taylor expansion

[helpplease]

What does "truncation of a taylor expansion" mean? 

 

 

 

 

 

Are someone here familiar with what an aperture is defined as in geophysics? 

[studiot]

15 minutes ago, helpplease said:

What does "truncation of a taylor expansion" mean? 

 

 

 

 

 

Are someone here familiar with what an aperture is defined as in geophysics? 

1) A taylor series, like any converging series (do you understand converging) may be used to estimate in numeric calculation or stand in for algebraically in formula an expression that is difficult to handle.
If the series convderges quickly enough (it often does) we can ignore all terms after the first n, for some small n.
There are criteria for deciding how many terms to use. The process of discarding the rest is called truncation.

Wolfram has a good explanation

https://mathworld.wolfram.com/TaylorSeries.html

 

2) An aperture is an opening, which is used to describe the 'size' of something or the size of something being measured or that can pass through.

So this term is widely used in geosciences.

Polyzoa (Fenestrilina) colonies form a basket like weave with apertures in the basket to filter and trap particles washing through.

Acidic groundwater percolating through susceptible rocks (eg limestone) solution weather apertures in the solid rock.

In photogeology the aperture is the camera opening that controls the amount of light entering.

Similary magnetic sensor arrays and seismic arrays have an aperture sensitivity.

These control the amplitude and frequency that can be accepted by the arrays.

Other wave phenomena (eg ultrasonic) have similar use of the term.

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

[helpplease]

Quote

A taylor series, like any converging series (do you understand converging) may be used to estimate in numeric calculation or stand in for algebraically in formula an expression that is difficult to handle.
If the series convderges quickly enough (it often does) we can ignore all terms after the first n, for some small n.
There are criteria for deciding how many terms to use. The process of discarding the rest is called truncation.

Hello, thank you for your reply.

Could you explan this image to simply understand this? What does this mean? 

Its also a sentence where it says: The truncation of this Taylor expansion leads to inaccuracies considering large apertures in the midpoint and offset domain.

This is what i asked earlier. 

Edited September 3, 2020 by helpplease

[HallsofIvy]

First, in mathematics a "series" is an infinite sum of numbers of functions.  A "power series" is an infinite sum of powers of the variable, usually "x".  An example is sum_{n=0 to infinity} a_n(x- x_0)^n= a_0+ a_1(x- x_0)+ a_2(x- x_0)^2+ \cdot\cdot\cdot.  A "Taylor's series" is a power series "representing" a given function derived in a particular way- a_n, the coefficient of x^n is the nth derivative of f evaluated at x_0 divided by n!.  (I say "representing" because the Taylor's series of a function is not necessarily equal to that function.)  An simple example is f(x)= e^x.  All derivatives of e^x are e^x again and its value at x_0= 0 is 1.  The Taylor's series for  e^x about x0= 0 is Sum x^n/n!.  It can be shown that this series does converge to e^x for all x.  In fact it converges fast enough that "truncating" it (cutting it short) at, say n= 2, gives a reasonable approximation.  That is e^x is approximately 1+ x+ x^2/2 for small x.

For example, if x= 0.1, e^x= e^0.1= 1.1051709180756476248117078264902... while 1+ 0.1+ 0.1^2= 1.11.

If x= 0.01, e^x= e^0.01= 1.0100501670841680575421654569029 while 1+ .01+ 0.01^2= 1.0101 

Edited October 2, 2020 by HallsofIvy