Double exponential function

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Double_exponential_function

 

A double exponential function (red curve) compared to a single exponential function (blue curve).

 

A double exponential function is a constant raised to the power of an exponential function. The general formula is ${\displaystyle f(x)=a^{b^{x}}=a^{(b^{x})}}$ (where a>1 and b>1), which grows much more quickly than an exponential function. For example, if a = b = 10:

• f(0) = 10
• f(1) = 10¹⁰
• f(2) = 10¹⁰⁰ = googol
• f(3) = 10¹⁰⁰⁰
• f(100) = 10¹⁰¹⁰⁰ = googolplex.

Factorials grow more quickly than exponential functions, but much more slowly than doubly exponential functions. However, tetration and the Ackermann function grow faster. See Big O notation for a comparison of the rate of growth of various functions. 

The inverse of the double exponential function is the double logarithm ln(ln(x)). 

Doubly exponential sequences

Aho and Sloane observed that in several important integer sequences, each term is a constant plus the square of the previous term. They show that such sequences can be formed by rounding to the nearest integer the values of a doubly exponential function in which the middle exponent is two. Integer sequences with this squaring behavior include

• The Fermat numbers

${\displaystyle F(m)=2^{2^{m}}+1}$

• The harmonic primes: The primes p, in which the sequence 1/2+1/3+1/5+1/7+....+1/p exceeds 0,1,2,3,....

The first few numbers, starting with 0, are 2,5,277,5195977,... (sequence A016088 in the OEIS)

• The Double Mersenne numbers

${\displaystyle MM(p)=2^{2^{p}-1}-1}$

• The elements of Sylvester's sequence (sequence A000058 in the OEIS)

${\displaystyle s_{n}=\left\lfloor E^{2^{n+1}}+{\frac {1}{2}}\right\rfloor }$

where E ≈ 1.264084735305302 is Vardi's constant (sequence A076393 in the OEIS).

• The number of k-ary Boolean functions:

${\displaystyle 2^{2^{k}}}$

More generally, if the nth value of an integer sequence is proportional to a double exponential function of n, Ionaşcu and Stănică call the sequence "almost doubly-exponential" and describe conditions under which it can be defined as the floor of a doubly exponential sequence plus a constant. Additional sequences of this type include

• The prime numbers 2, 11, 1361, ... (sequence A051254 in the OEIS)

${\displaystyle a(n)=\left\lfloor A^{3^{n}}\right\rfloor }$

where A ≈ 1.306377883863 is Mills' constant.

 

Applications

Algorithmic complexity

In computational complexity theory, some algorithms take doubly exponential time:

• Each decision procedure for Presburger arithmetic provably requires at least doubly exponential time 
• Computing a Gröbner basis over a field. In the worst case, a Gröbner basis may have a number of elements which is doubly exponential in the number of variables. On the other hand, the worst-case complexity of Gröbner basis algorithms is doubly exponential in the number of variables as well as in the entry size.
• Finding a complete set of associative-commutative unifiers 
• Satisfying CTL⁺ (which is, in fact, 2-EXPTIME-complete) 
• Quantifier elimination on real closed fields takes doubly exponential time (see Cylindrical algebraic decomposition).
• Calculating the complement of a regular expression 

In some other problems in the design and analysis of algorithms, doubly exponential sequences are used within the design of an algorithm rather than in its analysis. An example is Chan's algorithm for computing convex hulls, which performs a sequence of computations using test values h_i = 2²ⁱ (estimates for the eventual output size), taking time O(n log h_i) for each test value in the sequence. Because of the double exponential growth of these test values, the time for each computation in the sequence grows singly exponentially as a function of i, and the total time is dominated by the time for the final step of the sequence. Thus, the overall time for the algorithm is O(n log h) where h is the actual output size.

Number theory

Some number theoretical bounds are double exponential. Odd perfect numbers with n distinct prime factors are known to be at most

${\displaystyle 2^{4^{n}}}$

a result of Nielsen (2003). The maximal volume of a d-lattice polytope with k ≥ 1 interior lattice points is at most

${\displaystyle (8d)^{d}\cdot 15^{d\cdot 2^{2d+1}}}$

a result of Pikhurko.

The largest known prime number in the electronic era has grown roughly as a double exponential function of the year since Miller and Wheeler found a 79-digit prime on EDSAC1 in 1951.

Theoretical biology

In population dynamics the growth of human population is sometimes supposed to be double exponential. Varfolomeyev and Gurevich experimentally fit

${\displaystyle N(y)=375.6\cdot 1.00185^{1.00737^{y-1000}}\,}$

where N(y) is the population in year y in millions. 

Physics

In the Toda oscillator model of self-pulsation, the logarithm of amplitude varies exponentially with time (for large amplitudes), thus the amplitude varies as doubly exponential function of time.

Dendritic macromolecules have been observed to grow in a doubly-exponential fashion.

Geophysical survey

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Geophysical_survey

Geophysical survey is the systematic collection of geophysical data for spatial studies. Detection and analysis of the geophysical signals forms the core of Geophysical signal processing. The magnetic and gravitational fields emanating from the Earth's interior hold essential information concerning seismic activities and the internal structure. Hence, detection and analysis of the electric and Magnetic fields is very crucial. As the Electromagnetic and gravitational waves are multi-dimensional signals, all the 1-D transformation techniques can be extended for the analysis of these signals as well. Hence this article also discusses multi-dimensional signal processing techniques.

Geophysical surveys may use a great variety of sensing instruments, and data may be collected from above or below the Earth's surface or from aerial, orbital, or marine platforms. Geophysical surveys have many applications in geology, archaeology, mineral and energy exploration, oceanography, and engineering. Geophysical surveys are used in industry as well as for academic research.

The sensing instruments such as gravimeter, gravitational wave sensor and magnetometers detect fluctuations in the gravitational and magnetic field. The data collected from a geophysical survey is analysed to draw meaningful conclusions out of that. Analysing the spectral density and the time-frequency localisation of any signal is important in applications such as oil exploration and seismography.

Types of geophysical survey

There are many methods and types of instruments used in geophysical surveys. Technologies used for geophysical surveys include:

1. Seismic methods, such as reflection seismology, seismic refraction, and seismic tomography.
2. Seismoelectrical method
3. Geodesy and gravity techniques, including gravimetry and gravity gradiometry.
4. Magnetic techniques, including aeromagnetic surveys and magnetometers.
5. Electrical techniques, including electrical resistivity tomography, induced polarization, spontaneous potential and marine control source electromagnetic (mCSEM) or EM seabed logging.
6. Electromagnetic methods, such as magnetotellurics, ground penetrating radar and transient/time-domain electromagnetics, surface nuclear magnetic resonance (also known as magnetic resonance sounding).
7. Borehole geophysics, also called well logging.
8. Remote sensing techniques, including hyperspectral.

Geophysical signal detection

This section deals with the principles behind measurement of geophysical waves. The magnetic and gravitational fields are important components of geophysical signals. 

The instrument used to measure the change in gravitational field is the gravimeter. This meter measures the variation in the gravity due to the subsurface formations and deposits. To measure the changes in magnetic field the magnetometer is used. There are two types of magnetometers, one that measures only vertical component of the magnetic field and the other measures total magnetic field.

Measurement of Earth’s magnetic fields

Magnetometers are used to measure the magnetic fields, magnetic anomalies in the earth. The sensitivity of magnetometers depends upon the requirement. For example, the variations in the geomagnetic fields can be to the order of several aT where 1aT = 10⁻¹⁸T . In such cases, specialized magnetometers such as the superconducting quantum interference device (SQUID) are used.

Jim Zimmerman co-developed the rf superconducting quantum interference device (SQUID) during his tenure at Ford research lab. However, events leading to the invention of the SQUID were in fact, serendipitous. John Lambe, during his experiments on nuclear magnetic resonance noticed that the electrical properties of indium varied due to a change in the magnetic field of the order of few nT. However, Lambe was not able to fully recognize the utility of SQUID.

SQUIDs have the capability to detect magnetic fields of extremely low magnitude. This is due to the virtue of the Josephson junction. Jim Zimmerman pioneered the development of SQUID by proposing a new approach to making the Josephson junctions. He made use of niobium wires and niobium ribbons to form two Josephson junctions connected in parallel. The ribbons act as the interruptions to the superconducting current flowing through the wires. The junctions are very sensitive to the magnetic fields and hence are very useful in measuring fields of the order of 10^{^-18}T.

Seismic wave measurement using gravitational wave sensor

Gravitational wave sensors can detect even a minute change in the gravitational fields due to the influence of heavier bodies. Large seismic waves can interfere with the gravitational waves and may cause shifts in the atoms. Hence, the magnitude of seismic waves can be detected by a relative shift in the gravitational waves.

Measurement of seismic waves using atom interferometer

The motion of any mass is affected by the gravitational field. The motion of planets is affected by the Sun's enormous gravitational field. Likewise, a heavier object will influence the motion of other objects of smaller mass in its vicinity. However, this change in the motion is very small compared to the motion of heavenly bodies. Hence, special instruments are required to measure such a minute change.

Describes the atom interferometer principle

 

Atom interferometers work on the principle of diffraction. The diffraction gratings are nano fabricated materials with a separation of a quarter wavelength of light. When a beam of atoms pass through a diffraction grating, due the inherent wave nature of atoms, they split and form interference fringes on the screen. An atom interferometer is very sensitive to the changes in the positions of atoms. As heavier objects shifts the position of the atoms nearby, displacement of the atoms can be measured by detecting a shift in the interference fringes.

Existing approaches in geophysical signal recognition

This section addresses the methods and mathematical techniques behind signal recognition and signal analysis. It considers the time domain and frequency domain analysis of signals. This section also discusses various transforms and their usefulness in the analysis of multi-dimensional waves.

3D sampling

Sampling

The first step in any signal processing approach is analog to digital conversion. The geophysical signals in the analog domain has to be converted to digital domain for further processing. Most of the filters are available in 1D as well as 2D. 

Analog to digital conversion

As the name suggests, the gravitational and electromagnetic waves in the analog domain are detected, sampled and stored for further analysis. The signals can be sampled in both time and frequency domains. The signal component is measured at both intervals of time and space. Ex, time-domain sampling refers to measuring a signal component at several instances of time. Similarly, spatial-sampling refers to measuring the signal at different locations in space.

Traditional sampling of 1D time varying signals is performed by measuring the amplitude of the signal under consideration in discrete intervals of time. Similarly sampling of space-time signals (signals which are functions of 4 variables – 3D space and time), is performed by measuring the amplitude of the signals at different time instances and different locations in the space. For example, the earth's gravitational data is measured with the help of gravitational wave sensor or gradiometer by placing it in different locations at different instances of time. 

Spectrum analysis

Multi-dimensional Fourier transform

The Fourier expansion of a time domain signal is the representation of the signal as a sum of its frequency components, specifically sum of sines and cosines. Joseph Fourier came up with the Fourier representation to estimate the heat distribution of a body. The same approach can be followed to analyse the multi-dimensional signals such as gravitational waves and electromagnetic waves.

The 4D Fourier representation of such signals is given by

${\displaystyle S(K,\omega )=\iint s(x,t)e^{-j(\omega t-k'x)}\,dx\,dt}$

• ω represents temporal frequency and k represents spatial frequency.
• s(x,t) is a 4-dimensional space-time signal which can be imagined as travelling plane waves. For such plane waves, the plane of propagation is perpendicular to the direction of propagation of the considered wave.

Wavelet transform

The motivation for development of the Wavelet transform was the Short-time Fourier transform. The signal to be analysed, say f(t) is multiplied with a window function w(t) at a particular time instant. Analysing the Fourier coefficients of this signal gives us information about the frequency components of the signal at a particular time instant.

The STFT is mathematically written as:

${\displaystyle \{x(t)\}(\tau ,\omega )\equiv X(\tau ,\omega )=\int _{-\infty }^{\infty }x(t)w(t-\tau )e^{-j\omega t}\,dt}$

The Wavelet transform is defined as

${\displaystyle X(a,b)={\frac {1}{\sqrt {a}}}\int \limits _{\ }\Psi ({\frac {t-b}{a}})x(t)dt}$

A variety of window functions can be used for analysis. Wavelet functions are used for both time and frequency localisation. For example,one of the windows used in calculating the Fourier coefficients is the Gaussian window which is optimally concentrated in time and frequency. This optimal nature can be explained by considering the time scaling and time shifting parameters a and b respectively. By choosing the appropriate values of a and b, we can determine the frequencies and the time associated with that signal. By representing any signal as the linear combination of the wavelet functions, we can localize the signals in both time and frequency domain. Hence wavelet transforms are important in geophysical applications where spatial and temporal frequency localisation is important.

Time frequency localisation using wavelets

Geophysical signals are continuously varying functions of space and time. The wavelet transform techniques offer a way to decompose the signals as a linear combination of shifted and scaled version of basis functions. The amount of "shift" and "scale" can be modified to localize the signal in time and frequency. 

Beamforming

Simply put, space-time signal filtering problem can be thought as localizing the speed and direction of a particular signal. The design of filters for space-time signals follows a similar approach as that of 1D signals. The filters for 1-D signals are designed in such a way that if the requirement of the filter is to extract frequency components in a particular non-zero range of frequencies, a bandpass filter with appropriate passband and stop band frequencies in determined. Similarly, in the case of multi-dimensional systems, the wavenumber-frequency response of filters is designed in such a way that it is unity in the designed region of (k, ω) a.k.a. wavenumber – frequency and zero elsewhere.

Spatial distribution of phased arrays to filter geophysical signals

 

This approach is applied for filtering space-time signals. It is designed to isolate signals travelling in a particular direction. One of the simplest filters is weighted delay and sum beamformer. The output is the average of the linear combination of delayed signals. In other words, the beamformer output is formed by averaging weighted and delayed versions of receiver signals. The delay is chosen such that the passband of beamformer is directed to a specific direction in the space.

Classical estimation theory

This section deals with the estimation of the power spectral density of the multi-dimensional signals.The spectral density function can be defined as a multidimensional Fourier transform of the autocorrelation function of the random signal.

${\displaystyle P\left(K_{x},w\right)=\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }\varphi _{ss}\left(x,t\right)\ e^{-j\left(wt-k'x\right)}\,dx\,dt}$

${\displaystyle \varphi _{ss}\left(x,t\right)=s\left[\left(\xi ,\tau \right)s*\left(\xi -x,\tau -t\right)\right]}$

The spectral estimates can be obtained by finding the square of the magnitude of the Fourier transform also called as Periodogram. The spectral estimates obtained from the periodogram have a large variance in amplitude for consecutive periodogram samples or in wavenumber. This problem is resolved using techniques that constitute the classical estimation theory. They are as follows: 

1.Bartlett suggested a method that averages the spectral estimates to calculate the power spectrum. Average of spectral estimates over a time interval gives a better estimate.

${\displaystyle P_{B}\left(w\right)={\frac {1}{\mathrm {det} \,N}}\sum _{l}|\sum _{n}\ x\left(n+MI\right)\ e^{-j\left(w'n\right)}|^{2}}$ Bartlett's case

2.Welch's method suggested to divide the measurements using data window functions, calculate a periodogram, average them to get a spectral estimate and calculate the power spectrum using Fast Fourier Transform (FFT).This increased the computational speed.

${\displaystyle P_{W}\left(w\right)={\frac {1}{\mathrm {det} \,N}}\sum _{l}|\sum _{n}\ g\left(n\right)\ x\left(n+MI\right)\ e^{-j\left(w'n\right)}|^{2}}$ Welch's case 

4.The periodogram under consideration can be modified by multiplying it with a window function. Smoothing window will help us smoothen the estimate. Wider the main lobe of the smoothing spectrum, smoother it becomes at the cost of frequency resolution.

${\displaystyle P_{M}\left(w\right)={\frac {1}{detN}}|\sum _{n}\ g\left(n\right)\ x\left(n\right)\ e^{-j\left(w'n\right)}|^{2}}$ Modified periodogram 

 

Applications

Estimating positions of underground objects

The method being discussed here assumes that the mass distribution of the underground objects of interest is already known and hence the problem of estimating their location boils down to parametric localisation.Say underground objects with center of masses (CM₁, CM₂...CM_n) are located under the surface and at positions p₁, p₂...p_n. The gravity gradient(components of the gravity field) is measured using a spinning wheel with accelerometers also called as the gravity gradiometer. The instrument is positioned in different orientations to measure the respective component of the gravitational field. The values of gravitational gradient tensors are calculated and analyzed. The analysis includes observing the contribution of each object under consideration. A maximum likelihood procedure is followed and Cramér–Rao bound (CRB) is computed to assess the quality of location estimate.

Array processing for seismographic applications

Various sensors located on the surface of earth spaced equidistantly receive the seismic waves. The seismic waves travel through the various layers of earth and undergo changes in their properties - amplitude change, time of arrival, phase shift. By analyzing these properties of the signals, we can model the activities inside the earth. 

Visualization of 3D data

The method of volume rendering is an important tool to analyse the scalar fields. Volume rendering simplifies representation of 3D space. Every point in a 3D space is called a voxel. Data inside the 3-d dataset is projected to the 2-d space(display screen) using various techniques. Different data encoding schemes exist for various applications such as MRI, Seismic applications.

Gravity gradiometry

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Gravity_gradiometry

Gravity gradiometry is the study and measurement of variations in the acceleration due to gravity. The gravity gradient is the spatial rate of change of gravitational acceleration.

Gravity gradiometry is used by oil and mineral prospectors to measure the density of the subsurface, effectively by measuring the rate of change of gravitational acceleration (or jerk) due to underlying rock properties. From this information it is possible to build a picture of subsurface anomalies which can then be used to more accurately target oil, gas and mineral deposits. It is also used to image water column density, when locating submerged objects, or determining water depth (bathymetry). Physical scientists use gravimeters to determine the exact size and shape of the earth and they contribute to the gravity compensations applied to inertial navigation systems.

Measuring the gravity gradient

Gravity measurements are a reflection of the earth’s gravitational attraction, its centripetal force, tidal accelerations due to the sun, moon, and planets, and other applied forces. Gravity gradiometers measure the spatial derivatives of the gravity vector. The most frequently used and intuitive component is the vertical gravity gradient, G_zz, which represents the rate of change of vertical gravity (g_z) with height (z). It can be deduced by differencing the value of gravity at two points separated by a small vertical distance, l, and dividing by this distance.

${\displaystyle G_{zz}={\partial g_{z} \over \partial z}\approx {g_{z}{\bigl (}z+{\tfrac {l}{2}}{\bigr )}-g_{z}{\bigl (}z-{\tfrac {l}{2}}{\bigr )} \over l}}$

The two gravity measurements are provided by accelerometers which are matched and aligned to a high level of accuracy. 

Units

The unit of gravity gradient is the eotvos (abbreviated as E), which is equivalent to 10⁻⁹ s⁻² (or 10⁻⁴ mGal/m). A person walking past at a distance of 2 metres would provide a gravity gradient signal approximately one E. Mountains can give signals of several hundred Eotvos.

Gravity gradient tensor

Full tensor gradiometers measure the rate of change of the gravity vector in all three perpendicular directions giving rise to a gravity gradient tensor (Fig 1). 

Fig 1. Conventional gravity measures ONE component of the gravity field in the vertical direction Gz (LHS), Full tensor gravity gradiometry measures ALL components of the gravity field (RHS)

 

Comparison to gravity

Being the derivatives of gravity, the spectral power of gravity gradient signals is pushed to higher frequencies. This generally makes the gravity gradient anomaly more localised to the source than the gravity anomaly. The table (below) and graph (Fig 2) compare the g_z and G_zz responses from a point source. 

	Gravity (gz)	Gravity gradient (Gzz)
Signal	${\displaystyle {GM\,z \over \left(r^{2}+z^{2}\right)^{\frac {3}{2}}}\times 10^{5}\;\left[{\text{mGal}}\right]}$	${\displaystyle {GM\left(r^{2}-2z^{2}\right) \over \left(r^{2}+z^{2}\right)^{\frac {5}{2}}}\times 10^{9}\;\left[{\text{E}}\right]}$
Peak signal (r = 0)	${\displaystyle {GM \over z^{2}}\times 10^{5}}$	${\displaystyle {2GM \over z^{3}}\times 10^{9}}$
Full width at half maximum	${\displaystyle 1.53\,z}$	${\displaystyle \approx z}$
Wavelength (λ)	${\displaystyle 3.07\,z}$	${\displaystyle 2\,z}$

Fig 2. Vertical gravity and gravity gradient signals from a point source buried at 1 km depth

 

Conversely, gravity measurements have more signal power at low frequency therefore making them more sensitive to regional signals and deeper sources.

Dynamic survey environments (airborne and marine)

The derivative measurement sacrifices the overall energy in the signal, but significantly reduces the noise due to motional disturbance. On a moving platform, the acceleration disturbance measured by the two accelerometers is the same so that when forming the difference, it cancels in the gravity gradient measurement. This is the principal reason for deploying gradiometers in airborne and marine surveys where the acceleration levels are orders of magnitude greater than the signals of interest. The signal to noise ratio benefits most at high frequency (above 0.01 Hz), where the airborne acceleration noise is largest. 

Applications

Gravity gradiometry has predominately been used to image subsurface geology to aid hydrocarbon and mineral exploration. Over 2.5 million line km has now been surveyed using the technique. The surveys highlight gravity anomalies that can be related to geological features such as Salt diapirs, Fault systems, Reef structures, Kimberlite pipes, etc. Other applications include tunnel and bunker detection and the recent GOCE mission that aims to improve the knowledge of ocean circulation.

Gravity gradiometers

Lockheed Martin gravity gradiometers

During the 1970s, as an executive in the US Dept. of Defense, John Brett initiated the development of the gravity gradiometer to support the Trident 2 system. A committee was commissioned to seek commercial applications for the Full Tensor Gradient (FTG) system that was developed by Bell Aerospace (later acquired by Lockheed Martin) and was being deployed on US Navy Ohio-class Trident submarines designed to aid covert navigation. As the Cold War came to a close, the US Navy released the classified technology and opened the door for full commercialization of the technology. The existence of the gravity gradiometer was famously exposed in the film The Hunt for Red October released in 1990. 

There are two types of Lockheed Martin gravity gradiometers currently in operation: the 3D Full Tensor Gravity Gradiometer (FTG; deployed in either a fixed wing aircraft or a ship) and the FALCON gradiometer (a partial tensor system with 8 accelerometers and deployed in a fixed wing aircraft or a helicopter). The 3D FTG system contains three gravity gradiometry instruments (GGIs), each consisting of two opposing pairs of accelerometers arranged on a spinning disc with measurement direction in the spin direction. 

Other gravity gradiometers

Electrostatic gravity gradiometer

This is the gravity gradiometer deployed on the European Space Agency's GOCE mission. It is a three-axis diagonal gradiometer based on three pairs of electrostatic servo-controlled accelerometers.

ARKeX Exploration gravity gradiometer

An evolution of technology originally developed for European Space Agency, the Exploration Gravity Gradiometer (EGG), developed by ARKeX (a corporation that is now defunct), uses two key principles of superconductivity to deliver its performance: the Meissner effect, which provides levitation of the EGG proof masses and flux quantization, which gives the EGG its inherent stability. The EGG has been specifically designed for high dynamic survey environments.

Ribbon sensor gradiometer

The Gravitec gravity gradiometer sensor consists of a single sensing element (a ribbon) that responds to gravity gradient forces. It is designed for borehole applications.

UWA gravity gradiometer

The University of Western Australia (aka VK-1) Gravity Gradiometer is a superconducting instrument which uses an orthogonal quadrupole responder (OQR) design based on pairs of micro-flexure supported balance beams.

Gedex gravity gradiometer

The Gedex gravity gradiometer (AKA High-Definition Airborne Gravity Gradiometer, HD-AGG) is also a superconducting OQR-type gravity gradiometer, based on technology developed at the University of Maryland.