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# RMS to Interval Velocity¶

The rms (root mean square) velocity is related to the interval velocity through the Equation 1.17. This is a useful construct in reflection seismic data processing. Here we use this an an example to illustrate non-uniqueness and ill-conditioning that typifies inversion problems. For notation we use the symbol V to represent $V_{rms}$ and the symbol $v$ to represent $v_{int}$. The analytic expression Equation 1.18 allows us to uniquely compute $v(t)$ from the function $V(t)$. To invert a finite number of observations $V(t_j) j=(1,N)$ we first interpolate these data and then use the analytic inversion formula. The results depend upon how the data are interpolated. The effects of noise on the data are examined in the same manner. Gaussian noise is added to each datum, the data are interpolated, and then the analytic inverse is applied.

Equation 1.17 Integral equation to forward map the rms velocity $V_{rms}$ data on the interval $[0,t_{j}]$ given the interval velocity model $v_{int}$.

$V^2_{rms}(t)=\frac{1}{t_j}\int^{t_j}_0 v_{int}^2(t)dt$

Equation 1.18 Analytic inverse $\mathcal{F}^{-1}$to calculate the interval velocity model $v_{int}$ given the rms velocity data $V_{rms}$ and its time derivative $V_{rms}^\prime (t)$.

$v_{int}=V_{rms}\left(1+\frac{2tV_{rms}\prime(t)}{V_{rms}}\right)^{\frac{1}{2}}$

## Import Apps¶

from geoscilabs.seismic.RMS_to_interval_velocity_app import (
ForwardWidget, InversionBasicWidget, InversionSampledDataWidget, InversionNoisyDataWidget
)

## Forward Modeling App¶

The widget below is used to compute the RMS velocity as a function of time according to equation 1.17. The parameters used by this widget are as follows:

• ($v_0$) Average velocity for the interval velocity function
• ($a$) Amplitude of oscillation for the interval velocity function
• ($T$) Period of oscillation for the interval velocity function
• ($t_{max}$) Length of observation times
out = ForwardWidget()

## Analytic Inversion App¶

The widget below demonstrates that if we know the RMS velocity exactly, we can recover the interval velocity. For the parameters below, we used equation 1.17 to compute the RMS velocity from the interval velocity. Then we used equation 1.18 to recover the original interval velocity from the RMS velocity. The parameters used by this widget are as follows:

• ($v_0$) Average velocity for the interval velocity function
• ($a$) Amplitude of oscillation for the interval velocity function
• ($T$) Period of oscillation for the interval velocity function
• ($t_{max}$) Length of observation times
out = InversionBasicWidget()

## Inversion from Sampled Data App¶

In practice, the RMS velocity is only known at a set of sampled times. To estimate the interval velocity we must 1) interpolate the sampled RMS velocities to a finely gridded set of times, then 2) compute a discrete approximation to equation 1.18. The widget below demonstrates how strongly the interpolation method and density of sampling impacts the recovery of the interval velocity. The parameters used by this widget are as follows:

• ($v_0$) Average velocity for the true interval velocity function
• ($a$) Amplitude of oscillation for the true interval velocity function
• ($T$) Period of oscillation for the true interval velocity function
• ($t_{max}$) Length of observation times
• (n) Number of samples of the RMS velocity per period
• (Interpolation) The interpolation method
out = InversionSampledDataWidget()

## Inversion from Sampled Data with Noise App¶

In practice, sampled RMS velocity data are noisy. The interval velocity is still recovered by 1) interpolating the sampled RMS velocities to a finely gridded set of times, then 2) computing the discrete approximation to equation 1.18. Using the widget below, we can investigate the impact of noisy on the recovered interval velocity. The parameters used by this widget are as follows:

• ($v_0$) Average velocity for the true interval velocity function
• ($a$) Amplitude of oscillation for the true interval velocity function
• ($T$) Period of oscillation for the true interval velocity function
• ($t_{max}$) Length of observation times
• (n) Number of samples of the RMS velocity per period
• (std) Standard deviation of Gaussian noise added to each datum
• (Interpolation) The interpolation method
out = InversionNoisyDataWidget()
The RMS (Root Mean Square) velocity is related to the interval velocity through the equation XXX. This is a useful construct in reflection seismic data processing. Here we use this an an example to illustrate non-uniqueness and ill-conditioning that typifies inversion problems. For notation we use the symbol V to represent $V_{rms}$ and the symbol $v$ to represent $v_{int}$. The analytic expression XXX allows us to uniquely compute $v(t)$ from the function $V(t)$. To invert a finite number of observations $V(t_j) j=(1,N)$ we first interpolate these data and then use the analytic inversion formula. The results depend upon how the data are interpolated. The effects of noise on the data are examined in the same manner. Gaussian noise is added to each datum, the data are interpolated, and then the analytic inverse is applied.