Predict XRT with utu.dem.emission_measure in the DEM tutorial, and integrate its ranges exactly

[roytsmart]

This fixes two integrals of the DEM in the DEM tutorial. Neither one was the integral that utu.dem.plowman assumes.

The XRT intensity a DEM predicts

The tutorial predicted the XRT intensity of each DEM as the DEM times the response times the 0.05-dex step. plowman takes the DEM and each response to be linear in log T between the temperatures, and fits each intensity with the integral of their product, Rᵀ M D, where M is the mass matrix of those triangle functions. The rectangle sum weights each temperature by the sum of its row of M instead.

The tutorial now uses utu.dem.emission_measure (sun-data/utu#15, new in utu 0.5.0), which returns M D, so the predicted intensity is the one the inversion fit:

emission_measure = utu.dem.emission_measure(dem, "temperature")
predicted = (emission_measure * outputs_xrt).sum("temperature")

The change is small, and every number in the text still holds:

Ratio of predicted to observed XRT	Before	After
AIA + XRT, median	1.229	1.236
AIA + XRT, 5th–95th percentile	1.16–1.31	1.17–1.32
AIA alone, median	9.01	8.96
AIA alone, 10th–90th percentile ("3 to 18 times")	3.06–17.81	3.05–17.70

Per pixel, the ratio changes by a median of +0.6% for the DEM from AIA and XRT (at most +2.4%), and by −0.5% for the DEM from AIA alone (at most −8.7%). The χ² maps are unchanged.

The maps of emission measure

The four maps were left Riemann sums over [lower, upper), the nodes times the step, as in the DEM tutorial of sdo before sun-data/solar-dynamics-observatory#30. For a DEM linear between its nodes, the exact integral over a range is the trapezoid rule, which counts the two end nodes by half and includes the node at upper.

As in sdo, the tutorial now gives the DEM log₁₀ T as its inputs and integrates it with FunctionArray.integrate, from the node at lower to the node at upper. The text says why the trapezoid rule is exact here. The old maps, against the exact integral:

log T	Median ratio of old to exact
5.6–6.0	0.88
6.0–6.3	1.08
6.3–6.6	1.15
6.6–7.0	1.08

Other changes

• utu.dem.emission_measure is not used for the maps. Its sum over a range is not the integral of the DEM over that range, since each temperature stands for plasma up to a step either side of it.
• The logt of the response section, used only by the old maps, is now defined where the maps are made, from the DEM.
• The doc extras require utu>=0.5.

Checks

I ran every jupyter-execute cell of the edited tutorial in order, against utu 0.5.0, and the same cells of main for the "Before" numbers. All the cells ran. Each of the four maps equals np.trapezoid of the DEM over log T exactly, and the median reduced χ² is still 0.968, with one pixel above 3.

🤖 Generated with Claude Code

https://claude.ai/code/session_017xqpht97bF4FWsACAA4wQa

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1 file changed

± tutorials/dem.html