auto_fchp#

auto_fchp selects a high-pass corner frequency \(f_{chp}\) for a single component ground motion acceleration record, \(acc1\). The algorithm uses two criteria to select \(f_{chp}\):

criterion 1: The amplitude of a polynomial fit to the displacement record must be a specified multiple of the amplitude of the displacement record

criterion 2: The amplitude of the initial portion of the displacement record before the p-wave arrival must be less than or equal to a specified multiple of the amplitude of the displacement record

The algorithm first selects \(f_{chp}\) to satisfy criterion 1. It then checks criterion 2, and if necessary, increases \(f_{chp}\) to satify criterion 2. Criterion 2 is optional.

criterion 1#

The algorithm fits a polynomial of user-specified order to the filtered displacement record, and iterates on \(f_{chp}\) until the ratio of the amplitude of the polynomial to that of the displacement record is equal to a specified target. The acceleration record is pre-processed following these steps:

Define a Tukey window
Subtract the weighted mean from \(acc\), where the weights are equal to the Tukey window coefficients
Compute the Fourier transform, \(F_{acc}\), and frequency array, \(f\)
Compute the Fourier coefficients of the displacement record using Eq. (1)

After pre-processing, a check is performed to determine whether the optimal value of \(f_{chp}\) lies between \(f_{chp,min}\) and \(f_{chp,max}\). The check follows these steps:

Set \(f_{chp}\) equal to \(f_{chp,min}\)
Compute the filtered Fourier displacement coefficients \(Fdisp_{filt}\) using Eq. (2).
Compute the filtered displacement record by taking the inverse Fourier transform of \(Fdisp_{filt}\)
Compute the residual \(R1\) using Eq. (3)
Repeat steps 5 through 8 using \(f_{chp,max}\) instead of \(f_{chp,min}\)
If the sign of the residuals are equal, the root is not bracketed. Return \(f_{chp,max}\) if the sign is positive, and return \(f_{chp,min}\) if the sign is negative.

Finally, if the root is bracketed, use Ridders’ method to find the value of \(f_{chp}\) that satisfies Eq. (3). In this case, the Scipy package scipy.optimize.ridder is utilized.

If the root is bracketed, solve for \(f_{chp}\) using scipy.optimize.ridder()

(1)#\[Fdisp = \frac{Facc}{-\left(2\pi f\right)^2}\]

(2)#\[Fdisp_{filt} = \frac{Fdisp}{\sqrt{1+\left(\frac{f_{chp}}{f_u}\right)^{2\cdot order}}}\]

(3)#\[R1 = \frac{\left|disp_{fit}\right|}{\left|disp\right|} - target\]

criterion 2 (optional)#

If the user chooses to use criterion 2, the algorithm checks the ratio of the initial portion of the displacement record with duration of disp_ratio_time and compares it with the amplitude of the filtered displacement record. If the ratio is larger than disp_ratio_target, the algorithm iterates on \(f_{chp}\) until the ratio is equal to disp_ratio_target. Criterion 2 follows these steps:

Extract the initial portion of the displacmeent record \(disp_{init}\)
Using Eq. (4), compute the criterion 2 residual \(R2\) for the value of \(f_{chp}\) obtained for criterion 1.
If \(R2 \leq tol\), return \(f_{chp}\)
If \(R2 > tol\), follow the logic from criterion 1 using the scipy.optimize.ridder package to solve for \(f_{chp}\)

(4)#\[R2 = \frac{\left|disp_{init}\right|}{\left|disp\right|} - target\]

Installation#

pip install ucla_geotech_tools

Requirements#

numpy >= 1.22
scipy >= 1.8

Function#

get_fchp(**kwargs)
get_residual1(fchp *args)
get_residual2(fchp *args)

Input parameters#

get_fchp(**kwargs)#

Return the high-pass corner frequency value that stabilizes the displacement of a ground motion record.

Keyword Args:

parameter	type	description	required	default
`dt`	float	time step in seconds (required)	x
`acc`	numpy array, dtype=float	acceleration array in L/s/s (required)	x
`target`	float	desired ratio of polynomial fit amplitude to displacement amplitude		0.02
`tol`	float	tolerance for fchp		0.001
`poly_order`	int	order of polynomial to fit to filtered displacement record		6
`maxiter`	int	maximum number of iterations		30
`fchp_min`	float	minimum frequency in Hz to consider for fchp		0.001
`fchp_max`	float	maximum frequency in Hz to consider for fchp		0.5
`tukey_alpha`	float	alpha parameter for scipy.signal.windows.tukey		0.05
`apply_disp_ratio`	int	flag to indicate whether to apply criterion 2		0
`disp_ratio_time`	float	duration in seconds of the initial portion of the record before the p-wave arrival		30.0
`disp_ratio_target`	float	target ratio of amplitude of initial portion of displacement record to amplitude of displacement record		0.05

Example Commands:

fchp=get_fchp(dt=dt, acc=acc)

fchp=get_fchp(dt=dt,acc=acc,target=0.02,tol=0.001,poly_order=6,maxiter=30,fchp_min=0.001,fchp_max=0.5,filter_order=5.0,tukey_alpha=0.05,apply_disp_ratio=1,disp_ratio_time=2,disp_ratio_target=0.02)

get_residual1(fchp, *args):#

Return the residual defined as disp_fit/disp_filt - target.

Args:

parameter	type	description
`fchp`	float	High-pass corner frequency
`Fdisp`	numpy array, dtype = complex	Fourier coefficients of displacement record
`time`	numpy array, dtype = float	time array in seconds
`poly_order`	int	Order of polynomial to fit to filtered displacement
`target`	float	desired ratio of polynomial fit amplitude to displacement signal amplitude
`freq`	numpy array, dtype = float	frequency array

get_residual2(fchp, *args):#

Return the residual defined as disp_init/disp - target.