Solid Earth tides - time series and frequency aliasing¶

%matplotlib inline
import os
import datetime as dt
import numpy as np
from scipy import signal
from matplotlib import pyplot as plt, ticker, dates as mdates
import pysolid
from mintpy.utils import utils as ut, isce_utils
plt.rcParams.update({'font.size': 12})

work_dir = os.path.expanduser('~/Papers/2022_Geolocation/figs_src/SET')
os.chdir(work_dir)
print('Go to directory', work_dir)

## SET time-series
# inputs
lat, lon = 34.0, -118.0 # Los Angles, CA
step_sec = 60 * 30
dt0 = dt.datetime(2019,12,1,0,0,0)
dt1 = dt.datetime(2021,2,1,0,0,0)

# run
(dt_out,
 tide_e,
 tide_n,
 tide_u) = pysolid.calc_solid_earth_tides_point(lat, lon, dt0, dt1,
                                                step_sec=step_sec,
                                                display=False,
                                                verbose=False)

# tide: ENU2LOS
inc_angle = 42 * np.pi / 180.
az_angle = 100 * np.pi / 180.
tide_los = ut.enu2los(tide_e, tide_n, tide_u, inc_angle=inc_angle, az_angle=az_angle)

# SET time-series at SAR acquisitions
dt_sen12 = np.array([dt.datetime(2019, 9, 3, 13, 30, 0) + dt.timedelta(days=12) * i for i in range(-200,200)])
dt_sen06 = np.array([dt.datetime(2019, 9, 3, 13, 30, 0) + dt.timedelta(days=6)  * i for i in range(-400,400)])
tide_sen12 = np.array([tide_los[dt_out==d] for d in dt_sen12 if np.sum(dt_out==d) > 0], dtype=np.float32)
tide_sen06 = np.array([tide_los[dt_out==d] for d in dt_sen06 if np.sum(dt_out==d) > 0], dtype=np.float32)
# remove zero values
dt_sen12 = dt_sen12[np.multiply(dt_sen12>=dt0, dt_sen12<=dt1)]
dt_sen06 = dt_sen06[np.multiply(dt_sen06>=dt0, dt_sen06<=dt1)]

print('Done.')

# plot
fig, ax = plt.subplots(nrows=1, ncols=1, figsize=[7, 2.5])
ax.plot(dt_out, tide_u*100, lw=0.5, c='k')
ax.plot(dt_sen12, tide_sen12*100, 'o-', c='C0', label='Sampled every 12 days (by SAR)')
#ax.plot(dt_sen06, tide_sen06*100, 'o-', c='C1', mfc='none', label='Sampled every 6 days')

# axis format
ax.tick_params(which='both', direction='out', bottom=True, top=False, left=True, right=True)
ax.xaxis.set_major_formatter(mdates.DateFormatter('%Y-%m'))
ax.xaxis.set_major_locator(mdates.MonthLocator())
ax.xaxis.set_minor_locator(mdates.DayLocator())
ax.yaxis.set_minor_locator(ticker.AutoMinorLocator())
ax.set_ylabel('LOS displacement [cm]')
ax.set_xlim(dt.datetime(2020,9,10), dt.datetime(2020,12,15))
#ax.set_ylim(-22, 40)
ax.legend(loc='upper right', ncol=2)
fig.tight_layout()

# output
out_fig = os.path.abspath('SET_TS.pdf')
print('save figure to file:', out_fig)
fig.savefig(out_fig, bbox_inches='tight', transparent=True, dpi=600)
plt.show()

## calc PSD
fs = 1. / step_sec
freq, psd = signal.periodogram(tide_los, fs=fs, scaling='density')
# get rid of zero in the first element
psd = psd[1:] / 1e4
freq = freq[1:]
period = 1./3600./freq   # period (hour)

## plot
fig, axs = plt.subplots(nrows=1, ncols=2, figsize=[7.2, 2.8], sharey=True)
for ax in axs:
    ax.plot(period, psd, 'ko', ms=8, mfc='none', mew=1.5)

# axis format
for ax in axs:
    ax.tick_params(which='both', direction='out', bottom=True, top=True, left=True, right=True)
    ax.xaxis.set_minor_locator(ticker.AutoMinorLocator())
    ax.set_xlabel('Period [hour]')
axs[0].set_xlim(11.3, 13.2)
axs[1].set_xlim(22.0, 28.0)
ax = axs[0]
ax.set_ylabel('Power spectral density\n'+r'[$10^4$ '+r'm$^2/$ Hz]')
ax.set_ylim(0, ymax=axs[0].get_ylim()[1] * 1.1)
ax.yaxis.set_major_locator(ticker.FixedLocator([0,10,20]))
ax.yaxis.set_minor_locator(ticker.AutoMinorLocator())
fig.tight_layout()

# Tidal constituents
for ax in axs:  pysolid.add_tidal_constituents(ax, period, psd, min_psd=0.4)

# output
out_fig = 'SET_PSD.pdf'
print('save figure to file:', os.path.abspath(out_fig))
fig.savefig(out_fig, bbox_inches='tight', transparent=True, dpi=600)
plt.show()

t_m2 = 12.4206012 * 3600 # seconds, period of M2 tide
t_sen = 6 * 24 * 3600   # seconds, sampling interval of Sentinel-1 and NISAR
f_m2 = 1000. / t_m2      # mHz
f_sen = 1000. / t_sen    # mHz
print(f'Signal frequency [mHz]: {f_m2}')
print(f'Sampling rate    [mHz]: {f_sen}')

N = np.rint(f_m2 / f_sen)
f_alias = np.abs(N * f_sen - f_m2)
t_alias = (1000. / f_alias) / (24 * 3600)
print(f'Alias  frequency [mHz]: {f_alias} -> {t_alias:.2f} days')

tide_los_max_rate = np.max(np.abs(np.diff(tide_los) / step_sec)) * 1000 * 60
print('max LOS SET rate: {:.1f} mm/min'.format(tide_los_max_rate))

xml_file = os.path.expanduser('~/data/geolocation/ChileSenAT149/reference/IW2.xml')
meta, burst = isce_utils.extract_isce_metadata(xml_file)
Vs = float(meta['satelliteSpeed'])
Re = float(meta['earthRadius'])
Hs = float(meta['altitude'])
Vg = Vs * Re / (Re + Hs)
print('Satellite speed on orbit / ground: {:.1f} / {:.1f} m/s'.format(Vs, Vg))

t = 100e3 / Vg
ramp = tide_los_max_rate * (t / 60)
print('Time to image 100 km: {:.1f}'.format(t))
print('SET ramp in the along-track direction: {:.1f} mm / 100km'.format(ramp))