Name: Surface Mapping of Earth-like Exoplanets using Single Point Light Curves (Video) | JoVE Journal
Uploaded: 2020-05-10T12:30:00.000+00:00
Duration: 6 min 48 s
Description: Learn to map Earth-like exoplanet surfaces using single-point light curves and Earth as a proxy. This technique reconstructs two-dimensional maps to assess habitability of distant worlds.

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5 minutes

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: 
	$ conda create --name myenv python=3.7 
	$ conda activate myenv 
	$ conda install anaconda 
	$ conda install healpy 
	NOTE: These steps may take tens of minutes depending on the hardware and Internet speed. The environment name ‘myenv’ in the first two command lines can be changed to any other string.</li>
</ol>
2. Obtaining multi-wavelength light curves and viewing geometry from observations
<ol>
	<li>In the viewing geometry, include the longitude and latitude of the sub-stellar and the sub-observer points for each corresponding time frame. 
	To use the following attached code, ensure that these two files have the same format as LightCurve.csv and Geometry.csv.</li>
	<li>Run PlotTimeSeries.py to visualize the data and check their qualities. Two figures LightCurve.png and Geometry.png will be created (Supplemental Figure 1-2). Parameters in this and following plotting codes may need to be adjusted if applied to different observations. 
	$ python PlotTimeSeries.py LightCurve 
	$ python PlotTimeSeries.py Geometry</li>
</ol>
3. Extract surface information from light curves
<ol>
	<li>Center time-resolved multi-wavelength albedo light curves of an exoplanet and normalize them by corresponding standard deviation at each wavelength. This results in the equal importance of each channel. 
	<img alt="Standard score formula, R't,k=(Rt,k-μk)/σk; equation for data normalization and analysis." src="/files/ftp_upload/60951/60951equ01.jpg" data-altupdatedat="1788250015000" data-alt="Equation 1"> 
	where R’t,k and Rt,k are the scaled and observed albedo at the t-th time step and the k-th wavelength, respectively; μk and σk are the mean and standard deviation of the albedo time series at the k-th wavelength.
	<ol>
		<li>Run Normalize.py to normalize the light curves, Rt,k. The output is saved in NormalizedLightCurve.csv. 
		$ python Normalize.py</li>
	</ol>
	</li>
	<li>Run PlotTimeSeries.py to visualize the normalized light curves. A figure NormalizedLightCurve.png will be created (Supplemental Figure 3). 
	$ python PlotTimeSeries.py NormalizedLightCurve</li>
	<li>Apply SVD on the scaled albedo light curves to find dominant PCs and their corresponding time series. 
	<img alt="Matrix decomposition equation R'=UΣV^T, mathematical formulation, educational use." src="/files/ftp_upload/60951/60951equ02.jpg" data-altupdatedat="1788250015000" data-alt="Equation 2"> 
	On the left hand side, T and K are the total number of time steps and observation wavelengths; R’ is the matrix of scaled albedo observations, whose (t,k)-th element is R’t,k. On the right hand side, columns of V are PCs, orthonormal vectors that define the space SVD projects to; Σ is a diagonal matrix, whose (k,k)-th element is the standard deviation of scaled light curves along the k-th axis defined by the k-th column of V; columns of U are the corresponding time series of each PC in V.
	<ol>
		<li>Run SingularValueDecomposition.py to decompose R’. The resulting U, Σ, VT are saved in the output files U.csv, SingularValue.csv and V_T.csv, respectively. 
		$ python SingularValueDecomposition.py</li>
	</ol>
	</li>
	<li>Use PlotTimeSeries.py and PlotSVD.py to visualize the SVD result. Three figures U.png, Sigma.png and V_T.png will be created (Supplemental Figure 4-6). 
	$ python PlotTimeSeries.py U 
	$ python PlotSVD.py</li>
	<li>Analyze contributions and corresponding time series of PCs to determine the one that contains surface information.
	<ol>
		<li>Compare the singular values at the diagonal of Σ. An Earth-like partially cloudy exoplanet is expected to have two comparable dominant singular values. 
		NOTE: Σ may contain less or more than two dominant singular values, which is discussed below.</li>
		<li>Compare the time series patterns of the two dominant PCs. The PC that contains surface information tends to have more regular shape than the other. Due to the longitudinal asymmetry and the reappearance of surface with small changes in two consecutive days, the corresponding time series tends to have approximately constant daily variation.</li>
		<li>Compute the periodicities of the two dominant PCs using Lomb-Scargle periodogram9,10 to confirm the selection of PC. The PC that contains surface information tends to have higher peak corresponding to rotation period in the power density spectrum.</li>
		<li>Run Periodogram.py to obtain the power spectra of the time series of each PC. The power spectra are saved in Periodogram.csv. 
		$ python Periodogram.py</li>
		<li>Run PlotPeriodogram.py to visualize these periodograms and confirm the selection of PC. A figure Periodogram.png will be created (Supplemental Figure 7). The current plotting code adds in dashed lines representing annual, semi-annual, diurnal and half-daily cycles for reference, which may need to be changed when applied to other observations. 
		$ python PlotPeriodogram.py</li>
		<li>Select the PC, vj, that contains surface information and its time series, uj. 
		<img alt="Matrix equation for vector extraction; mathematical formula; educational use." src="/files/ftp_upload/60951/60951equ03v2.jpg" data-altupdatedat="1788250015000" data-alt="Equation 3"> 
		<img alt="Matrix equation $u_j = U[:,j]$, mathematics formula, linear algebra concepts." src="/files/ftp_upload/60951/60951equ04v2.jpg" data-altupdatedat="1788250015000" data-alt="Equation 4"> 
		where V[:,j] and U[:,j] are the j-th columns of V and U, respectively; j is the index of PC inferred at step 3.3 that contains surface information.</li>
	</ol>
	</li>
</ol>
4. Construct planetary surface map
<ol>
	<li>Use the Hierarchical Equal Area iso-Latitude Pixelization (HEALPix)11 method to pixelate the retrieving map. It divides spherical surface of a planet into pixels with the same area and uniform distribution. Denote the unknown value of the p-th pixel as xp.
	<ol>
		<li>Run HEALPixRandom.py to visualize the pixelization method. A figure HEALPixRandom.png will be created (Supplemental Figure 8). The parameter Nside at line 17 can be changed for different resolutions. This step may take a few seconds to minutes depending on the resolution. 
		$ python HEALPixRandom.py</li>
	</ol>
	</li>
	<li>Compute the weight of the p-th pixel in observations at the t-th time step, wt,p, using viewing geometry. 
	<img alt="Static equilibrium formula, wt,p = {ct cos(αt,p) cos(βt,p), 0}, conditional equation diagram." src="/files/ftp_upload/60951/60951equ05.jpg" data-altupdatedat="1788250015000" data-alt="Equation 5"> 
	where αt,p, βt,p are the solar and the spacecraft zenith angles at the p-th pixel at the t-th time step; ct is a normalization term of the t-th observation so that sum of the total weight at each time step is unity. 
	NOTE: Geometry is assumed to be known at this step, or can be derived from other analysis, which is discussed below.
	<ol>
		<li>Run ComputeWeight.py to compute wt,p. Change the value of Nside at line 23 for other resolutions of the retrieved map. The output is saved as W.npz due to its size. 
		$ python ComputeWeight.py</li>
	</ol>
	</li>
	<li>Use PlotWeight.py to visualize these weights. A number of figures, one at each time step, will be created in a folder Weight. Merging them results in Supplemental Video 1, which shows how the weight of each pixel changes with time. This step may take hours to finish due to the large number of visualizations. 
	$ python PlotWeight.py</li>
	<li>Combine geometry and observations to reach a linear regression problem. 
	<img alt="Matrix equation W[T×P]x[P×1]=u[T×1], mathematical formula, linear algebra concept." src="/files/ftp_upload/60951/60951equ06.jpg" data-altupdatedat="1788250015000" data-alt="Equation 6"> 
	where P is the total number of retrieving pixels; W is the weight matrix with wt,p as the (t,p)-th element; x consists of xp as the p-th element, which is the quantity to be solved in this problem. 
	Solve the linear regression problem with a regularization of L-2 norm. 
	<img alt="Linear regression equation for data fitting in experimental analysis." src="/files/ftp_upload/60951/60951equ07.jpg" data-altupdatedat="1788250015000" data-alt="Equation 7"> 
	where I is the identity matrix and λ is the regularization parameter. 
	NOTE: 10-3 is a good value for λ when T~104 and P~3*103. They should be adjusted by comparing the values of the two terms in the regularized square error, e, as shown below. 
	<img alt="Equation for optimization with norms, lambdas; mathematical expression." src="/files/ftp_upload/60951/60951equ08.jpg" data-altupdatedat="1788250015000" data-alt="Equation 8">
	<ol>
		<li>Run LinearRegression.py to solve this linear regression problem. The result of x is saved in the file PixelValue.csv. Change the value of λ at line 16 for different strengths of regularization. 
		$ python LinearRegression.py</li>
	</ol>
	</li>
	<li>Convert x to a 2D surface map according to the mapping rule of HEALPix.
	<ol>
		<li>Run PlotMap.py to construct the retrieved maps using different regularization parameters. Three figures Map_-2.png, Map_-3.png and Map_-4.png will be created with the current setting (Supplemental Figure 9). The relationship between the pixel indices and their locations on map is described in the HEALPix document11. This step takes tens of seconds. 
		$ python PolotMap.py</li>
	</ol>
	</li>
</ol>
5. Estimate uncertainty of the retrieved map
<ol>
	<li>Rewrite the linear regression problem at step 4.3 with the “true value” of x as z and the observation noise, ε. <img alt="Matrix equation $W_{[T \times P]}Z_{[P \times 1]} + \epsilon_{[T \times 1]} = u_{t[T \times 1]}$." src="/files/ftp_upload/60951/60951equ09.jpg" data-altupdatedat="1788250015000" data-alt="Equation 9">
	<ol>
		<li>Assume ε to follow a Gaussian Distribution N (0, σ2I[T*T]) and estimate its covariance. T-P is the degree of freedom of uj from observation when the retrieved map is fixed. 
		<img alt="Static equilibrium; variance formula σ²=(uj-Wx)ᵀ(uj-Wx)/(T-P); mathematical equation." src="/files/ftp_upload/60951/60951equ10.jpg" data-altupdatedat="1788250015000" data-alt="Equation 10"></li>
		<li>Combine equations in step 4.4 and 5.1. It results in a Gaussian vector of x. 
		<img alt="Regularization equation, x = (WᵀW + λI)⁻¹(WᵀW)z + (WᵀW + λI)⁻¹Wᵀε, mathematical formulation." src="/files/ftp_upload/60951/60951equ11.jpg" data-altupdatedat="1788250015000" data-alt="Equation 11"></li>
		<li>Compute the expectation and the covariance matrix of x. 
		<img alt="Regularized linear regression equations; statistical analysis method; formula for E[x] and Cov[x]." src="/files/ftp_upload/60951/60951equ12.jpg" data-altupdatedat="1788250015000" data-alt="Equation 12"></li>
		<li>Obtain the uncertainty of each element in x as the square root of the corresponding element on the diagonal of Cov[x]. 
		<img alt="equation static equilibrium ep=√Diag[Cov[x]]p mathematical formula analysis diagram" src="/files/ftp_upload/60951/60951equ13.jpg" data-altupdatedat="1788250015000" data-alt="Equation 13"> 
		where ep is the uncertainty of xp; Diag[Cov[x]]p is p-th element on the diagonal of Cov[x].</li>
		<li>Run Covariance.py to compute the covariance matrix of x. The result is saved in Covariance.npz due to its size. This step takes tens of seconds to minutes depending on the size of W. 
		$ python Covariance.py</li>
	</ol>
	</li>
	<li>Convert ep to the retrieved 2D map according to the mapping rule of HEALPix.
	<ol>
		<li>Run PlotCovariance.py to visualize Cov[x] and map the uncertainty ep to the retrieved map. Two figures Covariance.png and Uncertainty.png will be created (Supplemental Figure 10-11). 
		$ python PlotCovariance.py</li>
	</ol>
	</li>
</ol>

<ol><li>Schwieterman, E. W., et al. <a target="_blank" href="http://www.ncbi.nlm.nih.gov/pubmed?term=((Exoplanet+Biosignatures%3A+A+Review+of+Remotely+Detectable+Signs+of+Life%5BTitle%5D)+AND+%22Astrobiology%22%5BJournal%5D)">Exoplanet Biosignatures: A Review of Remotely Detectable Signs of Life.</a> Astrobiology. 18 (6), 663-708 (2018).</li>
<li>Campbell, B., Walker, G. A. H., Yang, S. <a target="_blank" href="http://www.ncbi.nlm.nih.gov/pubmed?term=((A+Search+for+Substellar+Companions+to+Solar-type+Stars%5BTitle%5D)+AND+%22The+Astrophysical+Journal%22%5BJournal%5D)">A Search for Substellar Companions to Solar-type Stars.</a> The Astrophysical Journal. 331, 902(1988).</li>
<li>NASA. <a target="_blank" href="http://scholar.google.com/scholar?hl=en&safe=off&q=%22NASA+Exoplanet+Archive+%282019%29+Confirmed+Planets+Table%22">NASA Exoplanet Archive (2019) Confirmed Planets Table</a>. , (2019).</li>
<li>Gillon, M., et al. <a target="_blank" href="http://www.ncbi.nlm.nih.gov/pubmed?term=((Seven+temperate+terrestrial+planets+around+the+nearby+ultracool+dwarf+star+TRAPPIST-1%5BTitle%5D)+AND+%22Nature%22%5BJournal%5D)">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/pubmed?term=((Global+Mapping+of+Earth-like+Exoplanets+from+Scattered+Light+Curves%5BTitle%5D)+AND+%22The+Astrophysical+Journal%22%5BJournal%5D)">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/pubmed?term=((Mapping+Earth+Analogs+from+Photometric+Variability%3A+Spin-Orbit+Tomography+for+Planets+in+Inclined+Orbits%5BTitle%5D)+AND+%22The+Astrophysical+Journal%22%5BJournal%5D)">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. B., Fujii, Y. Mapping Exoplanets. <a target="_blank" href="http://scholar.google.com/scholar?hl=en&safe=off&q=author%3aCowan%2C+NB+%22Handbook+of+Exoplanets%22">Handbook of Exoplanets</a>. , Springer, Cham. (2018).</li>
<li>Farr, B., Farr, W. M., Cowan, N. B., Haggard, H. M., Robinson, T. <a target="_blank" href="http://www.ncbi.nlm.nih.gov/pubmed?term=((exocartographer%3A+A+Bayesian+Framework+for+Mapping+Exoplanets+in+Reflected+Light%5BTitle%5D)+AND+%22The+Astronomical+Journal%22%5BJournal%5D)">exocartographer: A Bayesian Framework for Mapping Exoplanets in Reflected Light.</a> The Astronomical Journal. 156 (4), 146(2018).</li>
<li>Lomb, N. R. <a target="_blank" href="http://www.ncbi.nlm.nih.gov/pubmed?term=((Least-Squares+Frequency+Analysis+of+Unequally+Spaced+Data%5BTitle%5D)+AND+%22Astrophysics+and+Space+Science%22%5BJournal%5D)">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/pubmed?term=((Studies+in+astronomical+time+series+analysis.+II.+Statistical+aspects+of+spectral+analysis+of+unevenly+spaced+data%5BTitle%5D)+AND+%22The+Astrophysical+Journal%22%5BJournal%5D)">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>Górski, K. M., et al. <a target="_blank" href="http://www.ncbi.nlm.nih.gov/pubmed?term=((HEALPix%3A+A+Framework+for+High-Resolution+Discretization+and+Fast+Analysis+of+Data+Distributed+on+the+Sphere%5BTitle%5D)+AND+%22The+Astrophysical+Journal%22%5BJournal%5D)">HEALPix: A Framework for High-Resolution Discretization and Fast Analysis of Data Distributed on the Sphere.</a> The Astrophysical Journal. 622 (2), 759(2005).</li>
<li>Jiang, J. H., et al. <a target="_blank" href="http://www.ncbi.nlm.nih.gov/pubmed?term=((Using+Deep+Space+Climate+Observatory+Measurements+to+Study+the+Earth+as+an+Exoplanet%5BTitle%5D)+AND+%22The+Astronomical+Journal%22%5BJournal%5D)">Using Deep Space Climate Observatory Measurements to Study the Earth as an Exoplanet.</a> The Astronomical Journal. 156 (1), 26(2018).</li>
<li>Fan, S., et al. <a target="_blank" href="http://www.ncbi.nlm.nih.gov/pubmed?term=((Earth+as+an+Exoplanet%3A+A+Two-dimensional+Alien+Map%5BTitle%5D)+AND+%22The+Astrophysical+Journal+Letters%22%5BJournal%5D)">Earth as an Exoplanet: A Two-dimensional Alien Map.</a> The Astrophysical Journal Letters. 882 (1), 1(2019).</li>
<li>Cowan, N. B., Strait, T. E. <a target="_blank" href="http://www.ncbi.nlm.nih.gov/pubmed?term=((Determining+Reflectance+Spectra+of+Surfaces+and+Clouds+on+Exoplanets%5BTitle%5D)+AND+%22The+Astrophysical+Journal+Letters%22%5BJournal%5D)">Determining Reflectance Spectra of Surfaces and Clouds on Exoplanets.</a> The Astrophysical Journal Letters. 765 (1), 17(2013).</li>
<li>Fujii, Y., Lustig-Yaeger, J., Cowan, N. B. <a target="_blank" href="http://www.ncbi.nlm.nih.gov/pubmed?term=((Rotational+Spectral+Unmixing+of+Exoplanets%3A+Degeneracies+between+Surface+Colors+and+Geography%5BTitle%5D)+AND+%22The+Astronomical+Journal%22%5BJournal%5D)">Rotational Spectral Unmixing of Exoplanets: Degeneracies between Surface Colors and Geography.</a> The Astronomical Journal. 154 (5), 189(2017).</li>
<li>Kawahara, H., Fujii, Y. <a target="_blank" href="http://www.ncbi.nlm.nih.gov/pubmed?term=((Mapping+Clouds+and+Terrain+of+Earth-like+Planets+from+Photomertic+Variability%3A+Demonstration+with+Planets+in+Face-on+Orbits%5BTitle%5D)+AND+%22The+Astrophysical+Journal+Letters%22%5BJournal%5D)">Mapping Clouds and Terrain of Earth-like Planets from Photomertic Variability: Demonstration with Planets in Face-on Orbits.</a> The Astrophysical Journal Letters. 739 (2), 62(2011).</li>
<li>Kawahara, H. <a target="_blank" href="http://www.ncbi.nlm.nih.gov/pubmed?term=((Frequency+Modulation+of+Directly+Imaged+Exoplanets%3A+Geometric+Effect+as+a+Probe+of+Planetary+Obliquity%5BTitle%5D)+AND+%22The+Astrophysical+Journal%22%5BJournal%5D)">Frequency Modulation of Directly Imaged Exoplanets: Geometric Effect as a Probe of Planetary Obliquity.</a> The Astrophysical Journal. 822 (2), 112(2016).</li>
<li>Schwartz, J. C., Sekowski, C., Haggard, H. M., Pall ́e, E., Cowan, N. B. <a target="_blank" href="http://www.ncbi.nlm.nih.gov/pubmed?term=((Inferring+planetary+obliquity+using+rotational+and+orbital+photometry%5BTitle%5D)+AND+%22Monthly+Notices+of+the+Royal+Astronomical+Society%22%5BJournal%5D)">Inferring planetary obliquity using rotational and orbital photometry.</a> Monthly Notices of the Royal Astronomical Society. 457 (1), 926-938 (2016).</li>
</ol>

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

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...

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 ...

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

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This protocol involves transfecting cAMP sensors and bPAC-nLuc, an optogenetic protein, to accurately track its cellular distribution and response to light stimulation. The innovative approach of creating a cAMP response map using a point scanning system holds the potential for advancing research with optogenetic proteins in different fields.

Mapping the Cellular Distribution of an Optogenetic Protein Using a Light-Stimulation Grid

This article describes a protocol for mechanical characterisation of living spheroids within a 3D matrix using Brillouin micro-spectroscopy. This all-optical method enables spheroid visualisation with microscale resolution and quantitative mechanical properties. This approach has implications for mechanical phenotyping; for instance, determining pathological states of tumor spheroids within a 3D microenvironment.

Mechanical Mapping of Spheroids Using Brillouin Spectroscopy

This protocol covers a four-step large-area imaging survey methodology used to extract metrics of structural complexity, community composition, and population demographics for coral reef communities. The quality of imagery collected and integrated access to the source imagery are prioritized within each step of the protocol.

Measuring the Structure, Composition, and Change of Underwater Environments with Large-area Imaging

White light microscope interferometry is an optical, noncontact and quick method for measuring the topography of surfaces. It is shown how the method can be applied toward mechanical wear analysis, where wear scars on tribological test samples are analyzed; and in materials science to determine ion beam sputtering or laser ablation volumes and depths.

Characterization of Surface Modifications by White Light Interferometry: Applications in Ion Sputtering, Laser Ablation, and Tribology Experiments

Learn to map Earth-like exoplanet surfaces using single-point light curves and Earth as a proxy. This technique reconstructs two-dimensional maps to assess habitability of distant worlds.

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

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