In many ways, this should have been the first article I ever posted here. It goes without saying that we will not find the plane if we fail to look in the right place. And the only places we have looked or seriously considered so far are places that intersect the so-called “7th Arc”. But that Arc, first proposed by Inmarsat and variously updated by Australia’s CSIRO and others is at least 100 kilometers too far southeast. That means no part of the Arc has been searched as of today. None.

In this article I will go through the steps required to check the validity or lack thereof of the Seventh Arc, which I will refer to simply as “The Arc” in the future. Doesn’t hurt anything to continue calling it the 7th Arc, but that is another wrinkle: the 7th Arc does not exist. The 6th Arc is the closest we can reliably get to what Inmarsat dubbed the Seventh Arc. What we call it isn’t a big deal, but we still have to go to it to find the plane. It will not come to us no matter how hard we hope.

The sixth ping and the seventh ping occurred 8 minutes apart. The plane was either out of fuel, or it was in the water shortly after the 6th ping. We simply have to search enough seafloor width to eliminate the possibility we missed a few kilometers because the plane was still in the air by the time the abbreviated pings were picked up at 00:19. That has always been one of our challenges.

It is my belief now that the plane was indeed in the water prior to the seventh ping. I do not care how it got there. For me, that is the only explanation for a thoroughly garbled seventh ping. So yes, I suspect it was piloted to the end, but whether it was or wasn’t isn’t very important, either. We won’t know for sure until we find it.

**First: Correcting Inmarsat’s BTO Bias**

We begin by examining the BTO pings Inmarsat used to “calibrate” the BTO value. They were published as Table 3 in an article by Chris Ashton, et. al in the September issue of Journal of Navigation (JON).

Here is a copy of the part we want, taken from one of my worksheets. There are 17 BTO values Inmarsat selected to use for calibration purposes. They were not selected randomly. They were hand picked, which itself is problematic. They were all transmitted **from the tarmac** prior to the plane’s departure. That time frame was the only acknowledged criterion.

Rows 26 through 34 present my own central tendency calculations. While they are not a truly random sample, I nevertheless use a sample standard deviation metric for variability, shown in line B30: 58.84 microseconds (**µs**). If you’re not familiar with variability measures, this is a modest amount of variance. Nothing out of the ordinary.

That standard deviation is also sometimes called 1-sigma. We will use it to estimate how much seafloor we must search to be sure we find the plane where we believe it is, or to be sure our location is wrong if we don’t find it. It is a very simple and common technique that puts a feedback loop in our effort. If we find it, we get an “atta-boy”. If we don’t we get a “try harder”.

Now that we have a sample of pings, we can address the number one issue: average bias across that sample. We prefer to use discrete values for the bias associated with each ping, but Inmarsat and Malaysia have elected to withhold bias data across the board, except for these 17 “calibration pings”. Why? Who knows. It’s disgraceful, but we’re not in a position to do anything about it.

It turns out that the average bias value for all 17 pings in the sample is as you see it in row 24: -495679 **µs**. It becomes part of a formula used to calculate the plane’s distance from the satellite at each ping.

But wait a second. We see some additional information in the insert below, including the channel from which each calibration ping was taken. And sadly, three different channels sent that information. That’s important because the broadcast channel can have a lot to do with the magnitude of each BTO data value.

It also turns out that all of the “official pings” Inmarsat used came from Channel 4. ONLY Channel 4. It may be that Inmarsat conducted cross channel tests to make sure channels 4, 8, and 11 were comparable, but if they did that they didn’t note it anywhere.

So we’re not going to take a chance by using an average BTO Bias value that appears likely to be incorrect. (We know this was a problem with Channel 10 BTO data, so we aren’t going to take a chance. In fact, the last ping … the infamous 7th ping … came from Channel 10, and we will reject it for the same reason: it transmitted on its own scale and there is NO formal translation back to channel 4 data values. )

The following table shows us the nature of the problem: eight calibration BTO values came from channel 8; seven came from channel 11; and two came from channel 4.

So we have no choice. We reject all calibration BTO values that did not come from Channel 4. It is necessary partly because to the best of anyone’s knowledge, Inmarsat did no equivalence testing before it mixed and matched BTO values. Apples and oranges.

But we still require an average bias metric since discrete values are not available. That really isn’t a problem, we just won’t have a very large sample. In fact, now we are down to only two BTO pings prior to take off that can be used for the information we need. Both of those pings were sent right at the end of the calibration window Inmarsat used to select a sample. They meed out requirements, and average bias for those two values turns out to be 495651, which is a mere 28 **µs** smaller than Inmarsat used.

The difference is small, but we will have to tweak average bias one more time when we attempt to use it to predict locations for which we have independent GPS great circle measures. There are three of them at: 16:42, 16:55, and 17:07. All UTC, March 7, 2014.

Now let’s see how useful Inmarsat’s BTO bias average is in predicting location when compared to Great Circle GPS look-ups:

The prediction error is shown in column AF. It isn’t horrible-bad, but if we missed a runway by 11 or 21 or 17 kilometers, we and our passengers might call it a bad landing. Moreover, these are only land predictions where we have to assume Inmarsat’s estimate of BTO bias will perform better because it was calibrated with sitting-at-the-end-of-the-runway values.

So, let’s see if our BTO estimate fares better.

Well, that’s disappointing. The Chillit estimate is better than Inmarsat’s estimate, but neither of them is good enough to find even a maxed out Boeing 777, which is barely one-sixteenth of one kilometer wide if all in one piece. And the scan width of sidescan sonar units is maybe 3 km.

So we need to do better. Fortunately, we can.

It turns out that if we play with this problem long enough there are at least two BTO bias values for the MH370 flight. One while it was on the ground (495672) and one for flight at 35,000 feet (495587). These are important pieces of the puzzle and will help us find the plane faster.

Now let’s take another shot at predicting the plane’s distance from its satellite again (a ping ring).

The last column on the right shows we’ve now substantially reduced what is known as error-of-the-estimate (or mean) using a second BTO Bias estimate of -495587 for flight-level location estimation. It still isn’t perfect, but we’re not likely to improve on this much given the variance that occurs in any mechanical or electrical system. Not to mention in conjunction with a satellite that wobbles in orbit.

**Second: Establish Sub-Satellite Point for 3-F1 at 00:19
**

You need to call up NOAA’s ellipsoidal calculator at this link to do this, or you need to just take my word for it.

**http://www.ngs.noaa.gov/cgi-bin/xyz_getgp.prl**

The satellite’s location at 00:19, according to Inmarsat, was **0.53N, 64.464E**. That is simply the satellite’s location on the surface of the earth if you linked the satellite to the center of the earth with a straight line.

Here are all of the coordinates provided by Inmarsat:

I will mention that Whiz Kids among us have come up with a simple Excel spreadsheet that will permit calculation of the satellite at any time. It’s nifty and permits us to select any time on either side of the time values shown above.

**Third: Calculate BTO Variability, or Sigma
**

Once I decided to limit calibration to just Channel 4 values, calculation of sigma was set in stone. It is whatever sigma for a sample is for the two remaining BTO values: 14940 and 14920. Both were generated at 16:29 on the tarmac. The satellite was moving of course, so we like the fact that we have two values taken within a very brief period to minimize distortion caused by satellite movement. Using any standard deviation calculator for a sample, sigma is **10 µs**. That is a much friendlier sigma than the one we got earlier from the entire calibration sample (58.8 µs). Lower variability is good.

**Fourth: Calculate the Final Arc (formerly the Seventh Arc)
**

This is not difficult. The BTO and BFO values published in what are known as the supplemental log files make it very clear that there is no information at all in the seventh ping, beyond the fact that a time stamp was obtained. This is all that survived whatever the plane was trying to report at 00:19.

A BFO value of -2 means nothing. Same goes for a BTO value of 49660, although at least it appears to be on the right continuum for a Channel 10 transmission. We are better off without it. At best, it will mislead our analysis enormously. So we dump it.

That leaves us with the 6th Arc and that is enough. Where is it? This is where each of the arcs lies based on this revised data. Only the sixth is important (line 16). The rest may have value if we figure out how to connect them. For now, they may as well exist on other planets. They have no mutual links.

The table above shows that the 6th and final arc has a radius of 4,799 km. The margins of error on either side of it have radii of 4,739 km and 4,859 km, respectively. The spread is 120 km, which is the width of the swath we need to scan to be nearly certain it could not be farther from the mean. We can reduce the width of the swath by adopting a 3 Sigma standard, but we risk missing it if we do, and if it is really there. These are the trade-offs we face everyday in real life. We just have to be prepared to live with our decision.

Below is the visual for all of this. Obviously, we have put quite a bit of distance between the final arc and the area ATSB has actually scanned, shown in sand brown.

In fact, we have corrected Inmarsat’s original arc by 109 km, and ATSB’s arc by 129 km. Sadly, as noted at the outset, it also means no part of the area that has been searched is where the plane would now be if it flew south of Broken Ridge.

I am not proposing that the search simply move west 100+ kilometers. There is plenty of other evidence gathered by a number of respected agencies / individuals that show the plane simply did not fly that far south. All of the debris found to date almost certainly originated (if the Arc is correct) west of Exmouth, Shark Bay, and Geraldton. There is a 600 km strip of seafloor there that almost certainly has the plane. We need to go there.

And by the way, I hope we will all do a bit of remembering over the holidays.