These are some of my favourite online resources. I am also creating one of my own – follow @woollenwords on Twitter or Instagram, or sign up to my newsletter, to find out more in due course.
The Guardian and Observer Style Guide has never capitalised internet. Includes heavy metal umlauts.
Ravelry: it's all there. And if it isn't, you can ask about it.
Techknitting: "35 years of tricks want out of my mind & into yours": Simple or fiddly, pretty much every knitting technique can be found on Techknitter's site, with diagrams.
Stitchmastery is the software I use to create charts for knitting patterns; most of my clients do too.
Woolsack has all the information on British wool and breeds.
Knittingfool.com is an online stitch dictionary, where you can search by the name of a stitch pattern or the number of stitches in a repeat, among others.


"We wanted to be young farmers and have a profitable farm, so we decided to go with direct marketing".
Standing in a paddock with Berkshire sows following her every move, Eumungerie central west New South Wales farmer, Alex Hicks says she's delighted with the progress of her farming enterprise.
When Alex and her husband Michael were considering which strand of livestock production to get into, her lifelong affection for pigs made the choice quite easy.
Their plan has been to run pigs in as natural environment as possible, growing them out and fattening them up before having them slaughtered at a regional abattoir and then marketing the meat themselves.
In the couple of months they've been operational, Alex and Michael Hicks have built up a core herd of breeding sows and are developing a local clientele in and around Dubbo.
"That's as far afield as we want to go," says Michael Hicks.
"We don't want to be marketing any further, we want to be selling locally," she said.
The pigs spend their days, and nights, pottering around sections of the Hick's small farm, they're constrained to sectors of the farm by strips of electric fence which are regularly moved to give them access to fresh pasture.
And, like the pigs themselves, their infrastructure is portable as well.
The pigs need water supplies, both for drinking and for wallowing and they need shelter.
So, each time the pig's boundaries are moved, the water troughs and shelter sheds need to move as well, and that has resulted in some innovation in design.
Recycling, repurposing and reusing are watchwords at Extraordinary Pork, and the Hicks say their favourite shopping ground is the local recycling 'tip shop' in Dubbo.
It's here that they've found the 1,000 litre tanks they've cut down to make piggy swimming pools, it's the source of the bicycles and decommissioned wheelchairs which are stripped down to provide the wheels for movement around the farm.
Alex says the resulting infrastructure is not plush or glamorous but it's serviceable and it recycles discarded items.
© Extraordinary Pork 2016. www.extraordinarypork.com.au EMAIL michael@extraordinarypork.com.au PHONE 02 6888 5665.


text:
Q: After finishing the main quests, is Dishonored an open-world game? I want to buy Dishonored, but can I play open-world after finishing the main quest?
I mean playing free-roam, killing anyone like in Assassin's Creed.

A: Dishonored isn't an open world game. It is a linear series of missions set in large, fairly open areas which provide freedom to reach your objectives in a number of ways. This includes stealthy, entirely non-lethal approaches as well as wanton slaughter.
You can, however, replay any missions you have already finished. So you could play the game 'properly' the first time round (however you want to define that) to get the ending that you want, and then go back and replay each mission and kill everyone in your path.

redpajama_set_name: RedPajamaStackExchange


text:
Canadian Military History
New France, 1600 - 1730
Imperial Rivalry, 1731 - 1763
Colony to Confederation, 1764 - 1866
Young Nation, 1867 - 1898
Boer War, 1899 - 1902
Imperial Influences, 1903-13
First World War, 1914-18
Assassination of Archduke Ferdinand: 28 June 1914
Outbreak of the First World War: 28 July 1914
Schlieffen Plan and Plan XVII: Aug 1914
Canadian Mobilization: Aug. 1914.
Formation of the Canadian Automobile Machine Gun Brigade No. 1: Sept. 1914
First Canadian Contingent Embarks for Europe: 12-16 Oct 1914
Canadian Contingent Kept Intact: 12-16 Oct. 1914.
Interwar Years, 1919 - 1938
Second World War, 1939-45
Home >> First World War, 1914-18 >> 1914 >> Canadian Contingent Kept Intact: 12-16 Oct. 1914.
When Minister of Militia Sam Hughes arrived in London, Lord Horatio Herbert Kitchener, War Secretary of Britain, informed him that the Canadian regiments would be broken up and their officers and men dispersed among the British regiments. Kitchener justified this decision on the grounds that the Canadians lacked training and experience.
Hughes adamantly refused this proposal, even when Lord Kitchener gave him a direct order to comply. He took the position that the officers and men were in the pay of Canada, and the Canadian government should, therefore, control them while they were overseas. As a result of his obstinate refusal, the Canadian troops were kept together, and, once additional Canadian divisions arrived in France, they formed the nucleus of the Canadian Corps. Later, Canadian Deputy Chief Justice E.L. Newcombe prepared a memorandum that reinforced Hughes' position. He determined that, legally, the Canadian Expeditionary Force was composed of Canadian militia on active service overseas defending their country. For that reason, the Canadian government had the legal right to maintain control of Canadian troops serving overseas.
Edmonton web design by Chinook Multimedia Inc.
Copyright © 2018 The Loyal Edmonton Regiment Museum
Prince of Wales Armouries Heritage Centre
10440 - 108 Ave, Edmonton
Sponsors/Partners More »

redpajama_set_name: RedPajamaCommonCrawl


ATLANTA, Nov. 09, 2017 (GLOBE NEWSWIRE) -- Cumulus Media Inc. (NASDAQ:CMLS) (the "Company," "we," "us," or "our") today announced operating results for the three and nine months ended September 30, 2017.
For the three months ended September 30, 2017, the Company reported net revenue of $287.2 million, up 0.4% from the three months ended September 30, 2016, net income of $1.3 million and Adjusted EBITDA of $61.8 million which was up 40.7% from the three months ended September 30, 2016. For the nine months ended September 30, 2017, the Company reported net revenue of $841.8 million, net loss of $0.4 million and Adjusted EBITDA of $167.9 million which was up 12.7% from the nine months ended September 30, 2016. Adjusted EBITDA for the three and nine months ended September 30, 2016 included the impact of $14.4 million of expenses incurred to resolve previously disputed syndicated programming and network inventory expenses with CBS Radio Inc.
(1) Adjusted EBITDA is not a financial measure calculated or presented in accordance with accounting principles generally accepted in the United States of America ("GAAP"). For additional information, see "Non-GAAP Financial Measure and Definition".
(2) Term loan excludes debt issuance costs/discounts of $23,054 and $29,909 at September 30, 2017 and December 31, 2016, respectively.
(3) 7.75% senior notes exclude debt issuance costs/discounts of $4,335 and $6,200 at September 30, 2017 and December 31, 2016, respectively.
The Company operates in two reportable segments, the Radio Station Group and Westwood One. The Radio Station Group revenue is derived primarily from the sale of broadcasting time to local, regional and national advertisers. Westwood One revenue is generated primarily through network advertising.
Corporate and Other includes overall executive, administrative and support functions for both of the Company's reportable segments, including programming, finance, legal, human resources and information technology functions.
The following tables present our net revenue by segment (dollars in thousands).
The following tables present our net income (loss) by segment (dollars in thousands).
The following tables present our Adjusted EBITDA by segment (dollars in thousands).
Cumulus Media Inc. will host a teleconference today at 8:00 AM eastern time to discuss its third quarter 2017 operating results.
A link to the webcast of the conference call and the related earnings presentation will be available on the investor section of the Cumulus Media Inc. website (www.cumulus.com/investors). The conference call dial-in number for domestic callers is 877-830-7699, and international callers should dial 574-990-0924 for call access. If prompted, the conference ID number is 4899108. Please call five to ten minutes in advance to ensure that you are connected prior to the call. Following completion, a replay can be accessed until 11:30 PM EST on December 9, 2017. Domestic callers can access the replay by dialing 855-859-2056 or 404-537-3406, replay code 4899108. International callers should dial +44 (0)1452550000 for conference replay access. An archive of the webcast will be available beginning 24 hours after the call for a period of 30 days.
Certain statements in this presentation may constitute "forward-looking" statements within the meaning of the Private Securities Litigation Reform Act of 1995 and other federal securities laws. Such statements are statements other than historical fact and relate to our intent, belief or current expectations, primarily with respect to our future operating, financial and strategic performance. Any such forward-looking statements are not guarantees of future performance and involve risks and uncertainties. Actual results may differ from those contained in or implied by the forward-looking statements as a result of various factors including, but not limited to, risks and uncertainties relating to the need for additional funds to service our debt and to execute our business strategy; our need to restructure or refinance our debt and the terms on which any such restructuring or refinancing may be completed, including through any court-approved restructuring; our ability to access borrowings under our revolving credit facility; our ability from time to time to renew one or more of our broadcast licenses; changes in interest rates; changes in the fair value of our investments; the timing of, and our ability to complete, any acquisitions or dispositions pending from time to time; costs and synergies resulting from the integration of any completed acquisitions; our ability to effectively manage costs; our ability to effectively drive and manage growth; the popularity of radio as a broadcasting and advertising medium; changing consumer tastes; the impact of general economic conditions in the United States or in specific markets in which we currently do business; industry conditions, including existing competition and future competitive technologies and cancellation, disruptions or postponements of advertising schedules in response to national or world events; our ability to generate revenues from new sources, including local commerce and technology-based initiatives; the impact of regulatory rules or proceedings that may affect our business or any acquisitions; our ability successfully appeal the notice of delisting of our Class A common stock from the NASDAQ stock market; the write-off of a material portion of the fair value of our FCC broadcast licenses and goodwill from time to time; or other risk factors described from time to time in our filings with the Securities and Exchange Commission, including our Form 10-K for the year ended December 31, 2016 (the "2016 Form 10-K") and any subsequent filings. Many of these risks and uncertainties are beyond our control, and the unexpected occurrence or failure to occur of any such events or matters could significantly alter the actual results of our operations or financial condition. Cumulus Media Inc. assumes no responsibility to update any forward-looking statement as a result of new information, future events or otherwise.
A leader in the radio broadcasting industry, Cumulus Media (NASDAQ:CMLS) combines high-quality local programming with iconic, nationally syndicated media, sports and entertainment brands to deliver premium content choices to the 245 million million people reached each week through its 446 owned-and-operated stations broadcasting in 90 US media markets (including eight of the top 10), approximately 8,000 broadcast radio stations affiliated with its Westwood One network and numerous digital channels. Together, the Cumulus/Westwood One platforms make Cumulus Media one of the few media companies that can provide advertisers with national reach and local impact. Cumulus/Westwood One is the exclusive radio broadcast partner to some of the largest brands in sports, entertainment, news, and talk, including the NFL, the NCAA, the Masters, the Olympics, the GRAMMYs, the Academy of Country Music Awards, the American Music Awards, the Billboard Music Awards, Westwood One News, and more. Additionally, it is the nation's leading provider of country music and lifestyle content through its NASH brand, which serves country fans nationwide through radio programming, exclusive digital content, and live events. For more information, visit www.cumulus.com.
From time to time we utilize certain financial measures that are not prepared or calculated in accordance with GAAP to assess our financial performance and profitability. Adjusted EBITDA is the financial metric utilized by the Company to analyze the cash flow generated by our business. This measure isolates the amount of income generated by our core operations after the incurrence of corporate, general and administrative expenses. The Company also uses this measure to determine the contribution of our core operations to the funding of our corporate resources utilized to manage our operations and our non-operating expenses including debt service and acquisitions. In addition, consolidated Adjusted EBITDA, excluding the impact of local marketing agreement fees, is a key metric for purposes of calculating and determining our compliance with certain covenants contained in our Credit Agreement.
In deriving this measure, the Company excludes depreciation, amortization and stock-based compensation expense, as these do not represent cash payments for activities directly related to our core operations. The Company also excludes any gain or loss on the exchange or sale of any assets and any gain or loss on derivative instruments, early extinguishment of debt, and local marketing agreement fees as they are not associated with core operations. Expenses relating to acquisitions and restructuring costs are also excluded from the calculation of Adjusted EBITDA as they are not directly related to our ongoing core operations. The Company also excludes any costs associated with impairment of assets as they do not require a cash outlay.
The Company believes that Adjusted EBITDA, although not a measure that is calculated in accordance with GAAP, is commonly employed by the investment community as a measure for determining the market value of a media company and comparing the operational and financial performance among media companies. The Company has also observed that Adjusted EBITDA is routinely employed to evaluate and negotiate the potential purchase price for media companies. Given the relevance to our overall value, the Company believes that investors consider the metric to be extremely useful.
Adjusted EBITDA should not be considered in isolation or as a substitute for net income (loss), operating income, cash flows from operating activities or any other measure for determining the Company's operating performance or liquidity that is calculated in accordance with GAAP. In addition, Adjusted EBITDA may be defined or calculated differently by other companies, and comparability may be limited.


Once upon a time, Google Reader shut down, and everybody scrambled to write a replacement. I didn't actually use Reader or any RSS reader, but writing one seemed like a great idea. I'm quickly learning to regret that decision.
Let's consider just one terribly difficult task, extracting the link to a post. Maybe the <id> element? <id>tag:blogger.com,1999:blog-4341554630550651649.post-8843802384533935675</id> That doesn't look very clickable, but nobody said it should be, so let's move on.
Maybe it's one of the aptly named <link> elements?
Hurray! But, but, but... Nooooo, what's this?
WTF Google? Why are there two original links? What definition of original are we using here?


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text:
Estimated reading time: 3 minutes, 23 seconds.
There is no exact date as to when the County Durham came into existence, but the area has had an official administrative presence for more than 1300 years. In that time, it has seen the rise and fall of a number of monarchs and was once a vital stronghold against invading Scots. To help you understand a little more about this diverse and popular English county, here are 6 interesting facts about county Durham.
1. Home to one of the world's first World Heritage Sites
Originally built back in the 11th century, Durham Castle is one of the best-preserved examples of Norman motte and bailey castles. For over 900 years, the castle has been in constant use and was once a key strategic landmark for the ruling Normans. It's also interesting to note that the castle is the only one built in the UK that has never suffered a breach. With such a rich and lengthy history, it is almost unsurprising that it is a part of one of the world's first World Heritage Sites. The Durham Cathedral and Castle World Heritage Site was given its title around the same time as the palace of Versailles and the Taj Mahal.
2. Britain's tallest man was born in Durham
Neil Fingleton was crowned Britain's tallest man back in 2007. At 7 ft. 7 inches he beat out the reigning Guinness World Record holder at the age of 26. Since that time he played a number of rather exciting roles in Hollywood movies including 47 Ronin, Jupiter Ascending and Avengers: Age of Ultron. Most recently he played Mag the Great in the award-winning TV series, Game of Thrones. Sadly, Neil died in February 2017.
3. It's the birthplace of English Mustard
Few things are as British as English mustard, and County Durham was the official birthplace of this much-loved condiment. Its story begins back in the early 1900's, when one Mrs. Clements decided to grind up mustard seeds in order to get more flavour out of them. She developed a paste out of this process that quickly became popular all over the UK. After being passed down to her daughter, the family business was eventually sold to the Colman's of Norwich.
4. It contains the world's oldest surviving single-arch railway bridge
The Causey Arch was built back in 1725 and is officially the world's oldest surviving single-arch railway bridge. Upon completion, its main purpose was to transport coal and empty wagons to the River Tyne. At its peak, the bridge saw more than 900 horses a day dragging carts back and forth to serve this vital British industry. It is interesting to learn that, upon its completion, the bridge's designer committed suicide due to the fear he had that it would collapse. It would appear that his fears were rather unfounded as it's now stood for almost 300 years. If you fancy something nice to eat, there is a lovely pub right opposite that we would recommend Causey Arch Inn, you can make an afternoon out of it.
5. It's home to the first World Cup winning team
The World Cup as we know it today was founded in 1930. However, back in 1910, Sir Thomas Lipton organised an earlier edition of an international football tournament in Italy. He asked the English Football League to allow him to recruit an English side to compete and they denied him. After a strange turn of affairs, he ended up selecting a team from West Auckland, which was made up of miners. They went on to win that tournament 2-0, beating out Juventus 2-0. Two years later, they returned and once again beat Juventus in the final. As of writing, the whereabouts of the tournament trophy is unknown as it has been missing since the late 1900's.
6. It hosts the world's largest Socialist Trade Union event
Known locally as, "The Big Meeting", the Durham Miners Gala is now the world's largest Socialist Trade Union event that is still held. Each year, up to 150,000 descend on the county to take part in the event. It started back in 1871, and 2018 will mark the 134th gala. If you choose to attend, you can expect to find brass bands playing music through the streets accompanied by cheering clouds and bright coloured banners.
If you live in this beautiful part of the UK and need cleaning services get in touch.


Something that is special for you is hard to get. This is because that you have to build it up by yourself.
We'd like play a part of it with you. Not more than that, not less than that. So special being….
Everything will start from here.
Copyright© 2010 BenSopra offical web site[ベンソープラ] All Rights Reserved.


text:
\section{Introduction}

Accurate estimates of past sea surface temperatures (SSTs) are important for assessing historical climate states \citep{morice2012quantifying}, detecting and attributing changes in climate \citep{chan2015attributing}, and computing climate sensitivity \citep{gregory2002observationally}.  SST datasets are also used as boundary conditions to run general circulation models \citep{folland2005assessing,sobel2007longitude}, and are assimilated as part of generating atmospheric reanalysis data sets \citep{dee2011era}.  SST datasets are, however, known to have substantial errors \citep{kent2017call}, especially prior to the systematic satellite, drifters, and moored buoy temperatures that became routinely available in the 1980s \citep{kennedy2011reassessing}.   For SST, quantified errors include those associated with random errors of individual measurements \citep{kent2006toward,ingleby2010factors}, systematic errors associated with different measurement methods \citep{kennedy2011reassessingB,huang2017extended}, offsets amongst different groups of observers \citep{chan2019correcting} as well as those associated with individual ships \citep{kennedy2012using}.  Another important source of uncertainty involves mapping noisy and often sparse observations to infill unobserved locations \citep{kennedy2014review}.

Despite quantification of many contributors to SST uncertainties, we are unaware of previous studies having quantitatively assessed navigational uncertainties associated with historical ocean observations.  That is, errors in position associated with incorrectly recording or transcribing locations have been recognized \citep{woodruff1998coads}, as have errors introduced by rounding of positions \citep{kent1999statistical}, but the magnitude of navigational uncertainties prior to the widespread deployment of radio navigation in the 1930s \citep{fried1977comparative} appears not to have been quantified.  Prior to radio navigation, ship position in the open ocean was mainly estimated by dead reckoning and celestial techniques \citep{bowditch1906american}.  Dead reckoning involves updating ship position using estimates of heading and distance.  Celestial navigation involves estimating latitude from the zenith angle associated with various celestial bodies, including the sun, moon, and stars.  Longitude may be inferred using a chronometer method whereby the difference between a local apparent time and the time at some known longitude are determined from a clock carried onboard or some other method, such as the phase of Jupiter's moons.  Dead reckoning can potentially introduce both systematic and random uncertainties, whereas celestial correction is assumed free of systematic uncertainties.

Position errors have implications on the accuracy of mapped SSTs.  For example, if SST measurements are binned into gridboxes, misspecification of the appropriate box will influence the mean and higher-order moments \citep{director2015connecting}.  \cite{cervone2015gaussian} have shown that incorporating position uncertainties when averaging land-station data within gridboxes, which have typically been assumed to reside at the center of the gridbox, is important for valid inference of land surface temperatures, and we expect that the the additional uncertainties over the sea associated with ship positions are no less important.  In the following we propose a Bayesian model to quantify position errors for various ship tracks through estimating navigational uncertainties in dead reckoning and celestial correction.  We then translate position errors into SST uncertainties by sampling high-resolution SSTs using posterior samples of ship positions.

\section{Data description}{\label{sec:data}}

The ship data used in this study are from the International Comprehensive Ocean-Atmosphere Data Set (ICOADS3.0) \citep{freeman2016icoads}, which is the most comprehensive available historical dataset of ship-based measurements from the eighteenth century to the present. We use data in 1885 to demonstrate a Bayesian framework for estimating position errors associated with historical ship tracks. Individual ship tracks are identified using ICOADS identification (ID) information, and tracks with missing or non-unique IDs are excluded. Ship tracks traversing across open ocean are separated into shorter segments whenever they are close to islands.  Only ship tracks that have their positions reported at a resolution of every four hours or better and to a precision of better than one degree longitude and latitude are retained.  These high-resolution ship tracks are focussed on because it is otherwise difficult to identify positions errors and to distinguish between contributions from dead-reckoning and celestial navigation.

\begin{figure*}
\centering
\includegraphics[width=30pc]{figures/Fig_1.pdf}
\caption{Ship tracks used in this study. (a) Tracks are divided into high-quality, two-hourly ship tracks (HQ2) and low-quality, four-hourly ship tracks (LQ4). (b) Density of ship position reports at 2$^\circ$ resolution.}
\label{fig:AllTracks}
\end{figure*}

The highest resolution data comes from the U.S.~Marine Meteorological Journals Collection, which was a program sponsored by the U.S.~Navy's Hydrographic Office that enlisted the help of commercial vessels in compiling meteorological data.  Reports are primarily, albeit not exclusively, from U.S.~vessels and are provided every two hours at a resolution of 0.01$^\circ$ longitude and latitude.  In total there are 1,341 of these two-hourly ship tracks.  943 of these tracks are characterized by stable velocities that are episodically punctuated by jumps in position (see Figure \ref{fig:ThreeTypesRaw} and Table \ref{table:comparisonEmspJumpDist}).  Jumps in otherwise smooth ship tracks typically occur at midnight and are consistent with navigation using dead reckoning that is updated by a celestial positioning technique \citep{bowditch1906american}.  Ship tracks generally follow well-established trade routes that tend to be meridional in the tropics and zonal in the mid-latitudes, with the highest data density in the Atlantic, the Eastern Pacific, and the Southern Indian Ocean (see Figure \ref{fig:AllTracks}).

The remaining 398 two-hourly tracks show static positions followed by jumps averaging $84.6$ km in 2 hours, which is unphysical for a ship under sail.  We are unaware of metadata indicating how these positions were prescribed, and exclude these tracks from our present analysis.

\begin{figure*}
\centering
\includegraphics[width=30pc]{figures/Fig_2.pdf}
\caption{Two types of ship tracks. The blue track is the high-quality 2-hourly ship track No.30, and the yellow track is the low-quality 4-hourly ship track No.108. Jumps in position are seen in the 2-hourly track, whereas the 4-hourly track appears overly smooth.}
\label{fig:ThreeTypesRaw}
\end{figure*}

\begin{table*}
\centering
\caption{Empirical speed and the jumping distance.}
\begin{tabular}{|c|ccc|ccc|}
\hline
& \multicolumn{3}{c}{Empirical speed (km/hr)} & \multicolumn{3}{|c|}{Jumping distance (km)} \\
\hline
Quantiles &\hspace{0.15cm} 25\% \hspace{0.15cm} & 50\% & 75\% &\hspace{0.15cm} 25\%\hspace{0.15cm} & 50\% & 75\% \\
\hline
HQ2 & 6.4 & 10.4 & 14.6 & 15.0 & 22.8 & 37.0 \\
LQ4 & 15.8 & 18.3 & 19.6 & - - & - - & - - \\
\hline
\end{tabular}
\label{table:comparisonEmspJumpDist}
\end{table*}

There also exists a separate collection of 576 ship tracks that report position every four hours to a precision of 0.01$^\circ$ longitude and latitude.  These four-hourly tracks, referred to as LQ4 tracks, primarily track zonally between Europe and North America, and meridionally between Europe and South America.  Unlike HQ2 tracks, LQ4 tracks appear overly smooth, showing no discontinuities as would be expected from celestial navigational updates. We assume that the position reports of these tracks have been manually interpolated, implying that they contain less useful information for purposes of inferring navigational uncertainties.
Although LQ4 tracks are unreliable for inferring underlying navigational uncertainties, these more smoothly-varying tracks are more generally representative of position data available in ICOADS, and we develop a methodology for exploring their position uncertainties that leverages results obtained from HQ2 tracks.

It is necessary to define when celestial updates occur for HQ2 tracks.  Celestial navigational updates are presumed to occur when the ship track jumps. Using the speed and heading from neighboring ship positions, we predict the next position, and a jump is identified when the predicted and reported positions differ by at least 7 km in either longitude or latitude.  7 km is chosen on account of its being the 80th percentile of latitudinal differences between predicted and reported ship positions.  Note that longitude and latitude are treated independently because their respective methods of celestial positioning are distinct.  When several jumps are identified in a single day, only the largest jump is selected.

\section{Bayesian model}{\label{sec:bayesmodel}}

The proposed Bayesian model for estimating position errors contains three stages. First, position errors, uncertainties in ship speed and heading are inferred for each HQ2 track using a state-space time series model.  Second, navigational uncertainties are synthesized across different HQ2 tracks using a Bayesian hierarchical model.  Finally, uncertainties are modeled for LQ4 data using a forward navigation model based upon results obtained from HQ2 data.  Stages two and three utilize the posterior samples obtained from prior stages.
All the models described below are fitted using RStan \citep{rstan}.

We note that ideally, these stages would be integrated into a single, inclusive, hierarchical Bayesian model. However, as we will see, for each ship track, the number of parameters in the model is approximately two times the length of the track. Based on our experiences, it can take up to six hours using RStan to fit a single ship track with approximate 400 reported positions. Therefore, given the amount of data as well as the model complexity, we proceed with the multi-stage approach mentioned above to bypass the computational difficulty of implementing a full-Bayesian procedure.

\subsection{State-space model for HQ2 tracks}
\label{section:learnable 2-hourly}

The proposed model utilizes the reported HQ2 ship positions to empirically calculate ship speed and heading at two-hourly time steps.
Ship position and heading are in radians and speed is in km/hr.  Let $\left(q_t^x, q_t^y\right)$ be the displacements (km) that the ship travels from the starting position to the \textit{reported} position at time step $t$ using dead reckoning. Correspondingly, let $\left(p_t^x, p_t^y\right)$ be the displacements from the starting position to the \textit{true} position. $q_t^x, p_t^x$ ($q_t^y, p_t^y$) are positive if the current ship position is to the east (north) of the starting point. $q_t^x, q_t^y$ are calculated as accumulated sums following Equation \eqref{eq:qt}, where $\phi$ denotes the reported longitude, $\psi$ denotes the reported latitude, $r_a$ denotes the radius of earth, and the subscripts denote the time step. The cosine term accounts for changes in distance-longitude scaling with latitude. Definitions for parameters used in the model are listed in Table \ref{tab:parameters}.
\begin{equation}
q_t^x  = \sum_{i = 1}^t r_a \left(\phi_i - \phi_{i - 1}\right)\cos\left(\frac{\psi_i + \psi_{i - 1}}{2}\right),\ \ \
q_t^y = \sum_{i = 1}^t r_a (\psi_{i} - \psi_{i - 1}).
\label{eq:qt}
\end{equation}

\begin{table*}
\begin{minipage}{\textwidth}
\begin{center}
\caption{Definition of parameters}
\label{tab:parameters}
\begin{tabular}{l l}
\hline
Parameter\xdef\@thefnmark{}\@footnotetext{The subscripts or superscripts $x, y, s, \theta$ refer to the longitudinal direction, the latitudinal direction, ship speed and ship heading, respectively.} & Definition \\         \hline
$\left(\phi_t, \psi_t\right)$ & Reported ship position in longitude and latitude   \\
$\left(q_t^x, q_t^y\right)$ & Displacement from the starting position to the reported position\\
$\left(p_t^x, p_t^y\right)$ & Displacement from the starting position to the true position \\
$s_t$, $\hat{s}_t$, $\mu_s$ & True ship speed, empirical ship speed, mean of ship speed\\
$\theta_t$, $\hat{\theta}_t$, $\beta$ & True ship heading, empirical ship heading and its systematic bias \\
$\sigma_s, \sigma_\theta$ & Evolutionary uncertainties in ship speed and heading \\
$\tau_x, \tau_y$ & Track-based uncertainties in celestial correction\\
$\tau_s, \tau_\theta$ & Track-based uncertainties in dead reckoning \\
$\mu_{\tau_x}, \mu_{\tau_y}$ & Population median of uncertainties in celestial correction\\
$\mu_{\tau_s}, \mu_{\tau_\theta}$ & Population median of uncertainties in dead reckoning \\
\hline
\end{tabular}
\end{center}
\end{minipage}
\end{table*}

\subsubsection{Transition model}
For ship speed $s_t$, we assume a customary centered AR model described as below,
\begin{equation}
\label{eq:transition-sp}
s_t  =  \mu_s + \alpha_s(s_{t - 1} - \mu_s) + \epsilon^s_t,
\end{equation}
in which $\alpha_s \in (0, 1)$ denotes the drift parameter, and $\mu_s$ denotes the unknown \textit{track-based population} mean of $s_t$.
Conditioning on $\alpha_s$, $\mu_s$ and $s_{t - 1}$, we assume that $\epsilon_t^s$ follows a truncated normal distribution with mean 0, variance $\sigma_s^2$, and lower truncation at $-\mu_s - \alpha_s(s_{t - 1} - \mu_s)$, so as to guarantee that $s_t$ is non-negative.

For ship heading $\theta_t$, we assume a simple random walk model. That is,
\begin{equation}
\label{eq:transition-th}
\theta_t = \theta_{t - 1} + \epsilon_t^\theta,\ \ \ \ \ \ \epsilon_t^\theta \sim N(0, \sigma_\theta^2).
\end{equation}
Both $\epsilon^s_t$ and $\epsilon^\theta_t$ are assumed to be independent across time.
The transition model is essentially parametrized in terms of $s_t$ and $\theta_t$, while the aggregated displacements $p_t^x$ and $p_t^y$ are deterministic functions of $s_t$ and $\theta_t$. That is,
\begin{equation}
p_t^x  = p_{t - 1}^x + 2s_t\cos\theta_t,\ \ \ \ \ \ p_t^y  = p_{t - 1}^y + 2s_t\sin\theta_t,
\end{equation}
which captures the physical navigation process by using ship speed and ship heading to project its next position (the same as dead reckoning navigation).
We multiply $s_t$ by 2 in the model because we are modeling 2-hourly ship tracks, whereas $s_t$ is in unit of km/hr.

\subsubsection{Observation model}
Suppose there are in total $m$ celestial updates along the ship track. Let $\mathcal{C} = \{t_1, \cdots, t_m\}$ be the set of the corresponding time steps.
\begin{figure}[h]
\centering
\includegraphics[width=20pc]{figures/illustration.pdf}
\caption{A cartoon illustration of a celestial update along HQ2 tracks.  Reported ship positions are represented by circles, with the dashed circle representing a celestial correction.   The reported ship trajectory is indicated by the solid arrow, but it is systematically biased relative to the true ship trajectory indicated by the dashed arrow.}
\label{fig:illustration}
\end{figure}

For $t \not\in \mathcal{C}$ (e.g., position B in Figure \ref{fig:illustration}), we note that the reported ship trajectory is systematically biased, thus the ship position contains accumulated position errors.  Therefore, rather than basing the model on position information, empirical ship speed $\hat{s}_t$ and ship heading $\hat{\theta}_t$ are incorporated into the observation model, which are not expected to have persistence insomuch as dead-reckoning is utilized for navigation. Mathematically, for $k\in[1:(m - 1)]$ and $t \in (t_k, t_{k + 1})$, we assume
\begin{equation}
\label{eq:observation-dead-reckoning}
\hat{s}_t \sim N(s_t, (\tau_s s_t)^2), \ \ \ \ \hat{\theta}_t \sim N(\theta_t + \beta_k, \tau_\theta^2), \ \ \ \
\end{equation}
in which $\tau_s$ and $\tau_\theta$ denote the track-based uncertainties in dead reckoning.
It is worthwhile to note that in early navigations, ship speed was usually measured by a chip log that measures the length of a rope released as a ship travels, or by a patent log that counts the numbers of turns a turbine rotates when water passes though \citep{bowditch1906american}. Since both methods tend to have larger measurement errors at higher ship speed, we model the uncertainty in terms of the relative ship speed instead of the absolute ship speed.
Besides, we introduce $\beta_k$ to model the systematic bias in ship heading between two celestial updates at time step $t_k$ and $t_{k + 1}$. The prior on $\beta_k$ is assumed to be $\text{Unif}(-\pi, \pi]$.

For $t \in \mathcal{C}$ (e.g., position A in Figure \ref{fig:illustration}), empirical ship speed and heading are ignored as they are very inaccurate because of the large jumps (see Table \ref{table:comparisonEmspJumpDist}), whereas the reported ship positions are assumed to contain only celestial observational errors. Thus, for $k\in[1:(m - 1)]$ and $t = t_k$, we assume
\begin{equation}
\label{eq:observation-celestial}
q_t^x \sim N(p_t^x, \left({\tau_x}\cos\psi_t\right)^2),\ \ \ \ \ q_t^y \sim N(p_t^y, {\tau_y}^2),
\end{equation}
in which $q_t^x$ and $q_t^y$ denote the observed aggregated displacements. $\psi_t$ denotes the reported latitude. $\tau_x$ and $\tau_y$ denote the track-based uncertainties in celestial correction in the longitudinal and latitudinal direction, respectively.

\subsection{Synthesizing information across different HQ2 tracks}
\label{section:BHM}
The state-space model discussed in Section \ref{section:learnable 2-hourly} describes a single HQ2 track.  We assume consistent levels of accuracy in celestial correction and dead reckoning over HQ2 tracks, permitting for borrowing information on the uncertainty parameters across different tracks.  Let $\{\tau_x^{(j)}, \tau_y^{(j)}, \tau_s^{(j)}, \tau_\theta^{(j)}\}$ be the uncertainty parameters associated with HQ2 track $j$. We assume a hierarchical structure on the uncertainty parameters,
\begin{equation}
\label{eq:idealBHM}
\begin{aligned}
& \log\tau_x^{(j)} \sim N(\log \mu_{\tau_x}, \gamma_{\tau_x}^2), \ \ \ \log\tau_y^{(j)} \sim N(\log\mu_{\tau_y}, \gamma_{\tau_y}^2).
\end{aligned}
\end{equation}
$\log\mu_{\tau_x}$ and $\log\mu_{\tau_y}$ denote the population medians of $\log\tau_x^{(j)}$ and $\log\tau_y^{(j)}$, respectively. Since the log-transformation is monotone, $\mu_{\tau_x}$ and $\mu_{\tau_x}$ are the population medians of $\tau_x^{(j)}$ and $\tau_y^{(j)}$, respectively.
$\tau_s^{(j)}, \tau_\theta^{(j)}$ are modeled in the same fashion.
Conditioning on the population-level parameters, track-based navigational parameters are assumed to be independent.

To remedy the computational challenge, we consider a second-stage model, where we synthesize the uncertainty information across tracks based on the posterior samples obtained by fitting each HQ2 track. The variability of the navigational parameters across different tracks are summarized in Figure \ref{fig:variability} in the Appendix.

Given $n$ posterior samples $\{\tau_{x}^{(j, i)}, \tau_{y}^{(j, i)}\}_{i \in [1:n]}$ of HQ2 track $j$, we assume a hierarchical model,
\begin{equation}
\label{eq:BHM}
\begin{aligned}
\log\tau_{x}^{(j,i)} & \sim N\left(\log\mu^{(j)}_{\tau_x}, {\eta^{(j)}_{\tau_x}}^2\right),\ \ \ \log\mu^{(j)}_{\tau_x} \sim N(\log\mu_{\tau_x}, \gamma_{\tau_x}^2), \\
\log\tau_{y}^{(j,i)} & \sim N\left(\log\mu^{(j)}_{\tau_y}, {\eta^{(j)}_{\tau_y}}^2\right),\ \ \ \log\mu^{(j)}_{\tau_y} \sim N(\log\mu_{\tau_y}, \gamma_{\tau_y}^2),
\end{aligned}
\end{equation}
in which $\mu^{(j)}_{\tau_x}, \mu^{(j)}_{\tau_y}$ and $\eta^{(j)}_{\tau_x}, \eta^{(j)}_{\tau_y}$ denote the track-level medians and standard deviations for HQ2 track $j$. Conditioning on track-level parameters $\mu^{(j)}_{\tau_x}, \eta^{(j)}_{\tau_x}$ and $\mu^{(j)}_{\tau_y}, \eta^{(j)}_{\tau_y}$, we assume $\tau_{x}^{(j,i)}$ and $\tau_{y}^{(j,i)}$ are independent across the sample index $i$. Conditioning on the population-level parameters $\mu_{\tau_x}, \gamma_{\tau_x}$ and $\mu_{\tau_y}, \gamma_{\tau_y}$, we assume $\mu^{(j)}_{\tau_x}, \eta^{(j)}_{\tau_x}$ and $\mu^{(j)}_{\tau_y}, \eta^{(j)}_{\tau_y}$ are independent across the track index $j$. $\{\tau_{s}^{(j,i)}, \tau_{\theta}^{(j,i)}\}$ are modeled in the same fashion.

\subsection{Forward navigation model for LQ4 tracks}{\label{subsec:forward}}
The forward navigation model proposed in this section aims at representing dead reckoning and celestial correction contributions to position errors for LQ4 tracks.
We assume that the population-level uncertainties for ship speed and heading in LQ4 tracks are consistent with the population-level uncertainties in HQ2 tracks.  Speed and heading can then be represented as,
\begin{equation}
\hat{s}_t = s_t(1 + e_t^s),\ \ \
\hat{\theta}_t = \theta_t + e_t^\theta,
\end{equation}
where $e_t^s, e_t^\theta$ are assumed to be i.i.d. over time. We impose the following hierarchical priors on $e_t^s, e_t^\theta$,
\begin{equation}
\label{eq:ideal-forward-model}
\begin{aligned}
e_t^s & \sim N(0, \tau_s^2),\ \ \ \log\tau_s \sim N(\log\hat{\mu}_{\tau_s}, \hat{\gamma}_{\tau_s}^2),\\
e_t^\theta & \sim N(0, \tau_\theta^2),\ \ \ \log\tau_\theta \sim N(\log\hat{\mu}_{\tau_\theta}, \hat{\gamma}_{\tau_\theta}^2),
\end{aligned}
\end{equation}
in which $\hat{\mu}_{\tau_s}$, $\hat{\mu}_{\tau_\theta}$, $\hat{\gamma}_{\tau_s}$ and $\hat{\gamma}_{\tau_\theta}$ are the empirical posterior means of $\mu_{\tau_s}$, $\mu_{\tau_\theta}$, $\gamma_{\tau_s}$ and $\gamma_{\tau_\theta}$, respectively, obtained by fitting the hierarchical model discussed in Section \ref{section:BHM}. The uncertainties in celestial correction are incorporated as in Equation \eqref{eq:observation-celestial}, and the priors on $\tau_x$ and $\tau_y$ are specified similarly as in Equation \eqref{eq:ideal-forward-model}.

For celestial navigation we incorporate a term representing the probability that a celestial observation is employed.
In order to illustrate the sensitivity of our results to whether celestial observations are taken --- and because we have no direct evidence of whether celestial observations are employed on any given day --- we provide results assuming observations are taken every night ($p=1$), half of the time ($p=0.5$), and never ($p=0$).

\section{Results}{\label{sec:results}}

\subsection{High-quality two-hourly ship tracks}
A single track, HQ2 track No.30, is described for purposes of illustrating the results.  HQ2 track No.30 moved from West to East, where blue dots in Figure \ref{fig:ThreeTypesResults} represent the reported ship positions, black dots represent the posterior mean positions, and ellipses indicate posterior one-standard deviation uncertainties.
The trajectory of the posterior mean tends to diverge from the reported ship positions when celestial navigation updates are large because, unlike reported positions, posterior means take into account not only information from preceding celestial updates but also later ones. Figure \ref{fig:Good2hourlyPattern} demonstrates the global pattern of random position uncertainties for HQ2 tracks.

\begin{figure*}
\centering
\includegraphics[width=30pc]{figures/Fig_3.pdf}
\caption{The posterior distributions of ship positions. For HQ2 track No.30, blue dots are reported ship positions, black dots represent the posterior means, and circles show posterior uncertainties (one standard deviation). For LQ4 track No.108,  yellow dots are reported ship positions, red dots and circles are the posterior means and uncertainties when a celestial correction happens every midnight ($p = 1$), and black dots and gray circles are the posterior means and uncertainties when there are no celestial corrections ($p = 0$).}
\label{fig:ThreeTypesResults}
\end{figure*}

\begin{figure*}
\centering
\includegraphics[width=25pc]{figures/Fig_4-a.pdf}
\includegraphics[width=25pc]{figures/Fig_4-b.pdf}
\caption{Global pattern of random position uncertainties for HQ2 tracks. Individual panels are (a) longitude and (b) latitude.
Track-based estimates of random position uncertainties are gridded to 2$^\circ$ resolution for visualization. Binning is accomplished by averaging random position uncertainties within a grid box in quadrature.}
\label{fig:Good2hourlyPattern}
\end{figure*}

For celestial correction, the population median of the uncertainty in the longitudinal direction is approximately 33.1 km, or $0.30^\circ$ on the equator, while the population median of the uncertainty in the latitudinal direction is smaller, approximately 24.4 km, or $0.22^\circ$ (see Table \ref{table:BayesHierarResult}). This accords with expectation because celestial correction in the longitudinal direction is subject to errors of both celestial observations and chronometers, whereas celestial correction in the latitudinal direction is free of chronometer errors \citep{bowditch1906american}.  The population median of the uncertainty in the relative ship speed $\hat{s}_t/s_t$ is approximately 19.2\%, which is possibly attributed to the less reliable instruments, e.g., chip logs and patent logs, used in early navigations.
The population median of the uncertainty in ship heading is approximately 0.23 radian, or $13.2^\circ$.
Among all HQ2 tracks, the 25\%, 50\%, and 75\% quantiles of the absolute difference between the reported ship heading and the posterior mean heading are $2.9^\circ$, $7.9^\circ$ and $20.4^\circ$, respectively.

\begin{table*}
\begin{minipage}{\textwidth}
\centering
\caption{The population median of navigational parameters.}
\begin{tabular}{|l|c|c|c|c|c|c|c|}
\hline
Quantiles & 5\% & 25\% & 50\% & 75\% & 95\% & std\\
\hline
$\mu_{\tau_x}$\xdef\@thefnmark{}\@footnotetext{$\mu_{\tau_x}$ and $\mu_{\tau_y}$ denote the population median of the uncertainty in celestial correction along the longitudinal and the latitudinal directions, respectively. $\mu_{\tau_s}$ and $\mu_{\tau_\theta}$ denote the population median of the uncertainty in the relative ship speed $\hat{s}_t/s_t$ and ship heading, respectively.} (km) & 31.6 & 32.4 & 33.1 & 33.7 & 34.6 & 0.92\\
\hline
$\mu_{\tau_y}$ (km) & 23.2 & 23.9 & 24.4 & 24.9 & 25.6 & 0.74\\
\hline
$\mu_{\tau_s}$\hspace{0.1cm} (\%) & 18.3 & 18.8 & 19.2 & 19.6 & 20.1 & 0.5\\
\hline
$\mu_{\tau_\theta}$ \hspace{0.01cm}(rad) & 0.21 & 0.22 & 0.23 & 0.24 & 0.25 & 0.01\\
\hline
\end{tabular}
\label{table:BayesHierarResult}
\end{minipage}
\end{table*}

\begin{table}
\begin{minipage}{\textwidth}
\centering
\caption{Random and systematic position uncertainties.}
\begin{tabular}{ |l|c|c|c|c|}
\hline
\multicolumn{5}{c}{Random position uncertainty} \\
\hline
Quantiles & 25\% & 50\% & 75\% & mean \\
\hline
HQ2 longitude & $0.11^\circ$(12 km) &  $0.18^\circ$(20 km) & $0.23^\circ$(26 km) & $0.18^\circ$(20 km)\\
\hline
HQ2 latitude   & $0.10^\circ$(11 km) &  $0.15^\circ$(17 km) & $0.20^\circ$(22 km) & $0.16^\circ$(18 km)\\
\hline
LQ4 longitude (1.0)\xdef\@thefnmark{}\@footnotetext{1.0, 0.5 and 0.0 correspond to the the best-case scenario, the random-guess scenario, and the worst-case scenario for LQ4 tracks, respectively.\label{note1}} & $0.11^\circ$(12 km) &  $0.16^\circ$(18 km) & $0.20^\circ$(22 km) & $0.16^\circ$(18 km)\\
\hline
LQ4 longitude (0.5) & $0.16^\circ$(18 km) &  $0.22^\circ$(24 km) & $0.28^\circ$(31 km) & $0.23^\circ$(26 km)\\
\hline
LQ4 longitude (0.0) & $0.35^\circ$(39 km) &  $0.50^\circ$(56 km) & $0.66^\circ$(73 km) & $0.50^\circ$(56 km)\\
\hline
LQ4 latitude\ \ \ (1.0) & $0.09^\circ$(10 km) &  $0.13^\circ$(14 km) & $0.17^\circ$(19 km) & $0.13^\circ$(14 km)\\
\hline
LQ4 latitude\ \ \ (0.5) & $0.13^\circ$(14 km) &  $0.18^\circ$(20 km) & $0.24^\circ$(27 km) & $0.19^\circ$(21 km)\\
\hline
LQ4 latitude\ \ \ (0.0) & $0.29^\circ$(32 km) &  $0.44^\circ$(49 km) & $0.60^\circ$(67 km) & $0.44^\circ$(49 km)\\
\hline
\hline
\multicolumn{5}{c}{Systematic position uncertainty} \\
\hline
Quantiles & 25\% & 50\% & 75\% & mean \\
\hline
HQ2 longitude & $0.06^\circ$(6.7 km) &  $0.16^\circ$(18 km) & $0.34^\circ$(38 km) & $0.24^\circ$(27 km)\\
\hline
HQ2 latitude   & $0.05^\circ$(5.6 km) &  $0.14^\circ$(16 km) & $0.30^\circ$(33 km) & $0.21^\circ$(23 km)\\
\hline
LQ4 longitude (1.0) & $0.00^\circ$(0.0 km) &  $0.00^\circ$(0.0 km) & $0.01^\circ$(1.1 km) & $0.00^\circ$(0.0 km)\\
\hline
LQ4 longitude (0.5) & $0.00^\circ$(0.0 km) &  $0.01^\circ$(1.1 km) & $0.01^\circ$(1.1 km) & $0.01^\circ$(1.1 km)\\
\hline
LQ4 longitude (0.0) & $0.01^\circ$(1.1 km) &  $0.03^\circ$(3.3 km) & $0.06^\circ$(6.7 km) & $0.05^\circ$(5.6 km)\\
\hline
LQ4 latitude\ \ \ (1.0)  & $0.00^\circ$(0.0 km) &  $0.00^\circ$(0.0 km) & $0.01^\circ$(1.1 km) & $0.00^\circ$(0.0 km)\\
\hline
LQ4 latitude\ \ \ (0.5)  & $0.00^\circ$(0.0 km) &  $0.01^\circ$(1.1 km) & $0.01^\circ$(1.1 km) & $0.01^\circ$(1.1 km)\\
\hline
LQ4 latitude\ \ \ (0.0)  & $0.01^\circ$(1.1 km) &  $0.03^\circ$(3.3 km) & $0.06^\circ$(6.7 km) & $0.05^\circ$(5.6 km)\\
\hline
\hline
\multicolumn{5}{c}{Overall position uncertainty} \\
\hline
Quantiles & 25\% & 50\% & 75\% & mean \\
\hline
HQ2 longitude & $0.16^\circ$(18 km) & $0.26^\circ$(29 km) & $0.42^\circ$(47 km) & $0.32^\circ$(36 km)\\
\hline
HQ2 latitude   & $0.14^\circ$(16 km) & $0.23^\circ$(26 km) & $0.37^\circ$(41 km) & $0.28^\circ$(31 km)\\
\hline
LQ4 longitude (1.0) & $0.12^\circ$(13 km) & $0.16^\circ$(18 km) & $0.20^\circ$(22 km) & $0.16^\circ$(18 km)\\
\hline
LQ4 longitude (0.5) & $0.16^\circ$(18 km) & $0.22^\circ$(24 km) & $0.28^\circ$(31 km) & $0.22^\circ$(24 km)\\
\hline
LQ4 longitude (0.0) & $0.36^\circ$(40 km) & $0.51^\circ$(57 km) & $0.68^\circ$(75 km) & $0.51^\circ$(57 km)\\
\hline
LQ4 latitude\ \ \ (1.0) & $0.10^\circ$(11 km) & $0.14^\circ$(16 km) & $0.17^\circ$(19 km) & $0.14^\circ$(16 km)\\
\hline
LQ4 latitude\ \ \ (0.5) & $0.13^\circ$(14 km) & $0.18^\circ$(20 km) & $0.24^\circ$(27 km) & $0.19^\circ$(21 km)\\
\hline
LQ4 latitude\ \ \ (0.0) & $0.30^\circ$(33 km) & $0.46^\circ$(51 km) & $0.62^\circ$(69 km) & $0.46^\circ$(51 km)\\
\hline
\end{tabular}
\label{tab:uncertainty_offset}
\end{minipage}
\end{table}

\subsection{Low-quality four-hourly ship tracks}
As noted, we are unable to infer when LQ4 tracks have their positions updated by celestial observations and, therefore, explore three scenarios wherein celestial positions are made every midnight, with 0.5 probability, or never. Figure \ref{fig:ThreeTypesResults} shows the posterior distributions of LQ4 track No.108 under the best-case and worst-case scenarios.   Under the best-case scenario all the midnight positions along the track are considered being celestially corrected, and LQ4 track exhibits small Brownian-bridge uncertainty structures between consecutive midnight positions.
Under the worst-case scenario we assume no celestial corrections, and LQ4 track exhibits a Brownian-bridge uncertainty structure that spans the departure and arrival points.

Table \ref{tab:uncertainty_offset} summarizes position errors of all LQ4 tracks under the three scenarios. The overall position uncertainty (MSE) combines the random position uncertainty (standard deviation) and the systematic position uncertainty (bias) using the bias-variance decomposition. On average, random position uncertainties of LQ4 tracks are $0.50^\circ$ (56 km on the equator) in longitude and $0.44^\circ$(49 km) in latitude under the worst-case scenario, and $0.16^\circ$ (18 km) in longitude and $0.13^\circ$(14 km) in latitude under the best-case scenario. The half-probability scenario is similar to, albeit of course slightly more uncertain, than the best-case scenario.
The random position uncertainty is represented as better than HQ2 tracks, where the latter have only 87\% of nights associated with a celestial correction.
Because there are no apparent jumps, LQ4 tracks are inferred to have smaller systematic uncertainties than estimated for HQ2 tracks.

The Brownian-bridge uncertainty structure implies larger errors associated with longer journeys and being further away from departure and arrival points, such that positions in the interior of oceans are generally more uncertain (see Figure \ref{fig:OthersPattern}).
Note that position uncertainties depend not only on the distance from coasts or islands, but also on directions that ships are heading, which determines the relative magnitude of the uncertainties in longitude and latitude.

\begin{figure*}
\centering
\includegraphics[width=14pc]{figures/Fig_5-a.pdf}
\includegraphics[width=14pc]{figures/Fig_5-b.pdf}
\includegraphics[width=14pc]{figures/Fig_5-c.pdf}
\includegraphics[width=14pc]{figures/Fig_5-d.pdf}
\includegraphics[width=14pc]{figures/Fig_5-e.pdf}
\includegraphics[width=14pc]{figures/Fig_5-f.pdf}
\caption{Global pattern of random position uncertainties for LQ4 tracks.  Left panels show longitude uncertainties for (a) best-case, (b) random-guess, and (c) worst-case scenarios. Right panels are as the left but for latitude uncertainties.  In the best-case scenario, a celestial correction happens at each midnight.  In the random-guess scenario, midnight positions have a probability of 0.5 to be corrected, whereas in the worst-case scenario, celestial corrections never happen.  Maps are shifted to center on the Atlantic for the purpose of visualization.  Procedures of generating maps from individual measurements are as per Figure \ref{fig:Good2hourlyPattern}.}
\label{fig:OthersPattern}
\end{figure*}

\subsection{SST uncertainties}
\label{subsec:sst_results}

To quantify uncertainties in SSTs associated with errors in position, we sample a high-resolution SST dataset with position errors that mimic those expected from our analysis of HQ2 tracks.  We use the Multi-Scale Ultra-high resolution Sea Surface Temperature dataset (MURSST) \citep{chin2017multi} that incorporates infrared and microwave satellite retrievals and observations from ships and buoys.  Although the data is obviously more recent than the 1885 ship tracks that we analyze, MURSST has the advantage of having a 0.01$^\circ$ spatial resolution that is comparable to the HQ2 ship-track precision.  Estimated SST uncertainties are still meaningful because the basic SST patterns --- including those related to equator-to-pole temperature gradients, boundary currents, gyres, and upwelling regions --- are stable features of the ocean circulation \citep{wunsch2004gulf}.

SSTs in MURSST are repeatedly sampled in order to estimate uncertainties.
For each posterior ship track realized, we sample SSTs at the realized positions in MURSST on the corresponding high-resolution monthly climatology over 2003-2018. In total we obtain 1,000 posterior ship tracks and their corresponding SSTs are sampled. The uncertainties in SSTs are estimated by taking the standard deviation across these 1,000 samples.
For purposes of visual display, the uncertainties are re-gridded at 2$^\circ$ resolution.  An assumption associated with the usage of daily SST products is that diurnal SST signals are relatively uniform at small spatial scales.  This assumption is, in general, valid in cold seasons, when diurnal cycles are small \citep{morak2016climatological}, and over the extra-tropical ocean, where stratiform clouds associated with large-scale weather systems dominate.  We note that in the places where diurnal cycles are affected by small-scale convective clouds, such as in parts of the tropics, sampling from daily-average SST products leads to a larger underestimation of SST uncertainties.

Position errors induce uncertainties in SST in regions where position errors are large and SST gradients are strong.  On average, position errors in HQ2 tracks translate into $0.11^\circ$C SST uncertainties (see Table \ref{tab:SST_uncertainty_offset}), but can be as large as 0.33$^\circ$C in the Northwest Atlantic where the western boundary current detaches from the East Coast of the US (see Figure \ref{fig:SSTJan}).  Larger SST uncertainties are also found in regions where other boundaries currents detach in the Northwest Pacific and Southwest Atlantic, as well as in the vicinity of the Aghulas Current south of South Africa.  For LQ4 tracks (see Table  \ref{tab:SST_uncertainty_offset} and Figure \ref{fig:SSTJan4hr}), SST uncertainties average $0.22^\circ$C and reach $0.94^\circ$C in the mid-latitude Northwest Atlantic under the zero-celestial observation scenario, and are similar to HQ2 tracks in the other scenarios.
\begin{table}
\begin{minipage}{\textwidth}
\caption{A comparison of SST uncertainty and SST offset.}
\centering
\begin{tabular}{ |l|c|c|c|c||c|c|c|c|}
\hline
Global\footnote{The summary statistics are calculated using SSTs at all positions along ship tracks.}  & \multicolumn{4}{|c||}{SST uncertainty ($^\circ$C)} &  \multicolumn{4}{|c|}{SST offset ($^\circ$C)} \\
\hline
Quantiles & 25\% & 50\% & 75\% & mean &  25\% & 50\% & 75\% & mean\\
\hline
HQ2 & 0.03 & 0.06 & 0.12 & 0.11 & -0.05 & 0.01 & 0.07 & 0.02\\
\hline
LQ4 (1.0) & 0.02 & 0.05 & 0.09 & 0.08 & -0.01 & 0.00 & 0.01 & 0.00 \\
\hline
LQ4 (0.5) & 0.03 & 0.07 & 0.12 & 0.10 & -0.01 & 0.00 & 0.01 & 0.00\\
\hline
LQ4 (0.0) & 0.08 & 0.16 & 0.27 & 0.22 & -0.03 & 0.00 & 0.03 & 0.01\\
\hline

Regional\footnote{The summary statistics are calculated using SSTs restricted in the regions over the Gulf stream (280-320$^\circ$E, 40-50$^\circ$N and 280-300$^\circ$E, 35-40$^\circ$N).}  & \multicolumn{4}{|c||}{SST uncertainty ($^\circ$C)} &  \multicolumn{4}{|c|}{SST offset ($^\circ$C)} \\
\hline
Quantiles & 25\% & 50\% & 75\% & mean &  25\% & 50\% & 75\% & mean\\
\hline
HQ2 & 0.08 & 0.23 & 0.46 & 0.33 & -0.22 & -0.00 & 0.24 & 0.03 \\
\hline
LQ4 (1.0) & 0.11 & 0.26 & 0.47 & 0.32 & -0.05 & 0.00 & 0.04 & -0.01\\
\hline
LQ4 (0.5) & 0.17 & 0.37 & 0.62 & 0.43 & -0.07 & -0.01 & 0.04 & -0.01\\
\hline
LQ4 (0.0)  & 0.42 & 0.83 & 1.29 & 0.94 & -0.30 & -0.03 & 0.25 & -0.02\\
\hline
\end{tabular}
\label{tab:SST_uncertainty_offset}
\end{minipage}
\end{table}

\begin{figure*}
\centering
\includegraphics[width=25pc]{figures/Fig_6-a.pdf}
\includegraphics[width=25pc]{figures/Fig_6-b.pdf}
\caption{SST uncertainties associated with position errors for HQ2 tracks.  Individual panels are (a) random SST uncertainties and (b) systematic SST offsets.  Results are binned to $2^\circ$ grids for visualization. }
\label{fig:SSTJan}
\end{figure*}

\begin{figure*}
\centering
\includegraphics[width=14pc]{figures/Fig_7-a.pdf}
\includegraphics[width=14pc]{figures/Fig_7-b.pdf}
\includegraphics[width=14pc]{figures/Fig_7-c.pdf}
\includegraphics[width=14pc]{figures/Fig_7-d.pdf}
\caption{SST uncertainties associated with position errors for LQ4 tracks.  Panels (a), (b), and (c) demonstrate random SST uncertainties under the best-case scenario, the random-guess scenario, and the worst-case scenario, respectively.  Panel (d) shows SST offsets under the best-case scenario.  Results are binned to $2^\circ$ grids for visualization.}
\label{fig:SSTJan4hr}
\end{figure*}

\section{Concluding Remarks}{\label{sec:future}}
We are unaware of previous efforts to quantify the position uncertainty of historical observations from ships. The magnitude of uncertainties of a given quantity associated with position uncertainties will depend on the local gradient of that quantity, which we estimated for SSTs is on the order of $0.1^\circ$C over the globe but can be up to $0.3^\circ$C regionally (Figure \ref{fig:SSTJan}, \ref{fig:SSTJan4hr}, and Table \ref{tab:SST_uncertainty_offset}).  Expected SST errors from position uncertainties are small as compared with other error sources. For example, individual SST measurements have random measurement errors of approximately $1.0^\circ$C and are subject to biases that average approximately $0.4^\circ$C cool when measured using buckets and approximately $0.1^\circ$C warm when collected using the engine-room-intake method \citep{kennedy2011reassessing, kennedy2011reassessingB}. Recent studies also identified significant offsets up to $0.5^\circ$C associated with national groups by comparing collocated measurements \citep{chan2019systematic}.

We note, however, that position uncertainties are distinct from other errors in that they will influence all observations made from the same journey, leading to correlated errors along ship tracks. Moreover, position uncertainties will affect the estimation of biases and random SST errors since many existing approaches require pairing collocated measurements \citep{kent2006toward, chan2019systematic}. As noted, position errors can also lead to changes in tail behavior an high-order moments in grid-box averaged SSTs \citep{director2015connecting}.  We suspect that similar concerns arise with respect to other mapping procedures that interpolate using weighted averages of observations depending upon uncertain positions.

The HQ2 data provides sufficiently frequent and precise observations to characterize uncertainties, but may not be indicative of the overall accuracy of SST positions in 1885.  We speculate the vessels enrolled in the U.S. meteorological program represent a subset of ships wherein a higher priority was placed upon navigation.  Data that is reported with lower resolution and without distinct indications of celestial navigational updates may reflect cruises wherein navigation was a lower priority, or less feasible given limitations with regard to expertise, equipment, or labor.  Thus, estimated position errors may not reflect the overall uncertainty of position data in 1885.  There will also be heterogeneity in reports amongst ships that we have not fully accounted for. Some cruises presumably had lower need of precise navigation; for example, a zonal cross-Atlantic cruise would have less need of determining longitude for purposes of ensuring landfall than a cruise with a meridional heading whose intended port was an island.  In addition, there are also possibilities that longitude and latitude are not celestially corrected simultaneously for earlier navigators because of less-widely deployed ship-board chronometers \citep{bowditch1906american}.

We focus on a single year in developing and testing our procedure, but it would be useful to extend the analysis over a longer time horizon and to a greater fraction of the data.
In 1885, around 85\% of observations are associated with ship tracks.  Furthermore, \citet{carella2017probabilistic} have provided estimates of additional data belonging to individual ship tracks, bringing the percentage of observations associated with ship tracks in 1885 to 90\%.
In more data-rich intervals, however, distinguishing individual ship track becomes more difficult, such that between 1900-1940 only 60\% of observations, on average, are associated with tracks.

There is presumably a trend toward increasing accuracy of position with time, given technological improvements in marine navigation. This implies that errors and modification of SST distributions introduced through positional error will decrease through time, possibly having consequence for trend estimates, especially those in the vicinity of sharp SST gradients.  Position error may limit the spatial resolution over which trends can accurately be determined.  It would be useful to estimate uncertainties for a gridded SST product with global coverage that, in addition to accounting for observational SST errors \citep{kennedy2011reassessingB} and correcting for biases \citep{kent2017call, chan2019correcting}, also accounted for position errors.

\section*{Acknowledgements}
Natesh Pillai and Chenguang Dai were supported by the Office of Naval Research.
Peter Huybers and Duo Chan were supported by the Harvard Global Institute.

\section{Appendix}{\label{sec:check}}
To evaluate the fitness of the proposed models, we check the following aspects, including the posterior predictive distributions of HQ2 tracks and the posterior distributions of the navigational parameters.

\subsection{Posterior predictive check of HQ2 tracks}
The posterior predictive check is a self-consistency check in the sense that any replicated data simulated from the posterior predictive distribution should look similar to the observed data.

To generate posterior predictive samples of each HQ2 track, we first randomly sample a set of navigational parameters $\tau_x, \tau_y, \tau_s, \tau_\theta$ from their posterior samples. Recall that $\mathcal{C} = \{t_1, \cdots, t_m\}$ is the set of the time steps when celestial updates happen.
For $k\in[1:(m - 1)]$, if $t = t_k$, that is, the reported ship position contains only celestial observational errors, we randomly sample $p_t^x, p_t^y$ from their posterior samples, and sample $q_t^x, q_t^y$ from
\begin{equation}
q_t^x  \sim N(p_t^x, \left({\tau_x}\cos\psi_t\right)^2),\ \ \ q_t^y  \sim N(p_t^y, {\tau_y}^2).
\end{equation}
Otherwise if $t \in (t_k, t_{k + 1})$, we follow the dead reckoning navigation. That is, we randomly sample $s_t, \theta_t, \beta_k$ from their posterior samples, and sample $\hat{s}_t, \hat{\theta}_t$ from
\begin{equation}
\hat{s}_t  \sim N(s_t, (\tau_ss_t)^2),\ \ \ \ \ \hat{\theta}_t   \sim N(\theta_t + \beta_k, \tau_\theta^2).
\end{equation}
Then we generate $q_t^x, q_t^y$ as follows,
\begin{equation}
\begin{aligned}
q_t^x  = q_{t - 1}^x + 2\hat{s}_t \cos(\hat{\theta}_t), \ \ \ \
q_t^y = q_{t - 1}^y + 2\hat{s}_t \sin(\hat{\theta}_t).
\end{aligned}
\end{equation}

Figure \ref{fig:ppc} shows the posterior predictive distributions of HQ2 track No.30, generated based on 1,000 posterior predictive samples. We see that the empirical means of the posterior predictive samples imitate the observed data. As expected, the position uncertainty structure follows a quasi-daily pattern.

\begin{figure*}
\centering
\includegraphics[width=30pc]{figures/Fig_ppc.pdf}
\caption{Posterior predictive distributions of HQ2 track No.30. The blue dots represent the reported ship positions, the black dots represent the posterior predictive means, and the circles calibrate the posterior predictive uncertainties (one standard deviation).}
\label{fig:ppc}
\end{figure*}

\subsection{Check of navigational uncertainties via a linearized model}

The navigational parameter estimates play a crucial role in the downstream analysis.
We propose an independent linearized model to directly estimate these navigational parameters using HQ2 tracks, and compare the obtained estimates to the results in Table \ref{table:BayesHierarResult}.

We first linearize the dead reckoning process by Taylor expansion and omit the second order terms. For $t \notin \mathcal{C}$, we have
\begin{equation}
\begin{aligned}
\hat{\lambda}^x_t & = \hat{\lambda}^x_{t - 1} + 2s_t  (1 + e_t^s) \cos(\theta_t + e_t^\theta) \\
& \approx \hat{\lambda}^x_{t - 1} + 2s_t\cos\theta_t + 2s_t\cos\theta_t e_t^s - 2s_t\sin\theta_t e_t^\theta,\\
\hat{\lambda}^y_t & = \hat{\lambda}^y_{t - 1} + 2s_t  (1 + e_t^s)  \sin(\theta_t + e_t^\theta)\\
& \approx \hat{\lambda}^y_{t - 1} + 2s_t\sin\theta_t + 2s_t\sin\theta_t e_t^s + 2s_t\cos\theta_t e_t^\theta,\\
\end{aligned}
\end{equation}
where $\hat{\lambda}_t^x$ and $\hat{\lambda}_t^y$, in units of km, are the reported zonal and meridional displacements of a ship to its position since the last celestial correction.
$e_t^s$ and $e_t^\theta$ denote the errors in the relative ship speed and ship heading.
We assume that $e_t^s, e_t^\theta$ follow $N\left(0, \tau_s^2\right)$, $N\left(0, \tau_\theta^2\right)$, where $\tau_s, \tau_\theta$ calibrate the  magnitudes of uncertainties in the relative ship speed and ship heading, respectively.
$e_t^s, e_t^\theta$ are assumed to be independent across time. For $t \in \mathcal{C}$, we have
\begin{equation}
\hat{\lambda}^x_t \sim N(0, (\tau_x \cos\psi_t)^2), \ \ \
\hat{\lambda}^y_t \sim N(0, \tau_y^2),
\end{equation}
where $\tau_x$ and $\tau_y$ calibrate the magnitudes of uncertainties in celestial correction. Combining linearized dead reckoning and celestial correction, we can approximate the variances of jumps in the longitudinal and latitudinal direction by
\begin{equation}
\begin{aligned}
\text{Var}(J^x) & =\tau^{2}_{s} \Delta x^{2} + \tau^{2}_\theta \Delta y^{2} + 2 \left(\tau_{x}  \cos\psi\right)^2,\\
\text{Var}(J^y) & =\tau^{2}_{s} \Delta y^{2} + \tau^{2}_\theta \Delta x^{2} + 2 \tau^2_{y}.
\end{aligned}
\label{eq:linearized_variance}
\end{equation}
$J^x, J^y$ denote the jumping distances in the longitudinal and latitudinal direction, and $\Delta x^2$ and $\Delta y^2$ denote the sum of squared distances between consecutive reports from the last celestial update to the position right before the next celestial update. We drop the dependence on $t$ for notational convenience.

We use $20,694$ midnight jumps identified from 943 HQ2 tracks to estimate the navigational parameters $\tau_x, \tau_y, \tau_s, \tau_\theta$. All the jumps are binned by 20km$\times$20km grids. Within each bin, we obtain the sample variances, $\widehat{V}_x$ and $\widehat{V}_y$, of the jumping distances in the longitudinal and latitudinal direction. Approximately the sample variances in each bin follow
\begin{equation}
\label{eq:variancelikelihood}
(n-1)\frac{\widehat{V}_x}{\text{Var}(J^x)}\sim\chi^2_{n-1}, \ \ \ (n-1)\frac{\widehat{V}_y}{\text{Var}(J^y)}\sim\chi^2_{n-1},
\end{equation}
where $n$ is the sample size in that bin. We set up standard non-informative priors on all the navigational parameters, and combine the likelihood specified in Equation \eqref{eq:variancelikelihood} to obtain the posterior distributions.

The results are summarized in Table \ref{table:LinearResult}. For dead reckoning, due to the linearization, we obtain smaller estimates of the uncertainties in the relative ship speed and ship heading, compared to the results in Table  \ref{table:BayesHierarResult}. For celestial correction, we see that the two methods produce consistent estimates of the uncertainties.

\begin{table*}
\begin{minipage}{\textwidth}
\centering
\caption{Posterior distributions of the navigational parameters based on the linearized model.}
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
Quantiles & 5\% & 25\% & 50\% & 75\% & 95\%  & std\\
\hline
$\tau_{x}$\xdef\@thefnmark{}\@footnotetext{$\tau_x$ and $\tau_y$ denote the uncertainty in celestial correction along the longitudinal and the latitudinal direction, respectively. $\tau_s$ and $\tau_\theta$ denote the uncertainty in the relative ship speed and ship heading.} (km) & 30.91 & 31.12 & 31.27 & 31.44 & 31.67 & 0.23\\
\hline
$\tau_{y}$\ (km)& 24.17 & 24.37 & 24.50 & 24.64 & 24.85 & 0.21\\
\hline
$\tau_s$\hspace{0.12cm} (\%) & 15.07 & 15.25 & 15.38 & 15.49 & 15.65 &  0.17\\
\hline
$\tau_\theta$\ (rad) & 0.058 & 0.061 & 0.063 & 0.064 & 0.067 & 0.003\\
\hline
\end{tabular}
\label{table:LinearResult}
\end{minipage}
\end{table*}

\subsection{Additional checks}
Figure \ref{fig:speed} shows the posterior distribution of ship speed $s_t$ for HQ2 track No.30. The blue line represents the empirical ship speed, the black line represents the posterior median, and the gray shadow calibrates the first and the third quantiles of the posterior distribution. We note that the unphysical blue peaks correspond to the large celestial updates along the ship trajectory, and the proposed state-space time series model in Section \ref{section:learnable 2-hourly} helps reasonably smooth out $s_t$.

\begin{sidewaysfigure}
\includegraphics[width=55pc]{figures/Fig_speed.pdf}
\caption{The posterior distribution of ship speed $s_t$ for HQ2 track No.30. The blue line represents the empirical ship speed, the black line represents the posterior median, and the gray shadow calibrates the first and the third quantiles of the posterior distribution. The blue peaks correspond to the celestial corrections along the ship trajectory.}
\label{fig:speed}
\end{sidewaysfigure}

Figure \ref{fig:evolution-variance} demonstrates the distributions of the navigational parameters $\sigma_s$ and $\sigma_\theta$\footnote{See the definitions of $\sigma_s$ and $\sigma_\theta$ in Equations \eqref{eq:transition-sp}, \eqref{eq:transition-th} and the discussion therein.} across HQ2 tracks, which calibrate the evolutionary uncertainties in ship speed and heading, respectively.  We see that $\sigma_s$ is small enough so that the lower truncation on $\epsilon_t^s$ has little effect. In addition, although $\theta_t$ is a bounded quantity, $\sigma_\theta$ is also sufficiently small so that there are no unexpected
consequences to assume a normal distribution for $\epsilon_t^\theta$.
\begin{figure*}
\centering
\includegraphics[width=14pc]{figures/sig_sp.pdf}
\includegraphics[width=14pc]{figures/sig_th.pdf}
\caption{The histograms of the posterior means of the navigational parameters $\sigma_s$ and $\sigma_\theta$ across HQ2 tracks, which calibrate the evolutionary uncertainties in  ship speed and heading, respectively. }
\label{fig:evolution-variance}
\end{figure*}

We also summarize the variability of the navigational parameters across HQ2 tracks in Figure \ref{fig:variability}. In particular, we show the histograms of the posterior means of the navigational parameters $\tau_x$, $\tau_y$, $\tau_s$, and $\tau_\theta$ for HQ2 tracks.
\begin{figure*}
\centering
\includegraphics[width=14pc]{figures/tau_x_hist.pdf}
\includegraphics[width=14pc]{figures/tau_y_hist.pdf}
\includegraphics[width=14pc]{figures/tau_sp_hist.pdf}
\includegraphics[width=14pc]{figures/tau_th_hist.pdf}
\caption{The histograms of the posterior means of the navigational parameters across HQ2 tracks. $\tau_x$ and $\tau_y$ calibrate the uncertainties in celestial correction along the longitudinal and the latitudinal direction, respectively. $\tau_s$ and $\tau_\theta$ calibrate the uncertainties in the relative ship speed $\hat{s}_t/s_t$ and ship heading, respectively. }
\label{fig:variability}
\end{figure*}

\bibliographystyle{apalike}

redpajama_set_name: RedPajamaArXiv


Tell the EPA to get the lead out!
Almost three decades after the landmark Lead and Copper Rule went into effect, children and pregnant women are being poisoned by lead in our nation's drinking water in part because there is no requirement that the EPA be notified about where lead pipes are.
There are no safe levels of lead in drinking water. Low levels of exposure to lead in children are linked in hyperactivity, anemia, lower IQs, physical and learning disabilities and slowed growth. In pregnant women, lead can be transmitted to the bones of the developing fetus. In adults, lead can lead to memory loss and high blood pressure. Bones can retain lead for decades.
Let us sort the crap politics out and get something happening to clean up unsafe water. What century is this after all?


text:
У графічній перспективі, зникома точка () — точка, де сходяться паралельні лінії. У випадку одно- чи двоточкової перспективи зникомі точки лежать на лінії обрію, у випадку триточкової перспективи третя точка лежить у зеніті чи надирі, залежно від того, знизу чи згори ведеться спостереження.

Якщо в реальному світі дві або більше прямих паралельні одна одній, але не паралельні щодо площини зображення, тоді вони мають одну й ту саму зникому точку. Перспективне зображення цих прямих не буде паралельним. Якщо прямі повністю розгорнуті на малюнку, зображені лінії перетинатимуться у зникомій точці.

Означення
Формально зникома точка — це точка в площині зображення π, яка визначена прямою в просторі. Нехай задана точка розташування ока — O і пряма L не паралельна до π і нехай M буде прямою через O паралельною до L. Тоді зникома точка прямої L — це перетин M і π.

Криволінійна і зворотна перспективи
Криволінійна перспектива — це зображення з 4 або 5 зникомими точками, у разі 5-точкової перспективи зникомі точки складають коло з 4 зникомими точками на головних курсах N, W, S, E й однією в центрі кола.

Зворотна перспектива — це зображення зі зникомими точками розташованими поза картиною з ілюзією, що вони «перед» картиною.

Аксонометричні проєкції
Аксонометричні проєкції — ізометрична, диметрична та триметрична розроблені для зображення об'єктів задля передачі технічної інформації. Вони використовують паралельні лінії, але не мають зникомих точок. Отже, їх не можна використовувати для реалістичного тривимірного малюнка.

Примітки

Геометрія в комп'ютерному зорі

redpajama_set_name: RedPajamaWikipedia


Magnetism is based on theories of quantum mechanics and relativity devised 80 years ago, but it continues to throw up surprises. We are very interested in the magnetism of solids with no d- or f-electrons, and effects of magnetic fields in electrochemistry and biology.
This is a high-temperature phenomenon exhibited by thin films and nanoparticles of oxides that lack the d- or f-electrons normally required for ferromagnetism. We discovered the phenomenon in HfO2 thin films in 2004. The magnetization curves look ferromagnetic, but show no hysteresis. Surprisingly, the magnetism can be destroyed by dispersing the nanoparticles with an inert spacer. d0 magnetism is puzzling because the energy scale of the phenomenon is very high ~ 1000 K, the intrinsic length scale is > 100 nm, and the magnetization is very low, apparently involving only a small fraction of the sample volume. Defects are expected to play a key part in the explanation.
Examples of d0 magnetism. Magnetization curves of a 50nm of HfO2 film, exhibiting no temperature dependence (left). 4 nm CeO2 nanoparticles behave similarly, but dispersing them with 15 nm Al2O3 nanoparticles causes the magnetic moment to nearly disappear.
Claims that flowing hard water through a magnetic field gradient can influence the subsequent precipitation of CaCO3 up to a week later, are usually dismissed because there has been no plausible physical explanation. The recently-established presence of polymeric prenucleation clusters in equilibrium in calcium carbonate solutions provides a clue to how magnetic water treatment might possibly work.
Nucleation from the liquid phase is an essential first step for the growth of crystals or the evolution of gas bubbles from solution. Both processes appear to be sensitive to the presence of a uniform magnetic field, which changes the nucleation rate. A field influences the periodic growth and release of hydrogen gas bubbles produced by electrolysis of water, which can be followed by high-speed photography. The field may inhibit or enhance the bubble release, controlling the bubble size, or even destroy the periodic behaviour.
Hydrogen bubbles. Hydrogen bubble growth at a platinum microeletrode with the corresponding overpotential waveform. The panel on the right shows that a 5 T uniform magnetic field can reduce the releae frequency and double the bubble size.


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If elegance is your style, then this drop waist dress from Life is a perfect pick for you. Made from good quality fabric, it will provide you with immense comfort throughout the day. Featuring a round neck, short sleeves and printed pattern, this dress possesses a fashionable allure.
Fall in love with the soft texture as you adorn this drop waist dress. Complement this dress by teaming it with a pair of matching sandals.


Has that ever happened to you? Have you ever listened to the gospel reading and then said to yourself, "What is he talking about?" Have you ever read a piece of scripture and wondered, "What does this mean? And how does it apply to my life?" I have and I still do. Almost every week that's my starting place with the upcoming Sunday gospel.
This week was no different. Sheep and shepherd, gate and gatekeeper, thief and bandit, voices, abundant life. What's all that mean? How does it apply to our lives? I can't put it all together in a neat package that makes sense. There's too many contradictions and mixed metaphors.
At the beginning of today's gospel (John 10:1-10) it sounds like Jesus is suggesting that he is the shepherd who enters the gate, unlike the thieves and bandits who hop the fence. The sheep, he says, listen to and follow the voice of the shepherd but run from the voice of strangers. But then later in today's gospel Jesus says that he is the gate. So he is the shepherd who enters the gate and he is also the gate the shepherd enters? And then he says that we can enter him as a gate to abundant life?
It reminds me of when our younger son was little. He knew the sayings, "How about them apples?" and "Try that on for size." But that's not what he would say. He would say, "Try them apples on for size." I told him that doesn't make sense. It's either one or the other but not both at the same time. I want to tell Jesus the same thing. I imagine, though, that he would respond pretty much like my son would. He would just laugh and then say again, "Try them apples on for size." That's what I want us to do today with this gospel. I want us to "try them apples on for size." I really believe that is what Jesus is asking us to do.
I say that because St. John, today's gospel writer, clearly states that Jesus was using a "figure of speech." That's why this text, like so many of Jesus' sayings and teachings, is difficult to understand, and that's also the key to dealing with this text. This text cannot be taken literally. It makes no sense. The rational, logical, intellectual mind cannot understand a figure of speech. The figure of speech points beyond itself, in this case to abundant life. A figure of speech asks us to think, see, and listen differently. So let's take today's gospel and play with the images, dance with the ideas, ponder and muse over this figure of speech, and see what it brings up in us and where it takes us.
Let's start with abundant life. That, Jesus says, is why he came; that we might "have life, and have it abundantly." What does that mean for you? Where is your life abundant? And in what ways is it less than abundant? Keep in mind, however, that abundance is not something we get. It's a way of living and being.
The abundant life is not about quantity, wealth, success, approval, popularity, security, being number one, or any of the other things we often think it is. How many times have you gotten what you wanted, been what others said you should be, or done what was encouraged and rewarded by society only to discover your own emptiness and poverty? You had it all but you were not abundant. No, the abundant life is touching and living the divine life. It's a quality not a quantity. It's about meaning, integrity, purpose, creativity, relationship, and wholeness. The abundant life adds to the life of others and the world. It's life that leads to life, love that leads to love, joy that leads to joy, hope that leads to hope, kindness that leads to kindness, generosity that leads to generosity, beauty that leads to beauty, and gratitude that leads to gratitude. It does not add to the pain of the world but adds to and enhances life, our own as well as others'.
Isn't that what we really want? Isn't that what we want for ourselves and for each other? I do? Don't you? I want us to follow the shepherd into the pasture of abundance.
Who is the shepherd? The obvious answer is Jesus. That's certainly not a wrong answer. Later in John's account of the gospel Jesus will say, "I am the good shepherd" (John 10:11). But is that the only answer? Could there be other shepherds? And if so, who might that be? What about Psalm 23 that we read a few minutes ago? That said that the Lord, God, is our shepherd. He revives, he leads, he protects, he feeds, he waters, he pastures. What about you and me? Could we do those things? Could we be the shepherd? Who are the people that have shepherded, guided, nurtured, and protected your life? When have you shepherded the life of another? Maybe the shepherd here is anyone or anything that nourishes, fosters, empowers, and guards life. It could be God, Jesus, you, or me. Shepherding is not only an action but also a relationship. Ultimately, though it seems to be God's way of being in the world, in the life of Jesus, and in your life and my life.
What about those thieves and bandits? Jesus is not warning us about having our things stolen but about having our life stolen and our abundance depleted.
Thieves are stealthy. We often don't know they've been here until later. They come in the night when we are asleep and unaware. Then one day we wake up and wonder what happened to our life. How did we get to this point? Our life is a mess and we don't know when or how it happened. Where did our life go? Maybe time caught up with us, maybe we took a relationship for granted, maybe we procrastinated, maybe we were too comfortable, maybe we became complacent, maybe we were too busy or preoccupied, maybe we weren't paying attention, maybe we just gave up, maybe we opened our heart to the wrong person or thing.
Thieves are sneaky and they often wear the disguise of acceptance: work, money, productivity, knowledge, a relationship, meeting other's needs and expectations, saying yes and being available, striving for quality in what we do. There's nothing wrong with any of those things until they begin to thief our life. And sometimes they do.
Bandits, on the other hand, are overpowering. They come in the daylight. They don't even try to hide. We know when they are here and when we are being robbed. It's those times when we know our life is less than we want it to be, we recognize the patterns and behaviors that are destructive, we know there is a better and different choice to be made, but we continue down the same path. It's falling down and refusing to get back up. It's recognizing our brokenness and refusing to ask for help. It's choosing our resentments and anger over forgiveness. It's feeling the pain but refusing the treatment. It's knowing what will give life but giving in to fear.
In what ways has your life been stolen? Who or what are the thieves and bandits in your life? What will you do now? What have you learned? How will you guard your heart against the thieves and bandits of an abundant life? The abundant life is always a matter of the heart.
We are the gatekeepers of our lives. We are the keeper and guardian of our heart. Guarding our heart means staying awake, being watchful, and remaining diligent. Awareness of and reflection on what is happening within and outside us are the gatekeeper's key. Depending on who or what it is we either open our heart or keep it closed. Sometimes we need to open the gate and sometimes we need to keep the gate of our life closed. Look back on your life and you can see that.
To whom or what have you opened your heart in the past? Where did it take you? Did it lead to the pastures of abundance or was it a dead end, or maybe a detour? When have you opened your heart and been surprised by possibilities and a fullness of life you could never have imagined?
Who or what stands at your gates today? Who or what are the people, opportunities, possibilities, and choices that seek entry into your life? The gates in our life are threshold places, times of discernment, moments of transition, and decisions to be made. To whom or what do you need to open and to whom or what do you need to close? How might your life change if you open? What might you miss if you close? Will your life be enhanced and empowered by opening or will something be lost? What pain or wound might you prevent by closing? Where is fear in all this? Which will enhance and add to life today, opening or closing?
To what voices are we listening? The voices to which we listen form and shape our lives throughout our lifetime. And there are a myriad of voices in our world. What voices are rattling around in you? Vying for your attention? For better or for worse we eventually begin to speak with the voice that has formed and shaped our lives.
Who do you sound like when you speak? It could be your mom or dad, your spouse, your boss, your fear or anger, your wounds, the church, politicians, television, the internet, friends, money, success, popularity, expectations, failures, self-criticism or self-hatred. Is it the voice of thieves and bandits? Or is it the voice of wisdom, life, love, compassion, beauty, generosity, hope, joy? The voice of Jesus? The voice of a shepherd? Which voices are loudest in your life today? What voices have you invested in? Are they worthy of your trust and your life or are they bankrupt? Are they bankrupting you? To which voices do you need to close the gate of your heart and to which ones do you need to open?
In a few moments we will baptize Lane and James. When they come to the font they will stand at the gate and we will make vows on their behalf. Some vows will be about opening, and some about closing. Then we will vow to open our hearts to them, to support them, and to be their shepherds. What voices do you want them to listen to? How will you help them learn to recognize the shepherd's voice? To what would you tell them to open their hearts and to what would your tell them to close? And what if you followed your own advice? How might your life be different? Abundant? More whole and full?
Thank you Fr. Julius. I pray you and your congregation an abundant life.
Superior set of questions for GOOD SHEPHERD Sunday. Excellent reflections. Thank you Michael.
Thank you. May God bless you with peace and abundance.
Love how the "confusion" in a child's mind leads us to greater questions. I learn so much reading your posts!
PS My kids all got mixed up with doing something themselves. Each one said, "I did it my own very self!" with great pride. A wonderful memory on Mother's Day. Your post reminded me of those little lines.
Thank you Jeri. "I did it my own very self." I can hear the joy and excitement in that and I remembering seeing it in our boys. Yes, good memories. I hope you had a blessed and happy Mother's Day.


MANCHESTER, England -- Romelu Lukaku sat out Manchester United's final training session before their Champions League game against Juventus in Turin on Wednesday.
Lukaku missed Saturday's 2-1 win at Bournemouth after picking up a muscle injury in training.
Despite undergoing treatment over the weekend, the 25-year-old was not deemed fit enough to take part in a training session, open to the media, at Carrington on Tuesday.
The United squad are set to fly to Italy on Tuesday afternoon ahead of their meeting with Juventus.
The Serie A champions' 1-0 win at Old Trafford on Oct. 23 is the last game Lukaku started. The Belgium international was dropped to the bench for the win over Everton after going nine games without a goal.
Diogo Dalot was the only other first-team player to miss training on Tuesday.
Marouane Fellaini and Antonio Valencia, who have both missed the last five games through injury, took part in the session.
Marcos Rojo and Eric Bailly, who were not part of the squad at Bournemouth, were also involved.
Meanwhile, Shakhtar Donetsk manager Paulo Fonseca has backed United midfielder Fred to become "one of the most important players in England" despite his difficult start at Old Trafford.
"He's a big talent and he's still developing," he said, speaking exclusively to ESPN FC.


text:
Episode 7: Mario Riveira – Founder of Mayor of Fitness
Mario is a precise example of someone who knows what he is great at and took his talents to build a successful business...
WE VOTED FOR THE MAYOR OF FITNESS
PHOTO BY RICK SCOTCH / SCOTCH PARLOR
MARIO RIVEIRA
FOUNDER OF MAYOR OF FITNESS
MAYOROFFITNESS.COM
I want to be everywhere. I want to be a public figure and who's the public figure in the city? It's the Mayor...Mayor of Fitness.
We interviewed Mario Riveira aka Mayor of Fitness. He is the founder of a Fitness Company that provides top of the line fitness management and gym consulting services.
Mario is a precise example of someone who knows what he is great at and took his talents to build a successful business. Beyond his skill abilities, Mario is a determined, and ambitious entrepreneur who works extremely hard to achieve his dreams. Combine those skills sets and his entrepreneurial traits, and you instantly know why his fitness management company is taking over the city.
In this podcast, we talk about how it all began for him as an entrepreneur. From the evolution of where he is today, what motivates him to how important his health is for the foundation to everything he does. During our conversation with Mario, I discovered the many "hats" he needs to wear beyond just being a fitness trainer. Furthermore, on the lifestyle aspect Mario shares his insights on what healthy cocktails to order at the bar.
If you don't love what you do, you'll never succeed.
In the early stages of building his branding strategy, he realized the perfect public figure of San Francisco is the City Mayor. At that point, he decided he would base his business strategy and personal brand to emulate the city mayor. An individual who gets things done and someone you can count on for your fitness goals and aspirations. It's working too, he is now walking around the city and people are yelling "Mayor of Fitness."
As the public awareness of the Mayor of Fitness is expanding, he continues to hustle every day to build his company, brand and most importantly building relationships with his clients. Mario's passion to coach his clients and be part of their fitness journey is truly inspiring. You hear it in his voice; his dreams are big, he lives in a city he adores, he is doing what he loves and he is becoming the true Mayor of Fitness. He got our vote!
Enjoy the Podcast and keep being a "DOER."
Mario give his intro (1:08)
Mario decided he is going to build a business (3:15)
The finances of being a personal trainer vs having his own fitness company (4:10)
The evolution of where Mario is today. (5:20)
When did you know you were an Entrepreneur? (5:40)
Cannabis Industry Hustle. (6:14)
Always had the love for entrepreneurship. (7:00)
Saving money. (8:00)
Going All In (8:27)
What has been the evolution of the Mayor of Fitness? (8:55)
What is your typical day like? (10:35)
"The Hustle" (11:05)
Business to Business Fitness Services (11:35)
The Mayor of Fitness Process (12:55)
Mario explain the importance of his job as a trainer. (14:10)
Instant Gratification in health and business. (14:56)
Psychology & Counseling as a Fitness Trainer (15:32)
What has been Mario's biggest learning experience? (17:00)
Loving the process (18:10)
The benefits of being an entrepreneur (18:55)
What piece of advice for someone who is looking to start a business? (20:00)
Talks about constantly learning (21:25)
WIDWID...What's Your Inspiration? (22:12)
Mario being part of his clients journey of life. (23:30)
What's Mario's Next Move? (25:18)
Blogging & Photography(27:15)
How does Mario stay motivated? (29:03)
Podcasts & Influencers (30:25)
Cocktail of choice when living the healthy lifestyle. (31:45)
How does Mario want to define his legacy? (33:28)
Self-Branding (34:04)
Mario is... (34:51)
Contact Info (35:02)
FOLLOW THE MAYOR OF FITNESS
FITNESS MANAGEMENT INSTAGRAM
Editor's Note: This story was shared through a Scotch Parlor Podcast. Photography was done by Rick Scotch in San Francisco, CA
BEST DRESSED MAN: MIKE IRONS OF SITUATION CONTROL
In this podcast, you will learn how Mike has always been a real doer. He was once a DJ hustling to book gigs. He built a T-Shirt company called Irons Brand, which ended up being quite popular in Japan. Today, he finds himself doing his real passion which is helping clients look and feel good in his one of a kind suits.


What's that? Michael Beasley is in trouble again? No kidding!! From marijuana to speeding tickets and now an altercation with a fan during a game at the Dyckman Tournament in Washington Heights, Michael Beasley can't seem to stay out of the spotlight--for all the wrong reasons! This time, he got in an argument with a fan during the game which ended up delaying it 10 minutes.
Beasley began jawing at a group of heckling fans during a free throw, before walking over and shoving him in the face--which can be seen in the video below. Afterwards he went over to shake the fans hand, but the situation again escalated and security had to step in once more--not seen in the video.
Beasley's immaturity is the biggest obstacle in him stepping up to the next level, until he grows up a bit, we'll continue to hear stories like these. It could take a while too--just look at Memphis' Zach Randolph--he's 29 and his maturity he's developed--FINALLY was a big reason for this!