eyeprolog

Field nitrogen balance

Which fields get too little nitrogen, which get too much, and where is leaching worst?

Farmers balance the nitrogen a crop needs against what the soil and fertilizer supply, minus what is lost. This example turns that budget into rules for four fields, classifies each one, and names the field most at risk of nitrogen leaching into groundwater.

Run it in the playground · Source

The question

For each field, given soil nitrogen, fertilizer, the fraction lost, and the crop’s demand (all in kg per hectare):

How much nitrogen is actually available to the crop?
Is there a deficit or a surplus?
How large is the leaching risk?
Is the field under-supplied, balanced, or over-supplied?

The facts

field(low_input, 25, 40, 0.10, 110).
field(balanced_loam, 45, 80, 0.12, 110).
field(sandy_high, 30, 150, 0.35, 105).
field(clay_surplus, 70, 90, 0.08, 120).

field(Field, SoilN, FertilizerN, LossFraction, CropDemandN). Sandy soil loses a third of its nitrogen; clay holds on to most of it.

The rules

available_n(F, Avail) :-
  total_n(F, Total),
  field(F, _, _, Loss, _),
  (Retained is 1.0 - Loss),
  (Avail is Total * Retained).

leaching_index(F, Index) :-
  surplus_n(F, Surplus),
  field(F, _, _, Loss, _),
  (Index is Surplus * Loss).

status(F, under_supplied) :- deficit_n(F, D), (D > 10.0).
status(F, balanced) :- deficit_n(F, D), surplus_n(F, S), (D =< 10.0), (S =< 10.0).
status(F, over_supplied) :- surplus_n(F, S), (S > 10.0).

The leaching index weights the surplus by how leaky the soil is, so a small surplus on sand can matter more than a larger one on clay.

The answer

eyeprolog examples/field-nitrogen-balance.pl

status(low_input, under_supplied).
status(balanced_loam, balanced).
status(sandy_high, over_supplied).
status(clay_surplus, over_supplied).
availableN_kg_ha(low_input, 58.5).
availableN_kg_ha(balanced_loam, 110.0).
availableN_kg_ha(sandy_high, 117.0).
availableN_kg_ha(clay_surplus, 147.20000000000002).
...
leachingIndex(sandy_high, 4.199999999999999).
leachingIndex(clay_surplus, 2.1760000000000015).
...
highestLeachingRisk(field_nitrogen_balance, sandy_high).

21 answers in total. Clay has the bigger surplus (27.2 against 12.0), yet sand has the higher leaching risk. The trailing digits are ordinary floating-point rounding.

The reason

step(available_n(sandy_high, 117.0),
     rule(6),
     ['F' = sandy_high, 'Avail' = 117.0, 'Total' = 180, 'Loss' = 0.35, 'Retained' = 0.65],
     [total_n(sandy_high, 180),
      field(sandy_high, 30, 150, 0.35, 105),
      0.65 is 1.0 - 0.35,
      117.0 is 180 * 0.65]).

The sandy field receives 180 kg, keeps 65% of it, and so has 117 kg available. The proof shows the inputs and each arithmetic step, so the agronomy can be audited line by line.

The check

condition('C1', resolution, ok, 45).
condition('C5', re_decision, ok, 32).
steps(77).
verified(45).
recomputed(32).
trusted(0).
verdict(checked).

45 steps are rule or fact applications, replayed against their clauses. The other 32 are arithmetic and comparisons, which the checker recomputes itself (C5, “re-decision”). Nothing is trusted.

Try it

Raise the fertilizer on low_input from 40 to 100. The field becomes balanced, with a surplus of 2.5.
Set the loss fraction of sandy_high to 0.40. Its surplus falls to 3.0, and highestLeachingRisk disappears. Why does that rule only compare two named fields?
Write a general highest_leaching_risk/1 that compares all fields using aggregate_max/5.

Learn more in The Art of EyeProlog, Chapters 6 and 14.

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