Detecting transit and reflected light signals are classical signal-detection problems for deterministic signals in colored noise (Van Trees 1968). Essentially, the optimal detector whitens the observations and then correlates the whitened data with the signal resulting from passing a transit through the same whitening filter. An essential component of signal detection is the characterization of the observational noise, including stellar variability. The result is a time series of test statistics representing the likelihood that a transit was occurring at each point in time. Folding this time series at the orbital period of a planet coherently adds the test statistics for the bin containing the transit. The test statistics add incoherently if the wrong period or bin is chosen. The detection threshold is set so that no more than one false alarm occurs for the entire mission. Thus, the result is still meaningful even if terrestrial planets are infrequent or the mission had to be de-scoped. The detection rate depends on the SNR of a single event, the number of observed events and the detection threshold. The adaptive, nonparametric matched filter algorithm suggested by Kay (1999) is well suited to Kepler's needs, particularly when modified as per Jenkins (1999).

The number of independent tests depends on the signal to be detected and the range of values each signal parameter can have. In the case of transits, there are two basic parameters: period and phase (time to first transit). For orbital periods of less than 2 years, there are about 5x10⁶ tests per star required, for a total of 5x10¹¹ tests. In order to have no more than one false alarm for the entire experiment, a detection threshold of 7 sigma is required, assuming Gaussian statistics, which is important for being able to address a null result. We note that the measurement noise in the Tech Demo was well characterized as being Gaussian. For the point design of a 4 sigma single event SNR the combined SNR of a set of four transits is 8 sigma, yielding a detection efficiency of 84%.

The figure below shows SNRs for four transits of an Earth-size planet about an m_v=12 G2 dwarf star as a function of transit duration during periods of low, high and mean stellar variability. Here, low, high and mean stellar variability correspond to data collected by ACRIM 1 in 1985 and 1986, in the last half of 1988 and in 1989 and over the entire data set, respectively. For transits between 3 and 5 hours, 1 R_e planets are detected at least 84% of the time on average. If the transits are longer than 5 hours, 97% are detected on average, and at least 84% are detected even if all transits occur during high stellar activity. Planets of 1.3 R_e are detected virtually always out to m_v=12, and about 88% of the time for those orbiting m_v=14 G-dwarf stars. Planets yielding average SNRs below the threshold are also detected, but at a lower rate. For example, planets with SNRs of 7 sigma and 6.5 sigma are detected 50% and 30% of the time, respectively.

Note that a central Earth-analog passage lasts 13 hours. Since the chord length of a circle falls as the cosine with distance from the center >50% of transiting Sun-Earth-analogs have durations >11.3 hours, >70% are >9.2 hours. Quoting results for 6.5-hour periods represents a conservative estimate of the single event SNR.

Signal to Noise Ratio as a Function of Transit Duration for Various Stellar Variabilities
The three curves give the SNR for 4 combined transits about an m_v=12 solar-like star at times of low, medium and high stellar variability.

A simulation was performed to demonstrate the detectability of the transit signal for an Earth-size planet in the presence of all the noise that is expected to be in the data.

1. The stellar noise was generated to match the solar noise spectrum obtained from the ACRIM 1 instrument aboard the SMM spacecraft to form a 4 year data sequence.
2. Shot noise and pointing noise were added to the synthetic stellar noise to model the case for an m_v=12 G2V (solar-like) star.
3. A set of four 12 hour Earth-size transits (R=1.0 R_e) was added to the noise sequence.

A matched filter algorithm was applied to the data set to search for planets with periods between 100 and 400 days. The peak event of 8.5 sigma indicated a planet with the correct period of 365 days, as shown in the figure below. The probability of this event occurring by chance is less than 1x10^-17 for Gaussian noise, demonstrating the significance of the detection. The additional blue spikes result from folding signals composed of multiple pulses. The red points are the result for the same data set without the transits incorporated.

Probability of Occurrence by Chance for Simulated Data.
The strong minimum at 365 days in blue indicates the presence of a planet with a high confidence level. The red points are the result for the same data set without transits, with all events attributable to noise.

The figure below shows the sections of the data used in the above simulation. The individual transits are shown (left-hand scale), along with the result of folding and adding the data at the correct period derived with the detection algorithm (right-hand scale).

Transit Detection Simulations
The four sections of a light curve containing the transits of an Earth-size planet (1.0 R_e) are folded at the correct period, with the sum shown in red. The presence of the transit is unmistakable.

For the case of detecting giant-inner planets by phase modulation of reflected light, the number of tests depends only on the length of observation and on the largest planetary period to be considered. Stellar variability limits this search space to periods less than about 7 days as shown in the figure below. Thus, there are about 1,278 tests per star, for a total of 1.3x10⁸ tests (assuming the solar rotation period is typical of all stars in the sample). Jovian-size planets are detectable with periods less than 6 days and Uranus-size planets with periods less than 2.5 days. Because the signal amplitude depends on the planetary albedo, planets with low albedos are more difficult to detect than that shown here.

Detectability of Reflected Light from Short-Period Giant Planets
The blue, red and green curves represent the total noise and include expected stellar variability, shot noise, CCD noise and pointing noise appropriate for m_v=10, 12, and 14 stars, respectively.
The blue spike at 4.2 days is the reflected light signature of a 51 Peg-type planet with an albedo of 0.5 (assumed to match Jupiter) in orbit about a m_v=10 star. At other periods, the strength of this spike would vary, as given by the black dotted reflected light envelope. Since this envelope exceeds the measurement noise curves for periods less than about 7 days, giant planets with periods up to 7 days are detectable.

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