---
title: "Interferometric imaging from wave delay to EHT data"
description: "Learn Fourier sampling, Earth rotation and closure phases through three chapters, a real EHT triangle tutorial and an imaging-paper guide."
canonical_url: "https://sandboxphysics.com/stories/interferometric-imaging/"
language: "en"
page_type: "story"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/stories/interferometric-imaging/"
last_modified: "2026-10-28"
---

AC05

# Interferometric imaging from wave delay to EHT data

Sandbox Physics · 2026-10-28

A distant telescope pair measures one striped pattern across the sky. Derive that measurement, turn it with Earth, and learn what a triangle preserves when station phases drift. Then reconstruct a saved synthetic sky and fit a measured EHT triangle with its own calibration limits.

[Move a telescope and change the image →](https://sandboxphysics.com/labs/stellar-interferometer/)

[Concept How scattered telescopes make an image A telescope pair measures one pattern across the sky. Many directions and Earth’s rotation turn those patterns into an image.](https://sandboxphysics.com/stories/interferometric-imaging/how-scattered-telescopes-make-an-image/) [L1 From wave delay to complex visibility Derive a telescope pair’s Fourier measurement, fix its sign and units, and calculate the visibility of a point and a double source.](https://sandboxphysics.com/stories/interferometric-imaging/from-wave-delay-to-complex-visibility/) [L2 Earth rotation and closure phase Project real station separations onto the sky, count independent measurements, and derive the phase combination that cancels station errors.](https://sandboxphysics.com/stories/interferometric-imaging/earth-rotation-and-closure-phase/) [L3 Dirty beams and nonunique images Derive the dirty image, distinguish flux per beam from flux per pixel, and reconstruct one saved signal set with two explicit priors.](https://sandboxphysics.com/stories/interferometric-imaging/dirty-beams-and-nonunique-images/) [Tutorial Fit a real EHT closure triangle Reproduce a measured M87 closure phase from pinned EHT data, then fit several source models that explain the same three amplitudes and one angle.](https://sandboxphysics.com/stories/interferometric-imaging/fit-a-real-eht-closure-triangle/) [Paper guide Read EHT imaging and dynamic priors Read fixed EHT imaging figures, distinguish measured data from synthetic tests, and trace how spatial and temporal priors affect reconstruction.](https://sandboxphysics.com/stories/interferometric-imaging/read-eht-imaging-and-dynamic-priors/)

The browser reconstructs a finite-pixel synthetic sky from saved complex signals and a separate ideal calibrator. The measured EHT tutorial fits amplitudes and closure phase. The final guide follows fixed paper figures through calibration, synthetic tests and spatial or temporal priors.

## Sources and calculation scope

1. [NRAO Essential Radio Astronomy, Chapter 3, aperture synthesis](https://www.cv.nrao.edu/~sransom/web/Ch3.html)

   The narrow-field Fourier relation and Earth-rotation synthesis, especially equations 3.180–3.200. The browser uses a spherical Earth, a single wavelength and a finite pixel sky.

2. [EHT 2019-D01-01 M87 data release, pinned commit 27ae76a](https://github.com/eventhorizontelescope/2019-D01-01/tree/27ae76aacb9067a83388395c3826012b97aac371)

   Actual EHT-HOPS SR1 network-calibrated Stokes I from 10 April 2017, low band, 227.0707 GHz. The tutorial preserves the publisher CSV and selects one simultaneous AA–AZ–LM triangle. The repository licenses data under PDDL 1.0.

3. [EHT Collaboration, First M87 Results IV, arXiv:1906.11241v1](https://arxiv.org/abs/1906.11241v1)

   Fixed version of the 2019 imaging paper, DOI 10.3847/2041-8213/ab0e85. Read Figures 1, 3, 5 and 7, the station-gain equation 7, regularized imaging equation 14, and Appendix A. Figures remain with the publisher and authors; this guide links to them.

4. [Johnson et al., Dynamical Imaging with Interferometry, arXiv:1711.01286v1](https://arxiv.org/abs/1711.01286v1)

   Fixed 3 November 2017 version, DOI 10.3847/1538-4357/aa97dd. Sections 2.3–3 and equation 3 explain temporal regularization; Figures 7–10 use simulated black-hole observations. Figure 12 uses real 7 mm VLBA M87 data across epochs.