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We analyze the light spectrum of stars and distant planets to know characteristics like: chemical composition, temperature, mass, etc.

Starting from the same principle, a probe at the border of the solar system could analyze the spectrum of the light reflected by the Earth. In theory, comparing the spectra captured by the probe with what was happening here on Earth at a time, the light was reflected to the space we could discover new spectrographic patterns that would help us look for planets similar to Earth and maybe even life.

I wonder if any study like this has been done, where a whole-Earth spectrum from a distant spacecraft has been captured and analyzed, with the intent of working the problem as if it were a spectrum from an exoplanet, in order to correlate with known Earth conditions.

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TL;DR: (1) we don't need to go very far to measure the spectrum of Earth's reflected light: a satellite in orbit around the Earth could easily do that; however, (2) detecting the reflected light of an Earth-size planet is extremely difficult: we currently use other techniques with much greater success.

The Wikipedia article on how scientists detect exoplanets provides a good introduction and overview - well worth a quick read. In summary:

Most of the thousands of exoplanets detected so far have been found using the transit photometry method, where the measured brightness of the star temporarily dips because a planet is blocking some of the light as it transits across the face of the star.

The next most successful technique is to measure changes in the radial velocity of the star to deduce the presence of a planet (by the "wobble" its gravity causes). This was the most productive method until 2012, after which transit photometry became the dominant detection method.

However, neither of these methods are looking for reflected light from the planet.

It is possible to deduce the presence of a planet from periodic changes to a star's brightness that would correspond with a nearby planet going through its phases (like how the Moon is bright when full and "dark" when new), but we're typically talking about a Jupiter-sized exoplanet in a close orbit - definitely not a relatively "tiny" Earth-like planet further out from the star.

It's also possible to directly detect the reflected light, or the infrared light emitted by a "hot" planet, or light that has been polarised by the planet's atmosphere – but all these methods rely on a much-bigger-than-Jupiter planet orbiting a relatively small/dim star, and preferably in our local stellar neighbourhood, since the light we're trying to detect is exceedingly faint. Note that currently the smallest directly-imaged exoplanet is still more massive than Jupiter, and Jupiter itself is over 300 times more massive than the Earth (and ten times the Earth's radius, which is even more relevant).

Note also that Proxima b (an Earth-size planet orbiting our very closest neighbour, the red dwarf Proxima Centauri) was discovered using the radial velocity method, and while efforts are under way to observe it directly, no direct image has yet been obtained. This might give you an indication of just how hard it is to directly image such a (relatively) tiny planet!

Also, we don't need to send a probe to the edge of the solar system to work out what we're looking for, as we already know exactly what the reflected light from Earth looks like – the iconic Earthrise being the most famous example. And in 1990 Voyager 1 took a famous photo of the Earth from 6 billion kilometres (41 AU) away: see the Pale Blue Dot.

By Sept 2017, Voyager 1 was 21 billion km (140 AU) from the Sun - by far the most distant probe we've ever sent. Nonetheless, that's a long way from the "border of the solar system". Voyager has entered the interstellar medium but it hasn't yet left the Kuiper Belt, and it will be another 300 years before it reaches the Oort Cloud, and tens of thousands of years before it reaches the real edge of the Solar System.

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As an addendum to Chappo's very good answer, I'll note that people have indeed used satellite and spacecraft observations of the Earth to estimate what a similar exoplanet might look like (assuming we had better telescopes and instrumentation than we currently do).

For example, the NASA Deep Impact spacecraft (which had visited the comet Tempel 1 in 2005) was re-purposed as EPOXI for more comet and asteroid flybys, searches for extrasolar planet occultations of other stars -- and, along the way, observations of the Earth in 2008. A brief discussion of one of the papers analyzing the data can be found here.

(And in the early 1990s, spectroscopic observations made by the Galileo spacecraft during one of its gravitational-slingshot flybys of Earth were analyzed (Sagan et al. 1993) to see what indications of life might be detectable from space, such as atmospheric oxygen and methane.)

It's important to understand that there's no need to put something far away from the Earth (e.g., "at the border of the solar system"): you can take a nearby observation of the Earth and blur it and add noise to get the same effect. The important thing is to be able to look at the Earth from outside its atmosphere.

An alternate approach is to look at the effect of a planet's atmosphere on starlight passing through it. This has been done for the Earth by observing light reflected off the Moon during a lunar eclipse, since this sunlight passing through the Earth's atmosphere on its way to the Moon.

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I wonder if any study like this has been done, where a whole-Earth spectrum from a distant spacecraft has been captured and analyzed, with the intent of working the problem as if it were a spectrum from an exoplanet, in order to correlate with known Earth conditions.

Looking at your question from a different viewpoint, note that what you are suggesting is studying just one example of a planet with life to make general predictions about exoplanets.

This is fundamentally flawed and the way I would explain it to you is this :

If you have never seen some particular animal before and I give you one sample of that animal, does this tell you anything about other members of the same species ? No. You don't know if you have an extreme statistical outlier or even a mutant of that species. You certainly cannot extrapolate from that that all animals will be similar.

You cannot take a single example of anything and reliably or even usefully extrapolate anything from that about a larger population you know nothing about.

In theory, comparing the spectra captured by the probe with what was happening here on Earth at a time, the light was reflected to the space we could discover new spectrographic patterns that would help us look for planets similar to Earth and maybe even life.

This is not correct. Earth, as far as we can tell, has had life of some form on it for at least 3.5 billion years of it's 4.5 billion year history. We suspect life may have existed earlier, but there is no direct evidence for that.

During that long period conditions on Earth have radically altered. Atmospheric and surface chemistry have changed completely from what we have now. The spectrum of light from early Earth (with life) would be completely different because of this. Ocean temperatures on Earth are estimated to have been as much as $55-85^\circ C$ between 2 billion and 3.5 billion years ago (when life existed). Up until about 2.5 billion years ago Earth had no free atmospheric Oxygen to speak of, something that a spectrum would indicate. Now if we see an exoplanet with no apparent free oxygen can we say it has life or not ? Well it seems not from our own planet's history.

However the converse cannot be assumed either : just because we found free oxygen in the spectrum of an exoplanet it does not tell us anything about life on that planet, not even it's presence or absence.

So extrapolating from Earth's current spectrum is next to useless in terms of finding life on exoplanets.

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