with more
than three peaks (see Figure 1). There appear to be three eccentricity and perhaps three
precession peaks.
If you found a peak in the spectrum near 40 cycles per million years (precession),
you could extract it using a bandpass filter with low and high cut frequencies of 25 and 60
cpMy, respectively. Extending the pass band out to 60 allows you to see possible
influences from the higher frequency precession peak. If in another data set a peak is not
observed in this region of the spectrum. Perhaps, we might ask, is there something going
on there that we just can’t see because of other variation in adjacent bands of the
spectrum or because of noise? A bandpass filter can be applied to extract the variations in
this region of the spectrum, so that temporal variations in the same spectral band of each
data set can be compared. If precession effects are operating then shouldn't they be
observed in both datasets? They might not be in phase with each other, but wouldn't you
expect them to be similar in some respects? The additional figures presented below will
be discussed in class.
ECC
TILT
PREC
1.0
0.5
0.0
-0.5
-1.0
0.0
0.1
0.2
0.3
0.4
0.5
0.0
0.1
0.2
0.3
0.4
0.5
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.4
0.2
0.0
-0.2
-0.4
-0.6
1.0
0.5
0.0
-0.5
-1.0
SUM
Composite variations over time
3.0
2.0
1.0
0.0
-1.0
-2.0
-3.0
0.0
0.1
0.2
My
Figure 12: Simulated climate data.
0.3
0.4
0.5
The Signal
Simulated Climate Data
4.0
3.0
2.0
1.0
0.0
-1.0
-2.0
-3.0
-4.0
0.0
0.2
0.4
0.6
0.8
Time (millions of y ears)
1.0
Amplitude
The Spectrum
1.0
0.8
0.6
0.4
0.2
0.0
0.0
50.0
100.0
150.0
200.0
250.0
Amplitude
Frequency (cy cles/million y ears)
1.0
0.8
0.6
0.4
0.2
0.0
~100,000yrs
~20,000yrs
~40,000yrs
0
10
20
30
40
50
60
70
Figure 13: Amplitude spectrum of simulated Milankovich variations plus some
added noise. The spectra are shown in the bottom two plots.
Amplitude
Simulated Climate Data
4
3
2
1
0
-1
-2
-3
-4
0.0
0.2
0.4
0.6
0.8
1.0
Time (Ma)
Amplitude
Spectrum of Simulated Climate Data
1.0
0.8
0.6
0.4
0.2
0.0
Low Pass Filter
15 cycles/Ma Cutoff
0
20
40
60
80
100
Frequency
Amplitude
Low-Pass Output (Eccentricity Component + Local Noise)
2.0
1.5
1.0
0.5
0.0
-0.5
-1.0
-1.5
0.0
0.2
0.4
0.6
0.8
1.0
Time (Ma)
Figure 14: Simulated Milankovich cycles (top), amplitude spectrum with low-pass filter
(middle), and low-pass filter output (bottom). Compare filtered output with the eccentricity
input (top of Figure 12).
Amp litude Sp ectrum of Simulated Climate Data
Amplitude
1.5
30-60 cy cle/M a Bandp ass Filter
1.0
0.5
0.0
0
50
100
150
200
250
Frequency
Time Domain Smoother
Amplitude
30-60 cycles/Ma
60
40
20
0
-20
-40
-60
0.00 0.10 0.20
Time (Ma)
bppres
Filtered Data (p recessional comp onent)
1.5
1.0
0.5
0.0
-0.5
-1.0
-1.5
0.0
0.2
0.4
0.6
0.8
1.0
time
Figure 15: Amplitude spectrum of simulated climate data with bandpass filter (top), time
domain representation of the bandpass filter (effectively a smoother) (middle), and the filtered
output (bottom).
Antarctic Surface Temperatures
Amplitude
1.40
1.20
1.00
0.80
0.60
0.40
0.20
0.00
0
50
100
150
200
Frequency
Amplitude
Sealevel.txt
70
60
50
40
30
20
10
0
0
50
100
150
200
250
Frequency
Figure 16: Amplitude spectra of surface temperature variations in the Antarctic (top) and of sea
level fluctuations (bottom).
Amplitude
Antarctic Surface Temperatures
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0
80,000 yr period
0
20
40
60
80
100
80
100
Frequency (cycles/million years)
Amplitude
Sealevel.txt
70
60
50
40
30
20
10
0
93,000 yr period
0
20
40
60
Frequency (cycles/million years)
Figure 17: A look at the surface temperature and sea level variations out to 100 cycles per million
years. The average or approximate period of the major peak in each spectra is noted.
Temperature
Filtered Surface Temperature
13
12
11
10
9
8
7
6
5
4
0.0
0.1
0.2
0.3
0.4
0.5
time (My)
Sea Level (meters)
Filtered Sea Level Variations
0
-100
-200
-300
-400
-500
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
time (My)
Figure 18: Output after low pass filtering with 40 cycles/million year cutoff frequencies. Surface
temperature (top), sea level (bottom).
Coparison of Filtered Data
Filtered Sea Level
0
-100
-200
-300
r=0.57
-400
-500
4
5
6
7
8
9
10
11
12
13
Filtered Surface Temperatures
Normalized Temperature (black) and Sea Level (red) Data
Relative Value
1.2
1.0
0.8
0.6
0.4
0.2
0.0
0.0
0.1
0.2
0.3
0.4
0.5
Time (Mya)
Figure 19: Cross-correlation of filtered surface temperature and sea level data (top);
comparison of normalized outputs (bottom). Note shared periodicity at roughly 100,000 yr
intervals.
Normalized Temperature (black) and Sea Level (red) Data
Relative Value
1.2
1.0
0.8
0.6
0.4
0.2
0.0
0.0
0.1
0.2
0.3
0.4
0.5
Time (Mya)
Correlation
Cross-correlation between filtered sea level and surface temperature
0.60
SL shifted to left
SL shifted to right
0.40
0.20
0.00
-0.20
-0.40
-0.60
-0.3
-0.2
-0.1
0.0
0.1
Lag (My)
Figure 20: Filtered output (top); Cross correlation of filtered output (bottom).
0.2
0.3
Use the same data sets you picked last Friday and continue trying to answer that basic
question - Do the components observed in one area of the world correlate to those
observed in the other?
BASIC CHECK LIST –
Part 1:
1. Tabulate the different peaks in the two spectra you are working with. Note frequency
and corresponding period. Use labeled plots of the spectrum to reference your
identification.
2. Design and apply a low pass filter to isolate the region of the spectrum associated with
the orbital eccentricity variations in both of your data sets. Explain your design.
3. Plot the low pass output for both of your data sets.
4. Compare the low pass output from both of your data sets. How well do they correlate?
You can answer this question qualitatively using a visual comparison of your two plots
and also more quantitatively by computing a correlation coefficient between the two data
sets as illustrated in this handout.
5. Pick one additional component (tilt or precession) from your data set and use a band
pass filter to extract it.
6. Compare the band pass filtered outputs for your two data sets. How well do they
correlate? See comment in question 4 above.
7. State any conclusions you can make about whether variations in your area can be
associated with the Milankovich cycles.
Part 2:
Undertake similar analysis with the data you are working with. Present your results at our
next meeting.