Why correcting the distribution does not fix daily timing

Every stage lands daily Pearson r near 0.35 against CPC-UNI and near 0.24 against the independent BMKG stations. This is the expected behaviour of a marginal correction, not a shortcoming of this one. Each stage maps a pixel's value through a monotone non-decreasing function, so it introduces no rank reversals between days: if the satellite reported more rain on day i than day j, it still does after correction. Ordering is not preserved exactly, because dry-day matching ties about 43% of days at zero and ties can be created or broken there, but the effect on r is small. The day-by-day pairing is set by the satellite retrieval, and a marginal step cannot move rain from one day to its neighbour. You can reshape the distribution all you like - the timing does not move.

Key finding Against CPC-UNI, daily Pearson r moves just 0.005 across the three stages - from 0.343 (LS) to 0.345 (LSEQM) to 0.348 (LSEQM+DL), as a per-pixel spatial median averaged over the 36 dekads. Against the independent BMKG stations, under the identical recipe, it moves from 0.242 to 0.236 to 0.239. The correction reshapes the distribution; it does not touch the timing, against either reference.

See it on a synthetic pixel

A synthetic satellite-and-gauge day series (illustrative, as in the thesis bound schematic). Drag the quantile-mapping strength: the corrected series slides onto the gauge distribution - the distribution mismatch (KS) collapses - but its correlation with the gauge barely twitches.

Correlation with gauge

raw → corrected · pinned close to the raw value

Distribution mismatch (KS)

Slide λ from 0 to 1: KS falls toward 0 (the corrected distribution meets the gauge) while r holds near the whole way. Correcting the margin cannot buy timing that the satellite never measured.

The same thing on the real numbers

Daily r by stage, against both references. The recipe is the same in structure: a median taken across the reference's own units within each dekad - over the 19,393 land pixels for CPC-UNI, over the stations for BMKG - then averaged across the 36 dekads. CPC-UNI is the dataset the correction was fitted to, so it is the in-sample number; the 172 BMKG stations are held out of the fitting. Both are flat across the stages, which is the point of the page.

The monthly Taylor

The limit is not an artefact of one summary statistic. Every product's Taylor position - all six, in all twelve months - sits at a similarly low correlation. The correction slides the cloud along the standard-deviation axis (LS under-spread → LSEQM/LSEQM+DL on the reference circle) but never toward the correlation axis. Each grey dot is one BMKG station (all six products pooled); the coloured dots are the per-product medians. The cloud concentrates in the low-correlation wedge - product medians between r = 0.20 and 0.24, most stations below 0.5 - so the limit is a property of the whole network, not of how the summary was taken. A thin tail of stations reaches higher, but no product median does. Pick a month:

Headline proposition A marginal (per-cell) correction is a monotone remap of each day's value; it cannot move rain from one day to its neighbour. So Pearson r stays pinned close to the raw retrieval - and on the Taylor diagram all six products cluster between 0.20 and 0.24, taken as the per-station median of the dekadal correlation across the 171 stations that have one. What the correction fixes is the radius: on that same basis the standard-deviation ratio moves from 0.72 for LS to 1.00 for LSEQM+DL. The cloud moves out to the reference circle, never around toward the perfect corner.
The limit is real, and part of it is conventional rather than fundamental. Held to a fixed UTC-day pairing, no stage moves daily r away from against CPC-UNI or against BMKG. How much of that is a day-label artefact can only be measured against BMKG, because CPC-UNI dates its totals to the UTC day exactly as IMERG does and so carries no offset: re-pairing IMERG-L to the BMKG gauge day raises the pooled daily correlation there from to (see the window diagnostic). Beyond that, closing the remaining gap needs methods outside the marginal-correction family - the four paths.