Let’s pick this up right where the idea of apparent divergence left off — Page 282, Lesson 279

Let’s pick this up right where the idea of apparent divergence left off. That opening line is actually a warning about how you read IRS positions. If you have two inertial reference systems, and they disagree slightly in latitude, that tiny latitude difference gets multiplied by the secant of latitude when you convert it into longitude. Near the poles, secant latitude blows up, so a small physical separation looks like a huge longitude split. The actual distance between the two IRS positions is small, but the displayed longitude difference is enormous. That’s the trap — don’t mistake a big longitude number for a big physical error. Now I want to walk you through Kalman filtering, because this is the heart of how a navigation computer blends two very different kinds of accuracy. Kalman filtering is the process used within a navigation computer to combine the short-term accuracy of the IRS with the long-term accuracy of the external reference. Let me unpack that. The IRS is superb for a short while — its errors are small at first. But over time, inertial errors drift and grow. The external reference — that’s your DME, VOR, GPS, or similar — has errors that are roughly constant, so over a long flight it stays accurate. The Kalman filter’s job is to take the best of both. Here’s how the model works. It assesses the velocity and position errors from the IRS by comparing the IRS position with the external reference, and from that comparison it produces its own prediction of position and velocity. So it’s not just averaging two numbers. It’s watching how the IRS drifts relative to the trusted external fix, estimating the error, and then predicting where the aircraft really is and how fast it’s really moving. The weighting is dynamic. Initially, the IRS information is the most accurate, so the model favours the IRS. But as the ramp effect of IRS errors progresses — that steady build-up of drift — the external reference becomes the most accurate, and the weighting shifts toward it. So early in the flight the filter leans on the IRS; later it leans on the external reference. The consequence is that the position is most accurate right after the position update on the runway threshold, and then it gradually decays to the accuracy of the external reference. Then it improves again when the aircraft is on final approach using a precision system — either ILS or MLS. One more point: the more complex the model, meaning the more factors it includes, the better the quality of the system position and velocity. A richer model captures more of the real error behaviour, so the output is better. Now let’s look at DME–IRS accuracy, because that’s the practical example of this blending. The position accuracy of the IRS continually degrades throughout the flight. But note this contrast: the heading and ground speed maintain a high degree of accuracy. So the IRS stays good for attitude and velocity, but its position drifts. The measurement of position is subject to random errors which depend on two things: the range to the DME and the cut of the position lines. The cut is the angle at which two position lines intersect — a good cut, near 90 degrees, gives a sharp fix; a poor cut gives a fuzzy one. The computer solves the cut problem by selecting DMEs positioned so that a good cut will be obtained. So it picks the geometry for you. Slant range error is compensated for in the calculation. But the DME error itself is constant at plus or minus 0.25 nautical miles plus 1.25% of range. Let me make that concrete. At 100 nautical miles, the error is a maximum of 1.5 nautical miles. Check the arithmetic: 0.25 plus 1.25% of 100, which is 1.25, gives 1.5. So at the start of a flight, that DME error is large compared with the IRS error, because the IRS hasn’t had time to drift yet. But as the flight progresses, the IRS is degrading at around 1 nautical mile per hour. After several hours, since the DME error is constant, the DME fixing is significantly more accurate than the IRS. That’s exactly the crossover the Kalman filter is managing — it starts trusting the IRS, then gradually hands over to the DME as the hours pass. And that last bit you see there is a multiple-choice question about the FMC. The correct statement is that the FMC combines the short-term accuracy of the IRS with the long-term accuracy of the external reference. That’s the whole point of the Kalman filter — short-term IRS, long-term external. The other options mix up short and long term, and they’re wrong. So when you see that question, you’re really just restating the definition of Kalman filtering. Let me tie it together. The Kalman filter is the intelligence that blends the IRS’s short-term precision with the external reference’s long-term stability. It weights the IRS heavily at first, shifts to the external reference as the IRS drifts, and the result is a position that’s best on the runway, decays to external accuracy, and sharpens again on a precision final approach. And the DME example shows you the numbers behind that shift — a constant DME error of 0.25 NM plus 1.25% of range, versus an IRS that degrades at about 1 NM per hour. That’s the crossover you’re managing.