Fringe‑Fitting in VLBI: Theory, Practice, and Calibration Workflow

Name: L8: Fringe Fitting - Leah Morabito
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Channel: DARA - Development in Africa with Radio Astronomy
Description: Summary and key takeaways on Fringe‑Fitting in VLBI: Theory, Practice, and Calibration Workflow, covering Introduction This article explains fringe‑fitting, a

DARA - Development in Africa with Radio Astronomy

Jan 13, 2026

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Introduction

This article explains fringe‑fitting, a crucial step in calibrating Very Long Baseline Interferometry (VLBI) data. It follows the logical flow of the original lecture, covering the VLBI context, the mathematics of phase errors, baseline‑based versus global fringe‑fitting, and how these techniques are applied in practice using CASA.

What is VLBI?

Networks such as the EVN, VLBA, e‑MERLIN, LBA, and space‑based missions (e.g., RadioAstron) combine antennas that are separated by thousands of kilometres.
Each station has its own independent clock, requiring precise time alignment before correlation.
Long baselines give extremely high angular resolution but introduce challenges: clock offsets, geometric model errors, and atmospheric effects.

Why Resolved Sources Matter

Baseline sensitivity depends on system temperature, bandwidth, and integration time, not on baseline length.
For a point source the visibility amplitude is constant with uv distance, giving uniform signal‑to‑noise (S/N).
For resolved sources (e.g., a double source) the amplitude falls off with increasing uv distance, reducing S/N on long baselines.
Longer baselines also have faster fringe rates, demanding shorter solution intervals for phase calibration.

Phase, Delay, and Rate Fundamentals

The observed phase (\phi) can be expressed as: [ \phi = \phi_0 + 2\pi \, \nu \tau + 2\pi \, t \dot{\phi} ] where: - (\phi_0) – phase offset at a reference frequency and time. - (\tau) – delay (frequency‑dependent term). - (\dot{\phi}) – delay rate (time‑dependent term). Fringe‑fitting solves for all three terms simultaneously, unlike ordinary phase calibration which only estimates (\phi_0).

Baseline‑Based vs. Global Fringe‑Fitting

Baseline‑based: Solve the equation for each antenna pair (i‑j) independently. Requires detectable signal on every baseline; antenna‑based quantities are not directly recovered.
Global fringe‑fitting: Stack all baseline equations and solve for antenna‑based phases, delays, and rates. Advantages:
Works even if an antenna is not detected on every baseline.
Provides a consistent set of antenna‑based solutions.
Implemented in CASA’s fringefit task.

Practical Implementation in CASA

Choose a bright calibrator (often assumed to be a point source unless a model is supplied).
Set one antenna as the reference; all solutions are relative to it.
Run global fringe‑fitting to obtain phase, delay, and rate corrections.
Apply these corrections to the calibrator and later transfer them to the target (phase referencing).

Phase Referencing

Use a nearby calibrator with a known structure.
Alternate (nod) between calibrator and target; cycle time depends on atmospheric stability (≈10 min at 5 GHz, ≈5 min at 1.6 GHz).
Calibrator and target should be within ~1° to ensure similar atmospheric paths.
After fringe‑fitting the calibrator, interpolate the solutions to the target and image the target (e.g., a supernova remnant) without direct calibration on it.

Instrumental and Multi‑Band Fringe‑Fitting

Instrumental delays: Time‑independent offsets across spectral windows. Determined from a short scan on a bright calibrator; applied to all data.
Multi‑band fringe‑fitting: After removing instrumental delays, solve for residual atmospheric delays, rates, and phases using the full bandwidth. This step corrects time‑dependent atmospheric effects.

Amplitude Calibration

Bandpass calibration: Solves for amplitude (and phase) variation with frequency for each antenna. Requires a very bright calibrator because the solution is per channel.
Gain calibration: Corrects residual time‑dependent amplitude variations (e.g., amplifier gain changes) after bandpass correction. Usually performed on a secondary calibrator with a known model.
Polarization calibration is possible if a polarized calibrator model is provided, but it is not covered in depth here.

Closure Phases and Non‑Closing Errors

Closure phase: Sum of observed phases around a triangle of antennas; antenna‑based errors cancel, leaving only source structure information.
Ideal closure phases are flat; deviations indicate non‑closing errors arising from:
Incorrect source model (requires self‑calibration).
Excessive time averaging (rapid atmospheric changes).
Excessive frequency averaging (bandpass variations).
Mitigation strategies: use accurate models, limit averaging, or apply self‑calibration.

Summary of the Calibration Chain

A priori calibration (system temperature, gain curves, etc.).
Instrumental delay correction (short scan, primary calibrator).
Global fringe‑fitting (phase, delay, rate) on calibrators.
Bandpass calibration (amplitude vs. frequency).
Gain calibration (time‑dependent amplitude).
Phase referencing – transfer solutions to science target.
Optional self‑calibration to address non‑closing errors.

The lecture concludes with an invitation to Q&A sessions and the course Slack workspace for further discussion.

Fringe‑fitting simultaneously solves for phase, delay, and rate errors, dramatically improving signal‑to‑noise and enabling reliable calibration of VLBI data; combined with bandpass, gain, and phase‑referencing steps, it completes the calibration pipeline needed to turn raw interferometric measurements into high‑resolution astronomical images.

Frequently Asked Questions

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What is VLBI?

- Networks such as the EVN, VLBA, e‑MERLIN, LBA, and space‑based missions (e.g., RadioAstron) combine antennas that are separated by thousands of kilometres. - Each station has its own independent clock, requiring precise time alignment before correlation. - Long baselines give extremely high angular resolution but introduce challenges: clock offsets, geometric model errors, and atmospheric effects.

Why Resolved Sources Matter

- **Baseline sensitivity** depends on system temperature, bandwidth, and integration time, **not** on baseline length. - For a point source the visibility amplitude is constant with uv distance, giving uniform signal‑to‑noise (S/N). - For resolved sources (e.g., a double source) the amplitude falls off with increasing uv distance, reducing S/N on long baselines. - Longer baselines also have faster fringe rates, demanding shorter solution intervals for phase calibration.

Summarize another video

Full Transcript YouTube

Welcome to lecture seven. This lecture will cover fringe-fitting and it's a companion lecture to lecture six: calibration. So if you haven't watched the lecture six then go ahead and stop now, watch lecture six, and then come back to this lecture.
I'm Leah Morabito and I'll be walking you through this lecture today.
So hopefully you remember this slide from lecture six. Before we get started on actually fringe-fitting, let's talk a little bit about where we are in the calibration process.
So unless you're six we talked about a priori calibration and all the things you need to do there and we got up to the point where we have these Jones matrices that we need a calibrator source to solve for, and that's where this lecture picks up.
So, remember that these Jones matrices contain visibility-like information which is represented as complex numbers, that we often represent as amplitude and phases. So we'll talk about amplitude and phase calibration today, rather than just complex numbers.
So in today's lecture we'll give a very brief recap, of what very long baseline interferometry or VLBI actually is. And this is usually the context in which fringe-fitting is talked about.
We'll go through the fringe-fitting fundamentals and a little bit about fringe-fitting in practice, although you will get to do this in your tutorial, and so I won't cover all of the very small technical details here.
So what is VLBI anyway? Well, you probably know, some VLBI networks, for example, the EVN or maybe the Very Long Baseline Array in the US, e-MERLIN in the UK where I'm based, the Long Baseline Array in Australia.
And increasingly we're seeing that people are piecing together either individual antennas or arrays to into a network of
antennas and arrays like Tanami, and this is a VLBI network. And then you can also even do this with satellites -- so RadioASTRON was a radio receiver on a satellite
and you correlate the information with a radio telescope on earth, and this actually gives you the biggest baseline because if you're observing from earth the
largest baseline you can ever have that's looking in the same direction in the sky at the same time, is about the diameter of the earth. So you can increase that baseline length by have using a satellite and get super super great resolution.
Well, what do all of these things have in common, all of these arrays?
The common element here really is that the stations get to be so far apart that, unlike the VLA, which is a connected interferometer we talked about in the previous lecture
these stations are so far apart that they require separate non-distributed clocks. So each station has its own individual clock that's timing the information that's coming in to the telescope.
And so this drives non near-real time correlation: you have to correct for the clocks you have to make sure that all of the clocks are lined up before you even do corrections for clock drifts in time and frequency.
And there's a lower tolerance for errors in the geometric model of the array when your telescopes are so far apart.
And you have to cope with resolved sources. So there's no real hard and fast definition of what VLBI is, but generally when you're in the regime
where you have separate clocks you're really starting to talk about needing to use VLBI techniques.
And this comes with all of these problems that are listed on this slide which will we'll talk about here.
So the first thing that I want to talk about is how we cope with resolved sources. 
So what do I mean about having to cope with results sources?
This question really is: what part does resolution play? So let's think about this for a minute.
Let's look at the baseline sensitivity equation. So this is the sensitivity, S_ij, on a particular baseline between antenna i and antenna j.
You can see that this is dependent on the system temperature of each antenna, it's dependent also on the bandwidth of your observation, and the time of your observation.
Note, however, there is no dependence here on baseline length, so it doesn't matter how long your baseline is any two pairs of antennas will have the same sensitivity.
And sensitivity is important because, when we're doing calibration and we have to solve for these Jones matrices, we need to have good enough signal to noise that we can actually solve for the Jones matrix.
So what impact does source structure have? Well let's give an example.
If we think about a point source, if you look at a point source in the uv plane, you'll see amplitude is constant as a function of uv distance (phase is also constant as a function of uv distance but we're just going to stick with amplitude here for this example).
That means your signal, or your amplitude, your signal to noise does not change as a function of uv distance. Now what happens if you have a resolved source?
Here I'm giving an example for a double source. Because it's a double source, as the two components pass through the fringes you'll get
constructive and destructive interference and so you'll get this beating pattern. Now your amplitude as a function of UV distance looks quite different.
So at very short baselines, which have the poorest resolution, you will see this source as just a single source, a single point source,
and your amplitude will be very high, because it will contain all of the flux density for this source.
However, as you go out in baselines you'll see that the amplitude decreases as a function of uv distance.
And this is really a problem because this means we have smaller signal on longer baselines. So: your signal to noise on longer baselines is decreasing
not because your baseline sensitivity is changing, because that is staying the same, but because you just have less signal. So how do we calibrate and low signal to noise situations is really the question that we have to answer.
There's also other things, you can think about the fact that
as your baselines get longer your fringe rate gets faster, and that means that your phases change faster as a function of time, and so, then you have to have smaller time intervals to be able to calibrate for your phases.
Okay, so let's think back to the relationship between phase and delay. So you can see here that your phase is dependent on both frequency and the delay, which is dependent on time.
If you differentiate this, which gives you your error in the phase, then you can see, this is also dependent on frequency and time.
And if we take the first order expansion of this we end up with this equation here. So here what we're saying is that our error in phase is equal to a phase term plus a frequency dependent term and a time dependent term.
And this equation is really the basis for fringe-fitting. So again, here we have our phi_0, this first term, which is the phase error at a particular frequency and time.
And then we have the frequency dependent term, which we call the delay or the delay residual, which is how phase is changing in frequency.
And then we also have the time term here, where the d-phi/d-t is what we call the rate, the delay rate, or the rate residual, and this is how frequency is changing in time.
So, in normal phase calibration we say essentially that our error in phase is just this first term: the phase error 
at a particular frequency and time. Fringe-fitting is any process that also estimates the delays and rates, so then when you're finding your phase error
you're finding all three of these terms and anything that does that is referred to as fringe-fitting.
And this effectively increases your signal to noise, because it means you can increase your solution interval. So how does that work in practice,
increasing the solution interval and increasing the signal to noise? So let's give you a practical example. Here we have phase versus frequency.
If this is properly calibrated you would see that the phase is flat at zero, however, you can see it's not. You can see that there is a frequency dependence, there is a slope to these phases.
And that is actually the delay. So if we're going to find the phase correction that we need to make make this flat at zero, we need to find a phase at all the different frequencies here.
So typically in normal phase calibration what you would do is you would solve for phi_1 at frequency 1 and you would solve for phi_2 at frequency 2 and you would do this at all frequencies across here.
And the reason that you do this, you find phi_1 at frequency 1 and phi_2 at frequency 2 is because
phi_1 is only valid at frequency 1 and phi_2 is only valid frequency 2. So you can't find phi_1 at frequency 1 and apply it at frequency 2 and get the correction that puts the phase back to zero.
This means that you have to solve in very, very small frequency intervals, to get the right phases for that particular frequency. 
However, ff we now do fringe-fitting, which means that we also estimate the delays (and the rates, but this is a delay example)
then what we're solving for is phi_0 at whatever frequency and the slope here, and so by solving for these two terms, phi_0 and the slope, 
it's valid over a very large range of frequencies. All of a sudden, we only need these two values, and we can correct the entire system, which means we can use the entire frequency range to solve for phi_0 and the slope 
of the phases and it describes everything, and then we can correct all of the phases to zero.
And so that means that by using the entire frequency range, then we are increasing our signal to noise because we're increasing the amount of data that we're using which increases the signal to noise by driving down the noise.
So let's talk about baseline based fringe-fitting. The fringe-fitting equation can be written in terms of individual baselines, for example with antennas i and j.
So here we have the phase error on baseline ij and here you have the the phase, or the phase offset, on antenna i, the phase offset on antenna j, you have the delays for 
the baseline between antennas i and j, and the rates between antennas i and j.
So essentially what you do in baseline based fringe-fitting is you construct this equation for each pair of baselines and you solve for the phases delays and rates on each baseline. So you're not solving for the individual antenna quantities, you're solving for this term here.
The disadvantage here is that the source has to be detected on all baselines. You have to have good enough signal to noise on each baseline to be able to solve each of these equations.
And the antenna based quantities are not conserved because you're solving for this term, not each antenna quantity individually.
So what we usually do, and when people say fringe-fitting what they generally are talking about is what we call global fringe-fitting.
And so, global fringe-fitting is where you take the baseline based equations
but you stack them together and you solve the entire system simultaneously to find the antenna based solutions, so now we're solving for the individual antenna based solutions.
using the entire system of equations rather than the baseline quantities.
The major advantage here is that even if an antenna does not see a source on every baseline, the antenna based solutions can still be found, because as long as the source is seen on some baselines containing that antenna you can still find the solution for that antenna. So that's great.
So, in practice, global fringe-fitting is implemented in CASA and you'll get a chance to play around with this in your tutorial.
So in practice what we do is we set one antenna as the reference santana and then find the phases, delays, and rates for all of the other antennas. The solutions here are relative to the reference antenna; they're not absolute solutions, they're always relative to the reference antenna.
And again, you need a calibrator source to find the the fringe-fitting solutions.
So CASA just assumes this to be a bright point source unless you give it a model, and you can give it a model, and if you do have a model of your source it's always good to start with the model rather than just whatever generic model CASA uses.
Keep in mind that fringe-fitting gives you phase corrections only -- it does not give you an amplitude. I know we solve for the phase offset, the delays, and the rates, but the combination of all that, together, is your phase error: so that's your phase correction. 
So how this helps us out is we use what we call phase referencing. So we'll do global fringe-fitting to find the phases on a calibrator source that's near our target,
Where we know the structure, because, again, as I just mentioned CASA needs a model of your source to be able to find the fringe-fitting solutions.
So you use a calibrator source that you have a good model for, you find your fringe-fitting solutions and then you transfer them to your target
or you interpolate them to your target. So you want your calibrator and your target to be quite close together
so that the phase solutions are valid for that direction, because if you move to a completely different part of the sky you'll be looking through a completely different atmosphere.
There will be differences in your system because you're antennas are pointed a different direction, so you really want something that's close to your target.
And so, typically what we will do is we'll nod between the calibrator and target. So we'll go back and forth between the calibrator and target and this cycle time
is set by the time scale of the atmospheric fluctuations. So the more stable your atmosphere is the less you have to cycle; and the more rapidly your atmosphere is changing the more quickly you have to cycle back and forth to capture those time changes.
The general rule of thumb is that at five gigahertz, a cycle time about about 10 minutes is fine, and at 1.6 gigahertz about five minutes is usually okay.
So how close do these sources need to be? Generally, they have to be within about a degree. The biggest problem here that we're trying to solve for with our calibrator source is the atmosphere.
So again, this is the troposphere at high frequencies and the ionosphere low frequencies.
So, to give you a practical example of phase referencing.
This is a calibrator source and a target source,  and they don't look like much right? It doesn't look like there are sources there -- yet. However, we have a very good model of our calibrator source and we know what it looks like, and so we can do fringe-fitting on it.
And we get back an image of our source. Okay, great. So then what we do is we take those solutions and we transfer them to our target
and make an image and then you can see it pops out -- it's a very nice supernova remnant. So we haven't done any calibration on the science target itself here, we've only done calibration on the calibrator and then transferred the phase corrections to the target.
So in practice
we use fringe-fitting a couple of different ways. The first thing we do is we use it to solve for instrumental delays.
And so this is to account for different instrument responses across the spectral windows. This is time independent and so you'll take just a very short scan on a very bright calibrator source and you can find the the instrumental delays. This is phase versus frequency for
before the instrumental delay correction. The colors for all of these these phases are different baselines to the Effelsberg antenna, and this is just a single short scan.
And so you can see quite clearly that there is a delay here, again that slope as phase changes with frequency -- and that's what we're really trying to remove here.
We use fringe-fitting to find the delays here, and then we can correct for that.
So the colors again are baselines to Effelsberg and you can see that all of your phases are now pretty flat, there's maybe one or two antennas
that are having some issues but they're pretty flat at zero. So this was only on a single short scan on a calibrator source, so we transfer that
to all of the rest of our sources. So this instrumental delays is generally done on your primary calibrator and then you'll transfer it to your secondary calibrator and your target source.
So if you transfer it to your secondary calibrator, for example, then you might see something like this.
So this is the instrumental delays have been applied to a different scan, and you can see that your phases are fairly flat so they've had the delay the instrumental delay removed but
the phases are not corrected because the instrumental delays doesn't correct your time dependent phases so remember this is different scan at a different time.
So, then, we need to use fringe-fitting again for what we call multi band fringe-fitting .so now we've removed the instrumental effects, and we have to get rid of the atmospheric effects.
And so we use fringe-fitting again to do multi-band fringe-fitting to remove the delays and the rates and the phases.
So essentially to correct the phases on our secondary calibrator, and transfer that to our target source. And so once we've done that, we've removed the instrumental effects and multi-band fringe-fitting will get rid of the time dependent atmospheric effects.
Great! And so then that's all of that we've basically done to solve for this one Jones matrix. So this multi-band fringe-fitting.
gives you the corrections which go into this particular Jones matrix.
These are the a priori calibrations we talked about in the previous lecture and now we can check off another Jones matrix here. What about the rest of them? Well let's talk about the amplitudes. So up till now fringe-fitting only does phase corrections -- so what about the amplitudes?
Well, we do a bandpass calibration, and this is done on the primary calibrator and so when you do a bandpass calibration what you're doing is you're solving for amplitude as a function of frequency
gor each antenna and you have to use a very, very bright calibrator source because because you have to do this as a function of frequency.
You have to do it in each individual channel across your bandpass, which means that you have to have good enough signal to noise in each individual channel to find a solution.
So you do this on a very, very bright source and that's that's typically our primary calibrator source. So you can see here these are the
bandpasses for different frequency windows
and you can correct for this. So this is the bandpass Jones matrix and then also if your source is polarized, and you get polarization information 
then you can also correct for the polarization here. We're not really going to cover polarization in any depth, but just keep in mind that
if you're solving for these Jones matrices using a polarized model, then you can get back the polarization information.
Right, so we still have a couple more things to do, and so another thing that we need to do is what we call gain calibration.
So bandpass is again a time independent thing, we solve it on the primary calibrator and we transfer it to the rest of the rest of the sources but there might be slight variations in amplitude and also leftover phases.
And so we do gain calibration to take care of these residual amplitude effects. So this is mainly due to variable gain in the antenna amplifiers that weren't perfectly corrected by the gain curves in the a priori calibration.
Okay, and so we can use again a model of our secondary calibrator to solve for these gain calibrations. And then that's this Jones matrix here. So you'll note that we have one  Jones matrix left, this is for 
Non-closing errors which we'll talk about, but we have to talk about
what closure quantities are before we can talk about what non closing quantities are. Okay so let's talk about what we call closure phases.
So we have a triangle of antennas, and you have baselines between these antennas, and each antenna sees through a different part of the atmosphere, which means that each antenna has different antenna error associated with it. So the phase that you observe on a particular baseline
is the real phase on that baseline plus the combination of errors for each of the antennas.
And so, if you go around this triangle, and you can go around the triangle either way, but once you've picked a direction you have to keep going that direction, so if you go this way, you have to keep going around
You can also go around [the other way] but, once you start going this way, you have to keep going around.
And so you sum the phases in a direction around the triangle, and when you sum the phases in a direction around the triangle that gives you what we call the closure phase. So here we add
all of the observed phases together and what we get back is a combination of the real phases, because all of the errors cancel out. So here, you can see that if we add
the phase for antennas i,j to the phase for antennas j,k -- so we have plus the error in i and minus the error in j -- and if we add this then we have plus the error in j minus the error in k 
but the j's cancel out and then, when we add in the third baseline here, then the errors in k cancel out and the errors in i cancel out. 
And so what we're left with is something that should have no errors in it, right? Okay so then that's what a closure phase is, and if everything is calibrated perfectly, if there are no errors
then it should be completely flat. So let's take a look at a closure phase example. so here is phase on baseline 12, phase on baseline 31, and phase on baseline 32. 
And here's the closure phase, and you can see that it's not completely flat, so there are some variations here.
And these variations are what we call non-closing errors, it means that they haven't been cancelled out when you've taken the closure phase.
So where do these non closing errors come from? Well, they can come from baseline-based errors. And there's three main sources of this: The first one is that your source structure is not correct in the model.
And to correct for this, you really need to do self calibration which is covered in a different lecture.
You can also get non closing errors from time averaging. So your atmospheric conditions can
change very quickly on very short time scales, which means that if you average too much then your average value is not representative 
of the different quantities within that time range. You can't really solve for that, but what you can do is you can mitigate that by not averaging in time. And then also you can get
non-closing errors from frequency averaging, and so here your your bandpass response varies per telescope as a function of frequency, and if that varies a lot over a certain frequency range then again when you average your average value is not representative of all of
the values within that frequency range and so again, we cannot solve for this non-closing error, but what we can do is, we can not average in frequency.
So that kind of takes care of the last Jones matrix -- and I've put this in parentheses because it's not necessarily something that we always solve for,
it is sometimes something that we mitigate by not averaging in time and not averaging in frequency, but it can be solved for with self calibration which is handled in another lecture.
Alright, so we've come to the end. We started out with this equation at the beginning of the lecture.
We had all of these things that we didn't know about and we've talked through them today.
we've talked about fringe-fitting, which was the focus of this lecture, but then also how we correct the amplitudes with both the bandpass and also gain calibration.
We touched briefly on polarization which you can correct for using polarized calibrators, and then we also talked about what non-closing baseline based errors are.
So today we've gone through a brief recap of what very long baseline interferometry is, we talked about fringe-fitting fundamentals and fringe-fitting in practice. I look forward to seeing you in the
interactive Q&A sessions, and also on the Slack workspace, so enjoy the rest of the lectures.