Geology 659 - Quantitative Methods (Wilson, 2003)
Computer Lab - 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. Note that there are several
peaks in the spectrum - more than those commonly noted, usually including one for
precession as an ~ 21,000 yr period, axial tilt as having an ~ 41,000 year period, and
eccentricity with a period of ~100,000 yr period. When you look at the spectra assigned to
you, consider this more complicated perspective in your interpretation. In this lab, we will
learn how to use filtering methods to extract the temporal behavior associated with a
specified range of frequencies in the amplitude spectra of your data. 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 comparisons of the two data sets you are undertaking, you will be asked 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 a suspected Milankovich 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 in Psi-Plot’s MATH drop-down window. For starters
we will run through the procedures using the oxygen isotope data from the Caribbean and
Mediterranean seas. Compute the amplitude spectrum, and determine the regions of the
spectrum you wish to isolate. 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
1
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 always be 0.002
million years. If you are doing this kind of analysis for your project, make sure you pay
close attention to this since it is likely your sample rate will be different from 0.002.
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 with the
analysis of the DeloCar dataset. The design of a filter to extract the axial tilt component
will be illustrated (see also figures 5 and 6). You can follow this same general procedure
to extract components of interest to you.
The spectra 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 filter.
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 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 number of data points you will not want to
use the “FFT”. Click on NO and let the calculations proceed.
2
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 can also be applied to the filtering of
the O data from the Mediterranean. The results of lowpass and bandpass filtering of the
O data from the Mediterranean are illustrated in Figures 7 through 9. Note that we have
used the same cutoff frequencies for both the low and bandpass filters. Doing this ensures
that we are comparing the same region of the spectrum.
3
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 from the
popularized discussions of these influences.
4
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.
5
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.
6
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.
7
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.
8
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).
9
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.
10
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.
11
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.
12
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.
13
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.
14
Geology 659 - Quantitative Methods (Wilson 2003)
Problem Set - Filtering the Climate Data
As you work with the climate data assigned to you will notice that considerable
difference exists 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 may 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 you could argue that the Milankovich theory is
supported by the observations. 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 variations between 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 this problem set, you will
have to be judge. Based on the data from the Caribbean and Mediterranean seas (see
figures 10 and 11) you might argue that there is some similarity in the periodic behavior
over the range of periods associated with eccentricity variations but that no case can be
made for similar behavior over the range of periods or frequencies in which the axial tilt
influences should appear.
Note that in the case where we are dealing with simulated data we know what the
answer should be. But even in this case, note, as we did in class that the extracted
component and actual component can differ - in some cases significantly. These
differences are due, in part, to “edge effects” associated with the filtering process, and
also to noise in the data. In general distortions associated with the filtering process can be
minimized if excessively narrow filters are avoided.
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 is going on in the climatic 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 variations in this band of the spectrum or
because of noise? A bandpass filter can be applied to extract the variations in any region
of the spectrum, so that temporal variations over the same spectral band in different data
sets can be compared. If precession effects are operating then shouldn't they be observed
15
The Signal ECC
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.
0.0
-1.0 0.0
-2.0
-3.0
-4.0
0.0
TILT
0.6
0.4
0.2
0.0
-0.2
-0.4
-0.6
1.0
Amplitude
PREC
1.0
0.5
0.0
4.0
-0.5
3.0
2.0
-1.0
1.0
0.1
0.2
0.2
0.8 0.0
0.6
0.4
0.2
0.0
1.0 0.0
0.4
0.4
0.6
0.8
Time (millions of y ears)
0.1
0.2
50.0
100.0
0.3
150.0
0.5
0.0
-0.5
-1.0
1.0
0.8
0.6
3.0
0.4
2.0
0.2
1.0
0.0
0.0
0.3
0.5
1.0
The Spectrum
0.4
200.0
0.5
250.0
Frequency (cy cles/million y ears)
0.0
SUM
Amplitude
Simulated Climate Data
0.1
0.2
Composite
~100,000yrs
0.3
0.4
0.5
variations
over time
~20,000yrs
~40,000yrs
-1.0 0
10
20
30
40
50
60
70
-2.0
Figure-3.0
13: Amplitude spectrum of simulated Milankovich variations plus some
0.3plots. 0.4
0.5
added noise.0.0
The spectra0.1
are shown in0.2
the bottom two
My
Figure 12: Simulated climate data.
16
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).
17
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).
18
Use the same data sets you were assigned on Tuesday 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 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 you
identification.
2. Design and apply a filter to isolate the region of the spectrum associated with the
orbital eccentricity variations in both of your data sets.
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
bandpass filter to extract it.
6. Compare the bandpass 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.
19