Methods in Fourier Spectral Analysis

Fourier Transform

It was a significant discovery in mathematics that any function can be expanded as a sum of harmonic functions (sines and cosines) and the resulting expression is known as Fourier series. A harmonic of repeating signals such as sunusoidal wave is a wave with a frequency that is a positive integer multiple of the frequency of the original wave, known as the fundamental frequency. The original wave is called the first harmonic, the following harmonics are known as higher harmonics. Any function can also be expanded in terms of polynomials and the resulting expression is known as Taylor series. If the underlying forces are harmonic and there possibly exists some periodicity, then the use of harmonic series is more useful than using polynomials as it produces simpler equations. It is possible to discover a few dominating terms from such series expansion which may help identify the known natural forces with the same period.
Let the symbol h(t) represent a continuous function of time. The Fourier transform is a function of
frequency f.

H_T(f) = \int_{-\infty}^{\infty} h(t) e^{2 \pi i f t} dt
e^{2 \pi i f t} = cos(2\pi f t) + i sin(2\pi f t)

The amplitude and the phases of the sine waves can be found from this result. Given data h(t), we can find the Fourier transform H(f) using Inverse Fourier transform.

h(t) = \int_{-\infty}^{\infty} H_T(f) e^{-2 \pi i f t} dt

The spectral power P is defined as the square of the Fourier amplitude:

P_T(f) = |H_T(f)|^2

However, real data does not span infinite time and most likely be sampled only at a few discrete points over time. Suppose that, we received values of h(t) at times t_j, then an estimate of the Fourier transform is made by using summation. The inverse transform is also shown using the summation.

h_j \equiv h(t_j)
H(f) \equiv \sum_{j=0}^{N-1} h_j e^{2 \pi i f t_j}

The data are desired to be sampled from equally spaced time as nice statistical properties are available in such regular case. If the interval between equally spaced data points is \Delta t, then the highest frequency that will appear in the fourier transform is given by the Nyquist-Shannon sampling theorem. The theorem states “If a function f(t) contains no frequencies higher than f Hz, then it is completely determined by giving its ordinates at a series of points spaced \frac{1}{2f} seconds apart”. Therefore, the Nyquist frequency (highest frequency) is given by the following equation.

f_N = \frac{1}{2\Delta t}

The lowest frequency is the one that gives one full cycle in the time interval T. The other frequencies to evaluate is the multiples (f_k) of the low frequency f_L. And, also we can derive the symmetric pair of equations. Moreover, if h(t) is band-limited (no frequencies below f_L or above f_N), then there is a relationship between the continuous function h(t) and the discrete values H_k.

f_L = \frac{1}{T}
f_k = kf_L
H_k \equiv \sum_{j=0}^{N-1} h_j e^{2 \pi i jkf_L t_j}
h_j = \sum_{k=0}^{N-1} H_k e^{2 \pi i jkf_L t_j}
h(t) = \sum_{k=0}^{N-1} H_k e^{2 \pi i kf_L t} (when band limited)

Periodogram

Fourier transform give us the complex numbers and the square of the absolute value of these numbers represent the periodogram. This is the first form of numerical spectral analysis and is used to estimate spectral power. Even though the data points collected are at evenly spaced specific discrete time, it is possible to evaluate periodogram at any frequencies.

Fast Fourier Transform (FFT)

We can calculate the Fourier transform very efficiently by using FFT. It requires data at equally-spaced time points, and is most efficient when the number of points is an exact power of two. Interpolation is often used to produce the evenly-spaced data which may introduce additional bias and systematic eror. For real data consisting of N data points y_j, each taken at time t_j, the power spectrum outputs a set of N+1 data points. The first and the last data points are the same, and they represent the power at frequency zero. The second through to the N/2 + 1 data points represent the power at evenly-spaced frequencies up to the Nyquist frequency. The spectral power for a given frequency is distributed over several frequency bins, therefore an optimum determination of the power requires combining these information and proper investigation of leakage. FFT, generally, calculates the amplitude for a set of frequencies. N/2 complex amplitudes are calculated at N/2 different frequencies. Because, these may not be the true frequencies present in the record, we subtract the mean from the data and then pad it with zeros to overcome this challenge.

Aliasing

The time series consists of measurements made at a discrete, equally spaced, set of times on some phenomenon that is actually evolving continuously, or at least on a much finer time scale. For example, samples of Greenland Ice represent the temperature every 100 years, but if the sampling is not precisely spaced by a year, we will sometimes measure winter ice, and other times measure summer ice. Even without the existence of long-term variation in the temperature, fluctuations (jumping up and down) in the data can be noticed. So, there can be frequencies higher than the Nyquist frequency associated with the sampling interval. Thus a peak in the true spectrum at a frequncy beyond the Nyquist frequency may be strong enough to be seen(aliased) in the spectrum which may give the impression that a frequency is significant when it is not. Or, a peak may partly obscure another frequency of interest. This phenomenon is known as aliasing.

Tapering

Fourier transform is defined for a function on a finite interval and the function needs to be periodic. But with the real data set, this requirment is not met as the data end suddenly at t=0 and t=T and can have discontinuities. This discontinuity introduces distortions (known as Gibbs phenomenon) in fourier transform and generates false high frequency in the spectrum. Tapering (using data window) is used to reduce these artificial presence. The data y=f(t) is multiplied by a taper function g(t) which is a simple, slowly varying function, often going towards zero
near the edges. Some of the popular tapers are:

1. Sine taper g(t) = sin(\pi t/T)
2. Hanning (offset cosine) taper g(t) = \frac{1}{2}(1-cos(2\pi t/T))
3. Hamming taper g(t) = 0.54 - 0.46cos(2 \pi t/T)
4. Parzen or Bartlett (triangle) window g(t) = 1 - (t - T/2)/(T/2)
5. Welch (parabolic) window g(t) = 1 - (t - T/2)^2/(T/2)^2
6. Daniell (untapered or rectangular) window g(t) = 1

The frequency resolution in the spectrum of the tapered data is degraded. If the primary interest is the resolution of peaks, then the untapered periodogram is superior. However, tapering significantly reduces the sidelobes and also the bias applied to other nearby peaks by the sidelobes of a strong peak. Because, the taper functions are broad and slowly varying and their fourier transform FT(g) are narrow. The effect of tapering the data is to convolve the fourier transform of the data with the narrow fourier transform of the taper function which amounts to smoothing the spectral values.

FT(fg) = FT(f) * FT(g)

<

p style=”text-align:justify;”>
// Sine taper
t <- seq(0,1, by=0.01)
T <- 1
g <- sin(pi * t * T)
plot(t, g, t='l', col=1, ylab='g(t)')

// Hanning (offset cosine) taper
g2 <- 1/2 * (1-cos(2*pi*t/T))
lines(t, g2, t=’l’, col=2)

// Hamming
g3 <- 0.54 – 0.46 * cos(2*pi*t/T)
lines(t, g3, t=’l’, col=3)

// Parzen or Bartlett (triangle) window
g4 0.5, 1 – (t-T/2)/(T/2), 2*t)
lines(t, g4, t=’l’, col=4)

// Welch (parabolic) window
g5 <- 1 – (t-T/2)^2/(T/2)^2
lines(t, g5, t=’l’, col=5)

// Daniell window
g6 <- rep(0.5, length(t))
g6 <- ifelse(t <= 0.2, 0, g6)
g6 = 0.8, 0, g6)
lines(t, g6, t=’l’, col=6)

legnd = c(‘Sine’, ‘Hanning’, ‘Hamming’, ‘Bartlett’, ‘Welch’, ‘Daniell(20%)’)
legend(‘topleft’, legend=legnd ,col=1:6, lty=1, cex=0.75)

Multitaper Analysis

We apply taper or data window to reduce the side lobes of the spectral lines. Basically we want to minimize the leakage of power from the strong peaks to other frequencies. In multitaper method, several different tapers are applied to the data and the resulting powers then averaged. Each data taper is multiplied element-wise by the signal to provide a windowed trial from which one estimates the power at each component frequency. As each taper is pairwise orthogonal to all other tapers, the windowed signals provide statistically independent estimates of the underlying spectrum. The final spectrum is obtained by averaging over all the tapered spectra. D. Thomson chose the Slepian or discrete prolate spheroidal sequences as tapers since these vectors are mutually orthogonal and possess desirable spectral concentration properties. Multitaper method can suppress sidelobes but have higher resolution. If we use few tapers, the resolution won’t be degraded, but then sidelobe reduction won’t happen much. So, there is a trade-off which is often misunderstood.

Blackman-Tuckey Method

 

Blackman and Tuckey prescribed some techniques to analyze a continuous spectrum that was biased by the presence of sidelobes of strong peaks in the ordinary periodogram. Blackman-Tuckey(BT) method was developed before 1958, prior to the FFT(Fast Fourier Transform) method. A discrete fourier transform of N points would
require the calculation of N^2 sines and cosines. With the slower computer in the pre-FFT days, the calculation of fourier transform was thus expensive. BT method has reduced the time by reducing the size of the dataset by a factor of the lag in the autocorrelation calculation. BT method is based on a fundamental theorem of Fourier transform that the Fourier transform of a correlation is equal to the product of the Fourier transforms. The correlation of two functions g(t) and h(t) is given by the first equation below.

C(\tau)=g\otimes h= \int_{-\infty}^{\infty} g(t) h(t+\tau) d\tau
FT(g\otimes h) = FT(g) FT(h)

When g = h, it is called Wiener-Khintchine theorem. Here, P is the spectral power.

FT(g\otimes g) = |FT(g)|^2 = P

The algorithm in BT method calculates partial autocorrelation function, defined by

A_{BT}(\tau) = \int_{0}^{N/l} f(t+\tau)f(t) dt

Here, N is the length of the data set but we integrate only upto N/l. $l$ is associated with the lag. When l=3 (recommended by Blackman and Tuckey) is used, we say that “a lag of 1/3” is used. Now the fourier transform of partial autocorrelation function A_{BT} gives us the spectral power. Moreover, the symmetric property of the partial autocorrelation function (A(-\tau) = A(\tau)) saves half of the computation time.

FT(A_{BT}) = \int_{-\infty}^{\infty} e^{2\pi i ft } A_{BT}(\tau) d \tau = P_{BT}(f)
P_{BT}(f) = 2 \int_{0}^{\infty} cos(2\pi f) A_{BT}(\tau)

If l=1, then it is basically the full autocorrelation function A(\tau) and gives the same answer as the ordinary periodogram.

P(f) = 2 \int_{0}^{\infty} cos(2\pi f) A(\tau) = FT(A)

Because we are using partial correlation function instead of the full correlation, the spectral power function gets smoother. Therefore, we lose resolution in the BT method. However, it averages the sidelobes into the main peak, and thereby gives a better estimate of the true power. The smoothing in BT method is different from the smoothing when we use a taper. With a taper, the fourier transform is smoothed, where as with Blackman-Tukey, it is the spectral power which gets smoothed. A spectral amplitude that is rapidly varying will be averaged to zero with a taper. But in BT method, a rapidly varying amplitude does not necessarily average to zero, since the process of squaring can make the function positive over the region of smoothing. The tapering does not
average the sidelobes into the main peak. Because, shift in the time scale behaves like phase modulation. The sidelobes, when tapering is applied, will not have the same phase, and if averaged in amplitude, they can reduce the strength of the peaks. A major challenge in the BT method is that we will have to estimate the proper lag to use before doing all the calculations. Blackman and Tukey recommended starting with the value 1/3 for the lag.

Lomb-Scargle Periodogram

 

The classic periodogram requires evenly spaced data, but we frequently encounter with unevenly spaced data in paleoclimatic research. Lomb and Scargle showed that if the cosine and sine coefficients are normalized separately, then the classic periodogram can be used with unevenly spaced data. If we have a data set (t_k, y_k), we first calculate the mean and variance:

\bar{y} = \frac{1}{N} \sum_{k=1}^{N}y_k
\sigma^2 = \frac{1}{N-1} \sum_{k=1}^{N}[y_k - \bar{y}]^2

For every frequency f, a time constant \tau is defined by

\tau = \frac{ \sum_{k=1}^{N}sin(4\pi f t_k)}{\sum_{k=1}^{N}cos(4\pi f t_k)}

Then the Lomb-Scargle periodogram of the spectral power P(f) at frequency f is given by

$P(f) = \frac{1}{2\sigma^2}\frac{ \sum_{k=1}^{N}(y_k – \bar{y} ) [cos(2\pi f (t_k-\tau))]^2}{\sum_{k=1}^{N}cos^2(2\pi f (t_k-\tau))} +
\frac{ \sum_{k=1}^{N}(y_k – \bar{y} ) [sin(2\pi f (t_k-\tau))]^2}{\sum_{k=1}^{N}cos^2(2\pi f (t_k-\tau))}$

With evenly spaced data, two signals of different frequencies can have identical values which is known as Aliasing. That is why the classic periodogram is usually shown with the frequency range from 0 to 0.5, as the rest is a mirrored version. But with Lomb-Scargle periodogram, the aliasing effect can be significantly reduced.

Maximum Likelihood Analysis

In maximum likelihood method, we adjust the parameter of the model and ultimately find the parameters with which our model have the maximum probability/likelihood of generating the data. To estimate the spectral power, we first select a false alarm probability and calculate the normalized periodogram. We identify the maximum peak and test it against the false alarm probability. If the maximum peak meets the false alarm test, we determine the amplitude and phase of the sinusoid representing the peak. Then we subtract the sinusoidal curve from the data which also removes the annoying sidelobes associated with that peak. After peak removal, the variance in the total record is also reduced. Now, with the new subtracted data, we continue finding the other stronger peaks following the same procedure. We stop when a peak does not meet the false alarm test. We need to carefully choose the false alarm probability, as if it is too low, we can miss some significant peaks; it is too low, we can mislabel noise as peaks.

Maximum Entropy Method

It is assumed that the true power spectrum can be approximated by an equation which has a power series. This method finds the spectrum which is closest to white noise (has the maximum randomness or “entropy”) while still having an autocorrelation function that agrees with the measured values – in the range for which there are measured values. It yields narrower spectral lines. This method is suitable for relatively smooth spectra. With noisy input functions, if very high order is chosen, there may occur spurious peaks. This method should be used in conjuction with other conservative methods, like periodograms, to choose the correct model order and to avoid getting false peaks.

Cross Spectrum and Coherency

If a climate proxy a(t) is influenced or dominated by a driving force b(t), we can use cross spectrum to see if their amplitudes are similar. Cross spectrum is given by the product of the fourier transform.

C(f) = A(f) B^*(f)

where A is the Fourier transform of a and B is the complex conjugate of the fourier transform of b. If we want to know whether two signals are in phase with each other, regardless of amplitude, then we can take the cross spectrum, square it, and divide by the spectral powers of individual signals using the following equation for coherency. Coherency measures only the phase relationship and is not sensitive to amplitude which is a big drawback.

c(f) = \frac{|C(f)|^2}{P_a(f) P_b(f)}

Coherency is valuable if two signals that are varying in time, stay in phase over a band of frequencies instead of a single frequency. Therefore, a band of adjacent frequancies are used in the averaging process to compute coherency:

coherency(f) = \gamma^2(f) = \frac{|<C(f)>|^2}{<P_a(f)> <P_b(f)>}

Bispectra

In bispectra, coherency relationship between several frequencies are used. A bispectrum shows a peak whenever (1) three frequencies f_1, f_2 and f_3 are present in the data such that $f_1 + f_2 = f_3$ and (2) the phase relationship between the three frequencies is coherent for at least a short averaging time for a band near these frequencies. If the nonlinear processes in driving force (e.g. eccentricity or inclination of the orbit of earth) has coherent frequency triplets, then the response (i.e. climate) is likely to contain same frequency triplet. For example, \delta ^{18}O is driven by eccentricity, we should be able to find eccentricity triplet. Thus, by comparing the bispectrum plot of climate proxy with the bispectrum plot of the driving forces, we can verify the influences of driving forces.

## Monte Carlo Simulation of Background
Monte carlo simulation is extremely useful to answer the questions like whether the data is properly tuned or not, whether the timescale is incorrect, whether some spectral power is being leaked to adjacent frequencies, whether the peak has real structure and also to understand the structures near the base of the peak (a shoulder) in a spectral analysis. Generally monte carlo simulation is run multiple times. For each simulation, a real signal(sinusoidal wave) is generated, then random background signal is added, then the spectral power is calculated to look for shoulders. In this way, the frequency of the shoulder occurence can be measured and the randomness can be realized. It is important to create background that behaves similarly to the background in real data. Dissimilar background will cause false conclusion. We also need to estimate the statistical significance of the peaks very carefully.

(This article is a quick review of the fourier spectral analysis from the book “Ice Ages And Astronomical Causes- (Data, Spectral Analysis and Mechanics) by Richard A. Muller and Gordon J. MacDonald

Advertisements

Statistics and Data Exploration: Quantiles, probability distribution, Box plot and Q-Q (Quantile-Quantile) plot

Statistics and Data Exploration: Quantiles, probability distribution, Box plot and Q-Q (Quantile-Quantile) plot

Quantiles

What are quantiles in statistics?

If the data is sorted from small to big, Quantiles are the points which divide the data/samples into equal sized, adjacent subgroups. Every data sample has maximum value, minimum value, median value(the middle value after you sort the data). The middle value in the sorted data is the 50% quantile because half of the data are below that point and half above that point. A 25% quantile is the cut point in the data where 1/4 -th of the data is below the point. IQR is inter-quartile range which contains half of the data which contains the median and are higher than the 25% low-value data point but less than the 25% high-value data point.

Box Plot

A box-plot can be a good representation to show the quantiles. Box plot can take different shapes depending on the data. Here is an example:

Screen Shot 2018-04-16 at 10.05.34 AM

(image source: www.physics.csbsju.edu/stats/box2.html)

Example of Discrete/Continous Probability Distribution

In the figure below, you can see different frequency distribution. The blue data samples have most of it’s data near (0,1) interval, it’s left skewed. Check how the blue box is shifted to the left. The green data samples are normally distributed, meaning most of the data points are centered around zero. It also looks balanced. We find normal distribution in nature and in biological and social phenomena very often. The orange one shows almost a uniform distribution, where the data is spreaded across the range. And lastly a right skewed data. These are all discrete data points with discrete probability distribution. There are also very well known continuous probability distribution with continuous probability density function(https://en.wikipedia.org/wiki/List_of_probability_distributions#Continuous_distributions).

Screen Shot 2018-04-16 at 10.06.55 AM

  (Image source: https://www.otexts.org/node/620)

Below we can see the quantiles for the normal distribution- the cut points which divide the continuous range of points in equal probability area. The area over an interval (in x axis) under a continuous probability density function (like the normal distribution function below) represents the probability of the data falling into that range. In this case, the IQR is the blue box; data point in that interval has 50% probability of occurrence.

Screen Shot 2018-04-16 at 9.57.22 AM

Q-Q plot

We can use Q-Q plot to graphically compare two probability distributions. Q-Q plot stands for Quantile vs. Quantile plot. In Q-Q plotting, we basically compute the probabilities assuming a certain distribution (e.g. normal, gamma or poisson distribution) from the data and then compare it with theoritical quantiles. The steps used in Q-Q plotting is:

  1. Sort the data points from small to large
  2. For n data points, find n equally spaced points which serve as the probability using \frac{k}{n+1} where k=1, 2, ..., n
  3. Look at the data points, possibly plot it and assume the underlying probability distributions. Using the probabilities from the step 2, now you can calculate quantiles. Like in R language, you can use the quantile functions like qnorm or qgamma or qunif from the stats package.
  4. Now plot by putting the calculated quantiles in step 3 in x axis and putting the sorted data points in the y-axis. If the data points stay close to the y=x line, that means your assumption of the probability distribution was correct.

Below you can see one example, where the normal distribution is assumed for the ozone data. T

Screen Shot 2018-04-16 at 10.35.57 AM

Now you can see the gamma distribution fits better to the ozone data than the normal distribution.

Screen Shot 2018-04-16 at 10.37.45 AM

This is how you can check different probability distribution for your data using simple Q-Q plot. There is a fantastic Q-Q plot tutorial from which I collected the above image. For further reading, please check https://www.r-bloggers.com/exploratory-data-analysis-quantile-quantile-plots-for-new-yorks-ozone-pollution-data/ and https://www.r-bloggers.com/exploratory-data-analysis-quantile-quantile-plots-for-new-yorks-ozone-pollution-data/

 

Can Money Buy Everything?

My friend on Facebook posted..
Those who keep saying “Money Can’t Buy Everything” don’t have enough money.

And I argued:

“It’s very easy to disprove. Can a billionaire buy you with money? Yes or No? Can he buy to be the most empathetic person, can he buy to be the best scientist? I can go on and on what he can not buy. But it’s true that he can buy lots of things that most people can’t. But even if, what is it that most people want.. big house, big salary, luxary cars, jewelery, ability to travel, power and influence. I can give you hundreds of examples of enthusiastic people who are not rich, who are doing these way better than Donald Trump or some other boring billionaires. Just watch Instagram and YouTube mindfully, how beautifully people can live with less. Money can’t buy creativity, peace in mind; it can just help towards that path. It’s just a vehicle for comfort and survival but it itself is not the comfort and happiness. But look.. this kind of mentality that money is everything.. imprison the free mind and make your mind and then life a living hell. Do u really not think you can just leave every fucking luxary and go to a jungle and live peacefully eating fruits? I am telling you I can totally see myself doing that if that’s whats gonna give me happiness and peace. I’m not a prisoner of the society and the world where money is required. LoL. Money is just a peace of paper. It’s the idea and perception of living that’s important. Peace!! Haha. ;)”

What do you think and say? I would like to hear.

How do Paleoclimatologists investigate about ancient Earth? What are different Climate Proxies and what are their significance?

To know and understand about ancient climate, different climate proxies are generally used. We can measure the concentration of greenhouse gases by using entrapped air in the Greenland and Antarctic glaciers which give us samples of the atmosphere back to about 420 Kyr. The glaciers in North America and on mountains in tropical Andes can be estimated from scour marks, moraines and erratic boulders.Forams are microscopic organisms whose life cycles depend on local temperature and whose fossils preserve samples of ancient material. Some planktic forams (short for foraminifera) represent a “proxy” for sea surface temperature as they indirectly inform us about the temperature. One of the most remarkable proxies is the ratio of oxygen isotopes in benthic(bottom dwelling) forams in ancient sediment, which reflect the total amount of ice that existed on the Earth at the time the sea beds were formed. A scientist needs to be careful in their analysis as most proxies are dependent on more than one aspect of climate. Now I will discuss the primary proxies which have been used to investigate paleoclimate. Many of the samples come from seafloor cores, cores from Greenland or Anatarctic ice. The cores are named V22-174, RC13-110, DSXP-806 etc. In the geologic community, various of these prefixes are used some of which are enlisted below:

  • V: Vema, a converted yacht operated by Lamont-Doherty Earth Observatory of Columbia university.
  • RC: Research vessel Robert Conrad.
  • DSDP: Deep Sea Drilling Project operated from 1968 to 1983 by the Scripps Institution of Oceanography at University of California, San Diego.
  • ODP: Ocean Drilling Program as an international collaboration.\newline
  • GRIP: European based GReenland Ice-core Project.
  • GISP2: US-based Greenland Ice Sheet Project #2.
  • Vostok: Russian station on the East Antarctic ice plateau.
  • MD: The research vessel Marion Dufresne, operated by the French.

1. Oxygen Isotopes

The pattern of oxygen isotopes is remarkably similar in sea floor records around the world and this universality feature is very attractive for a climate proxy. The ratio of oxygen istopes found in ice, trapped air, benthic/planktic forams is widely used as a climate proxy. Oxygen consists of three stable istopes: 99.759% is ^{16}O, 0.037% is ^{17}O, and 0.204% is ^{18}O. The variation in the fraction of ^{18}Olatex can be measured with high accuracy. The fractional change, shown by the following equation, basically means that how much difference of the ratio of \frac{^{18}O}{^{16}O}latex exists in perts per thousand in the sample compared to the reference.

\delta^{18}O = \left(\frac{\left(\frac{^{18}O}{^{16}O}\right)_{Sample}}{\left(\frac{^{18}O}{^{16}O}\right)_{Reference}} - 1\right) \times 1000

 

Oxygen isotope separation occurs because of the isotopic differences in vapour pressure and chemical reaction rates, which depends on temperature. Some of the most important geophysical processes that lead to changes in \delta^{18}O are:

  1. Evaporated water is ligher than the remaining liquid. Water containing ^{16}O has higher vapor pressure than water containing ^{18}O, so it evaporates quickly.
  2. Precipitated water molecules are heavier than those in the residual vapor. H_2^{18}O condenses more readily than H_2^{16}O, so as water vapour is carried across to Greenland or to central Anatarctica, the residual becomes lighter.
  3. Oceanic \delta^{18}O in non-uniformly distributed. It means that the changes in the pattern of winds that carry vapor and change the source will also change \delta^{18}O. At present, the difference in surface water is 1.5% from pole to equator.
  4.  Biological activity enriches the heavy isotope. The \delta^{18}O in the calcium carbonate of shells is 40% greater, on average, than in the water in which the organism lives.

 

The net result of these effects is that glacial ice is light, with \delta^{18}O typically lower than seawater. So, in glacial ice containing more ^{16}O, \delta^{18}O is negative, where as in surface water containing more ^{18}O, \delta^{18}O is positive. However, when large volumes of ice are stored in ace-gage glaciers, then there can be considearable depletion of the light isotopes in the oceans.
In 1964, Dansgaard and colleageus showed that measurements of isotopic enrichment in ocean water as a function of latitude yield the following approximate relationship between temperature T and \delta^{18}O:

\delta^{18}O \equiv 0.7 T - 13.6
However, there can be other factors in the change of \delta^{18}O. Therefore, if we go back to earlier when the temperature was lower, \delta^{18}O might not be lower which contradicts the above equation. When several measurements are made at the same latitude, the effect is argued to depend on the amount of precipitation and not on temperature.
Moreover, depending on the source, we will have to consider other issues. In planktic fossils, we might expect \delta^{18}O to reflect surface conditions, and therefore be sensitive to temperature and salinity conditions. In benthic forams, \delta^{18}O must be more sensitive to global ice, since there is little temperature variation on the sea floor. In other samples (e.g. ice, trapper air or calcite), \delta^{18}O may represent the temperature, not ice volutme.
Several attempts have been made to extract the underlying \delta^{18}O signal that is common in the records. SPECMAP stack (Imbrie et al., 1984) was a combination of five $\delta^{18}O$ records from five cores: V30-40, RC11-120, V280238 and DSDP502b.

2. Deuterium – Temperature Proxy

Hydrogen generally contains only one proton in its nucleus and is lighter with atomic weight 1. Deuterim (D or ^2H), on the other hand, is one of the heavy isotopes of hydrogen which contains one proton and one neutron in its nucleus and thus the atomic weight is 2. Bonds formed with deuterium tend to be much more stable than those with light hydrogen. The deuterated water is more sensitive to temperature than that of ^{18}O. We can clearly see it in the “fractionation factor” which describes the equilibrium between liquid and vapour. The fractionation factor is defined to be the ratio of D/H in a liquid to the ratio of D/H in a vapor that is in equilibrium with that liquid. The fractionation factor for HDO is approximately 1.08 and it varies more rapidly with temperature compared to ^{18}O. Therefore, the condensation of the deuterised form of heavy water (HDO) is significantly more sensitive to temperature variation than is the ^{18}O form (H_2^{18}O). Therefore, deuterim is considered as a temperature proxy. A temperature scale was devised fro the Vostok ice core by assuming the equation:

\Delta T = \frac{\Delta \delta D_{ICE} - \Delta \delta^{18}O_{SW}}{9}
where, the $\delta^{18}O_{SW}$ refers to the sea floor isotope record.

3. Carbon-13

Carbon on the earth has two stable istopes, ^{12}C with an bundance of 98.9% and ^{13}C with an abundance of 1.1%. The ratio of these two isotopes is described by the quantity \delta^{13}C and defined by the equation below. The reference value is often taken to be a sample known as the “Peedee belemnite” (PDB); its \delta^{13}C value is very close to that of mean sea water.

\delta^{13}C = \left(\frac{\left(\frac{^{13}C}{^{12}C}\right)_{Sample}}{\left(\frac{^{13}C}{^{12}C}\right)_{Reference}} - 1\right) \times 1000

The lighter isotope, ^{12}C, is easily absorbed into the organic tissue of plants, leading to negative values for ^{13}C = -20% to -25%. In regions in which photosynthesis is active, this removes typically 10-20% of the dissolved inorganic carbon in seawater, leading to ^{13}C enrichment in surrounding water. Because different regions of the world have different activity, there is geographic variation. Warm surface water has the highest \delta^{13}C, where as deep Pacific water has the lowest \delta^{13}C. Thus \delta^{13}C can be used as a tracer for oceanic currents.
In contrast, there is only small separation of carbon istotopes that takes place in the formation of caclcium carbonate shells. Thus the measurement of \delta^{13}C reflects the composition of the ocean water at the time and location in which the shell grew.
^{13}C is extremely important isotope for paleoclimate studies, because it responds to the presence of life. \delta^{13}C can record climate change. During glacial periods, biological activity was reduced by advancing glaciers and colder temperature, and light carbon was released into the atmosphere and eventually mixed into the oceans. \delta^{13}C from benthic (bottom dwelling) forams is typically 0.35% lower during glacials than during interglacials. In contrast, planktic forms don’t show such changes.

4. Vostok

The ice core from the Vostok site in Antarctica (Petit et al., 1999) located at 78^oS and 107^oE, covers the longest period of time of any ice record. It reached a depth of 3623 metres. A untuned but unbiased timescale was derived based on ice accumulation and glacial flow models. Many proxies of climate interest were measured in the Vostok core, including atmospheric methane, atmospheric oxygen, deuterium in the ice, dust content and sea salt. Atmospheric methane is produced by the biological activity of anaerobic bacteria and it’s existence in paleoclimate data is presumed to reflect the area of the earth covered by swamps and wetlands. The observed dust (strong 100 Kyr cycle) in the Vostok dust record is beleived to reflect reduction in vegetation during those periods and accompanying increase in wind-blown erosion. Then, the sodium concentration reflects the presence of sea spray aerosols blowing over the Vostok region.

5. Atmospheric \delta^{18}O and Dole Effect

The atmospheric oxygen has a \delta^{18}O of +23.5% compared to that of mean ocean sea water due to the removal of lighter isotope ^{16}O from the atmosphere by biological activity. The difference is called the “Dole Effect” and it is assumed to be time-independent.

6. \delta^{18}O / $CO_2$ Mystery

The difference between ocean and atmospheric $\delta^{18}O$ is due to the biological activity. However, carbon dioxide, even though driven by biological processes, doesn’t show similar spectra. The strong peaks in the oxygen signal forced by precession parameter is absent in the carbon dioxide record which is mysterious and still under investigation.

7. Other Sea Floor Records

7.1 Terrigenous component

The terrigenous component of sea floor sediment is the fraction which has possibly come from land, in the form of wind-blown dust. The most significant frequencies which have been found in the spectrum of detuned terrigenous component Site 721 are marked with the periods: 41, 24, 22 and 19 Kyr. These periods indicate that the signals were dominated by solar insolation.

7.2 Foram size: the coarse, or “sand”, component

In the sea floor core, the main component of the sand is frequently large forams. Therefore, the coarse component reflects an interesting change in the ecology of the oceans. A clear eccentricity signal was detected in a core that already showed a clear absence of eccentricity in the \delta^{18}O component.

7.3 Lysocline: carbonate isopleths

Pressure varies in different depths of the ocean and which consequently influences the solubility of the calcium carbonate. At a certain depth, the shells of fossil plankton begin to dissolve, and this boundary is called lysocline. It can be quantified by the percentage of calcium carbonate in the sediment, as a function of depth. One can plot the depth at which the 60% lysocline is found, as a function of age and this depends on the depth of the oceans at that age. The signal apeears to be dominated by a 100 Kyr cycle, as would be expected if the primary driving force were the depth of the ocean, determined by the amount of ice accumulated on land.

Featured Image Courtesy: https://katiecoleborn.wordpress.com/5-proxy-climate-records-what-are-they-and-how-do-they-work/
Main Reference: Ice Ages and Astronomical Causes (Data, Spectral Analysis and Mechanisms) by Richard A. Muller and Gordon J. MacDonald.

Amazon book link:

Immigration problem

Just posting my Quora answer..
Intelligent, peaceful, skillful immigrants are needed not only in US but also in every other country. Because cultural assimilation is a dire need of our time. If you look at most of the problems we have in our society, they are born out of close mindedness of the people. Why racism is a problem, coz some people just deny or at least used to deny human rights to people who have darker skin. Why an immigrant can find job, but an American can not? You are born in America, but you haven’t explored much or expanded your horizons much to find a job. An immigrant is crossing the ocean or a desert and taking the risk and learning his way to live in a society by learning a different language. That is what a great great great grandparents of a jobless American might have done that the great great great grand son or daughter now doesn’t understand. I have met Americans in Japan when I lived there. When they couldn’t find job in America, they went to Japan and started teaching English to Japanese kids, married Japanese women and are now living there happily. My American friend now lives in Shikoku, Japan and I’m now living within 3 hours of his American home in Michigan. This is how the world should look like. We assimilate, we know, we challenge, we meet people, we make friends, we figure out what we like or dislike, we grow our potential and we live. Nobody is entitled to have anything for granted. Reading a little bit of history will give more persepective. Seeing some statistics on social well being, education, contribution to society by race, by country will open your mind. British have run the world for hundred years, where are they now? America in the sixties have had the highest and happiest middle class, what happened now? World, society will forever change and we should be resilient enough to adapt with the change. Instead of blaming immigrants taking away jobs, an American should rather unite with other Americans of all color and race and legal peaceful immigrants to see how politics and legislations in this country can be made so that people have more rights, more jobs and happy lives, how it’s possible to inspire people or communities rather than infuriating them. Yes, illegal immigration is a problem, but what is to do now to ensure the overall well being of everyone should be the focus or priority. Sometimes sentimental talks and arguments just divide us too much; rather constructive criticism and thoughts should unite us to find real solution to the real problem towards welfare. Putting people in suffering and building walls will never bring wealth and prosperity. So, we should all be critical thinker, empathetic and creative to create a great society. Let’s move our lazy ass, educate ourselves, do what needs to be done in our own space within our reach without complaining too much!

https://www.quora.com/Why-are-immigrants-needed-in-the-U-S-when-many-Americans-dont-have-jobs/answer/Zehady-Abdullah-Khan?share=f8d596b1&srid=X0U1

Sad statistics

So.. the pursuit of happiness. Some statistics just make me sad. People in America are suiciding more than ever. And I probably should not try to find the statistics in Japan. And in most developing and underdeveloped countries, these statistics are not even collected properly to see. Every day when a person wakes up, doesn’t he or she should want to make the today a little better than days before? But unfortunately, it seems we live in the vicious circle of disappointment, distrust, heartbreak, depression and potentially the desire to end ourselves. We are so solely focussed on our own life and all the pleasures but still we can’t see. The human mind is one of the biggest invention of nature. The immense power it possesses, the creativity and the imaginative nature! But still it seems like we live in a hell within our selves. And for some, it’s so terrible that the only way they can think of is to shut them down forever.

Screen Shot 2018-04-09 at 3.22.33 PM

Purpose

Sometimes I forget what I myself wrote.. And then when I read and realize again, I feel to slap myself for not remembering everything I told to myself. And I wake up.
“You don’t find purpose, because there is no purpose pre-ordained. You make your own purpose of your life. There is no ultimate purpose. I know, that feels like a huge slap on the face like most truths. Every morning you wake up, you will have to reinvigorate your purpose. You are free to change your purpose, you are free to not have any purpose. You are responsible for every thing your brain produces even though you have no control over most of them. You are mostly responsible for every action you make with your conscious, unconscious or subconscious mind. Yes.. Life is scary. But it’s better not to live delusionally but to face it. You were not alive on this mother earth for billions of years.. you won’t be alive on this universe filled with joy and distress for more than trillions of years.. So, the less than 100 years you got, don’t waste.. enjoy and live.. if you can have a purpose and propagate that in human culture, that’s a plus.. Even though there’s only one earth physically, but actually the number of perceived earth and the number of human brains are equal. If the purpose you think about is dictated by someone else, you are doomed. So.. You need to open your heart and reach other people who have purpose in their lives to learn and live and probably get a reference. And then.. you probably make your very own purpose. And be happy that you are alive. Pinch yourself to verify. You can either feel scared how little and insignificant you are amidst of the billions of exploding stars or you can think that you yourself are a stardust meaning you are part of everything. Every urge you have to eat, sleep, fuck are evolutionarily designed into you. Learn how to deal with it with good examples from others who lived meaningfully.. But eventually you will have to live on your own. Miserably if you choose wrong, meaningfully if you can make a purpose of your own and strive to be happy and productive. Existentialism at its core, but not ridiculing nihilists too. Hail to Sartre and Nietzsche. And blue is the warmest color.”
My other quora posts:

Diversity in the Past and Present

Well, a lot of us are fan of Khaleesi on Game of Thrones(GOT) and her fire-exhaling Dragons. It’s then a natural question 🙂 whether there existed creatures who could breathe or exhale fire. And science answers with a big “No” as there’s no evidence. Apparently the Bible thumpers and the wishful imaginative thinking of us! It feels exciting to see those dragons on GOT though!!

I am reading “The Ancestor’s Tale” by RD and within few hours, I have learnt so much about mesmerizing animals with their own features on the planet which survived, diverged, got trapped etc. The jurassic park movie did a great job on Jurassic period, but there’s so much more story in our ancestry and concestry.

I wish there were GOT like fictions based on the animals/creatures that exist, as most of us are not fans of animal planet like channels.

What is this obsession with us with non-existentence when there’s such beauty in the existence? 🤔

The Ancestor’s Tale by R.D.
https://www.goodreads.com/book/show/17977.The_Ancestor_s_Tale

Sharks on Bitcoin

What the big three sharks are saying. Somewhat dumb, somewhat great! Investor’s hat is definitely crucial, but not always the best. And I always love the entrepreneur’s hat. Who wouldn’t like to have the both hats with all different colors and taste!! 😉

Shark tank is a great platform that filters entrepreneurs, startup, growing business. If you haven’t known or watched shark tanks, you should definitely start. We human being are idea generators, everybody has ideas. Everytime you walk on the street or malls, you come up with ideas, it’s just the few who take the leap to materialize. Shark tank will give you some glimpse of what I just said, when you will see how innocuous, small ideas can go big if it’s under the right arm. Shark tank is one of the things that I like in both the fat, obese, dumb, racist in one hand and multicultural, smart, free, open, entrepreneurial North America on the other. LoL.;)
If you get the joke. Haha. You become what you choose to..
Here’s a great playlist for popular shark tank videos and roastings. Enjoy. 🙂

And subscribe to the Best of Shark Tank Channel to get more of these videos.

https://www.youtube.com/channel/UC8YLEQGzx-HpBuWRJbid8FA

How to Read Scientific papers/Text books?

Reading scientific textbook. Novels tell stories, but scientific text books try to inform with the fact, evidence, assumptions and logics.

Goal: Being informed and educated

1. Don’t always read front to back
2. ‎Read for Big Ideas + Connect Ideas
3. ‎Read for key details
4. ‎Take good enough notes. Read the book once but your notes multiple times.

Break the chronological order. Because too easy to get lost in the miniscule.
a) Read the questions at the end?
‎b) Read the final summary/conclusion
‎c) Headings and subdivisions of the chapter
‎d) Read the final intro
‎Finally have the view from front to back.