Second Round of Full Day RFI Flagging¶

by Josh Dillon, last updated July 31, 2023

This notebook is synthesizes information from individual delay_filtered_average_zscore notebooks to find low-level RFI and flag it. That notebook takes smooth_calibrated data, redundantly averages it, performs a high-pass delay filter, and then incoherently averages across baselines, creating a per-polarization z-score. This notebook then takes that whole night of z-scores and finds a new set of flags to both add to the smooth_cal files, which are updated in place, and to write down as new UVFlag waterfall-type .h5 files.

Here's a set of links to skip to particular figures and tables:

• Figure 1: Waterfall of Maximum z-Score of Either Polarization Before Round 2 Flagging¶

• Figure 2: Histogram of z-scores¶

• Figure 3: Waterfall of Maximum z-Score of Either Polarization After Round 2 Flagging¶

• Figure 4: Spectra of Time-Averaged z-Scores¶

• Figure 5: Summary of Flags Before and After Round 2 Flagging¶

In [1]:
import time
tstart = time.time()
In [2]:
import os
os.environ['HDF5_USE_FILE_LOCKING'] = 'FALSE'
import h5py
import hdf5plugin  # REQUIRED to have the compression plugins available
import numpy as np
import glob
import matplotlib.pyplot as plt
import matplotlib
import copy
import warnings
from pyuvdata import UVFlag, UVCal
from hera_cal import utils
from hera_qm import xrfi
from hera_qm.time_series_metrics import true_stretches

from IPython.display import display, HTML
%matplotlib inline
display(HTML("<style>.container { width:100% !important; }</style>"))
_ = np.seterr(all='ignore')  # get rid of red warnings
%config InlineBackend.figure_format = 'retina'
In [3]:
# get input data file names
SUM_FILE = os.environ.get("SUM_FILE", None)
# SUM_FILE = '/lustre/aoc/projects/hera/h6c-analysis/IDR2/2459861/zen.2459861.25297.sum.uvh5'
SUM_SUFFIX = os.environ.get("SUM_SUFFIX", 'sum.uvh5')

# get input and output suffixes
SMOOTH_CAL_SUFFIX = os.environ.get("SMOOTH_CAL_SUFFIX", 'sum.smooth.calfits')
ZSCORE_SUFFIX =  os.environ.get("ZSCORE_SUFFIX", 'sum.red_avg_zscore.h5')
FLAG_WATERFALL2_SUFFIX = os.environ.get("FLAG_WATERFALL2_SUFFIX", 'sum.flag_waterfall_round_2.h5')
OUT_YAML_SUFFIX = os.environ.get("OUT_YAML_SUFFIX", '_aposteriori_flags.yaml')
OUT_YAML_DIR = os.environ.get("OUT_YAML_DIR", None)

# build globs
sum_glob = '.'.join(SUM_FILE.split('.')[:-3]) + '.*.' + SUM_SUFFIX
cal_files_glob = sum_glob.replace(SUM_SUFFIX, SMOOTH_CAL_SUFFIX)
zscore_glob = sum_glob.replace(SUM_SUFFIX, ZSCORE_SUFFIX)

# build out yaml file
if OUT_YAML_DIR is None:
    OUT_YAML_DIR = os.path.dirname(SUM_FILE)
out_yaml_file = os.path.join(OUT_YAML_DIR, SUM_FILE.split('.')[-4] + OUT_YAML_SUFFIX)    

# get flagging parameters
Z_THRESH = float(os.environ.get("Z_THRESH", 5))
WS_Z_THRESH = float(os.environ.get("WS_Z_THRESH", 4))
AVG_Z_THRESH = float(os.environ.get("AVG_Z_THRESH", 1))
MAX_FREQ_FLAG_FRAC = float(os.environ.get("MAX_FREQ_FLAG_FRAC", .25))
MAX_TIME_FLAG_FRAC = float(os.environ.get("MAX_TIME_FLAG_FRAC", .1))

for setting in ['Z_THRESH', 'WS_Z_THRESH', 'AVG_Z_THRESH', 'MAX_FREQ_FLAG_FRAC', 'MAX_TIME_FLAG_FRAC']:
    print(f'{setting} = {eval(setting)}')
Z_THRESH = 5.0
WS_Z_THRESH = 4.0
AVG_Z_THRESH = 1.0
MAX_FREQ_FLAG_FRAC = 0.25
MAX_TIME_FLAG_FRAC = 0.1

Load z-scores¶

In [4]:
# load z-scores
zscore_files = sorted(glob.glob(zscore_glob))
print(f'Found {len(zscore_files)} *.{ZSCORE_SUFFIX} files starting with {zscore_files[0]}.')
uvf = UVFlag(zscore_files, use_future_array_shapes=True)
Found 1860 *.sum.red_avg_zscore.h5 files starting with /mnt/sn1/2460411/zen.2460411.16910.sum.red_avg_zscore.h5.
In [5]:
# get calibration solution files
cal_files = sorted(glob.glob(cal_files_glob))
print(f'Found {len(cal_files)} *.{SMOOTH_CAL_SUFFIX} files starting with {cal_files[0]}.')
Found 1860 *.sum.smooth.calfits files starting with /mnt/sn1/2460411/zen.2460411.16910.sum.smooth.calfits.
In [6]:
assert len(zscore_files) == len(cal_files)
In [7]:
# extract z-scores and correct by a single number per polarization to account for biases created by filtering
zscore = {pol: uvf.metric_array[:, :, np.argwhere(uvf.polarization_array == utils.polstr2num(pol, x_orientation=uvf.x_orientation))[0][0]] for pol in ['ee', 'nn']}
zscore = {pol: zscore[pol] - np.nanmedian(zscore[pol]) for pol in zscore}
In [8]:
freqs = uvf.freq_array
times = uvf.time_array
In [9]:
extent = [freqs[0] / 1e6, freqs[-1] / 1e6, times[-1] - int(times[0]), times[0] - int(times[0])]
In [10]:
def plot_max_z_score(zscore, flags=None):
    if flags is None:
        flags = np.any(~np.isfinite(list(zscore.values())), axis=0)
    plt.figure(figsize=(14,10), dpi=100)
    plt.imshow(np.where(flags, np.nan, np.nanmax([zscore['ee'], zscore['nn']], axis=0)), aspect='auto', 
               cmap='coolwarm', interpolation='none', vmin=-10, vmax=10, extent=extent)
    plt.colorbar(location='top', label='Max z-score of either polarization', extend='both', aspect=40, pad=.02)
    plt.xlabel('Frequency (MHz)')
    plt.ylabel(f'JD - {int(times[0])}')
    plt.tight_layout()

Figure 1: Waterfall of Maximum z-Score of Either Polarization Before Round 2 Flagging¶

Shows the worse of the two results from delay_filtered_average_zscore from either polarization. Dips near flagged channels are expected, due to overfitting of noise. Positive-going excursions are problematic and likely evidence of RFI.

In [11]:
plot_max_z_score(zscore)
All-NaN axis encountered
No description has been provided for this image
In [12]:
def plot_histogram():
    plt.figure(figsize=(14,4), dpi=100)
    bins = np.arange(-50, 100, .1)
    hist_ee = plt.hist(np.ravel(zscore['ee']), bins=bins, density=True, label='ee-polarized z-scores', alpha=.5)
    hist_nn = plt.hist(np.ravel(zscore['nn']), bins=bins, density=True, label='nn-polarized z-scores', alpha=.5)
    plt.plot(bins, (2*np.pi)**-.5 * np.exp(-bins**2 / 2), 'k:', label='Gaussian approximate\nnoise-only distribution')
    plt.axvline(WS_Z_THRESH, c='r', ls='--', label='Watershed z-score')
    plt.axvline(Z_THRESH, c='r', ls='-', label='Threshold z-score')
    plt.yscale('log')
    all_densities = np.concatenate([hist_ee[0][hist_ee[0] > 0], hist_nn[0][hist_nn[0] > 0]]) 
    plt.ylim(np.min(all_densities) / 2, np.max(all_densities) * 2)
    plt.xlim([-50, 100])
    plt.legend()
    plt.xlabel('z-score')
    plt.ylabel('Density')
    plt.tight_layout()

Figure 2: Histogram of z-scores¶

Shows a comparison of the histogram of z-scores in this file (one per polarization) to a Gaussian approximation of what one might expect from thermal noise. Without filtering, the actual distribution is a weighted sum of Rayleigh distributions. Filtering further complicates this. To make the z-scores more reliable, a single per-polarization median is subtracted from each waterfall, which allows us to flag low-level outliers with more confidence. Any points beyond the solid red line are flagged. Any points neighboring a flag beyond the dashed red line are also flagged. Finally, flagging is performed for low-level outliers in whole times or channels.

In [13]:
plot_histogram()
No description has been provided for this image

Perform flagging¶

In [14]:
def iteratively_flag_on_averaged_zscore(flags, zscore, avg_z_thresh=1.5, verbose=True):
    '''Flag whole integrations or channels based on average z-score. This is done
    iteratively to prevent bad times affecting channel averages or vice versa.'''
    flagged_chan_count = 0
    flagged_int_count = 0
    while True:
        zspec = np.nanmean(np.where(flags, np.nan, zscore), axis=0)
        ztseries = np.nanmean(np.where(flags, np.nan, zscore), axis=1)

        if (np.nanmax(zspec) < avg_z_thresh) and (np.nanmax(ztseries) < avg_z_thresh):
            break

        if np.nanmax(zspec) >= np.nanmax(ztseries):
            flagged_chan_count += np.sum((zspec >= np.nanmax(ztseries)) & (zspec >= avg_z_thresh))
            flags[:, (zspec >= np.nanmax(ztseries)) & (zspec >= avg_z_thresh)] = True
        else:
            flagged_int_count += np.sum((ztseries >= np.nanmax(zspec)) & (ztseries >= avg_z_thresh))
            flags[(ztseries >= np.nanmax(zspec)) & (ztseries >= avg_z_thresh), :] = True

    if verbose:
        print(f'\tFlagging an additional {flagged_int_count} integrations and {flagged_chan_count} channels.')

def impose_max_chan_flag_frac(flags, max_flag_frac=.25, verbose=True):
    '''Flag channels already flagged more than max_flag_frac (excluding completely flagged times).'''
    unflagged_times = ~np.all(flags, axis=1)
    frequently_flagged_chans =  np.mean(flags[unflagged_times, :], axis=0) >= max_flag_frac
    if verbose:
        print(f'\tFlagging {np.sum(frequently_flagged_chans) - np.sum(np.all(flags, axis=0))} channels previously flagged {max_flag_frac:.2%} or more.')        
    flags[:, frequently_flagged_chans] = True 
        
def impose_max_time_flag_frac(flags, max_flag_frac=.25, verbose=True):
    '''Flag times already flagged more than max_flag_frac (excluding completely flagged channels).'''
    unflagged_chans = ~np.all(flags, axis=0)
    frequently_flagged_times =  np.mean(flags[:, unflagged_chans], axis=1) >= max_flag_frac
    if verbose:
        print(f'\tFlagging {np.sum(frequently_flagged_times) - np.sum(np.all(flags, axis=1))} times previously flagged {max_flag_frac:.2%} or more.')
    flags[frequently_flagged_times, :] = True             
In [15]:
flags = np.any(~np.isfinite(list(zscore.values())), axis=0)
print(f'{np.mean(flags):.3%} of waterfall flagged to start.')

# flag largest outliers
for pol in ['ee', 'nn']:
    flags |= (zscore[pol] > Z_THRESH) 
print(f'{np.mean(flags):.3%} of waterfall flagged after flagging z > {Z_THRESH} outliers.')
    
# watershed flagging
while True:
    nflags = np.sum(flags)
    for pol in ['ee', 'nn']:
        flags |= xrfi._ws_flag_waterfall(zscore[pol], flags, WS_Z_THRESH)
    if np.sum(flags) == nflags:
        break
print(f'{np.mean(flags):.3%} of waterfall flagged after watershed flagging on z > {WS_Z_THRESH} neightbors of prior flags.')
        
# flag whole integrations or channels
while True:
    nflags = np.sum(flags)
    for pol in ['ee', 'nn']:    
        iteratively_flag_on_averaged_zscore(flags, zscore[pol], avg_z_thresh=AVG_Z_THRESH, verbose=True)
        impose_max_chan_flag_frac(flags, max_flag_frac=MAX_FREQ_FLAG_FRAC, verbose=True)
        impose_max_time_flag_frac(flags, max_flag_frac=MAX_TIME_FLAG_FRAC, verbose=True)
    if np.sum(flags) == nflags:
        break  
print(f'{np.mean(flags):.3%} of waterfall flagged after flagging whole times and channels with average z > {AVG_Z_THRESH}.')
23.999% of waterfall flagged to start.
26.043% of waterfall flagged after flagging z > 5.0 outliers.
26.402% of waterfall flagged after watershed flagging on z > 4.0 neightbors of prior flags.
Mean of empty slice
Mean of empty slice
	Flagging an additional 0 integrations and 11 channels.
	Flagging 20 channels previously flagged 25.00% or more.
	Flagging 278 times previously flagged 10.00% or more.
	Flagging an additional 0 integrations and 3 channels.
	Flagging 0 channels previously flagged 25.00% or more.
	Flagging 1 times previously flagged 10.00% or more.
	Flagging an additional 0 integrations and 0 channels.
	Flagging 0 channels previously flagged 25.00% or more.
	Flagging 0 times previously flagged 10.00% or more.
	Flagging an additional 0 integrations and 0 channels.
	Flagging 0 channels previously flagged 25.00% or more.
	Flagging 0 times previously flagged 10.00% or more.
32.766% of waterfall flagged after flagging whole times and channels with average z > 1.0.

Show results of flagging¶

Figure 3: Waterfall of Maximum z-Score of Either Polarization After Round 2 Flagging¶

The same as Figure 1, but after the flagging performed in this notebook.

In [16]:
plot_max_z_score(zscore, flags=flags)
All-NaN axis encountered
No description has been provided for this image
In [17]:
def zscore_spectra():
    fig, axes = plt.subplots(2, 1, figsize=(14,6), dpi=100, sharex=True, sharey=True, gridspec_kw={'hspace': 0})
    for ax, pol in zip(axes, ['ee', 'nn']):

        ax.plot(freqs / 1e6, np.nanmean(zscore[pol], axis=0),'r', label=f'{pol}-Polarization Before Round 2 Flagging', lw=.5)
        ax.plot(freqs / 1e6, np.nanmean(np.where(flags, np.nan, zscore[pol]), axis=0), label=f'{pol}-Polarization After Round 2 Flagging')
        ax.legend(loc='lower right')
        ax.set_ylabel('Time-Averged Z-Score\n(Excluding Flags)')
        ax.set_ylim(-11, 11)
    axes[1].set_xlabel('Frequency (MHz)')
    plt.tight_layout()

Figure 4: Spectra of Time-Averaged z-Scores¶

The average along the time axis of Figures 1 and 3 (though now separated per-polarization). This plot is useful for showing channels with repeated low-level RFI.

In [18]:
zscore_spectra()
Mean of empty slice
Mean of empty slice
No description has been provided for this image
In [19]:
def summarize_flagging():
    plt.figure(figsize=(14,10), dpi=100)
    cmap = matplotlib.colors.ListedColormap(((0, 0, 0),) + matplotlib.cm.get_cmap("Set2").colors[0:2])
    plt.imshow(np.where(np.any(~np.isfinite(list(zscore.values())), axis=0), 1, np.where(flags, 2, 0)), 
               aspect='auto', cmap=cmap, interpolation='none', extent=extent)
    plt.clim([-.5, 2.5])
    cbar = plt.colorbar(location='top', aspect=40, pad=.02)
    cbar.set_ticks([0, 1, 2])
    cbar.set_ticklabels(['Unflagged', 'Previously Flagged', 'Flagged Here Using Delayed Filtered z-Scores'])
    plt.xlabel('Frequency (MHz)')
    plt.ylabel(f'JD - {int(times[0])}')
    plt.tight_layout()

Figure 5: Summary of Flags Before and After Round 2 Flagging¶

This plot shows which times and frequencies were flagged before and after this notebook. It is directly comparable to Figure 5 of the first round full_day_rfi notebook.

In [20]:
summarize_flagging()
The get_cmap function was deprecated in Matplotlib 3.7 and will be removed two minor releases later. Use ``matplotlib.colormaps[name]`` or ``matplotlib.colormaps.get_cmap(obj)`` instead.
No description has been provided for this image

Save results¶

In [21]:
add_to_history = 'by full_day_rfi_round_2 notebook with the following environment:\n' + '=' * 65 + '\n' + os.popen('conda env export').read() + '=' * 65
In [22]:
tind = 0
always_flagged_ants = set()
ever_unflagged_ants = set()
for cal_file in cal_files:
    with warnings.catch_warnings():
        warnings.simplefilter("ignore")    
        
        # update cal_file
        uvc = UVCal()
        uvc.read(cal_file, use_future_array_shapes=True)
        uvc.flag_array |= (flags[tind:tind + len(uvc.time_array), :].T)[None, :, :, None]
        uvc.history += 'Modified ' + add_to_history
        uvc.write_calfits(cal_file, clobber=True)
        
        # keep track of flagged antennas
        for antnum in uvc.ant_array:
            for antpol in ['Jee', 'Jnn']:
                if np.all(uvc.get_flags(antnum, antpol)):
                    if (antnum, antpol) not in ever_unflagged_ants:
                        always_flagged_ants.add((antnum, antpol))
                else:
                    ever_unflagged_ants.add((antnum, antpol))
                    always_flagged_ants.discard((antnum, antpol))
                

        # Create new flag object
        uvf_out = UVFlag(uvc, waterfall=True, mode='flag')
        uvf_out.flag_array |= flags[tind:tind + len(uvc.time_array), :, None]
        uvf_out.history += 'Produced ' + add_to_history
        uvf_out.write(cal_file.replace(SMOOTH_CAL_SUFFIX, FLAG_WATERFALL2_SUFFIX), clobber=True)
        
        # increment time index
        tind += len(uvc.time_array)

print(f'Saved {len(cal_files)} *.{FLAG_WATERFALL2_SUFFIX} files starting with {cal_files[0].replace(SMOOTH_CAL_SUFFIX, FLAG_WATERFALL2_SUFFIX)}.')
Saved 1860 *.sum.flag_waterfall_round_2.h5 files starting with /mnt/sn1/2460411/zen.2460411.16910.sum.flag_waterfall_round_2.h5.
In [23]:
# write summary of entirely flagged times/freqs/ants to yaml
all_flagged_times = np.all(flags, axis=1)
all_flagged_freqs = np.all(flags, axis=0)
all_flagged_ants = sorted(always_flagged_ants)

dt = np.median(np.diff(times))
out_yml_str = 'JD_flags: ' + str([[times[flag_stretch][0] - dt / 2, times[flag_stretch][-1] + dt / 2] 
                                  for flag_stretch in true_stretches(all_flagged_times)])
df = np.median(np.diff(freqs))
out_yml_str += '\n\nfreq_flags: ' + str([[freqs[flag_stretch][0] - df / 2, freqs[flag_stretch][-1] + df / 2] 
                                         for flag_stretch in true_stretches(all_flagged_freqs)])
out_yml_str += '\n\nex_ants: ' + str(all_flagged_ants).replace("'", "").replace('(', '[').replace(')', ']')

print(f'Writing the following to {out_yaml_file}\n' + '-' * (25 + len(out_yaml_file)))
print(out_yml_str)
with open(out_yaml_file, 'w') as outfile:
    outfile.writelines(out_yml_str)
Writing the following to /mnt/sn1/2460411/2460411_aposteriori_flags.yaml
------------------------------------------------------------------------
JD_flags: [[2460411.168990614, 2460411.1773792217], [2460411.1790569434, 2460411.1791687915], [2460411.1812939057, 2460411.181405754], [2460411.1824123864, 2460411.182859779], [2460411.182971627, 2460411.183195323], [2460411.1842019563, 2460411.1845375006], [2460411.184761197, 2460411.184873045], [2460411.1864389186, 2460411.186662615], [2460411.18811664, 2460411.188340336], [2460411.189346969, 2460411.1894588172], [2460411.1957223113, 2460411.1958341594], [2460411.1959460075, 2460411.1960578556], [2460411.1967289443, 2460411.1969526405], [2460411.197176337, 2460411.1976237292], [2460411.201762109, 2460411.201873957], [2460411.2095914767, 2460411.209703325], [2460411.209815173, 2460411.209927021], [2460411.2156312745, 2460411.2158549707], [2460411.2161905146, 2460411.216637907], [2460411.216749755, 2460411.2169734514], [2460411.217532692, 2460411.2177563882], [2460411.21943411, 2460411.219545958], [2460411.219657806, 2460411.220217047], [2460411.2209999836, 2460411.22122368], [2460411.2242435785, 2460411.2243554266], [2460411.2244672747, 2460411.224690971], [2460411.2328558825, 2460411.233191427], [2460411.241691883, 2460411.2418037313], [2460411.2419155794, 2460411.2421392757], [2460411.2470605923, 2460411.2471724404], [2460411.2507515796, 2460411.251087124], [2460411.2517582127, 2460411.251870061], [2460411.252093757, 2460411.2523174533], [2460411.2525411495, 2460411.2527648457], [2460411.253883327, 2460411.253995175], [2460411.255001808, 2460411.2553373524], [2460411.257350618, 2460411.2576861624], [2460411.2579098586, 2460411.258133555], [2460411.260034973, 2460411.260146821], [2460411.266298467, 2460411.266522163], [2460411.2681998843, 2460411.2683117324], [2460411.272226416, 2460411.272338264], [2460411.2725619604, 2460411.2726738085], [2460411.2767003407, 2460411.276924037], [2460411.283187531, 2460411.283411227], [2460411.283523075, 2460411.283634923], [2460411.2868785183, 2460411.2871022145], [2460411.2879969995, 2460411.2882206957], 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ex_ants: [[3, Jnn], [7, Jee], [18, Jee], [18, Jnn], [21, Jee], [21, Jnn], [27, Jee], [27, Jnn], [28, Jee], [28, Jnn], [29, Jnn], [31, Jnn], [32, Jnn], [34, Jee], [35, Jnn], [37, Jnn], [40, Jnn], [45, Jee], [46, Jee], [47, Jee], [47, Jnn], [51, Jee], [53, Jnn], [61, Jee], [61, Jnn], [63, Jee], [63, Jnn], [64, Jee], [64, Jnn], [66, Jnn], [68, Jnn], [77, Jee], [77, Jnn], [78, Jee], [78, Jnn], [86, Jee], [86, Jnn], [87, Jee], [88, Jee], [88, Jnn], [90, Jee], [90, Jnn], [92, Jee], [93, Jee], [93, Jnn], [98, Jnn], [104, Jnn], [107, Jee], [107, Jnn], [109, Jnn], [111, Jee], [112, Jee], [115, Jee], [115, Jnn], [119, Jnn], [121, Jee], [130, Jee], [130, Jnn], [134, Jnn], [136, Jnn], [140, Jee], [161, Jnn], [162, Jee], [163, Jee], [163, Jnn], [170, Jee], [171, Jnn], [176, Jee], [176, Jnn], [177, Jee], [177, Jnn], [178, Jee], [178, Jnn], [180, Jnn], [182, Jnn], [183, Jee], [183, Jnn], [188, Jnn], [194, Jee], [194, Jnn], [196, Jee], [196, Jnn], [199, Jnn], [200, Jee], [200, Jnn], [204, Jnn], [212, Jnn], [217, Jee], [217, Jnn], [218, Jee], [218, Jnn], [229, Jee], [231, Jnn], [232, Jee], [241, Jee], [241, Jnn], [242, Jee], [242, Jnn], [243, Jee], [243, Jnn], [245, Jnn], [251, Jee], [255, Jee], [255, Jnn], [266, Jee], [269, Jnn], [270, Jee], [270, Jnn], [272, Jee], [272, Jnn], [281, Jee], [320, Jee], [320, Jnn], [321, Jee], [321, Jnn], [323, Jee], [323, Jnn], [324, Jee], [324, Jnn], [325, Jee], [325, Jnn], [326, Jee], [326, Jnn], [327, Jee], [327, Jnn], [328, Jee], [328, Jnn], [329, Jee], [329, Jnn], [331, Jee], [331, Jnn], [332, Jee], [332, Jnn], [333, Jee], [333, Jnn], [336, Jee], [336, Jnn], [340, Jee], [340, Jnn]]

Metadata¶

In [24]:
for repo in ['hera_cal', 'hera_qm', 'hera_filters', 'hera_notebook_templates', 'pyuvdata']:
    exec(f'from {repo} import __version__')
    print(f'{repo}: {__version__}')
hera_cal: 3.6.dev6+gf0cfd8d
hera_qm: 2.1.3.dev5+g3e71720
hera_filters: 0.1.5
hera_notebook_templates: 0.1.dev734+g90f16f4
pyuvdata: 2.4.2
In [25]:
print(f'Finished execution in {(time.time() - tstart) / 60:.2f} minutes.')
Finished execution in 36.25 minutes.