David Fox has written a very useful article about calculating the uncertainty in estimates of sediment and nutrient loads.

A standard expression for the variance of a variable is:

Mean of the square minus the square of the mean

In this case:

Where is the load and is the expectation.

Fox (2005) derives expressions for the required expectations based on the assumption that flow and concentration is described by the bivariate lognormal distribution.

An R function to calculate these values is as follows:

# mQ = mean log(flow) # mC = mean log(concentration) # sQ = standard deviation log(flow) # sC = standard deviation log(concentration) # r = correlation between log(flow) and log(concentration) Var_L = function(mQ, mC, sQ, sC, r){ EL = exp( (mC + mQ) + (1/(2*(1-r^2))) * ((r*sQ + sC)^2 + (r*sC + sQ)^2 - 2*r*(r*sQ + sC)*(r*sC + sQ))) EL2 = exp( 2*(mC + mQ) + (2/(1-r^2)) * ((r*sQ + sC)^2 + (r*sC + sQ)^2 - 2*r*(r*sQ + sC)*(r*sC + sQ))) EL2 - (EL)^2 }

### Sample calculation

Fox (2005) provides data for a sample calculation, we can use this data to test the functions to calculate these expectations.

Parameter | log(flow) | log(concentration) |
---|---|---|

mean | 2.5561 | -0.02834 |

standard deviation | 0.6706 | 0.8008 |

Correlation between log(flow) and log(concentration) = 0.482.

Using these, values Var[L] = 3132.297 which is consistent with the Fox’s calculations.

Code is available as a gist.

### References

Fox, D. R. (2005) Protocols for the optimal measurement and estimation of nutrient loads: error approximations. Australian Centre for Environmetrics. University of Melbourne. link