 # NDR calculations?

Hi there,
Sorry in advance for the long post, but I’m trying to understand the basics of how the NDR model calculates nutrient loss from a given pixel to the river. I’ve been looking at the user guide but I’m struggling a bit to make sense of the equations.
Say I use the conceptual figure in the NDR manual as basis (Fig 6) and I want to estimate the nutrient load that reaches the river from the pixel marked in red. In this figure, five downstream pixels (marked in blue) are separating the red pixel from the river. The upstream contributing area to the red pixel consists of 36 pixels (green pixels).
For simplicity, we can assume no leaching to the subsurface (p=0), that all cells have the same land use (and therefore the same load and retention), that the flow path from the red cell to the river is straight (meaning the distance from the red cell to the river is 5 x pixel length) and that the slope of the downstream pixels is constant.
Can I based on this calculate the NDR for the red pixel? I’ve tried to do some simple calculations below:
INPUT:
Pixel size: dx=50 m
Mean upstream slope: Sup=0.05 m/m
Downstream slope: Sdn=0.1 m/m
Upstream area: Aup=90000 m2
Distance to stream: dist=300 m
Effective (max) retention: E=0.48

CALCS
Dup = Sup*sqrt(Aup) = 15
Ddn = dist/Sdn = 3000
IC = log(Dup/Ddn) = -2.30

NDR = (1-E)/(1+exp((IC0-IC)/k))
(note: I think there is an error in Eq 34 in the user guide; I think IC and IC0 should be swapped around?). NDR is quite dependent on the calibration parameters IC0 and k. If I set IC0= -1 and k=1, I get NDR=0.11

Am I getting this right? In the above, I have assumed the effective retention for the red pixel is the equal to the maximum. I’m not sure I quite understand how the effective retention is calculated using the step factor (Eq 37). I think the step factor is supposed to depend on a “critical length” but I find it unclear where it fits in in Eq 37.

Any help appreciated.
Thanks

Thanks for pointing this out! You’re absolutely correct that it should have been IC_0 - IC_i . The model was already doing the correct calculation ( IC_0 - IC_i), it was just incorrect in the user’s guide. This has been fixed and will be updated in the online user’s guide chapter shortly.

The critical length is defined in your biophysical table as crit_len_n and/or crit_len_p and is represented in the User’s Guide’s equation 37 as \ell_{LULC_i}. I’m sorry about the confusion here! I’ve updated the user’s guide to clarify that \ell_{LULC_i} is the critical length.

As far as I can tell, your calculations for this example are correct. The effective retention for the red pixel is the maximum retention along the flowpath between the red pixel and the stream, which is calculated as a function of the step factor and the retention efficiencies of the pixels between the red pixel and the stream. But in your example, you’ve already defined E, so if you know that ahead of time, then your calculations are correct.

Effective retention is a little tricky to calculate because it involves walking the flowpath between the stream and the red pixel. For each pixel visited, equation 36 is applied to determine the effective retention. As a part of this calculation, the step factor is determined for each pixel visited along the flow path. What might not be clear from the User’s Guide is that \ell_{i_{down}} is the distance between the two cells, which could be the cell size adjusted for the diagonal if the pixels are diagonally adjacent. I’ll update the user’s guide to include this detail. The model also takes into account proportional flow, as flow paths may diverge and converge, though this isn’t an issue in the example we’re discussing where all water flows into the next pixel on the way to the stream.

Please let us know if anything isn’t clear here or if you have any further questions!
James

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Thanks for clarifying James, very helpful.
With respect to the effective retention, I think I understand how this is calculated. As I understand it, the effective retention of a pixel cannot exceed E_lulc (assuming all downstream pixels have the same land use)?
Thanks again

Wow, I am so sorry for the delay! This got buried in my email.

To the best of my knowledge, yes, the effective retention of a pixel cannot exceed E_lulc when all of those downstream pixels have the same land use.

If you’re seeing some different behavior, let us know and we an dig into it further.

James

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No worries James. Thanks again for clarifying