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The impact of interpolation in high-resolution spectroscopy -- The overlooked role of interpolation in radial velocity extraction

This study demonstrates that the choice of interpolation algorithm in template-based radial velocity extraction introduces systematic biases ranging from millimeters per second in noise-free data to tens of meters per second in low signal-to-noise or high-cadence observations, highlighting a critical, often overlooked source of error in high-precision spectroscopy.

Original authors: A. M. Silva, D. Doshi, K. Al Moulla, E. A. S. Cristo, É. Artigau, P. T. P. Viana, N. C. Santos, J. H. C. Martins, C. M. J. Marques, S. G. Sousa, C. San Nicolas Martinez, T. L. Campante, A. Cabral, S.

Published 2026-06-30

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: A. M. Silva, D. Doshi, K. Al Moulla, E. A. S. Cristo, É. Artigau, P. T. P. Viana, N. C. Santos, J. H. C. Martins, C. M. J. Marques, S. G. Sousa, C. San Nicolas Martinez, T. L. Campante, A. Cabral, S. Cristiani

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: A Digital Photo Puzzle

Imagine you are trying to measure the speed of a car by looking at its reflection in a mirror. To do this accurately, you need to take a series of photos. However, your camera doesn't take perfect, continuous photos; it takes snapshots at specific points (pixels).

Sometimes, the car moves slightly between snapshots, or the camera shifts. To figure out exactly where the car is, you have to guess what the image looks like between the pixels you actually captured. This guessing process is called interpolation.

This paper asks a very specific question: Does the way we "guess" the missing parts of the image (the interpolation) introduce fake errors into our speed measurements?

The authors found that yes, it does. Depending on how you guess the missing pixels, you might accidentally create a fake signal that looks like the star is wobbling, even if it isn't.

The Problem: The "Digital Zoom" Effect

Astronomers use powerful telescopes (like ESPRESSO) to look at stars. They want to detect Exoplanets (planets orbiting other stars). They do this by measuring the Radial Velocity (RV): how fast the star is moving toward or away from us.

The Method: They take a picture of the star's light (a spectrum). This light has dark lines (like barcodes) caused by elements in the star.
The Shift: If the star moves, those dark lines shift slightly left or right.
The Glitch: Because the telescope's detector is made of a grid of pixels, the dark lines don't always land perfectly in the center of a pixel. Sometimes they land between pixels.

To compare different photos taken at different times, astronomers must stretch or squeeze the images so they all line up on the same grid. This requires interpolation—mathematically filling in the gaps between the pixels.

The Analogy: Imagine you are drawing a smooth curve on a piece of graph paper, but you can only draw dots on the grid intersections. If you want to move your drawing slightly to the right, you have to guess where the new dots should go. If you guess wrong, your new drawing looks slightly different from the original, even though you just tried to move it.

What the Authors Did

The team ran two types of tests to see how bad this "guessing" problem is:

The Perfect Simulation (The "Toy" Test):
They created fake star light using perfect mathematical shapes (Gaussian curves). They knew exactly what the light should look like. They then simulated the telescope taking pictures, shifting the light slightly, and using different interpolation methods to fix the grid.
- The Result: Even with perfect, noise-free data, the "guessing" created a tiny fake wobble. It was very small (millimeters per second), but it was there.
- The Noise Factor: When they added "noise" (simulating a dimmer, fuzzier star), the fake wobble got much bigger. For dim stars, the error could reach 20 meters per second. That is huge in astronomy terms!
The Real World Test (The "ESPRESSO" Test):
They looked at real data from four different stars observed by the ESPRESSO telescope.
- Scenario A: The Sprint (Single Night): They looked at stars observed intensively over just one night. In this case, the star's position relative to the pixels didn't change much.
  - Result: The choice of interpolation algorithm mattered a lot. For a dimmer star, different math methods gave results that differed by 25 meters per second. This is a massive difference that could look like a fake planet.
- Scenario B: The Marathon (Many Nights): They looked at stars observed over many months. Here, the star's position shifted all over the place relative to the pixels.
  - Result: Because the errors happened in different directions over time, they mostly canceled each other out. The remaining error was tiny (less than 20 centimeters per second).

The Key Takeaways

The "Math" Matters: Just like using different map apps might give you slightly different routes, using different math formulas to fill in the gaps in star data changes the final speed measurement.
Dim Stars are Vulnerable: If a star is faint (low signal-to-noise ratio), the interpolation errors are much larger. This is dangerous because finding Earth-like planets often requires looking at fainter stars.
Time is the Cure: If you observe a star over a long period (many nights), the errors tend to average out. But if you observe a star for just one night (like when studying star "quakes" or oscillations), the interpolation error can look like a real signal.
It's Everywhere: This isn't just about finding planets. Any time astronomers analyze star light—whether studying the star's atmosphere, its internal vibrations, or the expansion of the universe—they are using interpolation. If they don't account for this "guessing" error, their results could be slightly biased.

In Summary

The paper warns astronomers that the mathematical tools they use to "smooth out" their data are not invisible. They leave a fingerprint. If you are looking for tiny signals (like an Earth twin), you need to be very careful about which "smoothing" method you use, especially if your data is a bit fuzzy or if you are only looking at a short period of time.

Technical Summary: The Impact of Interpolation in High-Resolution Spectroscopy

Problem Statement
The detection and characterization of Earth-twin exoplanets require radial velocity (RV) precision at the cm s $^{-1}$ level. While state-of-the-art spectrographs (e.g., ESPRESSO, EXPRES) achieve high instrumental stability, spurious signals often arise from data analysis pipelines rather than the instruments themselves. Specifically, template-based RV extraction methods require interpolating stellar spectra onto a common wavelength grid to construct high-SNR templates and to compare individual observations against them. This paper investigates the systematic biases introduced by the choice of interpolation algorithm during this process. The authors posit that interpolation implicitly assumes a specific shape for spectral lines; if these assumptions do not hold, the process can introduce flux residuals, line asymmetries, and systematic RV biases, particularly when the Barycentric Earth Radial Velocity (BERV) variation is small relative to the pixel size.

Methodology
The study employs a multi-faceted approach combining synthetic data generation and the analysis of real ESPRESSO observations:

Synthetic Data Generation:
- Gaussian Profiles: The authors generated synthetic datasets using Gaussian functions to isolate the effects of interpolation. They simulated a single Gaussian line and a "forest" of lines (summing parameters derived from real ARES-detected lines in $\epsilon$ Indi and HD40307).
- Sampling Variation: To mimic the effects of BERV motion and instrumental drifts, the synthetic spectra were Doppler-shifted across a range of 0 to 500 m s $^{-1}$ (covering the mean pixel size of ESPRESSO). The flux was then interpolated back to the original wavelength grid.
- Noise Injection: Simulations were performed in noise-free conditions and with injected Poisson noise to evaluate Signal-to-Noise Ratio (SNR) dependencies (ranging from SNR=10 to SNR=78).
Interpolation Algorithms:
The study compared five interpolation strategies implemented within the s-BART (Semi-Bayesian Approach for RVs with Template matching) pipeline:
- Cubic spline (default).
- Quadratic spline.
- Piecewise Cubic Hermite Interpolating Polynomial (PCHIP).
- Akima.
- Floater-Hormann.
Real Observations:
The s-BART pipeline was applied to ESPRESSO observations of four stars with distinct observing strategies:
- High-cadence, single-night: HD40307 (K2) and $\epsilon$ Indi (K6), used for asteroseismic campaigns.
- Multi-night/long-term: Proxima Centauri (M5) and HD39091 (G0), used for exoplanet detection and characterization.
- RVs were extracted using different interpolation algorithms and compared against the default cubic-spline results.

Key Results

Flux Residuals and Asymmetry: Interpolation introduces residuals in the flux of spectral lines, particularly when the line center falls between pixel nodes. The magnitude of these residuals depends on the sampling density (pixels per FWHM) and the algorithm used. Cubic spline and Floater-Hormann strategies generally yielded residuals below 100 ppm for single Gaussian lines. However, asymmetries in the line profile were observed, which are more pronounced in poorly sampled lines (low SNR or wide pixel spacing).
Simulated RV Biases (Single Line & Forest of Lines):
- In noise-free, perfectly symmetrical Gaussian scenarios, interpolation introduced a systematic RV signal with a peak-to-peak amplitude of $\sim$ mm s $^{-1}$ . This signal exhibited a sinusoidal shape correlated with BERV variation.
- In low-SNR scenarios (SNR=10), the amplitude of this systematic bias increased significantly, reaching peak-to-peak values of $\sim$ 20–40 m s $^{-1}$ .
- The bias amplitude is inversely correlated with SNR. As SNR increases, the systematic signal diminishes, converging toward the mm s $^{-1}$ regime.
- The study identified three regimes in the RV response to SNR: sinusoidal behavior at low and high SNR, and an almost flat profile at intermediate SNR, likely due to the destructive interference between deterministic and stochastic systematic effects.
Real Observations (Single-Night):
- For high-cadence observations with small BERV variation (comparable to pixel size), the choice of interpolation algorithm significantly impacts the RV time-series.
- In the lower-SNR case (HD40307), different algorithms produced peak-to-peak residuals of $\sim$ 15–25 m s $^{-1}$ . The Floater-Hormann algorithm, which performed well in symmetric simulations, actually increased the systematic signal in real data.
- In the higher-SNR case ( $\epsilon$ Indi), the differences between algorithms were smaller, with peak-to-peak residuals around 1 m s $^{-1}$ , agreeing within 1- $\sigma$ uncertainties.
Real Observations (Multi-Night):
- For observations spread over a large BERV window (Proxima Centauri and HD39091), the systematic signal caused by interpolation is significantly dampened.
- The peak-to-peak RV scatter due to interpolation choice was found to be an upper limit of 20 cm s $^{-1}$ , which is well below the 1- $\sigma$ uncertainty of the measurements.

Significance and Conclusions
The paper concludes that spectral interpolation is a non-negligible source of systematic error in RV extraction, particularly for specific science cases:

High-Cadence, Low-SNR Campaigns: Science cases involving intensive, short-baseline observations (e.g., asteroseismology or high-frequency monitoring) with small BERV variations are most susceptible to interpolation-induced biases, which can reach the m s $^{-1}$ level.
Broad Impact: While the bias is minimized in long-term, high-SNR exoplanet surveys (where BERV coverage is large), the authors argue that interpolation artifacts affect all science cases where spectra are placed on a common grid. This includes not only exoplanet detection but also atmospheric studies, asteroseismic analysis, and cosmological redshift determinations.
Algorithm Dependence: No single interpolation algorithm is universally optimal. The performance depends heavily on the SNR, the sampling density, and the intrinsic shape of the spectral lines. The study highlights that assumptions about line shape inherent in interpolation methods can lead to underestimation of flux and systematic biases in line centers.

The authors emphasize that while the mm s $^{-1}$ level bias in ideal conditions is below current instrumental precision, the m s $^{-1}$ level biases observed in low-SNR, high-cadence scenarios represent a critical systematic that must be accounted for in future high-precision RV analyses.

The Big Picture: A Digital Photo Puzzle

The Problem: The "Digital Zoom" Effect

What the Authors Did

The Key Takeaways

In Summary

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