harmonic mean vs arithmetic mean

 The harmonic mean is the appropriate average for ratios and rates because it gives equal weight to each "event" or "unit of work" rather than each individual number or time interval. Here's a deeper dive into why:

1. The Reciprocal (i.e. 1/x) Relationship

The harmonic mean is defined as the reciprocal of the arithmetic mean of the reciprocals of the data points. This "reciprocal of the average of the reciprocals" structure is key. When you're dealing with rates (like miles per hour, or words per minute), you're essentially looking at a ratio of two quantities (distance/time, words/time).

Consider the classic example of average speed:

 * Scenario 1: Equal Distances

   Imagine you drive from point A to point B at 60 km/h and return from point B to point A (the same distance) at 20 km/h. What's your average speed for the entire trip?

   * If you used the arithmetic mean: (60 + 20) / 2 = 40 km/h. This is incorrect.

   * Let's analyze it with the harmonic mean.

     * Distance (d) is constant.

     * Time for outbound trip = d/60

     * Time for return trip = d/20

     * Total distance = 2d

     * Total time = d/60 + d/20 = d(1/60 + 1/20) = d(4/60) = d/15

     * Average speed = Total Distance / Total Time = 2d / (d/15) = 2 \times 15 = 30 km/h.

   * Notice that the harmonic mean of 60 and 20 is: 2 / (1/60 + 1/20) = 2 / (4/60) = 2 \times 60 / 4 = 120 / 4 = 30 km/h.

   In this scenario, where the distance (the "work" done) is constant for each segment, the harmonic mean gives the correct average speed. The arithmetic mean would be too high because you spend more time traveling at the slower speed. The harmonic mean inherently accounts for the longer time spent at the lower rate.

 * Scenario 2: Equal Times

   If you drive for 1 hour at 60 km/h and then for another 1 hour at 20 km/h, your average speed would be the arithmetic mean: (60 \times 1 + 20 \times 1) / (1 + 1) = (60+20)/2 = 40 km/h. In this case, since the time (the denominator of the rate) is constant for each segment, the arithmetic mean is appropriate.

--gemini

IEEE AI consent

 "Authors understand that the use of artificial intelligence (AI)–generated content in an article (including but not limited to text, figures, images, and code) shall be disclosed in the acknowledgments section of any paper submitted to an IEEE Conference or Periodical. The sections of the paper that use AI-generated content shall have a citation to the AI system used to generate the content."

Many Photos with Short Exposures vs Fewer Photos with Longer Exposures

 In astrophotography stacking, the debate between taking many photos with short durations versus fewer photos with longer durations is complex, and the "better" option often depends on various factors. Both approaches have their distinct advantages and disadvantages.

Many Photos with Short Durations (Short Exposures):

Advantages:

• Mitigates Tracking Errors: Even with the best tracking mounts, minor deviations can occur. Shorter exposures minimize the impact of these errors, resulting in sharper stars and less trailing.

• Reduces Overexposure: Bright objects (like bright stars or the core of some nebulae) can easily be overexposed with long exposures, leading to a loss of detail. Shorter exposures help preserve detail in high-dynamic-range objects.

• Increased Flexibility and Error Tolerance: If a single short exposure is ruined by a plane, satellite trail, or sudden atmospheric turbulence, it's easier to discard that one frame without significantly impacting the overall data.¹ With many frames, you have more redundancy.

• Reduced Thermal Noise (for uncooled cameras): Shorter exposures mean the camera sensor heats up less, which can reduce thermal noise, especially in DSLRs or mirrorless cameras without active cooling.

• Better for Fast-Moving Objects or Poor Seeing: For objects like planets or the Moon, or in conditions with turbulent atmosphere (poor "seeing"), very short exposures are crucial to "freeze" the image and capture sharp details. While deep-sky objects are much slower, shorter exposures can still help mitigate atmospheric blurring.

• Easier on Mounts: Less demanding on tracking accuracy, especially for less expensive or less precisely aligned mounts.

Disadvantages:

• Higher Read Noise Contribution: Each time an image is read out from the sensor, it introduces a small amount of "read noise." With many short exposures, the cumulative read noise can become more significant.

• More Files and Processing Time: A large number of short exposures means more individual files to manage and process, which can be computationally intensive.

• May Not Capture Faint Details: If individual short exposures are too short, the signal from very faint objects might not be strong enough to rise above the read noise in a single frame, even with stacking.² You need enough signal in each sub-exposure to make the stacking effective for faint targets.

Fewer Photos with Longer Durations (Long Exposures):

Advantages:

• Better Signal-to-Noise Ratio (SNR) for Faint Objects: Longer exposures collect more light (signal) from faint deep-sky objects, allowing their signal to rise more prominently above the noise floor (especially read noise). This leads to clearer, smoother, and more detailed images, particularly in low-light areas.

• Less Read Noise: With fewer exposures, the overall read noise contribution is reduced because it's incurred once per frame.

• Less Processing Required: Fewer frames generally mean a simpler workflow and less computational demand.

• Captures More Photons per Frame: This directly translates to more signal from the target, which is essential for revealing dim structures and colors.

Disadvantages:

• More Susceptible to Tracking Errors: Any movement, drift, or periodic error in the mount becomes more apparent and can lead to star trailing or blurring.

• Higher Risk of Overexposure: Bright stars or nebula cores can easily be blown out, losing all detail and color.³

• Less Forgiving of Mistakes: If a single long exposure is ruined by a plane, wind gust, or mis-tracking, you lose a significant amount of valuable integration time.

• Increased Thermal Noise (for uncooled cameras): The sensor heats up more during longer exposures, which can increase thermal noise and hot pixels.⁴

• Requires More Precise Equipment: Demands a highly accurate and well-aligned equatorial mount, and often autoguiding, to prevent star trails.

Which is better?

There isn't a universally "better" option; the optimal choice often involves a balance and depends on:

1. Your Equipment:

   • Mount Accuracy: If you have an excellent, precisely aligned, and well-guiding mount, longer exposures become more feasible.

   • Camera Sensor: Cameras with very low read noise can benefit more from shorter exposures, as the read noise penalty is less significant. Cooled astrophotography cameras help mitigate thermal noise in longer exposures.⁵

2. The Target:

   • Bright Objects: For bright nebulae or star clusters, shorter exposures can prevent saturation.⁶ You might even combine different exposure lengths (HDR stacking) for objects with extreme dynamic range (e.g., Orion Nebula).⁷

   • Faint Objects: For very faint galaxies or nebulae, longer individual exposures are often preferred to ensure enough signal is captured in each frame to rise above the read noise.⁸

3. Sky Conditions:

   • Light Pollution: In light-polluted skies, shorter exposures might be necessary to avoid quickly saturating the sky background.⁹

   • Atmospheric Seeing: If the atmosphere is turbulent (poor seeing), shorter exposures can "freeze" the turbulence, resulting in sharper stars. In excellent seeing, longer exposures are less problematic.

General Rule of Thumb:

Many experienced astrophotographers aim for individual sub-exposure lengths that are long enough for the signal from the faintest details of the target to be significantly above the camera's read noise, but not so long that bright stars are saturated or tracking errors become apparent. Then, they take as many of these optimal-length exposures as possible to maximize the total integration time and further improve the signal-to-noise ratio.

In essence, stacking is always beneficial as it averages out random noise. The question is about the length of each individual exposure ("sub-exposure"). For deep-sky objects, it's generally a compromise between minimizing read noise (favors longer) and mitigating tracking errors/overexposure/hot pixels (favors shorter).