A Schomburg rule deterministically relates 1 or more predictors to the anomaly of 1 predictand with respect to the spline interpolation of step 1 of the Schomburg scheme. Coarse variables from (near) the surface are taken to be predictors, but they are joined by external fine fields. The predictands are fine fields of variables to be downscaled. Only 1st principles determine 1 rule, for short-wave net irradiance at the surface. Another 2, for temperature and long-wave net irradiance, result solely from a statistical study of the training data. Schomburg et al. (2010) attempted the same for gravimetric[10] humidity, wind speed and precipitation, but deemed the findings insignificant. Lastly, the rule for pressure relies on a 1st principle, the hydrostatic equation, but statistically infers its coefficient.

The careful conservation of the average in all steps of the Schomburg scheme reverts the addition of a constant to the end result of a Schomburg rule. If an addend stays constant for the entire set of fine pixels associated with 1 coarse pixel, it is deemed constant for this purpose - This includes the conserved average.

Table of contents

• Pressure
• Temperature
• Irradiance
• Long-wave net irradiance
• Short-wave net irradiance
• Footnotes

Pressure

Pressure anomalies are always derived from the finite-differences hydrostatic equation, but with a constant air density and constant gravitational acceleration. The code studied in Schomburg et al. (2010) features these inferred coefficients:

• –11.3825 for ‘PS’, presumably surface pressure
• –11.6939 for ‘P0’, presumably the pressure on COSMO-DE’s lowest airborne layer

Elevation anomaly is the fine predictor here. Schomburg et al. (2010), as well as the more extensive Schomburg (2011), subtly suggest that these two anomalies are to be taken with respect to the spline-interpolated fields. This means that the elevation has to undergo upscaling.

Temperature

Among 25 similar rules, this one applied to the largest part of the training data, 64%:

If the vertical gradient of T of the lowest 105m falls short of 0.0058K/m, then
T = –0.0084K/m H .
Else there is no rule.

The symbol T stands for temperature anomaly with respect to the spline-interpolated field, and H is the fine orographic elevation. Because of the average conservation, H can be exchanged with its anomaly with respect to its upscaled field.

Schomburg et al. (2010), and the more extensive Schomburg (2011) derive this rule from the calculation of the correlation between fine temperature anomalies and fine elevation anomalies, both with respect to the coarse field. By the construction of the Schomburg scheme, T has to be taken with respect to the spline-interpolated field, though. What that means for H is left unsaid.

Irradiance

The available active data does not share the definition of irradiance with the training data from the study. The latter used net irradiances, the former differentiates upwelling and downwelling radiation. Therefore, namcouple prevents these wrong rules from application by giving wrong names to OASIS3 which correspond to variables unambiguously without rules like precipitation.

Long-wave net irradiance

Here, L and l mean long-wave net irradiance on the coarse and fine scale, respectively. The predictor, ground temperature anomaly, is G.

If L falls short of –82.5 W m^(–2), then
l = –3.878 W m^{(–2) K}(–1) G .
Else there is no rule.

Short-wave net irradiance

The shortwave net irradiance N is a linear combination of direct downwelling irradiance S, the diffuse downwelling irradiance C and the diffuse upwelling irradiance R:

N = S + C - R

The definition of the latter 2 variables ensures them not to be negative, physical considerations ensure the same for the former 2.
In turn, R is a linear combination of S and C, given the direct albedo a and the diffuse albedo u:

R = a S + u C

This leads to:

N = (1-a) S + (1-u) C

The albedos stem from external high resolution data, the irradiances from coarse active data. The sun emits the direct radiation, an unambiguous coarse-scale effect, and the dominant addend in a clear sky day. The diffuse downwelling irradiance C may vary on the fine scale, though. Unfortunately, clouds cause this variability and not surface variables. Schomburg rules cannot access the free troposphere, nor anything higher than some 30 m. Therefore, the Schomburg scheme will see this variability as noise and address it in its 3rd step.

Footnotes

fn1^. The source calls this only specific humidity, or QVS. There are 4 reasons ‘specific’ here means mass-specific, or gravimetric, instead of volume-specific, or volumetric:

1. Canonically, ‘specific’ means mass-specific unless otherwise specified.
2. Its table A1 lists its unit as kg/kg, a common way to hint at mass-specificity, though of course, kg/kg technically just means no unit.
3. The symbol q usually substitutes gravimetric humidity and rarely anything volumetric.
4. More applications for gravimetric humidity exist than for volumetric humidity.