This work was partly supported by the Jet Propulsion Laboratory, California Institute of Technology, under contract with NASA. YLY acknowledge support by the Virtual Planetary Laboratory at the University of Washington.

1. Programming setup
<ol>
	<li>Set up the programming environment for the attached code. A computer with Linux operating system is required, as the healpy package is not available on Windows. The code is not computationally expensive, so a normal personal computer can handle the protocol.</li>
	<li>Follow the instruction (https://docs.anaconda.com/anaconda/install/linux/) to install Anaconda with Python 3.7 onto the system, then use the following commands in terminal to set up the programming environment: 
	$ cond...

<ol>
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H., Yang, S. <a target="_blank" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?db=PubMed&cmd=Search&doptcmdl=Citation&defaultField=Title+Word&term=A+Search+for+Substellar+Companions+to+Solar-type+Stars.">A Search for Substellar Companions to Solar-type Stars.</a> The Astrophysical Journal. 331, 902 (1988).</li><li>NASA. <a target="_blank" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?db=PubMed&cmd=Search&doptcmdl=Citation&defaultField=Title+Word&term=.">.</a> NASA Exoplanet Archive (2019) Confirmed Planets Table. , (2019).</li><li>Gillon, M., et al. <a target="_blank" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?db=PubMed&cmd=Search&doptcmdl=Citation&defaultField=Title+Word&term=Seven+temperate+terrestrial+planets+around+the+nearby+ultracool+dwarf+star+TRAPPIST-1.">Seven temperate terrestrial planets around the nearby ultracool dwarf star TRAPPIST-1.</a> Nature. 542 (7642), 456-460 (2017).</li><li>Kawahara, H., Fujii, Y. <a target="_blank" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?db=PubMed&cmd=Search&doptcmdl=Citation&defaultField=Title+Word&term=Global+Mapping+of+Earth-like+Exoplanets+from+Scattered+Light+Curves.">Global Mapping of Earth-like Exoplanets from Scattered Light Curves.</a> The Astrophysical Journal. 720 (2), 1333 (2010).</li><li>Fujii, Y., Kawahara, H. <a target="_blank" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?db=PubMed&cmd=Search&doptcmdl=Citation&defaultField=Title+Word&term=Mapping+Earth+Analogs+from+Photometric+Variability:+Spin-Orbit+Tomography+for+Planets+in+Inclined+Orbits.">Mapping Earth Analogs from Photometric Variability: Spin-Orbit Tomography for Planets in Inclined Orbits.</a> The Astrophysical Journal. 755 (2), 101 (2012).</li><li>Cowan, N. 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R. <a target="_blank" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?db=PubMed&cmd=Search&doptcmdl=Citation&defaultField=Title+Word&term=Least-Squares+Frequency+Analysis+of+Unequally+Spaced+Data.">Least-Squares Frequency Analysis of Unequally Spaced Data.</a> Astrophysics and Space Science. 39 (2), 447 (1976).</li><li>Scargle, J. D. <a target="_blank" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?db=PubMed&cmd=Search&doptcmdl=Citation&defaultField=Title+Word&term=Studies+in+astronomical+time+series+analysis.+II.+Statistical+aspects+of+spectral+analysis+of+unevenly+spaced+data.">Studies in astronomical time series analysis. II. Statistical aspects of spectral analysis of unevenly spaced data.</a> The Astrophysical Journal. 263, 835 (1982).</li><li>G&#243;rski, K. 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The authors have nothing to disclose.

One critical requirement of the protocol is the feasibility of extracting surface information from light curves, which depends on the cloud coverage. In step 3.5.1, the relative values of the PCs may be different among exoplanets. In the case of Earth, the first two PCs dominate the light curve variations, and correspond to surface-independent clouds and surface (Fan et al. 2019)13. They have comparable singular values so that the surface information can be separated following steps 3.5.2 and 3.5....

Identifying habitable worlds is one of the ultimate goals in astrobiology1. Since the first detection2, more than 4000 exoplanets have been confirmed to date3 with a number of Earth analogs (e.g., TRAPPIST-1e)4. These planets have orbital and planetary properties similar to those of Earth, and therefore are potentially habitable. Evaluating their habitability from limited observations is essential in this context. Based on the knowledge of life on Earth, geological and climate systems are critical to habitability, which can therefore serve as biosignatures. In principle, features of these systems could be observed from a distance even when a planet could not be spatially resolved better than one single point. In this case, identifying geological features and climate systems from single-point light curves is essential when assessing the habitability of exoplanets. Surface mapping of these exoplanets becomes urgent.
Despite the convolution between viewing geometry and spectral features, information of an exoplanet&#8217;s surface is contained in its time-resolved single-point light curves, which can be obtained from a distance, and derived with sufficient observations. However, two-dimensional (2D) surface mapping of potentially habitable Earth-like exoplanets is challenging due to the influence of clouds. Methods of retrieving 2D maps have been developed and tested using simulated light curves and known spectra5,6,7,8, but they have not been applied to real observations. Moreover, in the analyses of exoplanet observations now and in the near future, assumptions of characteristic spectra may be controversial when the planetary surface compositions are not well-constrained.
In this paper, we demonstrate a surface mapping technique for Earth-like exoplanets. We use SVD to evaluate and separate information from different sources that is contained in multi-wavelength light curves without assumptions of any specific spectra. Combined with viewing geometry, we present the reconstruction of surface maps using timely resolved but spatially convoluted surface information. For the purpose of demonstrating this method, two-year multi-wavelength single-point observations of Earth obtained by the Deep Space Climate Observatory/Earth Polychromatic Imaging Camera (DSCOVR/EPIC; www.nesdis.noaa.gov/DSCOVR/spacecraft.html) are analyzed. We use Earth as a proxy exoplanet to assess this method because currently available observations of exoplanets are not sufficient. We attach the code with the paper as an example. It is developed under python 3.7 with anaconda and healpy packages, but the mathematics of the protocol can also be done in other programming environments (e.g., IDL or MATLAB).

<table>
	<tbody>
		<tr>
			<td>Python 3.7 with anaconda and healpy packages</td>
			<td></td>
			<td></td>
			<td>Other programming environments (e.g., IDL or MATLAB) also work.</td>
		</tr>
	</tbody>
</table>

surface mapping of earth-like exoplanets using single point light curves

Spatially resolving exoplanet features from single-point observations is essential for evaluating the potential habitability of exoplanets. The ultimate goal of this protocol is to determine whether these planetary worlds harbor geological features and/or climate systems. We present a method of extracting information from multi-wavelength single-point light curves and retrieving surface maps. It uses singular value decomposition (SVD) to separate sources that contribute to light curve variations and infer the existence of partially cloudy climate systems. Through analysis of the time series obtained from SVD, physical attributions of principal components (PCs) could be inferred without assumptions of any spectral properties. Combining with viewing geometry, it is feasible to reconstruct surface maps if one of the PCs are found to contain surface information. Degeneracy originated from convolution of the pixel geometry and spectrum information determines the quality of reconstructed surface maps, which requires the introduction of regularization. For the purpose of demonstrating the protocol, multi-wavelength light curves of Earth, which serves as a proxy exoplanet, are analyzed. Comparison between the results and the ground truth is presented to show the performance and limitation of the protocol. This work provides a benchmark for future generalization of exoplanet applications.

We use multi-wavelength single-point light curves of Earth to demonstrate the protocol, and compare the results with the ground truth to evaluate the quality of surface mapping. Observation used here is obtained by DSCOVR/EPIC, which is a satellite located near the first Lagrangian point (L1) between Earth and Sun taking images at ten wavelengths of the sunlit face of Earth. Two years (2016 and 2017) of observations are used for this demonstration, which are the same as those in Jiang et al. (2018)12</s...

Watch this Scientific Journal Video about Surface Mapping of Earth-like Exoplanets using Single Point Light Curves at JoVE.com

Surface Mapping of Earth-like Exoplanets using Single Point Light Curves

The protocol extracts information from light curves of exoplanets and constructs their surface maps. It uses light curves of Earth, which serves as a proxy exoplanet, to demonstrate the approach.

Spatially resolving exoplanet features from single-point observations is essential for evaluating the potential habitability of exoplanets. The ultimate ...

This protocol can be used to spatially resolve exoplanet features from single-point observations and is essential for evaluating the potential habitability of exoplanets. This technique can be used to reconstruct two-dimensional surface maps of Earth-like exoplanet. And this is the first technique being tested with real observations using the Earth as a proxy.The mathematics of this technique is straightforward and can be easily adjusted for other observations. One does not need to strictly follow the coding scripts. Visual demonstration of this technique is important because one picture is worth a thousand words.After setting up the programming environment for the attached code, enter the command to install Anaconda with Python 3.7 onto the system. After setting up the programming environment, obtain multi-wavelength light curves, view the geometry from the observations, and run the plot time series. py command to visualize the data and check their qualities.Then enter the command to generate a geometry figure. To extract the light curve surface information, run the normalize. py command.The output is saved in normalized light curve.csv. Enter the command to visualize the normalized light curves. A normalized light curve figure will be created.Enter the command to decompose the normalized light curves. The resulting time series, singular values, and principal components will be saved in the appropriate output files in csv format. Use the commands to visualize the singular value decomposition result.Figures for the singular values and principal components will be generated. Analyze the contributions and corresponding time series of the principal components to determine the one that contains surface information and compare the singular values at the diagonal of singular value matrix. An Earth-like partially cloudy exoplanet is expected to have two comparable dominant singular values.To confirm the selection of the principal component, enter the command to obtain the power spectra of the time series of each principal component. The power spectra will be saved in periodogram.csv. Enter the command to visualize the periodograms and confirm the selection of the principal component.A periodogram figure will be generated. The current plotting code adds dashed lines that represent the annual, semi-annual, diurnal, and half-daily cycles for reference. Select the principal component that contains the surface information and its corresponding time series.To construct a planetary surface map, use the HEALPix random command to visualize the pixelation method. A HEALPix random figure will be created. The parameter n subside at line 17 can be changed for different resolutions.To determine the weight of each pixel, enter the command. The output will be saved as w. npz due to its size.Chang the n subside value at line 23 as appropriate for the other resolutions of the retrieved map. Use the plot weight. py command to visualize the weight.A number of figures will be created in the weight folder. Merging the figures will allow illustration of how the weight of each pixel changes with time. Use the linear regression.py command to solve the linear regression problem. The result of pixel values will be saved in the pixel value. csv file.The value of Lambda at line 16 can be changed for different strengths of regularization as appropriate. Then run the plot map. py command to construct the retrieved maps using different regularization parameters.Three maps will be generated. The relationship between the pixel indices and their locations on each map are described in the HEALPix document. To compute the covariance matrix of each pixel, run the covariance.py command. The result will be saved in covariance. npz due to its size.To visualize the covariance matrix and to map the uncertainty to the retrieved map, run the plot covariance. py command. One covariance and one uncertainty figure will be created.Here, sample multi-wavelength observations of the Earth at 927 coordinated universal time, February 8th, 2017 are shown. Here, the time series of the two dominant principal components of the multi-wavelength light curves can be observed. The time series for the second principal component demonstrates a more regular morphology with an approximately constant daily variation and stronger diurnal cycle in its power spectrum than the first principal component.A surface map of this proxy exoplanet consisting of the value of the second principal component at each pixel can then be constructed. Compared to the ground truth of Earth, the reconstructed map recovers all major continents after separating the surface information from the clouds. The results for the Southern Hemisphere are worse than those observed for the Northern Hemisphere due to the cloud cover over the Southern oceans.The uncertainty of each pixel value is on the order of 10%of that in the retrieved map, suggesting a good quality of the surface mapping and a positive result. The critical requirement for applying this protocol to a future analysis is confirming that the surface information can be extracted from light curves. This technique serves as a benchmark in the surface mapping of exoplanets and may be improved with other decompensation and the regularization methods for new observations.

surface-mapping-earth-like-exoplanets-using-single-point-light

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3.4K Görüntüleme.  California Institute of Technology. Bu protokol, dış gezegen özelliklerini tek noktalı gözlemlerden mekansal olarak çözmek için kullanılabilir ve dış gezegenlerin potansiyel yaşanabilirliğini değerlendirmek için gereklidir. Bu teknik, Dünya benzeri dış gezegenin iki boyutlu yüzey haritalarını yeniden oluşturmak için kullanılabilir. Ve bu, Dünya'yı vekil olarak kullanarak gerçek gözlemlerle test edilen ilk tekniktir.Bu tekniğin matematiği basittir ve diğer gözlemler için kolayca ayarlanabi...