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Transfer function noise (TFN) model
The transfer function noise model (TFN) simulates an observed hydrograph by weighting the historic input forcing data, such as rainfall or groundwater pumping, and estimating the random noise as an exponential decay function, which allows irregular water level observations to be simulated (von Asmuth et al., 2005). That is, for each water level observation the entire historic daily forcing data is obtained and a weight is applied to each day's past forcing data. This weighted forcing is then integrated over time (i.e. summed) to give the contribution of the historic forcing to the water level at the time point of interest. This is then repeated for every water level observation. Importantly, the weights are not uniform back in time but are defined by a simple function and can have, say, a decayed exponential distribution, a Guassian-like skewed distribution or a distribution derived from the Theis drawdown equation. In applying the TFN model, the parameters for the weighting functions are adjusted using a <doc_Calibration.html global calibration scheme> (Peterson and Western, 2014).
A weakness of simply weighting past rainfall is, however, that the nonlinear partitioning of rainfall is not accounted for. This is addressed within the Toolbox by the ability to transform forcing data prior to the weighting (Peterson and Western 2015). For example, a simple 1-D vertically lumped soil model can be included in the time-series modelling and calibrated to account for the often nonlinear response of the water table to rainfall.
Lastly, the model TFN time-series model can also be calibrated using Xeon Phi co-processor cards. If a co-processor card is available then the numerical intergration step of the TFN model will be undertaken using the co-processor card(s). To use this features, the computer requires Intel compiler ICC >=2013.
The toolkit allows the user to select one of the following weighting functions for each forcing data time series:
- responseFunction_Bruggeman : used for simulating the streamflow influence on head (von Asmuth et al. 2008). Note, this function is still to be tested and must be used with caution.
- responseFunction_FerrisKnowles : used for simulating the drawdown from pumping with an instantaneous version for the Theis drawdown equation (Shapoori et al. 2015a, 2015b).
- responseFunction_FerrisKnowlesJacobs : as for responseFunction_FerrisKnowles but with Jacob's correction added to approximate an unconfined aquifer (Shapoori et al. 2015b).
- responseFunction_Hantush : used for simulating drawdown from pumping assuming a leaky aquifer response (Shapoori et al. 2015a).
- responseFunction_Pearsons : used for simulating climatic influences that increase the head, for example recharge (Peterson and Western 2014).
- responseFunction_PearsonsNegative : as for responseFunction_Pearsons but with a sign change to simulate climatic influences that lower the head, for example ET (Peterson and Western 2014).
The above weighting functions can also be used as inputs to the following derived weighting functions. This allow, for example, the impact from ET to be simulated not as an additional weighting function but by simply re-scaling the Pearson's weighting function used for recharge, and hence eliminating two model parameters from the model.
- derivedweighting_UnconstrainedRescaled : rescales an input weighting function whereby the recsaling can be positive or negative. This function can be used to simulate the impacts of, say, revegetation by rescaling responseFunction_Pearsons used to simulate recharge. For details, see the GUI example TFN model - Landuse change.
- derivedweighting_PearsonsNegativeRescaled : similar to derivedweighting_UnconstrainedRescaled but only to be used to rescale responseFunction_Pearsons. The function uses a normalised Pearson's function and then applies a scaling parameter. This may reduce the parameter covariance between the input weighting function and the rescaling and hence contribute to reliable calibration.
- derivedweighting_PearsonsPositiveRescaled : as for responseFunction_Pearsons but using a negative rescaling.
The toolkit allows the user to select the following forcing transformation function:
An alternative approach for simulating the impacts of land cover change is to use the forcing transformation functions as inputs to the following derived transformation functions. This allow, for example, the impact of revegetation to be simulated by creating a new flux from the soil moisture model that is scaled by the fraction of revegetation. That is, the free drainage from the soil moisture model could be scaled by a time series of the fraction of revegetation to produce an estimate of the change in recharge from the revegetation. For details, see the GUI example TFN model - Landuse change.
- derivedForcing_linearUnconstrainedScaling : rescales a transformation function flux whereby the rescaling can be positive or negative. Additionally, in the calibration the rescaling is initially assumed to be +- 0.2 of the transformation function flux.
- von Asmuth, J. R. and Bierkens M. F. P. (2005). Modeling irregularly spaced residual series as a continuous stochastic process, Water Resources Research, 41, W12404, DOI: <http://dx.doi.org/10.1029/2004WR003726 10.1029/2004WR003726>.
- von Asmuth J. R., Mass K., Bakker M., Peterson J., (2008). Modeling time series of ground water head fluctuations subject to multiple stresses. Groundwater, 46(1), 30-40. DOI: <http://dx.doi.org/10.1111/j.1745-6584.2007.00382.x 10.1111/j.1745-6584.2007.00382.x>
- Kavetski, D., G. Kuczera, and S. W. Franks (2006), Bayesian analysis of input uncertainty in hydrological modeling: 1. Theory, Water Resources Research, 42, W03407, DOI: <http://dx.doi.org/10.1029/2005WR004368 10.1029/2005WR004368>.
- Peterson, T. J. and Western A. W. (2014). Nonlinear time-series modeling of unconfined groundwater head. Water Resources Research, 50, 8330-8355, DOI: <http://dx.doi.org/10.1002/2013WR014800 10.1002/2013WR014800>. <papers/Peterson_Western_2014.pdf PDF Copy>
- Shapoori V., Peterson T.J. , Western A.W. and Costelloe J. F. (2015a). Decomposing groundwater head variations into meteorological and pumping components: a synthetic study, Hydrogeology Journal, DOI: <http://dx.doi.org/10.1007/s10040-015-1269-7 10.1007/s10040-015-1269-7>. <papers/Shapoori_2015A.pdf PDF Copy>
- Shapoori V., Peterson T.J. , Western A.W. and Costelloe J. F. (2015b). Top-down groundwater hydrograph time-series modeling for climate-pumping decomposition, Hydrogeology Journal, 23(4), 819-83, DOI: <http://dx.doi.org/10.1007/s10040-014-1223-0 10.1007/s10040-014-1223-0>. <papers/Shapoori_2015B.pdf PDF Copy>