It is almost the end of October, and I only added one blog entry so far this month. One reason is that we had a minor health scare: when my wife traveled to Hungary last month, she landed in a hospital on arrival, as she had an unexplained seizure during her flight.

She is doing well, thankfully. She returned home safely, with no lasting effects. There will still be a few more tests to be sure, but the scare is largely over.

This unfortunate incident, however, allowed us to experience first-hand the state of the health care system in Hungary, about which we read so much in recent years. Yes, as it is well known, the system is badly underfunded: salaries are miserably low, and sometimes, even basic supplies are lacking.

But we cannot utter a bad word about the health care professionals that my wife encountered. They were impeccably professional and helpful, going out of their way to assist us, even beyond the call of duty. When my wife’s cell phone was acting up, one nurse volunteered to help fix it with my assistance. When I explained to a doctor that I cannot visit her in person because I happen to be a continent away, he handed the phone to my wife, allowing us to have a conversation (finally!) using a hospital line for a few minutes. They helped with insurance matters, too, and they issued a very thorough discharge report, complete with a CD-ROM containing the results of a CT scan.

All in all, we have nothing but praise for these overworked and underpaid health care workers, some minor mishaps notwithstanding. They were kind, they were helpful, and every one of them that I spoke with was ready to assist, forthcoming, and flawlessly polite. Thank you for your kind care. Köszönjük szépen.

Tonight, the two most important women in my life are both in the stratosphere.

My wife is en route to Hungary, to visit her Mom. At this moment, she is about halfway across the North Atlantic Ocean.

Meanwhile, my Mom is about to land in Beijing, on a memorable adventure I hope: a one-week trip to China with a friend.

And I am stuck here with three cats, fending for ourselves in the Ottawa wilderness. I would say that life is not fair, but I actually enjoy staying at home for a change. I like my home, and travel is such a chore.

CNN reports that ACARS messages received from MS804 in the minutes before its disappearance indicate a fire on board.

I was able to find online a copy of the screenshot discussed by CNN:

CNN’s talking heads are still discussing the terrorism theory, but in light of these data, it looks increasingly unlikely. It is very difficult to imagine a terrorism scenario that begins with one of the cockpit windows.

This iconic photograph was snapped 60 years ago today from the window of a Boeing 707 prototype.

In case it’s unclear, that airplane is flying upside down. And no, it is not about to crash. It was just in the middle of an aerobatic maneuver called a “barrel roll”.

It was, at least for its test pilot, Alvin “Tex” Johnston, a perfectly normal demonstration of what his new airplane is capable of doing. His company’s CEO, Bill Allen, rather disagreed, but the 707 went on to become a great commercial success and the famous Seattle barrel roll became a matter of legend. I first heard about this barrel roll incident more than 30 years ago from my long gone friend Ferenc Szatmári, physicist, private pilot and storyteller extraordinaire.

Reportedly, Allen never really got over this incident; decades later, when he received a framed copy of the photograph above at his retirement banquet, the memento was accidentally left behind. Then again, if someone gave me a minor heart attack like that (after all, for Allen, the future of his entire company was at stake), I, too, may be inclined to hold a grudge.

I have been neglecting my blog in the past two weeks, but I had a good reason: I was traveling again.

I was once again in Dubai, where I spent some time on the 14th floor of an office building. From that building, I noticed another, very strange edifice with barely a window on its first few floors, and some absolutely giant fans on its roof. I was wondering what it was: a telephone exchange? A data center? No… a central air conditioning plant for a city block. Wow.

Later that evening, we walked to a restaurant in downtown Dubai, and I had a chance to stare at one of the city’s newest attractions (I don’t know how long it has been around, I only just noticed it now), a red streetcar. Lovely.

I only spent a day in Dubai before heading off to Abu Dhabi in a rental car. Rather than purchasing a new GPS (or an expensive map for my old Garmin) I decided to rely on Google Maps on my phone. Not that I really needed to; once I found my way out of Dubai, I was quite familiar with the route and the location of my Abu Dhabi hotel. Indeed, I found the hotel without difficulty, and early next morning, I found myself staring at a surprisingly faint desert sunrise, the Sun obscured by sand and fog:

I spent four days in Abu Dhabi, working hard. Having my own transportation really made things easier, not just to commute back and forth between my hotel and my client’s office, but also to do mundane things like going to a supermarket for some fresh fruit or a bottle of Coke purchased at a tiny fraction of the hotel price.

Four days fly by like nothing, and all too soon, I found myself on the highway again, heading back to Dubai. This time around, I used my phone not just for navigation but also as a dash cam:

This turned out to be a big mistake. Barely more than ten miles from my hotel, just as I was entering the thickest parts of the freeway spaghetti in Dubai, my phone went dark. I couldn’t for the life of me figure out why a phone’s battery gets depleted when the phone was plugged in just fine. (I solved the mystery today through some experimentation: the use of two applications, but especially the continuous HD video recording overheated the phone, and it stopped charging the battery as a precaution.) Anyhow, I was left to my own devices, so it was with no small amount of pride when, maybe 20 minutes later, I got out of the rental car in front of my hotel, having found the place from memory, driving through downtown Dubai like a native.

The same evening, I took the rental car back to the Dubai Mall, once again navigating flawlessly to the right parking lot, finding the rental agency’s kiosk, and completing the return of the vehicle exactly as planned, nailing the timing spot on for my next meeting.

The next evening, I once again went to the Dubai Mall, but this time on foot. The hotel is about a mile from the Mall. Normally, walking a mile is no big deal, but try doing it in 36°C weather! I felt very brave. Not only did I walk to the Mall, I also walked back from the Mall, after I did some shopping (some gifts and once again, some groceries). On the way back, I paused for a moment, looking at (and listening to) the cacophony of non-stop large scale construction right across the Mall:

I spent two more days in Dubai, again very busy. On Sunday, I checked out of my hotel, heading to that same 14th floor office again… and en route, I got wet. There was rain! Not a lot of rain to be sure but still, any rain is a delightful experience when you are in one of the driest places of the entire planet.

Finally, Sunday evening arrived and once again I was heading back to Dubai airport. Heading home.

Halfway through the flight, I looked out my window and saw a half moon over a deep blue sky. It was beautiful.

And then, after a grueling 14.5 hour nonstop flight, I was at Washington’s Dulles airport again, going through a surprisingly quick, efficient, and painless border and customs inspection (the concept of international transit does not exist at North American airports.) As I was walking towards my Ottawa departure gate, I encountered a delightful little facility:

Yes, a rest room for canines! As the door was open, I was even able to take a peek, although for some reason, my phone camera was badly out of focus:

Even in this blurry picture though, perhaps it is clear that the room contains a piece of a fence, some fake grass, and a plastic fire hydrant replica, offering maximum comfort for canine companions. It really gave me a good laugh!

And then, after a brief delay (our aircraft was late inbound), I was flying again. Less than 90 minutes later, I was at Ottawa airport, and soon in a taxi, heading home.

And I am still playing catch-up. So many little things pile up in two weeks!

I was quite doubtful about the theory that Germanwings 9525 was brought down intentionally by its co-pilot. I was more inclined to bet on a health problem. Perhaps I could argue that in a sense, I was right: mental illness can be just as debilitating as any physical condition. And a person suffering from severe mental illness can be quite capable of doing unspeakable, irrational things, whether it is due to “voices”, an exaggerated sense of self, or whatever.

In response, airlines around the world are now instituting the “two person rule”, requiring two members of the crew to be present in the cockpit at all times. Like the reinforced cockpit door, I wonder if this recommendation is based on the best scientific analysis available, or if it is another example of knee-jerk “security theater”.

I am especially concerned because in response to a very low probability event (a pilot going berserk and locking the other pilot out), we introduce another possibility, the probability of which may in fact be higher: that a non-pilot member of the crew gains access to the cockpit with nefarious intent.

Arguably, this is replacing one black swan with another black swan but still… think of the reinforced cockpit door itself. It was supposed to prevent another 9/11-like scenario, however unlikely it is for one to occur again; instead, it introduced its own unique dangers, and now claimed 150 lives. Not only that, but I just found out that the door mechanism itself has been responsible for at least one unscheduled landing, when it began smoldering and the resulting smoke caused pilots to divert.

Wouldn’t it make more sense to listen to the professionals, say, at the American NTSB and ask for their informed analysis and opinion before introducing these regulations?

It’s all over the news this morning: the young, inexperienced copilot (or maybe not that inexperienced: he was recognized by the FAA in September 2013) was in the cockpit, the pilot was locked out, and the copilot deliberately set up the plane for descent with the intent to commit mass murder-suicide.

However, I have lingering doubts. Specifically, I think it’s too early to discard concerns about the copilot’s health.

We learned that in the first 20 minutes, the conversation between the two pilots was cheerful, but during the last 10 minutes, as they reviewed the landing checklist, the copilot’s responses were “laconic”.

Could it not be that the copilot was experiencing a severe health problem, for instance, a stroke?

Nah, how could that be, he was only 28!

Except that I personally knew a 28-year old young man, an engineer in seemingly perfect health, no destructive habits, nice, intelligent, soft spoken, gentle: only to be found dead in his bedroom one day, after he suffered a fatal brain aneurysm.

Confusion caused by the onset of a stroke or aneurysm may also explain why the copilot may have entered incorrect information into the plane’s autopilot. It may also explain why he never said a word.

One thing it does not explain is why the pilot was unable to enter the cockpit using an emergency code. It is not clear to me what procedures were available to the pilot at this point, but if the copilot had to actively prevent the pilot from entering the cockpit, then obviously the suicide theory must prevail.

Regardless of the outcome, I also consider these 150 (or 149) innocent lives victims of post-9/11 hysteria. In our fear over the possibility that hijackers with the ability to pilot a plane might enter the cockpit, we made sure that it is impossible to enter to cockpit in a legitimate emergency… something that is far more likely to happen than another 9/11-style hijacking. Sadly, I fear that this is not the last time that people get killed as a consequence of our thoughtless, knee-jerk response to terrorism that is based on ignorance, not facts and scientific analysis.

Why would a perfectly good airplane, flying along its designated route in broad daylight over Europe, suddenly decide to descend in an orderly fashion until it flew into the French Alps?

Maybe we will find out when the black boxes are analyzed. (One reportedly has been found already.) Given the controlled nature of the descent and the fact that the aircraft never deviated from its planned route, I fear that it will turn out to be something blatantly stupid and trivial, such as the autopilot’s altitude hold being accidentally disengaged, or the wrong value entered.

I am still digesting the news, which I received while in Hungary, that Leonard Nimoy, aka. Mr. Spock, is no longer with us.

And now there is breaking news that Harrison Ford, aka. Han Solo, aka. Indiana Jones, crashed while flying solo in a vintage WW2 aircraft in California. The good news: according to the LA Fire Department, his injuries were moderate and he was alert as he was transported to hospital.

I wish him a speedy recovery.

Five days ago, I was sitting on an Emirates Airlines flight from Dubai to Budapest.

Our flight took an unusual route. Normally (well, at least within my limited experience) such flights take a route north of Iraq, flying over Iranian airspace towards Turkey. Not this time: We flew across the Saudi desert instead, then turned north over the Sinai peninsula before entering Turkish airspace and turning northwest again. I was wondering about that kink in our trajectory: was it weather or perhaps some airspace over the Mediterranean was closed for military reasons?

As a service to business class customers, Emirates provides a limo service to the destination of your choice on arrival. I was wondering how I would find the limo pick-up location, but it was easier than I thought: the chauffeur was waiting for me at the customs exit, holding up a sign bearing my name. During the journey to my hotel, he told me about his son who wishes to become a particle physicist at CERN. So for a while, we were discussing the Higgs boson and teraelectronvolts, instead of more customary topics, like Hungarian politics.

I rented a car in Budapest, for my mother and I to take a short trip to southern Hungary, to visit my mother-in-law. As we had the car for a whole weekend, on Sunday we decided to take another small trip, this time to the north of Budapest, the small but historical city of Visegrád.

I used to live in Visegrád, from 1974 to 1977, mostly in this building:

At the time, this building served as a resort owned by the Hungarian Industrial Association. As a member of a crafts artisan cooperative, my mother was entitled to vacation in this place, which we did in the spring of 1974. This is how she came to meet my stepfather who at the time was the manager of this facility. To make a long story short, we lived in the manager’s apartment for several years, while my parents built a new house in the same town. I have fond memories of this place.

Today, it serves as a home for the elderly. It seems to be well taken care of. Much to my surprise, one of its terraces appears to have been converted into a chicken coop, complete with a rather loud rooster:

Other than these two excursions and a brief visit to a 91-year old friend who recently had a serious health crisis, I spent most of my time at my parents’ place, a small apartment on the Buda side, nearly filled by a giant dog and his favorite toy:

My parents are very fond of this animal. He is nice, but I remain committed to cats. They are quieter, smell nicer, and require a whole lot less maintenance.

And all too soon, I was on another airplane, flying “business class” on British Airways to London. I had to put “business class” in quotation marks, as there was ridiculously little legroom on this middle-aged A320:

At least, the middle seats were converted into an extra tray instead.

And the flight left Budapest nearly an hour late. The reason? The air crew arrived in Budapest late the previous night, and they had to have their mandatory rest. This presented a potentially serious problem for me: the possibility that I would miss my connecting flight, which, to make things worse, was purchased separately. I probably broke some records at Heathrow Airport as I managed to make it from the arrival gate in Terminal 3 to the Terminal 2 departure gate in only 32 minutes, which included a bus ride between terminals and going through security. I made it with about 10 minutes to spare. I checked and I was told that my suitcase made it, too.

I have to say, while I like both Air Canada and British Airways, their service doesn’t even come close to the quality of service I enjoyed on Emirates or Etihad. And I am not just referring to legroom or the age of the aircraft (the Emirates flight to Budapest was a really aged A330 and the seats, while a great deal more comfortable than these British Airways seats, were nonetheless a little cramped) but also the attentiveness of the staff on board.

Still, the flight was pleasant (except for some rather severe turbulence near the southern trip of Greenland), and some eight hours later, I was back in sunny snowy Ottawa. The land of deep freeze, where the Rideau Canal is breaking all kinds of records, having been open for well over 50 consecutive days already.

I am back from another round of globe-trotting. I spent a week in the UAE, in Dubai and Abu Dhabi, followed by a very brief detour to Budapest to visit my parents.

On arrival to Dubai, we landed nearly an hour late. The reason: a massive sandstorm, which still blanketed the city the next day. Sandstorms can be just as scary and dangerous as snowstorms here in Ottawa: they can reduce visibility to near zero, and the sand can cover roadways in no time. In fact, sand is worse than snow in one respect: it does not melt away.

I was taken to Abu Dhabi by car and during that trip, I saw the rarest of desert wonders: rain.

I was unable to book a room in the hotel of my choice in Abu Dhabi, as most hotels were full due to some defense exhibition. I did, however, find not a room, but a whole apartment at a very modest price in downtown Abu Dhabi, in a brand new apartment hotel called Bin Majid Hotel Apartments (second building from the left):

I had a one bedroom apartment, fully equipped, with a beautiful view of the sea:

Around the hotel, I befriended several cats. OK, maybe befriended is too strong a term, but one of them was certainly friendly for a while. It was a mostly white cat with a tabby tail; young, probably, but also rather small and skinny. When I saw this cat for the second time and let her (?) sniff my fingers, she would not stop following me. She rubbed against my leg, dropped on her side onto my shoes, and even when I picked her up and placed her back on a safe sidewalk, she continued to come after me. Only after I picked her up and put her onto a stone ledge did she take offense: she jumped off the ledge and disappeared under a parked car. I hope I did not offend her too deeply. I have not seen her afterwards. However, on my last morning in Abu Dhabi, I went looking for her and found instead another, equally tiny and skinny male cat: obviously an adult male, it was a tabby with a heavy limp and only one functioning eye. I felt heartbroken for these cats, although I have been assured that in the UAE, cats are generally not treated badly.

As always, the food in the UAE was both tasty and spectacular. One evening, a colleague took me to a restaurant recently opened by one of his friends. Here, we were treated with a spectacular feast:

Yum! My only regret is that I couldn’t eat it all. (Actually that’s not true: my other regret is that I came back heavier than I left, which is most unfortunate but not at all unusual when visiting the UAE.)

Then all too soon, it was time to leave. I boarded a direct flight from Dubai to Budapest, a recently introduced route by Emirates Airlines. After take-off, I enjoyed a spectacular view of downtown Dubai, dominated by the world’s tallest skyscraper, the Burj Khalifa, from my airplane window. Sadly, my attempts to take a picture were not very successful.

And just like that, this part of my journey under the desert sun, a few very busy workdays in Dubai and Abu Dhabi, was over.

It appears that an Indonesian AirAsia flight with over 160 souls on board vanished a few hours ago.

Here is the last track of the flight from flightradar24.com:

I don’t know if the tracking ended because the flight vanished at that point, or perhaps it just flew out of range of ground-based facilities and had no appropriate satellite service subscription like ADS-B. I guess we shall find out in the coming hours or days.

On my way back from sunny Abu Dhabi to autumn Ottawa. My wife asked me to bring some warm weather. I’ll try…

When you fly over trouble spots, the flight path can get interesting.

Our flight carefully avoided Iraqi, Syrian and Ukrainian airspace. We also spent as little time in Iranian skies as possible.

Soon, we’ll be flying over Hungary. Maybe I should try to wave to my Mom, in case she sees me…

To sum up:

• MH17 was shot down over a region of Ukraine controlled by pro-Russian separatists.
• Said separatists have shot down several Ukrainian military aircraft in recent weeks, including an AN-26 transport aircraft.
• Said separatists recently bragged about having acquired a powerful truck-mounted anti-aircraft missile battery.
• Said separatists posted publicly on social media, complete with eyewitness video, about having shot down another AN-26 at the time when MH17 crashed, and in the same region.
• The posts came from a source that was accepted as credible in the past, and were widely circulated in Russian media.
• The posts were removed a few hours after it became clear that the aircraft in question was in fact a civilian airliner.
• The Ukrainian government published intercepted cellular telephone calls, in which alleged separatist leaders, and possibly their Russian contacts, were discussing the incident. Although the source (Ukraine) is not impartial, no reason was presented so far to question the authenticity of the intercepts.

In light of all these facts, even if the black boxes can never be analyzed, even if the wreckage has been disturbed, even if the site has been looted, even if no investigators are allowed on the scene, I think that the basic conclusion is rock solid: MH17 was shot down by pro-Russian separatists who mistook it for a Ukrainian military airplane, and who were possibly not aware of the fact that international air traffic was allowed to fly over the disputed region at altitudes greater than 32,000 feet.

The only real questions that remain, in my opinion, concern the extent of Russia’s involvement. Where did the missile come from? Was it captured by the separatists themselves or was it provided by Russia? Who operated the missile? Was it Ukrainian rebels who may have received relevant training while serving in the military, or was it personnel provided by Russia?

From The Times of London.

The answers to these questions determine the degree of Mr. Putin’s culpability.

The Ukrainians also presented a video, showing the alleged missile truck, with one of four missiles missing, en route to Russia. There were also reports according to which the airplanes black boxes were found and have since been removed to Russia. If true, these facts only increase the culpability of Mr. Putin’s government.

Meanwhile, Russia’s propaganda machine is now in high gear, full of insinuations, including these:

• Ukrainian air defense forces were tracking MH17. (Of course they were. That’s what air defense forces are supposed to do. I know… I served in one of them 33 years ago as an unhappy conscript, trained as a radar operator.)
• Ukraine is responsible for a crime that happened over her territory. (Except when said territory is controlled by separatists who are supported, overtly and covertly, by a foreign government.)
• MH17 earlier flew near Mr. Putin’s presidential airplane, which had similar markings, and NATO may have been trying to murder the Russian president. (Mr. Putin’s airplane never entered Ukrainian airspace. NATO does not usually miss its target by several hundred miles. And the actual locations of civilian aircraft in European airspace are available to anyone with a Web browser, which includes NATO officers but not necessarily ragtag paramilitaries in the fields of eastern Ukraine.)

So there. Tensions are never a good thing, especially with a nuclear superpower like Russia, but I nonetheless believe that at the very least, Mr. Putin should be held accountable for being an accomplice in a crime against humanity.

Here is the Google Maps location of the last ADS-B coordinate transmitted by MH17:

And here is a (since deleted) social media release by Igor Ivanovich Strelkov, commander of the separatist Donbass People’s militia:

In the area Torrez just downed plane An-26, lying somewhere in the mine “Progress.” As we warned – do not fly in “our sky.” And here is the video confirmation of the “birdfall.”

Bravo, murderers. I hope you are proud. And congratulations are also due to Mr. Putin; after all, if you give a lit match to a child, you cannot really disclaim responsibility when the house burns down.

Malaysian airlines confirms that it lost contact with MH17, a scheduled flight from Amsterdam to Kuala Lumpur.

According to Interfax, the flight went down near the Ukrainian-Russian border. As this is a known conflict zone, it raises the possibility of foul play, especially considering that a few days ago, Ukraine lost a cargo plane and it may have been shot down by Russia.

On the other hand, whatever the cause, it cannot be good news for Boeing either.

So a few days ago, Inmarsat and the Malaysian government released what they call the “raw” data about the so-called handshakes between Inmarsat’s Indian Ocean satellite and the ill-fated airliner.

I looked at the data. It was presented in a most inconvenient form, a PDF file with little explanation and with copy-and-paste disabled. Nonetheless, I was able to copy the data to a spreadsheet with only a moderate effort, using Firefox’s built-in PDF viewer instead of Acrobat.

The data set is very detailed, but really, only two columns seem to be of interest: the so-called “beat frequency offset” (BFO) and “burst timing offset” (BTO).

To make a long story short: I am able to confirm the approximate location of the southern search area that was initially explored on March 18 or thereabouts. I am not able to find any rationale for preferring the southerly route over the northern route. The data are symmetrical in this regard. If the airplane flew north, it would have ended up somewhere in Kazakhstan, again in agreement with published predictions for the northerly route.

So here is what I did to arrive at these conclusions.

The little explanation that is offered suggests to me that the BFO is essentially a one-way Doppler frequency shift $$(\Delta f)$$; whereas the BTO is a two-way propagation delay $$(\Delta t)$$, with some constant, unknown bias.

The raw data are pretty confusing:

First, there is a lot of stuff happening before and around take-off, around 16:41 UTC. Then something clearly changes around 17:07, the time of the last reported ACARS contact; and once again, there is an anomaly around 18:15, the time of the last reported primary radar contact.

So rightly or wrongly I am summarily dismissing these early data points, as I cannot make heads or tails of them.

Similarly, I am dismissing the very last data points, when clearly some other anomaly was taking place.

Which leaves me with a total of 5 “clean” pairs (both BTO and BFO) of data points between 19:41 and 00:11 UTC. There are two additional data points around 18:15: one is BFO only, the other is BTO only as the corresponding BFO is ambiguous.

My basic assumptions are as follows:

I accept the published time and location of the last civilian (secondary) radar contact (6°55’15” N, 103°34’43” E at 17:21 UTC) and the last military (primary) radar contact (6°49’38” N, 97°43’15” E at 18:15 UTC) as valid. I am assuming that the aircraft flew between these points in a straight line (i.e., along a great circle) at constant speed.

I accept the validity of the aforementioned “clean” Inmarsat data points, 7 in total (2 partial). I am assuming that during this phase of the flight, the aircraft was again traveling in a straight line (that is, along another great circle) at constant speed.

For the sake of simplicity, I assume that the satellite is above the equator, exactly stationary relative to the Earth’s surface, and that the aircraft is not affected by wind. (A more refined calculation might take into account the slight motion of the Inmarsat IOR satellite relative to the ground, and also the known wind patterns at the time of the aircraft’s disappearance. Of course such an analysis is necessarily fraught with significant uncertainties, as not knowing the altitude of the aircraft means we don’t know how it is affected by wind. So the resulting error may not be much smaller than the error I introduced into my calculations by these simplifications.)

The first order of the day was to find a formulation that characterizes the airplane’s trajectory using a minimal number of parameters. For this, I drew the following diagram:

Here, $$R\simeq 6,371~{\rm km}$$ is the Earth’s radius; $${\bf s}$$ is the geocentric position vector (of length $$R+h$$) of the Inmarsat satellite, which is hovering at an altitude of $$h\simeq 35,800~{\rm km}$$; and $$\phi$$ is the geocentric angle between the position of the satellite and the initial position of the airplane, which is moving with velocity $${\bf v}$$. The angle between the initial velocity vector of the airplane and the direction of the satellite ground track position is given by $$\theta$$.

In a Cartesian coordinate system in which the initial position, $${\bf r}(t=t_0)$$ of the airplane is given by $${\bf r}_0=[0, 0, R]$$ and $${\bf s}$$ is in the $$y-z$$ plane, these vectors are given by

\begin{align}
{\bf s}&=[0,(R+h)\sin\phi,(R+h)\cos\phi],\\
{\bf r}&=[R\sin\theta\sin\omega t,R\cos\theta\sin\omega t,R\cos\omega t],\\
{\bf v}&=\dot{\bf r}=[v\sin\theta\cos\omega t,v\cos\theta\cos\omega t,-v\sin\omega t],
\end{align}

where the angular velocity is given by $$\omega=v/R$$ and of course $$v=|{\bf v}|$$.

The distance between the satellite and the airplane is then given by $$d=|{\bf s}-{\bf r}|$$. The line-of-sight velocity of the airplane, as seen from the satellite, is given by $$v_{\rm LOS}=({\bf s}-{\bf r})\cdot{\bf v}/d$$. This leads us directly to a model of the two Inmarsat observables:

\begin{align}
|\Delta t|&=\left|\frac{2d}{c}\right|,\\
|\Delta f|&=\left|\frac{v_{\rm LOS}}{c}f_0\right|,
\end{align}

where $$c$$ is the velocity of light and $$f_0\simeq 1.6435~{\rm GHz}$$ is the transmitter frequency on board the aircraft. Absolute values are used to account for the inherent sign ambiguity in the reported observables.

This is not quite the end of the story, however. We know from the description of the Inmarsat data that $$\Delta t$$ includes an unknown constant bias $$\Delta t_0$$. Furthermore, $$\Delta f$$ not only includes an unknown constant bias $$\Delta f_0$$ but also includes partial compensation for the Doppler effect by the aircraft transmitter, which we can represent by a scaling factor $$\xi$$. Thus the model for the observables must be revised as follows:

\begin{align}
|\Delta t-\Delta t_0|&=\left|\frac{2d}{c}\right|,\\
|\Delta f-\Delta f_0|&=\xi\left|\frac{v_{\rm LOS}}{c}f_0\right|,
\end{align}

This, then, represents our final model for the Inmarsat observables. The model has several adjustable parameters, namely $$v$$, $$\phi$$, $$\theta$$, $$\Delta t_0$$, $$\Delta f_0$$ and $$\xi$$, which can be used to fit the scarce BFO and BTO data points. One question remains, however: shall these be fitted together or separately? Fitting them together would require weighting data of different physical dimenions (seconds for BTO, inverse seconds, or Hz, for BFO). These weights would normally come from estimates of standard deviation on the data, but we have no such estimates. The unknown weights can completely alter the resulting parameter fits, and produce nonsensical results.

Instead, I opted to fit the BFO and BTO data points separately. First, I fitted BTO against $$\Delta t$$ by varying only $$v$$, $$\phi$$, $$\theta$$ and $$\Delta t_0$$, as the remaining two parameters have no effect on $$\Delta t$$. This yielded an estimate for the trajectory of the aircraft. Next, I checked if I can fit $$\Delta f$$ against BFO by varying only $$\Delta f_0$$ and $$\xi$$. (NB: So I am really not fitting the trajectory to the Doppler frequency shift, I am merely checking the validity of the fit. If the model I use is not appropriate to estimate the transmitter frequency bias and Doppler compensation, it might explain why Inmarsat are confident that the northerly route can be excluded.)

Without further ado, the following plot demonstrates the result:

In this plot, the vertical axis on the left is in Hz (for $$|\Delta f|$$ and BFO) whereas the vertical axis on the right is in seconds (for $$\Delta t$$ and BTO). Yes, I used Microsoft Excel… when you have a nail, a hammer will do nicely.

The values that I obtained for this fit are given by:

\begin{align}
v&=867.16~{\rm kph},\\
\phi&=32.12^\circ,\\
\theta&=-86.55^\circ,\\
\Delta t_0&=0.234572~{\rm s},\\
\Delta f_0&=121.7882~{\rm Hz},\\
\xi&=-0.18199.
\end{align}

This fit by itself does not tell me where the aircraft flew. That is because the model is rotationally symmetric around the ground track position of the Inmarsat satellite. What it does tell me is that at $$t=t_0$$ (I arbitrarily chose $$t_0$$ to be 19:41 UTC, the time of the first clean Inmarsat data point with both BFO and BTO) the aircraft was on a circle from which the satellite could be seen at 52.6° above the horizon; whereas at the time of the last data point, at 0:11 UTC, the satellite-to-horizon angle was 38.6°.

This is encouraging, as it agrees with published estimates of the infamous “Inmarsat arcs” that position the aircraft along an arc from which the satellite viewing angle is approximately 40°.

But I can do better than this. Let me take a closer look at the two radar data points at 17:21 and 18:15 UTC. These correspond to a great circle trajectory with a velocity of 718.6 kph, almost exactly in the direction of the Inmarsat satellite. By simultaneously extrapolating this trajectory forward and the just calculated Inmarsat-based trajectory backward (for details, see the Wikipedia article on great circle navigation), I can find the time $$t$$ when the two trajectories have matching satellite viewing angles. This happens just a few seconds after the last military radar contact at 18:15.

This means that I have an actual starting position for the Inmarsat track at 6°49’34” N, 97°40’47” E, at 18:15:22.7 UTC. If my calculations are valid, at this spot the aircraft would have turned to follow a trajectory that is nearly at a right angle to the line that connects its position to the satellite ground track position.

One question, however, remains and it is a nagging one. Did the aircraft turn north or south? Malaysian authorities and Inmarsat insist on the southern track. But I see nothing in the data that they released to support this assertion. If the aircraft turned south, it would have ended up at 38°36’30″S, 88°17’27″E by my calculation, a position that is pretty close to the original search location southwest of Perth, Australia in the Southern Indian Ocean. Therefore, I can confirm the validity of this part of Inmarsat’s analysis.

But what if the airplane flew north instead? That would put it at 44°34’1″N, 66°55’42″E, somewhere in eastern Kazakhstan.

My conclusions, then: the original search area that was designated on March 18, 2014, seems to be validated by the recently released data and my analysis. The assertion that the airplane flew south does not appear to be supported by anything in this data set. If the northern trajectory is not excluded, the airplane may have ended up in Kazakhstan, after flying over places like India, Pakistan, maybe parts of Afghanistan, Uzbekistan and Tajikistan. Is it plausible that such a flight took place without being noticed by these countries’ air defense systems? I’ll leave that for others to decide.

For what it’s worth, here are the two locations that I calculated, presented using Google Maps:

On a final note, I was hesitating before making this blog post public. This is not just a theoretical exercise of matching aircraft trajectories and radio-metric data. There were 239 souls on board, and their relatives still have no idea what happened to them. Is publishing my less than well-informed speculation about the fate of the airliner the right thing to do? I would probably not be publishing anything if my calculations were in contradiction with the “official” analysis. But as I was able to confirm that analysis (again, except for the exclusion of the northern track) it is perhaps not irresponsible for me to publish these results.

The spreadsheet that I used for storing the transcribed Inmarsat data and for my calculations is available for download.

Remarkable news from Australia: a U.S. Navy ship* observed an acoustic signal for over two hours that appears to be from two separate “ping” transmitters. This would be consistent with a lost aircraft’s flight data and cockpit voice recorders, both with still operating acoustic transmitters.

This is amazing.

Speaking of pingers, yesterday the ridiculousness on CNN reached new limits. The pingers have batteries that are supposed to last approximately 30 days. The actual duration depends on the age of the battery, its maintenance and storage. And when the battery dies, the process may be a gradual process (i.e., the ping signal weakens but does not necessarily stop immediately.)

None of this prevented CNN from showing a countdown clock, accurate to the second, showing the remaining life of the pinger batteries.

*Actually, a U.S. Navy device towed by an Australian ship.

Some details have been released (leaked?) by Inmarsat and the AAIB about their analysis of the flight path of the missing Malaysian airliner. Some details remain frustratingly absent.

Relying on the measured frequency of the signal received from the missing jet, they plotted possible courses of the aircraft and they concluded that only the route that took MH370 to the southern Indian Ocean is consistent with the data. Here are the two critical slides from the annex of their released material:

They are clearly quite confident about the validity of their analysis, and they may be right. Still, there are a few potential issues with which I am not comfortable.

The analysis obviously relies on two key assumptions: first, that the aircraft traveled at a constant speed and second, that its transmitter had good frequency stability. I am not familiar with Inmarsat equipment used on board aircraft, but I do know that a frequency drift of a couple of hundred Hz, over a period of time of several hours and under changing environmental conditions, is not at all unusual [Update (2014/03/28): I now know (thanks, Craig!) that Inmarsat equipment uses an oven-controlled oscillator, with a frequency stability of a few 10 Hz or better over the course of a year, so this is a non-issue] for an oscillator that is running at around 1.6 GHz (which, I believe, is the frequency range used by Inmarsat.)

The analysis also relies on the estimated range at the time of final transmission, which is what was used to generate the infamous “arcs” along which the airplane is expected to be found. Presumably, similar estimated ranges are available for all the intermediate data points. However, this range information was not published in the currently released document. [Update (2014/03/28): Intermediate range arcs were, however, published by the Washington Post on March 21 (thanks again, Craig!).]

It is also unclear to me why the northern route can be excluded, as the top slide shows. If the satellite was stationary with respect to the ground, the northern and southern routes would have identical Doppler signatures. Presumably the difference is due to the fact that the satellite, though geostationary, still moves with respect to the Earth’s surface, e.g., because its orbit is inclined. [Update (2014/03/28): The orbital inclination of the satellite in question is 1.6° (once again, thanks, Craig!)] But this is not explained.

Finally, I am also concerned about the large deviations in the early stages of flight between the predicted and observed values and what it says about the validity of the analysis.

Just to be clear, I do not subscribe to conspiracy theories. I do believe that it may have been premature to exclude the possibility that the aircraft made an emergency landing and remained intact in a remote area not far from the location of its last transponder signal, but I may very well be wrong about this. However, I do think that a little more transparency would be useful.

These are the dreaded words no pilot wants to see engraved on his or her tombstone: “Controlled flight into terrain”.

Yet this is precisely what happened when a 737 was flying into Resolute Bay, Nunavut in August 2011, only to miss the runway on approach and collide with terrain instead, killing 12 of the 15 souls on board.

The reasons are aptly summarized in this brief video from the Transportation Safety Board of Canada.

The sad story of this flight reminds me of something I first learned when I was 17 or so, working on my first ever paying software development contract, for Hungary’s equivalent of TSB Canada. I was taught that it is rare, almost unheard of, for an air accident to have a single cause. Most of the time, accidents happen as a result of a chain of mistakes, and if only one link in the chain is missing, the accident would not occur. In this case, the links of the chain included an inadvertent autopilot setting; a compass alignment issue; bad visibility; and breakdown in communications between the two pilots on board. At any time before the final few seconds, the situation was still salvageable, if only the pilots became aware of what was happening. But by the time the ground proximity warning alarm sounded, it was too late.