Location spread centre position error, in units of the hump width σ
- truth
- 100 members
- + obs P
- sample dist.
- Gaussian fit
- mean — · std — · skew — · kurt — → —
- member samples
- vortex center dist.
Chapters 1–6 built every analysis scheme on two linear-Gaussian assumptions: the observation operator is (locally) linear and the prior error is Gaussian. Both fail for coherent structures like tropical cyclones. This chapter starts with the most common failure mode — position error — and will build up to particle filters, which replace the Gaussian assumption altogether.
A Gaussian hump — the exact same warm blob as Chapter 2's own data generator (peak 8 K, radius scale σ = 50 km, on the same 500 × 500 km / 10 km grid) — is the simplest model of a coherent anomaly. The ensemble below has 100 members that differ only in where the hump sits: each centre is drawn from a Gaussian with standard deviation Lsprd around the truth location (given in units of the hump radius scale σ). The observation point P (marked +) sits at the same grid offset from the truth centre as Chapter 2's own observation from its blob peak — not on the 4 K contour, same as in Chapter 2 — and the state we watch is the temperature T at P. Click P on panel (a) to move it anywhere; click a dot in panel (c) to highlight that member's ring in panel (a). Slide Lsprd up and down: at small spread the distribution of T at P (panel (b)) is Gaussian; once the spread reaches the hump size, members' centres pass P and the histogram skews hard against the T = 0 bound — no Gaussian prior can represent it.
Why the bend. The temperature at a fixed point is the hump evaluated at the member centre's distance from that point: a bell curve in the centre position. Near P that map is locally linear, so a small position spread keeps the error at P Gaussian. As the spread grows, members' centres sweep across P: the value at P peaks when a member's centre sits exactly on P and falls to zero on both sides (panel (c)) — a non-monotone map pushed through a Gaussian cloud of centres. The histogram in panel (b) skews and fattens, piling against the hard T = 0 bound, and the readout's skewness leaves 0. An ensemble that is Gaussian in where the anomaly is is emphatically non-Gaussian in what the value is at any fixed point.
This is the same mechanism that produced the correlation dipole in Chapter 2 — position uncertainty at work — and it is exactly the situation section 2's comparison has to deal with when the observation at P is assimilated.
When the Gaussian assumptions fail, Bayes' rule is solved directly on the ensemble: the particle filter turns the prior members into a weighted sample, each particle reweighted by the observation likelihood p(y | centre). The EnKF cannot do that — its analysis is an affine combination of the prior members, so the posterior keeps the Gaussian shape of the prior. The three panels below show the 4 K contour spaghetti of the same Gaussian hump ensemble as section 1 — where every member's rings sit — before and after assimilating one observation of the temperature T at P (the +, same grid offset from the truth centre as Chapter 2's own observation — not on the 4 K contour): the prior, the EnKF analysis, and the particle-filter posterior, whose members light up with their weight. The observed value is 4 K, with error σo (tunable, default 1 K). Every member keeps its own colour in all three panels, so its rings can be traced from prior to posterior.
Try it: slide Lsprd up — the prior band widens, the EnKF band only tightens where the regression points, and the particle filter lights up just the members whose rings are consistent with the observation. Watch the readout: the weighted centre distance shows how far the posterior is pulled toward the truth, and Neff = 1/Σwm2 collapses as the observation becomes unlikely under the prior. Nens changes how many contours the filters share.
Reading the panels. (a) The prior: every member's 4 K rings around its centre — a wide band of where the 4 K line could be; the shading is the ensemble mean field, broader and weaker than the truth ring since position uncertainty averages the blob out (same convention as Chapter 3's background panel). (b) The EnKF analysis: the SAME per-grid-cell formula as Chapter 3, applied independently at every cell — it does not move any member's centre, so instead of sliding the rings tighten and distort into the classic position-error dipole shape (the readout's centre distance can even go up: a linear per-cell update cannot represent a rigid shift); the shading is the analysis mean, on the same colour scale as (a) so the two are directly comparable. (c) The particle filter: the same prior members, reweighted by the observation likelihood — the rings light up only where the observed temperature is plausible, and the shading is the weighted mean (Σ wm·fieldm, same colour scale as (a)/(b)): near-zero-weight particles contribute essentially nothing to it, so it concentrates on wherever the surviving members sit — a direct picture of importance-sampling degeneracy, which the readout's Neff quantifies as how few particles effectively survive.
→ Evensen et al., Data Assimilation Fundamentals (2022), Ch 9 — nonlinear filters; van Leeuwen (2009), Mon. Wea. Rev., particle filters in geoscience.