Last year, throughout the month of November, I vowed to think of at least one thing I was grateful for, throughout the month. Too often we concentrate on those things which are sad and stressful, without remembering all those things which bless us in our lives. I decided I will keep the November daily gratitude journal this year too. Since you already know that my husband and my children mean the world to me, it's not necessary to mention them in these daily entries - as I am thankful for them every day of my life!
The last few days went by very quickly. Steve was off from work and I loved having him home with me. We took care of a lot of errands together, and went for a drive up the Steese Highway on Tuesday. We also had a most wonderful dinner at Lavelle's Bistro later that evening, using a $100 gift card that Steve got from his company, as a reward for not losing any time for accidents for the last quarter. Every employee received this gift card (a Visa card that could be used anywhere), and we decided to use it on a splurge, rather than something sensible.
Wednesday , November 14th - Busy Busy!
I have been very busy over the last week. (I can't believe it's been almost a week since I've updated this journal!) I've spent a lot of time mounting, matting, and framing photos, and also working on my gallery website - which needs a total makeover since I have new photos to add to it, and photos to remove from it.
On Friday (the 8th), I originally made plans to run errands, but then seemed to lose my energy - opting instead to stay inside and do house chores and photo work. However, the day turned out even better than I anticipated, and I didn't have to look any further than my yard.
Wednesday , November 21st - Still very, very busy!
November 15th Gratitude: Today I am thankful for HEAT! It's currently -8F here in North Pole! BRRRRR!!!! When I opened the sliding glass doors to feed the birds, it was frozen and I had to give it a yank to get it to open. The garage doors have been making a *POP* sound as the panels fold when the door's on its way up, when the ice breaks loose between them. There is a layer of ice at the bottom of the garage doors where the water that has dripped off of Steve's thawing truck has frozen. This is only the beginning... But how thankful that my house is nice and warm! I'm also thankful for radiant heat in my wood floors. I love walking barefoot on a warm floor.
November 16th Gratitude: Today I am thankful for STASH Licorice Spice Herbal Tea. My friend Celeste gave me two tea bags to sample, and I'm hooked. I just love black licorice (Steve hates it) and this tea is really tasty and caffeine free, so I can enjoy it in the evening too. STASH teas are delicious! My other favorite is the Pomegranate Raspberry Green Tea.
November 17th Gratitude: Today I am thankful for the talented folks at the FLOT (Fairbanks Light Opera Theater). What a fabulous show we saw last night! (We saw the musical, Oklahoma) It never ceases to amaze me that Fairbanks - for being a relatively *small* town - can put on such entertaining performances. This is the fourth musical I've attended here, and all have been top-notch.
November 18th Gratitude: Today I am thankful for having a husband who likes to cook. If not for him, we'd both starve!
November 19th Gratitude: Today I am thankful for butterscotch cocoa. (It's one of the flavors in the Land O' Lakes assortment carton of cocoas) I stayed up all night working on photos and it kept my energy levels up. It is really, really, good! I think I like it more than the raspberry cocoa.
November 20th Gratitude: I've had several invites to Thanksgiving dinner from local friends - a few who were worried that Steve is working and I'll be alone. Getting the invites via email and telephone made me feel loved. I'm blessed to have such wonderful people in my life.
November 21st Gratitude: Today I am thankful that my husband is off tomorrow (last minute decision from his boss!) and we'll be able to eat Thanksgiving dinner at 3pm instead of 7pm. I'm really looking forward to having him here while I prepare dinner. But mostly I'm looking forward to sleeping in with him tomorrow morning.
I apologize immensely for my absence. It has been a rough couple of weeks for me. I have literally stayed up all night long working on my new web gallery and organizing photos to display at First Friday. For all intents and purposes, my Gallery is finally published to the web!
If you find any glitches, or broken links, please let me know about them. Also, if you're on dialup, you're going to find some of the gallery pages slow-loading due to all the thumbnails there. I'm using a feature called "lightbox" to have my photos display in a slideshow. Unfortunately this feature won't work if all of the thumbnails haven't loaded. In that case, you can still see enlarged versions of the thumbnails, but they will open up in another window, rather than in a neat and automated slide show.
I've eliminated some photos from my old gallery, and added some more to this gallery. I hope to keep this one better updated as time goes on. However, if you see a photo here in this journal, but it's not in the gallery, that doesn't mean it's not available for print. Just ask!
Anyone local who reads me, I would love if you could stop by WILLOW CREEK CLOTHING, located on 12th - between Lacey and Cushman. Just stop in and say hello, and introduce yourself if we haven't met. My photos will be on display from 5pm - 8pm, but Steve and I will be there setting up at 3pm.
I still have tons to do between now and next Friday, and I still have to compose my Christmas newsletter for friends and family, as well as get holiday cards and packages ready to go to the post office. I fear I will never be done!
Thank you for your patience. Once January 1st comes I hope to be back to regular journaling and photo posts.


For some reason, the dll(s) that i place within my se1\bin folder doesn't seem to load. Is there any way of telling the engine to load the extra dll?
Sorry for double posting, i thought I should explain my problem a bit.
I'm trying to get se1 to look better, by that i need to have the game load an extra dll on startup, but it doesnt want to load it automaticly like other source based games. Is there a way to tell the game to load the extra dll? like in a config file, gameinfo or something. Please help me.
Last edited by Senast : 04-06-2011 at 09:01 PM.


text:
Q: guess_dates() missing from linelist package I'm trying to use the guess_dates() function from the linelist package and I'm receiving this error:

Error: 'guess_dates' is not an exported object from 'namespace:linelist'

I've tried restarting R and updating all packages without luck. When I query the function using ??guess_dates, the Help page returns

"no results found."

This package::function() combo linelist::guess_dates() is widely referenced by other R users.
Any ideas?? Thanks in advance!

A: Try installing the package using devtools::install_github("reconhub/linelist")  mentioned on the GitHub. Here is a reproducible example which worked when installing the package:
devtools::install_github("reconhub/linelist")
#> Skipping install of 'linelist' from a github remote, the SHA1 (cae034de) has not changed since last install.
#>   Use `force = TRUE` to force installation
library(linelist)
linelist::guess_dates(c("03 Jan 2018", "07/03/1982", "08/20/85"))
#> [1] "2018-01-03" "1982-03-07" "1985-08-20"

Created on 2022-08-30 with reprex v2.0.2

redpajama_set_name: RedPajamaStackExchange


text:
Q: JS element innerHTML fails in IE 11 using neptune plugin Description: I'm using neptune plugin to open IE from other browsers (chrome, firefox etc).
Assume that when user clicks a button then embedded IE screen is opened.
The problem: Javascript fails to execute getElementById innerHTML by using neptune
The error:   Browser prompts an error that script fails to execute: LINE 1 CHAR 1 (it looks weird, because in this line i haven't any code o.O)
Snippet:
Javascript
var container = document.getElementById('container');
container.innerHTML = '<embed id="emb" type="application/x'
+'-meadco-neptune-ax"' + 'pluginspage="http://www.meadroid.com/neptune/download/"\n'
+ 'param-location='
+ "localhost:8080/test/embeddedFromIE"
+ '\n' + 'style="border: none; margin: 0; padding: 0; width: 100%; height: 100%; frameborder: 0; overflow: hidden; overflow-x: hidden; scrolling: no"></embed>'

HTML

Any ideas?


SCARLETT Moffatt has already annoyed her new neighbours with her "eyesore" new home, according to reports.
The angry residents have allegedly blasted the Saturday Night Takeaway star's newly-built £300,000 pad for failing to fit in with their picturesque village.
Scarlett, 27, moved into the detached four-bedroom house in her native North East a few months ago.
The new build is set on a T-junction in the village, and 'annoyed' locals believe it doesn't fit into the look of their town.
The former reality star's posh pad comes with it's own cinema and games rooms, a large open-plan kitchen, complete with a central staircase.
One villager told The Mirror: "It's an eyesore that sticks out like a sore thumb and lets the village down.
"It'd look better in a field with more land around it. It's far too modern.
Another local, who has lived there 54 years, thought Scarlett's home was going to be a bungalow.
Meanwhile, Scarlett faced another backlash after she was accused of "cheating" during a competition on this week's Saturday Night Takeaway.
Game Of Phones saw viewers at home call in if they knew someone in the studio audience, giving them the opportunity to fly to win £500 for both themselves and the caller at home.
Buckling under the pressure, Scarlett seemed to grab phones from the same row, causing Twitter users to suggest she wasn't being fair.
Scarlett hurried to answer as many phones as possible but when she struggled with a bad connection, she hung up on the caller, prompting shocked gasps from the live audience.
One fan tweeted: "Scarlett Moffatt lifted every phone from the same shelf!! Give everyone a chance."
Another vented: "Scarlett picks up phones upside down and back to front and wonders why no one answers her."
However Scarlett posted a cryptic tweet about 'positivity and kindness' after the comments.
Choosing to ignore the comments about it being a fix, Scarlett instead took to Twitter to thank the fans who had said nice things about her latest stint on the show.
As well as retweeting lots of positive messages, she wrote: "Thanks for all the amazing tweets about game of phones and takeaway last night.
"A day of board games and pizza for me, enjoy your sunday everybody...remember to sprinkle your positivity and kindness like confetti."


Travelling south from Lillooet, follow highway #12 to Lytton. Cross the Chief Spintlum Bridge over the Thompson River, then turn right at the first stop sign. You are now on Main Street.
Travelling from either Hope or Cache Creek, follow Highway #1 and turn into Lytton which takes you onto Main Street (aka Highway #12).
At the Village Office, turn one block down 4th Street. At the end of 4th Street is the Information Centre, and the Lytton Museum is the next building to the right!


text:
Multiple Railcars Overturn During Derailment in Southwest Detroit
Twelve railcars overturned as a Canadian National freight train derailed in southwest Detroit, Michigan, on the morning of December 12, authorities said. No one was injured, the city said. The Detroit Fire Department said five of the 12 overturned cars were marked as containing flammable material, but a hazmat team found they were empty at the time of the derailment. Several nearby roads were closed as authorities worked to clear the tracks, which they said could take up to eight hours. Credit: Ideal Group via Storyful
Trump Addresses World Economic Forum in Davos

redpajama_set_name: RedPajamaCommonCrawl


text:
WCW Blog
Education & Child Care
Youth and Adolescent Development
Society & Leadership
Jean Baker Miller Training Institute
Justice and Gender-Based Violence Research Initiative
National SEED Project
National Institute on Out-of-School Time
Open Circle
Work, Families and Children
Research & Action Report
Women's Review of Books
The WCW Fund
Scholars & Trainers
WCW's Women Change Worlds Blog
Bullying Prevention Starts with Adults
Women Change Worlds
Nancy MacKay
Policies, procedures, and protocols for bullying prevention and intervention are now a requirement for most schools across the country. Yet policies that are developed and implemented in isolation are insufficient to address the challenges of bullying behavior. It is also critical to create a school culture and climate of communication, collaboration, and trust where children and adults feel safe and supported to speak up about bullying.
Building a safe environment is a key element to preventing and addressing bullying in schools. New research from ChildTrends found that bullying prevention programs that use a whole-school approach to foster a safe and caring school climate – by training all adults to model and reinforce positive behavior and anti-bullying messages – were generally found to be effective.
The Open Circle Curriculum, an evidence-based social and emotional learning program, focuses on both proactively developing children's social and emotional skills (like calming down, speaking up, and problem solving) and building a school community where children and adults feel safe, cared for and engaged in learning. We encourage a unique whole-school approach that includes training all adults in the school community – teachers, administrators, counselors, support staff, and families – to learn, model, and reinforce pro-social skills throughout the school day and at home.
Students are always watching. They are watching adults at their best and they are particularly watching adults when they are in conflict. While emphasis and expectations of behavior is often placed on the students, adults in schools should remember to take a step back and look at themselves, their relationships, and the behaviors students see them model. It's imperative that adult communities in schools reflect the same expectations of behavior that we have for students. Otherwise a climate may develop where students and adults may not feel safe to identify, report, and effectively address bullying behavior.
When a consistent culture and climate is created both on the student and the adult level, bullying prevention efforts will be strengthened along with creating the best possible environment for learning.
Nancy MacKay, B.A., and Nova Biro, M.B.A. are Co-directors of Open Circle, a leading provider of evidence-based curriculum and professional development for social and emotional learning (SEL) in Kindergarten through Grade 5. Open Circle, a program of the Wellesley Centers for Women at Wellesley College, is at the end of its 25th anniversary year.
How do you feel about this post?
Project: Open Circle Bullying Child & Adolescent Development Education People: Nova Biro People: Nancy MacKay Social-Emotional Learning Prevention WCW
WCW admin on Thursday, 28 November 2013 12:50
Thanks for the great post ....Bullying Prevention Starts with Adults! I do agree with 3 P's which is Policies, procedures, and protocols are responsible for bullying prevention & intervention are now a requirement for schools in the country. Weight Loss counselling Adelaide, Relationship counselling, Teenagers counselling Adelaide
0 Cancel Reply
Cancel Update Comment
I have read and agreed to the Terms & Conditions
Connections Are at the Core of Social Justice
Creating Space for More Than Tolerance
November 26, 2013 Bullying Prevention Starts with Adults
Canada Steps Out Front on Funding Feminist Futures Worldwide
Tackling Inter-generational Poverty through Education
More than the Gender Wage Gap
Enough with the Excuses—Corporate Boards Need Women
Poverty and the Rural African Girl
Women Change Worlds Blog
Lively debate and opposing opinions are welcome, but please communicate courteously and responsibly, adhering to the guidelines.
WCW Website
CONTACT: news-wcw@wellesley.edu
Seeking Help/Services?
Subscribe to receive Women Change Worlds blog updates!
Youth & Adolescent Development
Fact Sheets and Briefs
Wellesley Centers for Women
Wellesley, MA 02481-8203 USA
WCW@wellesley.edu
Copyright © 2019 Wellesley Centers for Women, Wellesley College | Privacy Policy
Jean Baker Miller Training Institute | National SEED Project | National Institute on Out-of-School Time | Open Circle
Our website uses cookies to enhance your experience. By continuing to use our site, or clicking "Continue", you are agreeing to our privacy policy.
Continue Privacy Policy


The school is held in the "golden" hills near Lyon, in St Germain au Mont d'Or, in the Domaine des Hautannes. The domain offers all facilities for meetings: meeting rooms, lodging facilities, a restaurant, a small bar, a courtyard, a park, and WIFI everywhere. For the school, we have access to 3 small meeting rooms (for working groups or nightly talks), 1 large meeting room (for the main courses), and 35 bedrooms (that can be converted into double rooms).
If some effort is made to share rooms, all participants should be able to stay at Domaine des Hautannes. Otherwise, some participants can still stay in hotels nearby (in Quincieux or even in Lyon). In this case, shuttles may be organized.
The Domaine des Hautannes is at walking distance from the train station of St Germain au Mont d'Or (1.2 km, but it is not flat!), which is connected to the Lyon Part-Dieu and Lyon Vaise train stations (10 minutes). Shuttles will be available for the arrivals on Monday (around lunch time) and departures on Friday (after lunch time).
To get to Lyon from abroad by plane, the easiest is often to flight to Paris (Aéroport Roissy Charles de Gaulle) and to take the direct TGV (fast train) to Lyon Part-Dieu station (2 hours). It is also possible to get flight connections directly to Lyon (Aéroport St Exupéry), then to take the Rhônexpress express tram to Lyon Part-Dieu station (30 minutes). If you don't find any suitable flight connections to Paris or Lyon, you may find interesting flights to Geneva, then take the train.
For any question relative to the polyhedral school, please contact Alain Darte (organizer) or Laëtitia Lécot (administrative staff) at polyhedral-school@ens-lyon.fr.


From September 23rd to 25th, the 2016 Shanghai Food &Wine Festival (also called Hippo Living Festival) was held in the Shanghai Expo Park. The festival was organized by Shanghai News Group and its associated Shanghai Morning Post, a daily newspaper in Shanghai with 700.000 circulations. During the festival, visitors enjoyed exquisite cuisines and fine wines, and watched celebrity chefs' demonstrations.
As one of the co-sponsors, CMC had its logo on the 30 sun umbrellas which were placed in the national pavilions, including Japan, South Korea, and France etc. CMC tent cards were also put on each table. CMC hired promoters to station at pavilions to encourage visitor to try cranberry juice and scan CMC Wechat account's QR code. Each visitor who did so got a 28-g pack of dried cranberries. Many of them mentioned that the dried cranberries tasted more delicious than those they bought from the stores and wanted to buy the products on-site.
In the afternoon on September 24th, CMC invited a celebrity food blogger, Four Season Barbie, to demonstrate food preparations, after a short speech on US Cranberry by CMC's China representative. Barbie prepared a cranberry salad and shared it with all the audiences. She then showed the audience how to make cranberry green bean cake and invited 8 consumers to make the cake on stage following her recipe and methods.
All the national pavilions had good visitors' traffic, giving the CMC's signage very good exposure. Apart from the activities at the pavilions, dried cranberries and cranberry juice were also served in the VIP room to distinguished guests.
A specially designed poster showing various cranberry applications was placed near the main entrance and CMC's logo was also printed on a sponsors' bulletin board which was very visible to the visitors.
After the three-day event, CMC's Wechat account acquired more than 2000 new followers. Through this event US Cranberry gained exposure to close to 10,000 visitors.


Baking is a past time and hobby loved by many. It can be a happy past time or a lucrative, money making hobby. Baking is also a hobby and past time that can be a bonding activity between parents and kids or between friends. Baking can easily be learned and taught. There are many self help baking books in the bookstores and many step by step articles on the World Wide Web.
Baking books with recipes usually provide a step by step guide on how to do certain things related to baking like making icing, separating the egg white and egg yolk, etc. There are so many different baking ideas to choose from and here is a quick over view of what you can make.
One of the best baking ideas is to make cookies. This is also one of the easiest to do. Just look for a good and easy to do cookie batter recipe and select the flavor of cookies that you would like to do. You can make a big base cookie batter and separate it into different batches and add the flavor you like. You can make chocolate chip, white chocolate with macadamia nuts, peanut butter, m&m's, oatmeal or whatever kind of cookies you like. All you have to do is arrange them on a baking tray and put them in the oven. You can also be creative and make different sizes and shapes of cookies by using cookie cutters. Take note of how you want the cookies to come out: soft and moist or hard and dry.
Another baking idea is to make cakes. You can create the cakes you bring to a birthday party or you can gift friends' with cakes on their special day. You just have to find out what type of cake they like and make sure to follow what they like about the cake: the rich chocolate icing or the not so sweet dark chocolate center. You can scan the web for different cake recipes and for tips on how to make the best cake. You can also put more time into the cake and tap your creative side by decorating the cake.
Another baking idea is to make baked goodies that are low in sugar or that contain sugar substitutes and are low fat too. A handful of people are diabetic and many are watching their diet. You can make sure you include them when making your baked goods by using substitutes or by limiting the amount of sugar used. Just look for items you can substitute with healthier goodies and you're good to go.


This knife was forged from 1095 spring steel and differentially hardened using clay on the spine and quenching in a commercial quenchant. I then polished and etched the steel which makes the hamon very prominent. This 1095 spring steel is very pure with 1.0% carbon and just a touch of manganese to help the hardening process. The edge it takes is very fine. The handle is desert ironwood burl and an aerospace carbon fiber ferrule. The spacers are mammoth tooth and copper. This knife has a convex grind.


text:
\section{Introduction}

Very long baseline interferometry (VLBI) now provides angular resolution of tens of microarcseconds, both at millimeter wavelengths with the Event Horizon Telescope \citep[EHT;][]{Doeleman_2009} and at centimeter wavelengths with {\it RadioAstron} \citep{Kardashev_2013}. As these and future interferometers push to ever higher angular resolution, they must contend with a fundamental limitation: radio-wave scattering in the inhomogeneous, ionized interstellar medium (ISM). It has generally been assumed that the effects of scattering on resolved images of AGN are simply to ``blur'' the images with an approximately Gaussian kernel that grows roughly with the squared observing wavelength. However, within individual observing epochs scattering has the opposite effect, introducing substructure into the scattered image \citep{NarayanGoodman89,GoodmanNarayan89,Johnson_Gwinn_2015}. Unlike the time-averaged blurring of scattering, the substructure becomes stronger at shorter wavelengths and does not correspond to a convolution of the unscattered image. The scattering substructure predominantly affects visibilities on long baselines and can significantly influence high-resolution VLBI data.

Scattering is especially important for studies of the Galactic Center supermassive black hole, Sagittarius~A$^\ast$ (Sgr~A$^{\ast}$), with the EHT and for studies of extremely high brightness temperatures (${\gg}\,10^{12}~{\rm K}$) in active galactic nuclei (AGNs) with {\it RadioAstron}. For Sgr~A$^{\ast}$\ at 1.3~cm wavelength, the effects of scattering substructure have been conclusively detected, apparent as a ${\sim}10$~mJy noise floor on long baselines \citep{Gwinn_2014}. At 3~mm, non-zero closure phases (indicative of structural asymmetry) in Sgr~A$^{\ast}$\  are consistent with being introduced by scattering rather than being intrinsic to the source  \citep{Ortiz_2016}. At 1.3~mm, non-zero closure phases in Sgr~A$^{\ast}$\ have now been measured by the EHT \citep{Fish_2016}; however, these have persistent sign over four years and so they must predominantly reflect intrinsic source structure. Even so, the epoch-to-epoch variations of these closure phases may be dominated by scattering substructure \citep{Broderick_2016}.  For {\it RadioAstron}, scattering can explain apparent brightness temperatures in excess of $10^{14}~{\rm K}$ in 3C\,273 inferred at 18~cm wavelength \citep{Johnson_2016}, and can dominate long-baseline detections between 6 and 92~cm for sources with high brightness temperatures \citep{Johnson_Gwinn_2015}.  For all these reasons, it is essential to have a straightforward and comprehensive framework to understand the effects of scattering on VLBI observations.

Currently, analytic expressions for scattering effects are limited to quantities such as the root-mean-square (RMS) fluctuations in flux density and complex visibility. Consequently, previous efforts have relied on scattering simulations to estimate effects on realistic VLBI observables such as closure phase and closure amplitude. However, these simulations are computationally expensive and make it difficult to analyze scattering effects over a wide range of source and scattering models. Simulations also provide little insight into how to develop scattering mitigation strategies.

Here, we derive analytical tools to estimate scattering effects by extending a simplified scattering formalism that was developed by \citet{Blandford_Narayan_1985} to understand the scattering of pulsars. This formalism separates large scale (``refractive'') effects of scattering from small scale (``diffractive'') effects, greatly simplifying computations. It enables a wide range of calculations and can even be used to estimate the covariance between different scattering effects \citep[see also][]{Romani_1986}. We extend the formalism in two directions: to interferometry, and to sources with arbitrary intrinsic structure.  We show how to obtain exact estimates of refractive scattering effects on VLBI observables using a Monte Carlo method and we also derive an approximate form for the closure phase fluctuations when refractive fluctuations are small.

\clearpage

\section{Theoretical Background}

We first summarize the basic prescription for interstellar scattering of radio waves. For a more detailed discussion and review, see \citet{Rickett_1990} or \citet{Narayan_1992}. When possible, our treatment and notation mirror those of \citet{Blandford_Narayan_1985}.

\subsection{Interstellar Scattering}
\label{sec::ISM}

The local index of refraction $n$ of the ionized interstellar medium (ISM) is determined by its plasma frequency, $\nu_{\rm p} = \sqrt{\frac{n_{\rm e} e^2}{\pi m_{\rm e}}} \approx 9.0 \times \sqrt{\frac{n_{\rm e}}{1~{\rm cm}^{-3}}}\,{\rm kHz}$, where $n_{\rm e}$, $e$, and $m_{\rm e}$ are the local electron number density, electron charge, and electron mass \citep{Jackson_1999}. At frequencies $\nu$ significantly higher than $\nu_{\rm p}$, $n \approx 1 - \frac{1}{2} \left( \frac{\nu_{\rm p}}{\nu} \right)^2$. A density fluctuation $\delta n_{\rm e}$ along a path length $dz$ then introduces a corresponding phase change $\delta \phi = -r_{\rm e} \lambda \times dz \times \delta n_{\rm e}$, where $r_{\rm e} = e^2/(m_{\rm e} c^2) \approx 2.8 \times 10^{-13}~{\rm cm}$ is the classical electron radius and $\lambda$ is the wavelength. Consequently, inhomogeneities in the density of the ionized ISM scatter radio waves. Note that the inhomogeneities have the opposite action of conventional lenses because the phase velocity in a plasma increases with density (and is superluminal).

In many cases, the density fluctuations are well-described as a turbulent cascade, injected at scales ${\mathrel{\raise.3ex\hbox{$>$\kern-.75em\lower1ex\hbox{$\sim$}}}}100~{\rm AU}$ and dissipated at scales ${\mathrel{\raise.3ex\hbox{$<$\kern-.75em\lower1ex\hbox{$\sim$}}}}1000~{\rm km}$ \citep{Armstrong_1995}.
Moreover, the scattering can often be well-described as being confined to a single thin screen, located a distance $D$ from the observer and $R$ from the source, which only affects the phase of the incident radiation via an additive contribution $\phi(\mathbf{r})$, where $\mathbf{r}$ is a transverse two-dimensional vector on the screen. This screen is typically assumed to be ``frozen,'' so that the scattering evolution is deterministic, depending only on the relative transverse motions of the observer, screen, and source with a characteristic velocity $\mathbf{V}_{\perp}$ \citep{Taylor_1938}.

Geometrical effects of the scattering are then described via the Fresnel scale, $r_{\rm F} \equiv \sqrt{ \frac{ D R }{D + R} \lambdabar }$ (where $\lambdabar \equiv \frac{\lambda}{2\pi}$), while the screen phase statistics are described via the phase decorrelation length, $r_0$, which is the transverse scale on the scattering screen over which the RMS phase difference is 1 radian. Another relevant quantity is the magnification parameter $M=D/R$ of the scattering screen. In the strong-scattering regime, defined by $r_0 \ll r_{\rm F}$, the refractive scale $r_{\rm R} \equiv r_{\rm F}^2/r_0$ is also important, roughly defining the extent of the scattered image of a point source. The present paper is primarily focused on the strong-scattering regime, although our results are also applicable in the weak-scattering regime if the angular size of the unscattered source is larger than $r_{\rm F}/D$ (see \S\ref{sec::RefractiveSteering}).

In the strong scattering limit, scattering effects are dominated by phase fluctuations on two widely separated scales. ``Diffractive'' scattering arises from small-scale fluctuations, dominated on scales of ${\sim}r_0$. Diffractive effects decorrelate over a small fractional bandwidth, $(r_0/r_{\rm F})^2$, and have a coherence timescale of only ${\sim}r_0/V_{\perp}$ (typically seconds to minutes). Diffractive effects are quenched for a source that has an angular size larger than $r_0/D$, so they are typically only seen in pulsars. In contrast, ``refractive'' scattering arises from large-scale fluctuations, dominated on scales of ${\sim}r_{\rm R}$. Refractive effects have a decorrelation bandwidth of order unity and a coherence timescale of ${\sim}r_{\rm R}/V_{\perp}$ (typically days to months). Refractive effects are only quenched for a source that has an angular size larger than $r_{\rm R}/D$, so they are present in compact AGNs \citep{RCB_1984}.

\citet{NarayanGoodman89} and \citet{GoodmanNarayan89} showed that there are three distinct averaging regimes for images in the strong-scattering limit: a ``snapshot'' image, an ``average'' image, and an ``ensemble-average'' image. The snapshot image occurs for averaging timescales less than $r_0/V_{\perp}$ and exhibits both diffractive and refractive scintillation. The average image occurs for longer averaging timescales that are still shorter than $r_{\rm R}/V_{\perp}$, and this regime only exhibits refractive scintillation. The ensemble-average image reflects a complete average over a scattering ensemble (or, equivalently, an infinite average over time), and the scattering effects are a deterministic image blurring via convolution of the unscattered image with a scattering kernel. Because AGNs almost always quench diffractive scintillation, only the average and ensemble-average regimes are of significance in most cases. Throughout this paper, we will use subscripts ``ss'', ``a'', and ``ea'' to denote quantities in the snapshot, average, and ensemble-average regimes, respectively.

\subsection{Statistics of the Screen Phase}
\label{sec::ScreenPhaseStatistics}

We now introduce some properties, notation, and results related to the screen phase. We assume that the function $\phi(\mathbf{r})$ defines a Gaussian random field that is statistically homogeneous but not necessarily isotropic. This field is most intuitively characterized by the phase structure function:
\begin{align}
D_{\phi}(\mathbf{r}) \equiv \left \langle \left[ \phi(\mathbf{r}' + \mathbf{r}) - \phi(\mathbf{r}')  \right]^2 \right \rangle.
\end{align}
Here and throughout the paper, $\langle \dots \rangle$ denotes an ensemble average over realizations of the screen phase. In practice, the ensemble-average can be approximated by averaging over time. The phase varies smoothly up to some ``inner scale,'' which is physically associated with the scale on which turbulence is dissipated. Consequently, $D_{\phi}(\mathbf{r}) \propto |\mathbf{r}|^2$ up to this scale, though it need not be isotropic \citep{Tatarskii_1971}. The phase structure function is then described by a power law in an inertial range until saturating at some ``outer scale,'' which is physically associated with the scale on which the turbulence is injected.

Another important representation is the two-point correlation function of phase:
\begin{align}
C(\mathbf{r}) \equiv \left \langle \phi(\mathbf{r}' + \mathbf{r}) \phi(\mathbf{r}') \right \rangle.
\end{align}
We can also define the power spectrum of the phase fluctuations:
\begin{align}
\label{eq::Q_eq}
Q(\mathbf{q}) &= \frac{1}{\lambdabar^2} \int d^2\mathbf{r}\, C(\mathbf{r}) e^{-i \mathbf{q} \cdot \mathbf{r}}.
\end{align}
Note that $Q(\mathbf{q})$ is dimensionless and is independent of wavelength, and $\lambdabar^2 |\mathbf{q}|^2 Q(\mathbf{q})$ gives the mean-squared phase fluctuations on a spatial scale $2\pi/|\mathbf{q}|$. Because $D_{\phi}({\bf r}) = 2\left[ C({\bf 0}) - C({\bf r}) \right]$, we can also express Eq.~\ref{eq::Q_eq} as $Q( \mathbf{q} ) \equiv -\frac{1}{2\lambdabar^2}\tilde{D}_{\phi}(\mathbf{q})$. In this expression and throughout this paper, we use a tilde to denote a Fourier transform and adopt the convention that
\begin{align}
\tilde{f}(\mathbf{q},\lambdabar) &= \int d^2\mathbf{r}\, f(\mathbf{r},\lambdabar) e^{-i \mathbf{q} \cdot \mathbf{r}}.
\end{align}
For an isotropic power-law structure function, $D_{\phi}(\mathbf{r}) = \left| \mathbf{r}/r_0 \right|^\alpha$, we obtain\footnote{For 3D Kolmogorov turbulence, $\alpha=5/3$. The present paper focuses on ``shallow'' spectra: $0 < \alpha < 2$.}
\begin{align}
Q(\mathbf{q}) &= 2^{\alpha} \pi \alpha \frac{\Gamma\left(1+\alpha/2\right)}{\Gamma\left(1-\alpha/2\right)} \lambdabar^{-2} r_0^{-\alpha} \left| \mathbf{q} \right|^{-(2+\alpha)}.
\end{align}
These expressions can easily be generalized to include inner and outer scales or anisotropic scattering. One convenient form is,
\begin{align}
Q(\mathbf{q}) &= 2^{\alpha} \pi \alpha \frac{\Gamma\left(1+\alpha/2\right)}{\Gamma\left(1-\alpha/2\right)} \lambdabar^{-2} \left( r_{0,x} r_{0,y} \right)^{-\alpha/2} \left[ \left(\frac{r_{0,x}}{r_{0,y}}\right) q_{x}^2 + \left(\frac{r_{0,y}}{r_{0,x}}\right) q_{y}^2 + q_{\rm min}^2 \right]^{-(1 + \alpha/2)} e^{-(q/q_{\rm max})^2},
\end{align}
where the phase decoherence scales can now differ in two orthogonal directions, $r_{0,x}$ and $r_{0,y}$, so that the scattering has an axial ratio of $r_{0,x}/r_{0,y}$ (and the smaller of the two determines the major axis of the scattering disk).

The above results allow us to derive an important computational tool that we will employ throughout the remainder of this paper. Namely, given two complex functions $f_i(\mathbf{r}_i,\lambda_i)$,
\begin{align}
\label{eq::BN_Reduction}
\nonumber \left \langle \int d^2\mathbf{r}_{1} d^2\mathbf{r}_{2}\, \phi(\mathbf{r}_1,\lambda_1) \phi(\mathbf{r}_2,\lambda_2) f_1(\mathbf{r}_1,\lambda_1) f_2(\mathbf{r}_2,\lambda_2) \right \rangle
&= \int d^2\mathbf{r}_{1} d^2\mathbf{r}_{2}\, \frac{\lambda_2}{\lambda_1}  \left \langle \phi(\mathbf{r}_1,\lambda_1) \phi(\mathbf{r}_2,\lambda_1) \right \rangle f_1(\mathbf{r}_1,\lambda_1) f_2(\mathbf{r}_2,\lambda_2)\\
\nonumber &= \frac{\lambda_2}{\lambda_1} \int d^2\mathbf{r}_{2} \frac{1}{(2\pi)^2} \left[ \int d^2\mathbf{q}\, \lambdabar_1^2 Q(\mathbf{q}) e^{-i \mathbf{q} \cdot \mathbf{r}_2} \tilde{f}_1(-\mathbf{q},\lambda_1) \right] f_2(\mathbf{r}_2,\lambda_2)\\
&= \frac{\lambda_1 \lambda_2}{(2\pi)^4} \int d^2 \mathbf{q}\, Q(\mathbf{q}) \tilde{f}_1(-\mathbf{q},\lambda_1) \tilde{f}_2(\mathbf{q},\lambda_2).
\end{align}
The second equality follows from the general identify $\int d^2\mathbf{r}\, g_1(\mathbf{r}) g_2(\mathbf{r}) = \frac{1}{(2\pi)^2} \int d^2\mathbf{q}\, \tilde{g}_1(\mathbf{q}) \tilde{g}_2(-\mathbf{q})$.
Eq.~\ref{eq::BN_Reduction} was given by \citet{Blandford_Narayan_1985} (their Eq.~4.4) for the special case that the functions $f_i$ are real. Eq.~\ref{eq::BN_Reduction} can also easily be extended to the useful cases where one or both of the functions $f_i$ are conjugated:
\begin{align}
\label{eq::BN_Reduction_Conj}
\left \langle \int d^2\mathbf{r}_{1} d^2\mathbf{r}_{2}\, \phi(\mathbf{r}_1,\lambda_1) \phi(\mathbf{r}_2,\lambda_2) f_1(\mathbf{r}_1,\lambda_1) f_2^\ast(\mathbf{r}_2,\lambda_2) \right \rangle &= \frac{\lambda_1 \lambda_2}{(2\pi)^4} \int d^2\mathbf{q}\, Q(\mathbf{q}) \tilde{f}_1(\mathbf{q},\lambda_1) \tilde{f}_2^\ast(\mathbf{q},\lambda_2),\\
\nonumber \left \langle \int d^2\mathbf{r}_{1} d^2\mathbf{r}_{2}\, \phi(\mathbf{r}_1,\lambda_1) \phi(\mathbf{r}_2,\lambda_2) f_1^\ast(\mathbf{r}_1,\lambda_1) f_2^\ast(\mathbf{r}_2,\lambda_2) \right \rangle &= \frac{\lambda_1 \lambda_2}{(2\pi)^4} \int d^2\mathbf{q}\, Q(\mathbf{q}) \tilde{f}_1^\ast(\mathbf{q},\lambda_1) \tilde{f}_2^\ast(-\mathbf{q},\lambda_2).
\end{align}

\subsection{Separation of Diffractive and Refractive Scattering}
\label{sec::RefractiveSteering}

The framework of \citet{Blandford_Narayan_1985} greatly simplifies the scattering by separating diffractive and refractive effects. We simply sketch the key ideas here but provide formal justification for this separation of scales in Appendix~\ref{sec::ScaleSeparation}.

Diffractive effects reflect stochastic fluctuations on small scales, so their influence on an image can be approximated through an ensemble average. The ensemble average effect is a convolution of the intensity distribution of the unscattered image, $I_{\rm src}(\mathbf{r})$, with a deterministic kernel, resulting in the ensemble-average image $I_{\rm ea}(\mathbf{r})$, which is a ``blurred'' version of the unscattered image \citep[for details and discussion, see][]{Fish_2014}. We again emphasize that $\mathbf{r}$ is a transverse coordinate at the distance $D$ of the scattering screen, so the corresponding angular coordinates for the images are $\boldsymbol{\theta} = \mathbf{r}/D$.

Refractive effects arise from fluctuations on much larger scales, and their stochastic effects persist in the average-image regime. Gradients in the refractive component of the phase screen, $\phi_{\rm r}(\mathbf{r})$, steer and distort brightness elements of the unscattered image while preserving the surface brightness \citep{Born_Wolf}. The Fresnel scale, $r_{\rm F}$, determines how these phase gradients affect the average image, such that the observed brightness received at some location $\mathbf{x}=0$ in the observing plane can be expressed as
\begin{align}
\label{eq::RefractiveSteering_Prelim}
I_{\rm a}(\mathbf{r}) &\approx I_{\rm ea}(\mathbf{r} + r_{\rm F}^2 \nabla \phi_{\rm r}(\mathbf{r})).
\end{align}
Here, $\nabla$ denotes a transverse (two-dimensional) spatial gradient operator. On diffractive scales, the phase gradient is $\left| \nabla \phi(\mathbf{r})) \right| \sim 1/r_0$, whereas the gradient from phase fluctuations on a larger, refractive scale $L$ is $\left| \nabla \phi(\mathbf{r})) \right| \sim \sqrt{D_{\phi}(L)}/L \sim L^{\alpha/2-1}/r_0^{\alpha/2}$. Consequently, refractive steering angles are smaller than diffractive steering angles by a factor of ${\sim}(r_0/r_{\rm R})^{1-\alpha/2} = (r_0/r_{\rm F})^{2-\alpha}$, and this factor roughly determines the relative strength of refractive and diffractive effects for metrics such as flux modulation \citep{Narayan_1992}. Moreover, because the refractive steering is smaller than the diffractive blurring, we can approximate the average image by expanding to leading order:
\begin{align}
\label{eq::RefractiveSteering}
I_{\rm a}(\mathbf{r}) &\approx I_{\rm ea}(\mathbf{r}) + r_{\rm F}^2 \left[\nabla \phi_{\rm r}(\mathbf{r})\right] \cdot \left[ \nabla I_{\rm ea}(\mathbf{r}) \right].
\end{align}
Note that this approximation is valid even in the weak-scattering regime ($r_0 > r_{\rm F}$) if the intrinsic source does not have significant spatial variations in its structure on scales finer than the typical scattering angle (i.e., if the interferometric visibility of the unscattered source is negligible on baselines that resolve the typical scattering angle). For this reason, there is a unified description of refractive scattering effects in the strong- and weak-scattering regimes for an extended source \citep[see, e.g.,][]{Coles_1987}.

We are interested in describing VLBI observations in the average-image regime, for which diffractive scintillation is replaced by its ensemble-average effects. Moreover, because we are interested in VLBI observations that resolve intrinsic source structure, we will assume that the source is sufficiently extended to quench diffractive scintillation, so no additional averaging in time is required (in \S\ref{sec::Interferometry}, we will show how an extended source acts to filter contributions from wavenumbers $\mathbf{q}$ on diffractive scales). For these reasons, we can substitute $\phi(\mathbf{r})$ for $\phi_{\rm r}(\mathbf{r})$ in Eq.~\ref{eq::RefractiveSteering}, and we will use this replacement throughout the remainder of the paper for simplicity.

\subsection{Interferometry of Scattered Sources}
\label{sec::Interferometry}

Using the approximate form for the scattered image given in Eq.~\ref{eq::RefractiveSteering} and the statistical properties of the screen phase $\phi$ derived in \S\ref{sec::ScreenPhaseStatistics}, it is straightforward to estimate how refractive scattering affects interferometric visibilities. Recall that the average image in Eq.~\ref{eq::RefractiveSteering} is defined at the location of the scattering screen, located a distance $D$ from the observer. The van~Cittert-Zernike Theorem\footnote{The van~Cittert-Zernike Theorem only strictly applies when a source is spatially incoherent. Scattering introduces spatial coherence, but because the correlation length is $r_0$, the source appears spatially incoherent to interferometric baselines $|\mathbf{b}| \ll \lambda (r_0/D)^{-1} \sim r_{\rm R}$, as holds for all cases of immediate interest (see Appendix~\ref{sec::ScaleSeparation} for details).} then gives the interferometric visibility on a baseline $\mathbf{b}$ \citep{TMS}:
\begin{align}
\label{eq::Vavg}
\nonumber V_{\rm a}(\mathbf{b}) &= \int d^2\mathbf{r}\, I_{\rm a}(\mathbf{r}) e^{-i \mathbf{r} \cdot \mathbf{b}/(D \lambdabar)}\\
\nonumber &\approx V_{\rm ea}(\mathbf{b}) + r_{\rm F}^2 \int d^2\mathbf{r}\,  e^{-i \mathbf{r} \cdot \mathbf{b}/(D \lambdabar)}\nabla \phi(\mathbf{r}) \cdot \nabla I_{\rm ea}(\mathbf{r})\\
\nonumber &= V_{\rm ea}(\mathbf{b}) + r_{\rm F}^2 \int d^2\mathbf{r}\,  e^{-i \mathbf{r} \cdot \mathbf{b}/(D \lambdabar)} \left[ \frac{i}{D\lambdabar} \mathbf{b} \cdot \nabla I_{\rm ea}(\mathbf{r}) - \nabla^2 I_{\rm ea}(\mathbf{r})  \right] \phi(\mathbf{r})\\
&\equiv V_{\rm ea}(\mathbf{b}) + \int d^2\mathbf{r}\,  f_{\rm V}(\mathbf{r};\mathbf{b},\lambda) \phi(\mathbf{r}).
\end{align}
The third line was obtained by integrating by parts (we assume that $I_{\rm ea}(\mathbf{r})$ is restricted to a finite domain, so the boundary term vanishes), and we have defined
\begin{align}
\label{eq::fV_def}
f_{\rm V}(\mathbf{r};\mathbf{b},\lambda) &\equiv r_{\rm F}^2 e^{-i \mathbf{r} \cdot \mathbf{b}/(D \lambdabar)} \left[ \frac{i}{D\lambdabar} \mathbf{b} \cdot \nabla I_{\rm ea}(\mathbf{r}) - \nabla^2 I_{\rm ea}(\mathbf{r}) \right].
\end{align}
Note that $f_{\rm V}$ depends on the ensemble-average image. This dependence is important when comparing the refractive noise at different frequencies (with different intrinsic structure), when studying the refractive noise for an image that varies more rapidly than the scattering (as may be the case for Sgr~A$^{\ast}$), or when comparing the refractive noise for images corresponding to different Stokes parameters. When convenient, we will write quantities in terms of the ensemble-average visibility, $V_{\rm ea}(\mathbf{b})$, which is easily related to the interferometric visibility of the unscattered source, $V_{\rm src}(\mathbf{b})$, through the convolution action of the scattering in this regime \citep[e.g.,][]{Coles_1987}:
\begin{align}
\label{eq::Vea}
V_{\rm ea}(\mathbf{b}) = V_{\rm src}(\mathbf{b}) e^{-\frac{1}{2} D_{\phi}\left( \frac{\mathbf{b}}{1+M} \right)}.
\end{align}

From Eq.~\ref{eq::Vavg}, it is apparent that refractive noise on interferometric visibilities is Gaussian because it is simply a weighted sum of (correlated) Gaussian random variables, $\phi(\mathbf{r})$. Because the refractive noise is Gaussian, it is completely characterized by its spatial covariance.  However, the real and imaginary parts of the Gaussian noise may have different standard deviations and different spatial covariance structure; for example, on the baseline $\mathbf{b}=\mathbf{0}$, the refractive noise is purely real since the zero-baseline visibility is real (and equal to the total flux density of the source). On long baselines (i.e., those baselines that completely resolve the ensemble-average image) the refractive noise is drawn from a circular complex Gaussian distribution.

To calculate the full spatial covariance of refractive noise, we must therefore derive expressions for the real and imaginary parts of the noise separately. These are simplified by the observation that $f_{\rm V}$ obeys the necessary conjugation symmetry for interferometric visibilities: $f_{\rm V}^\ast(\mathbf{r};\mathbf{b},\lambda) = f_{\rm V}(\mathbf{r};-\mathbf{b},\lambda)$ (see Eq.~\ref{eq::Vavg}). Consequently,
\begin{align}
\label{eq::fVreim}
f_{\rm V,re}(\mathbf{r};\mathbf{b},\lambda) &\equiv \mathrm{Re}\left[ f_{\rm V}(\mathbf{r};\mathbf{b},\lambda) \right]\\
\nonumber &= \frac{1}{2} \left[ f_{\rm V}(\mathbf{r};\mathbf{b},\lambda) + f_{\rm V}(\mathbf{r};-\mathbf{b},\lambda) \right] \\
\nonumber &= r_{\rm F}^2 \left\{ -\cos\left[ \mathbf{r} \cdot \mathbf{b}/(D \lambdabar) \right] \nabla^2 I_{\rm ea}(\mathbf{r}) + \sin\left[ \mathbf{r} \cdot \mathbf{b}/(D \lambdabar) \right] \frac{1}{D\lambdabar} \mathbf{b} \cdot \nabla I_{\rm ea}(\mathbf{r}) \right\}, \\
\nonumber f_{\rm V,im}(\mathbf{r};\mathbf{b},\lambda) &\equiv \mathrm{Im}\left[ f_{\rm V}(\mathbf{r};\mathbf{b},\lambda) \right]\\
\nonumber &= \frac{1}{2i} \left[ f_{\rm V}(\mathbf{r};\mathbf{b},\lambda) - f_{\rm V}(\mathbf{r};-\mathbf{b},\lambda) \right] \\
\nonumber &= r_{\rm F}^2 \left\{+\sin\left[ \mathbf{r} \cdot \mathbf{b}/(D \lambdabar) \right] \nabla^2 I_{\rm ea}(\mathbf{r}) + \cos\left[ \mathbf{r} \cdot \mathbf{b}/(D \lambdabar) \right] \frac{1}{D\lambdabar} \mathbf{b} \cdot \nabla I_{\rm ea}(\mathbf{r}) \right\}.
\end{align}

The corresponding Fourier conjugate quantities are then
\begin{align}
\label{eq::fVtilde}
\tilde{f}_{\rm V}(\mathbf{q};\mathbf{b},\lambda) &= \int d^2\mathbf{r}\, f_{\rm V}(\mathbf{r};\mathbf{b},\lambda) e^{-i \mathbf{q}\cdot \mathbf{r}}\\
\nonumber &= r_{\rm F}^2 \int d^2\mathbf{r}\, e^{-i \mathbf{r} \cdot \left[\mathbf{q} + \mathbf{\mathbf{b}}/(D\lambdabar) \right]} \left[ \frac{i}{D\lambdabar} \mathbf{b} \cdot \nabla I_{\rm ea}(\mathbf{r}) - \nabla^2 I_{\rm ea}(\mathbf{r})  \right]\\
\nonumber &= r_{\rm F}^2 \left[ \mathbf{q} + \mathbf{b}/(D\lambdabar) \right] \cdot \left\{ \left[ \mathbf{q} + \mathbf{b}/(D\lambdabar) \right] - \frac{\mathbf{b}}{D \lambdabar} \right\} V_{\rm ea}\left( D\lambdabar\mathbf{q}  + \mathbf{b} \right)\\
\nonumber &= r_{\rm F}^2 \mathbf{q} \cdot \left[ \mathbf{q} + (1+M)^{-1} r_{\rm F}^{-2} \mathbf{b} \right] V_{\rm ea}\left( (1+M)r_{\rm F}^2 \mathbf{q}  + \mathbf{b} \right),\\
\nonumber \tilde{f}_{\rm V,re}(\mathbf{q};\mathbf{b},\lambda) &= \frac{1}{2} \left[ \tilde{f}_{\rm V}(\mathbf{q};\mathbf{b},\lambda) + \tilde{f}_{\rm V}(\mathbf{q};-\mathbf{b},\lambda) \right],\\
\nonumber \tilde{f}_{\rm V,im}(\mathbf{q};\mathbf{b},\lambda) &= \frac{1}{2i} \left[ \tilde{f}_{\rm V}(\mathbf{q};\mathbf{b},\lambda) - \tilde{f}_{\rm V}(\mathbf{q};-\mathbf{b},\lambda) \right],
\end{align}
where we have again integrated by parts, dropping the boundary term because we expect a source image with finite domain, and we have made the substitution $D \lambdabar = (1 + M) r_{\rm F}^2$.

When coupled with Eq.~\ref{eq::BN_Reduction}, Eq.~\ref{eq::fVtilde} shows how the source, scattering, and observing baseline all act to filter different wavenumbers $\mathbf{q}$ of the screen phase. For instance, on long baselines the ensemble-average visibility filters wavenumbers that are significantly different than $-(1+M)^{-1} r_{\rm F}^{-2}\mathbf{b} = -\mathbf{b}/(\lambdabar D)$. Thus, the contributing wavenumbers are those with a corresponding angular scale, $\left(2\pi/\mathbf{q}\right)/D$, that is matched to the vector baseline resolution, $\lambda/\mathbf{b}$. On short baselines, the ensemble-average visibility filters wavenumbers with $|\mathbf{q}| \mathrel{\raise.3ex\hbox{$>$\kern-.75em\lower1ex\hbox{$\sim$}}} r_0/r_{\rm F}^2 = r_{\rm R}^{-1}$. In both cases, this term filters the wavenumbers that produce diffractive scintillation because they have $|\mathbf{q}| \sim r_0^{-1} \gg r_{\rm R}^{-1}$, and the ensemble-average visibility falls to zero for baselines $|\mathbf{b}| \gg r_0$ (see Eq.~\ref{eq::Vea}). The term $r_{\rm F}^2 \mathbf{q} \cdot \left[ \mathbf{q} + (1+M)^{-1} r_{\rm F}^{-2} \mathbf{b} \right]$ filters power at low wavenumbers because of wrapping of the phase from geometrical path length. The characteristic cutoff for this term is $|\mathbf{q}| \mathrel{\raise.3ex\hbox{$<$\kern-.75em\lower1ex\hbox{$\sim$}}} 1/r_{\rm F}$, so this term is called the Fresnel filter \citep{Cronyn_1972}. The Fresnel filter regulates power on refractive scales, so the strength of refractive scintillation falls as the scales $r_0$, $r_{\rm F}$, and $r_{\rm R}$ become more widely separated.

\section{Signatures of Refractive Noise}

From the results derived in \S\ref{sec::Interferometry}, it is straightforward to compute the effects of refractive scattering on interferometric observables. We begin by calculating the RMS fluctuations of interferometric visibilities, showing that our formalism exactly reproduces the results of \citet{GoodmanNarayan89} and \citet{Johnson_Gwinn_2015} but requires significantly less effort than in their approach. We then show how to calculate arbitrary observables to any desired accuracy using a Monte Carlo framework, and we derive approximations for the refractive fluctuations in closure phase. We conclude this section by discussing the coherence timescale of refractive noise.

\subsection{Refractive Fluctuations of the Complex Interferometric Visibility}
\label{sec::sigma_ref}

To make contact with previous results and illustrate our computational methodology, we will first estimate the RMS fluctuation of the complex visibility introduced by refractive substructure on a fixed baseline $\mathbf{b}$. Combining Eqs.~\ref{eq::BN_Reduction_Conj}, \ref{eq::Vavg}, and \ref{eq::fVtilde}, we obtain
\begin{align}
\label{eq::sigmaref_Q}
\sigma_{\rm ref}^2(\mathbf{b}) \equiv \left \langle \left| \Delta V_{\rm a}(\mathbf{b}) \right|^2 \right \rangle &= \frac{\lambda^2}{(2\pi)^4} \int d^2\mathbf{q}\, \left| \tilde{f}_{\rm V}(\mathbf{q};\mathbf{b},\lambda)  \right|^2 Q(\mathbf{q})\\
\nonumber &= \frac{\lambda^2 r_{\rm F}^4}{(2\pi)^4} \int d^2\mathbf{q}\,  \left( \mathbf{q} \cdot \left[ \mathbf{q} + \mathbf{b}/(D\lambdabar) \right] \right)^2 \left|V_{\rm ea}\left( D\lambdabar\mathbf{q}  + \mathbf{b} \right) \right|^2 Q(\mathbf{q}),
\end{align}
where $\Delta V_{\rm a}(\mathbf{b}) \equiv V_{\rm a}(\mathbf{b}) - \langle V_{\rm a}(\mathbf{b}) \rangle = V_{\rm a}(\mathbf{b}) - V_{\rm ea}(\mathbf{b})$.
Performing the substitutions $Q(\mathbf{q}) = -\frac{1}{2\lambdabar^2} \tilde{D}_\phi(\mathbf{q})$, $\mathbf{y} = \left( \mathbf{q} + \mathbf{b}/(D\lambdabar) \right) r_{\rm F}^2$, and $\lambdabar D = (1 + M) r_{\rm F}^2$ then gives
\begin{align}
\label{eq::sigmaref}
\sigma_{\rm ref}^2(\mathbf{b}) = -\frac{1}{8 \pi^2 r_{\rm F}^8} \int d^2\mathbf{y}\, \left( \mathbf{y} \cdot \left[ \mathbf{y} - (1+M)^{-1} \mathbf{b}  \right] \right)^2 \left|V_{\rm ea}\left( (1+M)\mathbf{y} \right) \right|^2 \tilde{D}_{\phi}\left(r_{\rm F}^{-2} \left[\mathbf{y} - (1+M)^{-1}\mathbf{b}\right] \right).
\end{align}
Apart from an opposite convention for the sign of $\mathbf{b}$ and choice of units for the argument of $\tilde{D}_{\phi}$, Eq.~\ref{eq::sigmaref} exactly matches Eq.~32 of \citet{Johnson_Gwinn_2015}, which was derived using a more general formulation \citep[based on][]{GoodmanNarayan89}. This agreement strongly reinforces the utility of the present approach and demonstrates its capability for quickly estimating results that formerly required arduous derivations. \citet{Cronyn_1972} considered visibility scintillation in the weak-scattering regime and derived a result that is similar to Eq.~\ref{eq::sigmaref_Q} \citep[see also][]{Vedantham_2015}.

Although Eq.~\ref{eq::sigmaref} requires numerical integration in the most general case, it can be evaluated in closed form for a Gaussian source in the limit of long baseline (specifically, a baseline that resolves the ensemble-average image). For isotropic scattering and a circular Gaussian intrinsic source with FWHM $\theta_{\rm src}$, we obtain
\begin{align}
\label{eq::sigmaref_approx}
\sigma_{\rm ref}(\mathbf{b}) &\approx \sqrt{\frac{\Gamma(4/\alpha)}{2^{2-\alpha}}\frac{\Gamma\left(1+\frac{\alpha}{2}\right)}{\Gamma\left(1-\frac{\alpha}{2}\right)}} \left( \frac{r_0}{r_{\rm F}} \right)^{2-\alpha} \left(\frac{\left| \mathbf{b} \right|}{\left(1+M\right)r_0}\right)^{-\frac{\alpha}{2}} \left( \frac{\theta_{\rm scatt}}{\theta_{\rm img}}\right)^2,
\end{align}
where $\theta_{\rm scatt} \approx \frac{\sqrt{2\ln{2}}}{\pi}\frac{\lambda}{(1+D/R)r_0}$ is the scattered angular size of a point source and $\theta_{\rm img} \approx \sqrt{\theta_{\rm src}^2 + \theta_{\rm scatt}^2}$ is the ensemble-average angular size. In the limit of a small intrinsic source, $\theta_{\rm img} \rightarrow \theta_{\rm scatt}$, Eq.~\ref{eq::sigmaref_approx} reproduces the result of \citet{GoodmanNarayan89} (their Eq.~5.1.2).

\begin{figure}[t]
\centering
\includegraphics[width=\textwidth]{f1.pdf}
\caption
{
Examples showing the correlation $\left \langle \Delta V_{\rm a}\left(\mathbf{b}_1\right) \Delta V_{\rm a}^\ast\left(\mathbf{b}_2\right) \right \rangle/\left[\sigma_{\rm ref}(\mathbf{b}_1) \sigma_{\rm ref}(\mathbf{b}_2)\right]$ between refractive noise on a fixed interferometric baseline $\mathbf{b}_1$ (indicated by a cross) and refractive noise on another arbitrary baseline $\mathbf{b}_2$. The source (FWHM: $130~\mu{\rm as}$) and scattering (FWHM: $160~\mu{\rm as}$) are both taken to be isotropic and correspond roughly to the major axis of Sgr~A$^{\ast}$\ at 3.5~mm wavelength \citep[see][]{Ortiz_2016}. Three choices of $\mathbf{b}_1$ are shown: a zero-baseline (measuring the total flux density), a short baseline (which partially resolves the source), and a long baseline (which entirely resolves the ensemble-average image). As expected, the refractive noise is correlated with a scale of approximately $r_{\rm img}$, which is the baseline length at which the ensemble-average visibility amplitude falls to $1/\sqrt{e}$ of its zero-baseline value. Note that there is also a region of anti-correlation before the noise becomes uncorrelated on baselines that are significantly distant from $\mathbf{b}_1$. Also, note that the correlation in this case is real because the unscattered image is point symmetric.
}
\label{fig::RefractiveNoise_Correlation}
\end{figure}

We can similarly calculate the covariance of the visibility noise among different baselines, the covariance of the real or imaginary parts of visibility, or the covariance for multiple wavelengths or different underlying images (e.g., among different polarizations). Importantly, none of these was straightforward to estimate in the formulation of \citet{GoodmanNarayan89} or \citet{Johnson_Gwinn_2015}. For example,
\begin{align}
\label{eq::ComplexCovariance}
\left \langle \Delta V_{\rm a}\left(\mathbf{b}_1\right) \Delta V_{\rm a}^\ast\left(\mathbf{b}_2\right) \right \rangle &= \frac{\lambda^2}{(2\pi)^4} \int d^2\mathbf{q}\, \tilde{f}_{\rm V}(\mathbf{q};\mathbf{b}_1,\lambda) \tilde{f}_{\rm V}^\ast(\mathbf{q};\mathbf{b}_2,\lambda) Q(\mathbf{q})\\
\nonumber &= \frac{\lambda^2 r_{\rm F}^4}{(2\pi)^4} \int d^2\mathbf{q}\, \left(\mathbf{q} \cdot \left[ \mathbf{q} + (1+M)^{-1} r_{\rm F}^{-2} \mathbf{b}_1 \right]\right) \left(\mathbf{q} \cdot \left[ \mathbf{q} + (1+M)^{-1} r_{\rm F}^{-2} \mathbf{b}_2 \right]\right)\\
\nonumber &\qquad \qquad \qquad \times V_{\rm ea}\left( (1+M)r_{\rm F}^2 \mathbf{q}  + \mathbf{b}_1 \right) V_{\rm ea}^\ast\left( (1+M)r_{\rm F}^2 \mathbf{q}  + \mathbf{b}_2 \right) Q(\mathbf{q}).
\end{align}
Although this covariance is complex, it is real when $V_{\rm ea}(\mathbf{b}) \in \mathbb{R}$ for all baselines, a condition that is met in the special case of a point-symmetric source. Baselines that are separated by a distance less than ${\sim}r_{\rm img}$ will have correlated refractive fluctuations, where $r_{\rm img}$ is the baseline length needed to resolve the ensemble-average image, as can be derived using more general arguments \citep[see \S3.2 of][]{Johnson_Gwinn_2015}. Note that $r_{\rm img}$ depends on the baseline orientation if the image is anisotropic. Figure~\ref{fig::RefractiveNoise_Correlation} shows the refractive noise correlation structure for isotropic scattering, and Figure~\ref{fig::RefractiveNoise_Correlation_Anisotropic} shows how anisotropic scattering affects the correlation structure.

\begin{figure}[t]
\centering
\includegraphics[width=\textwidth]{f2.pdf}\vspace{-0.3cm}
\caption
{
Examples showing the correlation between refractive noise on a fixed interferometric baseline $\mathbf{b}_1$ (indicated by a cross) and another arbitrary baseline $\mathbf{b}_2$, as in Figure~\ref{fig::RefractiveNoise_Correlation} but here with anisotropic scattering. The source (FWHM: $130~\mu{\rm as}$) and scattering (FWHM: $160~\mu{\rm as} \times 78~\mu{\rm as}$) again correspond roughly to Sgr~A$^{\ast}$\ at 3.5~mm wavelength, but we have set the major axis of the scattering to be East-West for clarity. Three choices of $\mathbf{b}_1$ are shown: a zero-baseline (left) measuring the total flux density, and two long baselines located along the minor (center) and major (right) axes of the scattering disk. As in Figure~\ref{fig::RefractiveNoise_Correlation}, the refractive noise is correlated with a scale of approximately $r_{\rm img}$, which is now anisotropic.
}
\label{fig::RefractiveNoise_Correlation_Anisotropic}
\end{figure}

\subsection{A General Monte Carlo Procedure to Estimate Refractive Effects on Interferometric Observables}
\label{sec::MC}

We now describe a general approach to estimate refractive effects on arbitrary interferometric observables. Although this approach does not produce closed-form expressions, it can easily be computed to any desired accuracy and is not limited to cases in which the refractive noise on a given measurable is small.

To begin, recall that the refractive fluctuations of a complex visibility about its ensemble-average value correspond to a zero-mean Gaussian random process (see \S\ref{sec::Interferometry}). Consequently, the statistical properties of refractive noise on a set of visibilities (possibly with different sampled times and even different underlying images) are entirely described by the corresponding set of ensemble-average visibilities on those baselines $\{ V_{{\rm ea},i} \}$ and the covariance matrix $\boldsymbol{\Sigma}$ for the set of refractive fluctuations $\{ {\rm Re}\left( \Delta V_{{\rm a},i}\right), {\rm Im}\left( \Delta V_{{\rm a},i}\right) \}$. Explicitly, the probability density function (PDF) of $N$ visibilities is
\begin{align}
\label{eq::PDF}
P(\mathbf{V} \equiv \{ {\rm Re}\left(V_1\right), {\rm Im}\left(V_1\right), \ldots, {\rm Re}\left(V_N\right), {\rm Im}\left(V_N\right) \}) &= \frac{1}{(2\pi)^N \sqrt{|\boldsymbol{\Sigma}|}} \exp\left[ -\frac{1}{2} \left( \mathbf{V} - \mathbf{V}_{\rm ea} \right)^\intercal \boldsymbol{\Sigma}^{-1} \left( \mathbf{V} - \mathbf{V}_{\rm ea} \right) \right].
\end{align}
Each element of the covariance matrix $\boldsymbol{\Sigma}$ must be computed using Eqs.~\ref{eq::BN_Reduction} and \ref{eq::fVtilde}. For example,
\begin{align}
\label{eq::ReRe}
\left \langle {\rm Re}\left[\Delta V_{\rm a}\left(\mathbf{b}_1;\lambda_1 \right)\right] {\rm Re}\left[ \Delta V_{\rm a}\left(\mathbf{b}_2; \lambda_2\right) \right] \right \rangle &= \frac{\lambda_1 \lambda_2}{(2\pi)^4} \int d^2\mathbf{q}\, \tilde{f}_{\rm V,re}(-\mathbf{q};\mathbf{b}_1,\lambda_1) \tilde{f}_{\rm V,re}(\mathbf{q};\mathbf{b}_2,\lambda_2) Q(\mathbf{q}).
\end{align}
This equation is similar to Eq.~\ref{eq::ComplexCovariance}, which instead gave the covariance of the complex visibility.

To evaluate the refractive fluctuations of any quantity derived from a discrete set of visibilities, one can then use a Monte Carlo method by generating complex Gaussian random variables with the appropriate means (the ensemble average) and covariance matrix (the refractive noise).\footnote{Many software packages can generate samples from the multivariate normal distribution for a specified mean and covariance matrix \citep[e.g., {\tt random.multivariate\_normal} in {\tt numpy};][]{SciPy}.} For example, to generate an ensemble of closure phase measurements (see \S\ref{sec::Closure_Phase} below) on a particular triangle requires repeated sampling of $N=3$ complex Gaussian random variables after calculating their $6\times 6$ covariance matrix. To construct an ensemble of closure amplitude realizations on a particular quadrangle requires $N=4$ complex Gaussian random variables and an $8\times 8$ covariance matrix. One could also numerically integrate the desired quantity (e.g., $\mathrm{arg}(V_1 V_2 V_3)$ for closure phase) over the PDF specified by Eq.~\ref{eq::PDF} to estimate properties such as the closure phase variance from scattering. We again emphasize that this approach can be used to calculate the covariance among different images, as is needed to generate an ensemble of measurements for different polarizations, or at different observing wavelengths.

\subsection{Refractive Fluctuations of Closure Phase}
\label{sec::Closure_Phase}
A fundamental interferometric observable is closure phase: the phase of a directed product of three complex visibilities for a closed triangle of baselines (i.e., the phase of the ``bispectrum'') \citep{Rogers_1974,TMS}. Because phase information in VLBI is typically only accessible through closure phase, interferometric imaging algorithms often operate directly on closure phases or even on the bispectrum \citep[see, e.g.,][]{Buscher_1994,Baron_2010,Bouman_2015,Chael_2016}. Consequently, understanding how refractive scattering affects closure phase is critical for VLBI imaging efforts.

To proceed, we will write the bispectrum $\mathcal{B}$ on a fixed baseline triangle $\left\{ \mathbf{b}_1, \mathbf{b}_2, \mathbf{b}_3 \right\}$ in simplified notation:
\begin{align}
\mathcal{B} = \left( V_1 + \Delta V_1 \right) \left( V_2 + \Delta V_2 \right) \left( V_3 + \Delta V_3 \right),
\end{align}
where $V_i$ denotes an ensemble-average visibility and $\Delta V_i$ denotes the refractive noise for a particular realization of the scattering. To estimate the fluctuations of the phase of the bispectrum, we define a normalized bispectrum:
\begin{align}
\hat{\mathcal{B}} \equiv \mathcal{B}/\mathcal{B}_{\rm ea} =  \left( 1 + \frac{\Delta V_1}{V_1} \right) \left( 1 + \frac{\Delta V_2}{V_2} \right) \left( 1 + \frac{\Delta V_3}{V_3} \right),
\end{align}
where $\mathcal{B}_{\rm ea} \equiv V_1 V_2 V_3$ is the bispectrum of the ensemble-average image.
When $|\Delta V_i/V_i| \ll 1$ for each $i$, the RMS closure phase fluctuation is then $\sigma_{\rm CP} \approx \sqrt{\left \langle {\rm Im}\left( \hat{\mathcal{B}} \right)^2 \right \rangle}~{\rm radians}$. Keeping only terms that are quadratic in $\Delta V_i/V_i$, we obtain:
\begin{align}
{\rm Im}\left( \hat{\mathcal{B}} \right)^2 = -\frac{1}{4}\left[ \hat{\mathcal{B}} - \hat{\mathcal{B}}^\ast \right]^2 \approx -\frac{1}{4} \left[ \sum_{i=1}^3  \left( \frac{\Delta V_i}{V_i} - \frac{\Delta V_i^\ast}{V_i^\ast} \right) \right]^2.
\end{align}
Taking the ensemble-average of this expression (via Eqs.~\ref{eq::BN_Reduction}, \ref{eq::BN_Reduction_Conj}, and \ref{eq::Vavg}) then yields
\begin{align}
\label{eq::ClosurePhase}
\sigma_{\rm CP}^2(\{ \mathbf{b}_\ell \}) &\approx -\frac{1}{4} \left \langle \left[ \sum_{i=1}^3  \left( \frac{\Delta V_i}{V_i} - \frac{\Delta V_i^\ast}{V_i^\ast} \right) \right]^2 \right \rangle \\
\nonumber &= -\frac{1}{4} \left \langle \sum_{i,j=1}^3  \left( \frac{\Delta V_i}{V_i} \frac{\Delta V_j}{V_j} + \frac{\Delta V_i^\ast}{V_i^\ast} \frac{\Delta V_j^\ast}{V_j^\ast} -  \frac{\Delta V_i^\ast}{V_i^\ast} \frac{\Delta V_j}{V_j} -  \frac{\Delta V_i}{V_i} \frac{\Delta V_j^\ast}{V_j^\ast} \right) \right \rangle \\
\nonumber &= -\frac{1}{4} \frac{\lambda^2}{(2\pi)^4} \int d^2 \mathbf{q}\, Q(\mathbf{q}) \sum_{i,j=1}^3  \left[
\left( \frac{\tilde{f}_{\rm V}(\mathbf{q};\mathbf{b}_i) \tilde{f}_{\rm V}(-\mathbf{q};\mathbf{b}_j)}{V_{\rm ea}(\mathbf{b}_i) V_{\rm ea}(\mathbf{b}_j)}
+ \frac{\tilde{f}^\ast_{\rm V}(\mathbf{q};\mathbf{b}_i) \tilde{f}^\ast_{\rm V}(-\mathbf{q};\mathbf{b}_j)}{V_{\rm ea}^\ast(\mathbf{b}_i) V_{\rm ea}^\ast(\mathbf{b}_j)}  \right)
\right. \\
\nonumber & \left. \hspace{0.3\linewidth}
- \left( \frac{\tilde{f}^\ast_{\rm V}(\mathbf{q};\mathbf{b}_i) \tilde{f}_{\rm V}(\mathbf{q};\mathbf{b}_j)}{V_{\rm ea}^\ast(\mathbf{b}_i) V_{\rm ea}(\mathbf{b}_j)}
+ \frac{\tilde{f}_{\rm V}(\mathbf{q};\mathbf{b}_i) \tilde{f}^\ast_{\rm V}(\mathbf{q};\mathbf{b}_j)}{V_{\rm ea}(\mathbf{b}_i) V_{\rm ea}^\ast(\mathbf{b}_j)} \right)
\right]    \\
\nonumber &= \frac{1}{2}\frac{\lambda^2}{(2\pi)^4} \int d^2 \mathbf{q}\, Q(\mathbf{q})
\sum_{i,j=1}^3  {\rm Re}\left[
\frac{\tilde{f}_{\rm V}(\mathbf{q};\mathbf{b}_i) \tilde{f}^\ast_{\rm V}(\mathbf{q};\mathbf{b}_j)}{V_{\rm ea}(\mathbf{b}_i) V_{\rm ea}^\ast(\mathbf{b}_j)}
-
\frac{\tilde{f}_{\rm V}(\mathbf{q};\mathbf{b}_i) \tilde{f}_{\rm V}(-\mathbf{q};\mathbf{b}_j)}{V_{\rm ea}(\mathbf{b}_i) V_{\rm ea}(\mathbf{b}_j)}
\right].
\end{align}
This final representation can be readily evaluated numerically. This representation does not require that the baselines close.

\begin{figure}[t]
\centering
\includegraphics[width=0.8\textwidth]{f3.pdf}
\caption
{
Closure phase fluctuations from refractive scattering as a function of baseline length using our approximate expression (Eq.~\ref{eq::ClosurePhase}) and using the more general Monte Carlo technique (\S\ref{sec::MC}), which is accurate on all baselines. The source and scattering are taken to be isotropic with scales that are comparable to Sgr~A$^{\ast}$\ at $\lambda=3.5~{\rm mm}$ \citep{Ortiz_2016}. Specifically, the source is a circular Gaussian with a FWHM of $130~\mu{\rm as}$, and the scattering is isotropic and Kolmogorov with a FWHM of $160~\mu{\rm as}$ (note that the scattering of Sgr~A$^{\ast}$\ is anisotropic). The left plot shows results for an equilateral triangle of baselines. The two estimates for closure phase noise agree well almost until the closure phase fluctuations saturate at $60\sqrt{3}$~degrees (corresponding to entirely randomized phase); this transition corresponds to the baselines for which refractive noise is becoming dominant so that the approximation of Eq.~\ref{eq::ClosurePhase} breaks down. For longer baselines, only the Monte Carlo method gives the correct result. The right panel shows the closure phase fluctuations for a long, narrow triangle of baselines with an opening angle of $10^\circ$ (the x-axis baseline length corresponds to the long legs). As for the left panel, the small-noise approximation of Eq.~\ref{eq::ClosurePhase} breaks down when refractive noise on the long legs becomes dominant (near 3500~km), even though the closure phase fluctuations are still small, because the two long legs have highly correlated refractive noise that largely cancels in the closure phase.
}
\label{fig::SgrA_CPhase2}
\end{figure}

\begin{figure}[t]
\centering
\includegraphics[width=0.48\textwidth]{f4.pdf}
\caption
{
Expected refractive fluctuations of closure phase on the SMT-CARMA-SMA triangle of EHT baselines as a function of Greenwich Sidereal Time (GST) \citep[for recent EHT closure phase measurements on this triangle, see][]{Fish_2016}. As for Figure~\ref{fig::SgrA_CPhase2}, we compare estimates calculated using our small-noise approximation (Eq.~\ref{eq::ClosurePhase}) with the exact values calculated using the Monte Carlo approach outlined in \S\ref{sec::MC}. Two source models are plotted, corresponding to the best-fit Gaussian (FWHM: $52~\mu{\rm as}$) and annulus (inner
diameter: $21~\mu{\rm as}$, outer diameter: $97~\mu{\rm as}$) shown in Figure~S5 of \citet{Johnson_2015}. Note that only the annulus model provides an acceptable fit to the EHT visibility amplitudes reported in \citet{Johnson_2015}. For the annulus, the closure phase RMS fluctuations become large at the end of the track because the SMA-CARMA baseline is approaching a ``null'' in the visibility amplitude. The RMS noise even exceeds the limit of completely random phase because the distribution of closure phase is bimodal, with peaks at $0^\circ$ and $180^\circ$. Because the current visibility amplitudes follow the annulus model, we expect large refractive fluctuations in the closure phase on this triangle (and others that include CARMA-SMA) for GST greater than ${\sim}3.5$.
}
\label{fig::SgrA_CPhase}
\end{figure}

Although Eq.~\ref{eq::ClosurePhase} provides a computationally efficient method to estimate closure phase ``jitter'' from refractive scattering, it requires $|\Delta V_i / V_i| \ll 1$ for \underline{each} of the three participating baselines. Note that there are cases for which the approximation is poor even though the closure phase fluctuations are small. For instance, in a long narrow triangle of baselines, the phase fluctuations on each long baseline may be large individually but will be highly correlated so that the closure phase fluctuations may be small. In these cases, it is advantageous to use the more general approach that we have outlined in \S\ref{sec::MC} to estimate closure phase fluctuations. Although that approach requires a factor of 36 more computations (to evaluate the covariance matrix for the three complex noise terms), it does not require $|\Delta V_i / V_i| \ll 1$. Figures~\ref{fig::SgrA_CPhase2} and \ref{fig::SgrA_CPhase} compare these two approaches to estimate the refractive fluctuations of closure phase for Sgr~A$^{\ast}$\ at $\lambda=3.5$ and 1.3~mm, respectively.

Lastly, we caution that refractive fluctuations in closure phase are highly sensitive to the detailed structure of the unscattered image, especially near visibility ``nulls'' of the unscattered image, because they depend on the ratio of refractive noise to the ensemble-average visibility (see Figure~\ref{fig::SgrA_CPhase}). Thus, epoch-to-epoch fluctuations in closure phase can potentially be utilized for model discrimination. In contrast, the RMS fluctuation of visibilities (\S\ref{sec::sigma_ref}) depends on the overall image extent but is not sensitive to the detailed intrinsic structure. Consequently, metrics other than closure phase fluctuations may provide more reliable connections between theory and observations for single-epoch observations, especially when the intrinsic structure is not well-known a priori. Alternatively, if the phase fluctuations are small then one could use measured visibilities as estimates of the ensemble average visibilities $V_{\rm ea}(\mathbf{b}_i)$ in Eq.~\ref{eq::ClosurePhase} to avoid estimates of $\sigma_{\rm CP}$ that are overly model-specific.

\subsection{The Coherence Timescale for Refractive Noise}

Refractive fluctuations in complex visibilities will change over time from relative motions of the Earth, scattering screen, and source, as discussed in \S\ref{sec::ISM}. The statistical properties of this evolution in time $t$ can be estimated using the frozen-screen approximation: $\phi(\mathbf{r},t+\Delta t) = \phi(\mathbf{r} - \mathbf{V}_\perp \Delta t,t)$. The modification to Eqs.~\ref{eq::BN_Reduction} and \ref{eq::BN_Reduction_Conj} for a pair of phase screens separated by a time $\Delta t$ is straightforward -- the integrands simply obtain extra factors of $e^{-i \mathbf{q} \cdot \mathbf{V}_\perp \Delta t}$. It is then straightforward to estimate the temporal correlation function of refractive metrics \citep[see, e.g.,][]{Blandford_Narayan_1985,Romani_1986}.

Even without detailed computations, we can estimate the relevant coherence timescales from this additional factor; the coherence timescale will be determined by the condition $ |\mathbf{q} \cdot \mathbf{V}_\perp \Delta t| = 1$ for the particular scales $\mathbf{q}$ that dominate the refractive metric of interest. For a point source, refractive metrics such as flux modulation are determined by fluctuations on scales $q \sim 1/r_{\rm R}$, so they evolve on the refractive timescale $t_{\rm R} = r_{\rm R}/V_\perp$. Refractive noise on a baseline of length $\mathbf{b}$ is sensitive to power in the image on scales corresponding to the baseline resolution (see \S\ref{sec::Interferometry}), so they will evolve on a shorter timescale of $\frac{(1+M)^{-1}r_0}{|\mathbf{b}|} t_{\rm R}$. An extended source \emph{increases} the coherence timescales on short baselines by a factor of $\theta_{\rm img}/\theta_{\rm scatt}$ but does not affect the coherence timescale for baselines that resolve the ensemble-average scattered image. Note that the coherence length for displacing a baseline (${\sim} r_{\rm R}$) is much larger than the coherence length for changing a baseline length or orientation (${\sim} r_{\rm img}$; see Figures~\ref{fig::RefractiveNoise_Correlation} and \ref{fig::RefractiveNoise_Correlation_Anisotropic}).

\section{Summary}
\label{sec::Summary}

With rapid increases in angular resolution and sensitivity, effects of interstellar scattering are newly apparent in a variety of VLBI observations \citep[e.g.,][]{Gwinn_2014,Johnson_2016,Ortiz_2016}. For the EHT, imminent additions, notably the Atacama Large Millimeter/submillimeter Array (ALMA) \citep{Fish_2013}, will enable images of Sgr~A$^{\ast}$\ at resolution comparable to its event horizon, and understanding how scattering affects these images is essential. For {\it RadioAstron}, the detection of brightness temperatures $T_{\rm b} \mathrel{\raise.3ex\hbox{$>$\kern-.75em\lower1ex\hbox{$\sim$}}} 10^{13}~{\rm K}$ at wavelengths from 1.3 to 18~cm \citep{Gomez_2016,Kovalev_2016,Johnson_2016} likewise necessitates a detailed understanding of the role of scattering in these measurements.

The framework that we have developed enables straightforward calculation of refractive scattering effects for these observations. In particular, we have shown how to estimate refractive fluctuations of VLBI observables such as closure phases and amplitudes --- estimates that previously required expensive numerical simulations. Our results accommodate arbitrary source structure and anisotropic scattering. And although we have assumed thin-screen scattering, our results could be generalized to a thick scattering screen or uniform medium, following the approach of \citet{Romani_1986}.

Our framework also points to new scattering mitigation strategies for these projects. Specifically, because our approach represents the scattered image entirely in terms of the ensemble-average image and its large-scale, refractive perturbations, it cleanly decouples the deterministic ``blurring'' of scattering from the stochastic large-scale distortions. We will explore new mitigation strategies in detail in a subsequent paper.

\acknowledgements{MDJ thanks the National Science Foundation and the Gordon and Betty Moore Foundation (GBMF-3561) for financial support of this work. RN's research was supported in part by NSF grant AST1312651 and NASA grant TCAN NNX14AB47G. MDJ thanks Lindy Blackburn, Katie Bouman, Katherine Rosenfeld, and Andrew Chael for many valuable discussions. We thank Charles Gammie for helpful comments on the manuscript. We thank the referee for many comments that improved the clarity and presentation of these results, and for pointing out connections to literature focused on the weak scattering regime.
}

redpajama_set_name: RedPajamaArXiv


text:
Mandy Moore Shows Behind-the-Scenes Transformation Into This Is Us Character
By D. Elizabeth
Mandy Moore has been charming us with her career-revitalizing role on NBC's This Is Us for the past year, but, if it's even possible, we think we just fell in love with her a little more. The actress just walked the Internet through her radical (and frankly, unpleasant) transformation to play her beloved character, cool mom Rebecca Pearson, and we've got to admit: She's just so damn alluring.
Moore, who's currently 33 years old but frequently has to portray a 66-year-old character on the small screen — double her age, folks — recently shared a few videos on Instagram Live, which showcase exactly how she undergoes her beauty transformation to play a sexagenarian. (Spoiler alert: It takes a lot of prosthetics.)
Per People, the aging process understandably takes quite a bit of time in the makeup trailer. "This is 66," Moore says in the live video. "Look how incredible all of this is. We haven't even done any of the painting and stuff yet."
In the videos, People reports, Moore elaborates on what exactly the show's makeup team does to her face — a slow process that includes creating noticeable wrinkles, more demure dark spots along her neck and cheek, and a "creepy" appearance that surrounds her eyes. (Just curious, but does Milo Ventimiglia gets the same treatment?) The makeup team actually avoids doing anything aging-related with Moore's forehead, since she wears a wig that covers that part of her face, anyway. "Getting older is a lot of work," Moore says. "The hardest part is the removal process. It's pretty tricky and it's really hard on my skin. I have really sensitive skin."
This Is Us has proved to be a smash melodramatic success for Moore and her fellow actors since it debuted last fall. It proved to be so popular, in fact, that NBC gave it an unprecedented two-season renewal in the midst of its first season. Get ready for more Moore and her beauty secrets on aging beyond gracefully.
More on Mandy Moore:
Watch Mandy Moore Get Red-Carpet Ready for the 2017 Met Gala
Mandy Moore's Lipstick at the 2017 Golden Globes Is #Goals
Real Talk: Mandy Moore Has the Best Bangs
Now, see 100 years of movie makeup:
Keywordssndmandy moorethis is us


In the ever-changing world of Search Engine Optimisation (SEO), being in-the-know is crucial for success. It is a well-known fact that Google uses over 200 ranking factors in their algorithm. While important, taking them all into consideration when building your SEO strategy could be time-consuming and headache-inducing.
With Google constantly tweaking and refining its algorithm to provide the most relevant search results, you must be able to build a website that will benefit your users and ensure it is optimised to provide visitors with friendlier user experience.
Getting help from an SEO services company will enable you to build a strong SEO strategy so your website can rank higher in search engine results pages (SERPs).
Does it provide real value to the readers?
Does it answer questions that the readers have?
Does it provide solutions that will help make their life easier?
Does it give readers a reason to share or link back to your site?
Does it have engaging visual content?
Rankings and content would always belong together – your website's content is the very reason that people will visit your site.
Ever since Google rolled out its freshness algorithm in 2010, it has had a huge impact on SEO. If you search for a certain topic, notice that the search results that would appear are those which are published recently. To make the most out of this ranking factor, make sure that your posts are fresh and updated. If your topics are out-of-date and haven't been updated for some time, it will simply be surpassed by newer, more interesting and engaging content.
Backlinks, or links pointing back to your site, have long been and still are a critical factor that determines site authority. Take note that Google appreciates quality more than quantity so it is better to have fewer backlinks with high authority rather than linking to so many sites which are of low quality. Google prefers to see links from sites that have real user engagement and quality content. Also, make sure that the anchor text used to link to your site is semantically relevant to your topic.
When Google introduced mobile-first indexing, its aim is to use the mobile version of your website as the starting point in determining rankings. This means that Google is drawing results from mobile-optimised sites first. Mobile-optimised means that users can easily access and navigate and read texts on the mobile version of your website. This also includes page speed or load time as it is also a factor that affects user experience.
Note that it's called mobile-first, and not mobile-only, meaning, even if your website doesn't have a mobile version yet, the desktop version of your website will still be included in the index. However, the lack of a mobile version and the user experience that comes with it could negatively impact your website's search rankings.
If your website URL is still using HTTP, you might want to consider migrating to a secure HTTPS version. Google considers security as a top priority so if your website is secure, it can receive a boost in search rankings. Converting to an HTTPS version protects you and your visitors' data from phishing and other scam practices.
With these factors in mind, you, together with your SEO consultant, will be able to build a stronger SEO strategy for your campaign. These factors focus on quality over short-term benefits, which will help your website perform better in the SERPs.


Mortgage rates for credit score 716 on Lender411 for 30-year fixed-rate mortgages are at 4.22%. That dropped from 4.23% to 4.22%. The 15-year fixed rates are now at 3.67%. The 5/1 ARM mortgage for 716 FICO is now at 4.17%.