I would love to know what exactly we can do with the extracted wavelet. What kind of information extracted wavelet can contain?
We're usually not interested in the wavelet per se, but we need to know it in order to understand how the seismic trace relates to geological properties we are interested in.
According to the so-called convolutional model of seismic, a trace $s(t)$ (the amplitude is a function of time $t$) has three main components, so to speak:
$$ s(t) = r(t) \ast w(t) + n $$
where $r$ is the reflectivity of the earth, $w$ is the wavelet, $n$ is random noise, and $\ast$ denotes a mathematical operation called convolution.
The reflectivity of the earth is very useful geological information. Reflectivity is a function of the impedance contrast at a boundary (e.g. between two beds) and impedance is, in turn, a function of important rock properties, especially density and the speed of sound in the rock. Both of these properties can help us deduce the rock type, porosity, fluid type, and so on, and other related properties can tell us about thin beds and fractures and other interesting things.
The problem with getting at the rock property information from the seismic trace is that it's all tangled up with the wavelet via the convolutional operator (there's noise too, but we can't do much about that.)
So why do we need the wavelet?
- Forward modeling. With a wavelet, we can try to forward model the seismic. For example we can measure, or guess, the impedance, calculate the reflectivity, and convolve with a wavelet to have a good guess at what the seismic might look like. This will result in synthetic seismic. We can use this to, for example, try to figure out which bed in a borehole is which event in a seismic volume (we call this process 'well tying'.)
- Inversion. Or, we can use the wavelet to get at the geology from the seismic data we measured. If we can estimate the wavelet, we can try to untangle it, or deconvolve it, from the seismic trace and get at the juicy geological stuff.
So, having the wavelet, or being able to make a good guess at it, is useful. There are several ways to do it, e.g. see the SEG wiki. But it's a bit of an art and can be frustrating to get right... so a lot of people give up and use a standard off-the-shelf wavelet like a Ricker.