Target Location Error (TLE)

Target Location Error (TLE)#

When two independent measurements of the same target disagree, the difference between them is the Target Location Error (TLE). In test-pilot work this is the canonical accuracy metric — survey-truth versus sensor estimate, designated coordinate versus what the inertial system reports, target-truth versus weapon impact. The standard process is always the same four steps:

  1. Express both measurements in latitude, longitude, and height (LLH).

  2. Convert each to Earth-Centered Earth-Fixed (ECEF) coordinates.

  3. Subtract in ECEF, then rotate the difference into a local North-East-Down (NED) frame anchored at the truth point.

  4. Reduce to two operational scalars: \(e_H = \sqrt{e_N^2 + e_E^2}\) (horizontal) and \(e_V = |e_D|\) (vertical).

The math chain is the same regardless of where on Earth the scenario is, but the meaning of a given LLH offset is not — a one-arcsecond longitude error produces about 30 m of east error at the equator, 15 m at 60° latitude, and barely a meter near the pole. The cos(latitude) factor is one of the central facts to internalize from this demo.

Why not just subtract LLH directly?#

Tempting, but wrong. Latitude and longitude are angles, not distances, and a degree of longitude shrinks toward zero as you approach the poles while a degree of latitude is roughly the same everywhere. The conversion through ECEF and back into NED is what bakes the WGS-84 ellipsoid geometry into the answer. The DCM \(\mathbf{C}_e^n\) at the truth point turns the metric ECEF difference into a meaningful local horizontal/vertical decomposition.

Interactive demo#

Open in full screen

Walkthrough#

The demo opens at the Edwards default scenario — Edwards AFB at \(34.92°\text{N}, 117.90°\text{W}\), with a small simulated sensor error of about 30 microdegrees of latitude, 20 microdegrees of longitude, and 6 m of altitude. Try the following:

  1. Read the headline metrics. \(e_H \approx 3.80\) m, \(e_V = 6.000\) m. Open the Conversion chain panel at the bottom to see the four steps explicitly: each LLH converted to ECEF, the dECEF difference, the \(\mathbf{C}_e^n\) matrix evaluated at truth, and the resulting NED.

  2. Drag the orange E pin in the NE plot. This sets the sensor location graphically; the Sensor LLH inputs update live to whatever LLH gives that NED offset. Use the zoom chips above the plot to pick a sensible scale; the zoom doesn’t auto-change while you drag, so the cursor and the pin stay locked together.

  3. Try each scenario chip. “Lat error only” applies a 1-arcsecond latitude offset and produces \(e_N \approx 30.8\) m, \(e_E \approx 0\) at the latitude of Edwards. “Lon error only” gives \(e_E \approx 25.3\) m, \(e_N \approx 0\) — fewer meters per arcsecond because \(\cos(34.9°) \approx 0.82\). “Alt error only” gives \(e_H = 0\), \(e_V = 10\) m.

  4. Run the cos(lat) experiment. With “Lon error only” loaded, slide the truth latitude slider from \(-85°\) to \(+85°\). Both pins translate together — the same arcsecond of longitude offset stays applied — but the metric east-error changes dramatically. Watch the curve at the bottom: \(e_H\) traces \(\cos(\phi)\), peaking near the equator and falling toward zero at the poles. Meanwhile \(e_V\) stays flat — vertical doesn’t care about latitude. The dot on the curve marks the current state.

  5. Run the same sweep with “Lat error only.” Now \(e_H\) stays nearly constant across all latitudes; the small variation comes from the WGS-84 ellipsoid (the meridian radius of curvature isn’t quite constant). Latitude offsets translate to north-error roughly the same way everywhere on Earth; longitude offsets do not. That asymmetry is the headline.

  6. Edit the LLH inputs directly. The two LLH coordinates are the primary inputs to the chain — everything else (NE plot, altitude view, conversion chain readout) is derived from them. The Δ row beneath the inputs shows the offset in arcseconds, plus a per-arcsecond sensitivity readout for the current truth latitude.

Key observations#

  • Arcseconds are the right operational unit for LLH offsets. A degree is huge (\(\approx 111\) km), a microdegree is awkward, an arcsecond is roughly 30 m everywhere — close to the precision of GPS-grade civilian equipment. The Δ readout uses arcseconds for that reason.

  • \(e_V\) is essentially \(|\Delta h|\). The vertical chain has very little geometric machinery; altitude offsets pass through almost untouched. All the geometric subtlety of TLE lives in the horizontal channel.

  • Same arcsecond, different meters. A test point at the equator and the same test point at \(80°\text{N}\) produce wildly different \(e_E\) values for the same \(\Delta\text{lon}\). Test programs at high or low latitudes need to factor the cos(lat) effect into measurement-precision requirements.

  • The \(\mathbf{C}_e^n\) matrix at truth depends only on the truth latitude and longitude, not on the sensor offset. That’s why the chain evaluates it once at the truth point and reuses it.

Source#

MATLAB · code/TargetLocationError.m↓

With the LLH↔ECEF and ECEF↔NED utilities in code/navutils/.

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