'''
How might you implement a divergence free hatch filter in python for GPS data A divergence-free Hatch filter can be used to remove noise from GPS data
'''

import numpy as np

def divergence_free_hatch_filter(positions, velocities, dt, sigma):
    # Compute the finite differences of the velocities to obtain the velocity gradients
    gradients = np.diff(velocities, axis=0) / dt

    # Compute the divergence of the velocity gradients
    divergence = np.sum(gradients, axis=1)

    # Compute the correction factor based on the divergence
    correction = np.zeros_like(gradients)
    correction[1:-1] = 0.5 * sigma * (divergence[:-2] - 2 * divergence[1:-1] + divergence[2:])
    correction[0] = sigma * (divergence[0] - divergence[1])
    correction[-1] = sigma * (divergence[-1] - divergence[-2])

    # Apply the correction to the velocity gradients
    corrected_gradients = gradients - correction

    # Integrate the corrected gradients to obtain the corrected velocities
    corrected_velocities = np.cumsum(corrected_gradients, axis=0) * dt + velocities[0]

    # Integrate the corrected velocities to obtain the corrected positions
    corrected_positions = np.cumsum(corrected_velocities, axis=0) * dt + positions[0]

    return corrected_positions, corrected_velocities