Methodology

Beginning with the advective flux term of Penman (1948) [kg/m²/s]:

\[ E_a=\rho_a \frac{\varepsilon}{p_a} d_a \cdot f(u) \]

where \(d_a=(1-r)e_s\) is the vapour pressure deficit [Pa], and the wind-function \(f(u)=a+ub\) [m/s] leaves a pair of empirical parameters: \(a\) and \(b\). With \(\varepsilon=0.622\) and using standard values of \(\rho_a=1.2\text{ kg m}^{-3}\) and \(p_a=10^5\text{ Pa}\), the above is reduced to (Novák, 2012):

\[ E_a=7.46\times 10^{-6} \cdot (a+ub) d_a \]

has the units [mm/s] for water and requires as input:

  • Air temperature \((T_a)\) [°C] to compute \(e_s\) [Pa]
  • Relative humidity \((r)\) [-]
  • Wind speed \((u)\) [m/s]

Pan Evaporation Data

Daily Pan evaporation was measured by the Meteorological Services Canada (MSC) at a variety (17) of locations near the ORMGP jurisdiction from 1962-04-04 to 1996-10-31 (12629 days). The file 199611010000_exportPanEvap.nc was exported from FEWS.

Local MSC pan evaporation stations
station_id aes_id station_name lat long elev
4534 6119325 WASHAGO 44.75000 -79.33333 221.6
4703 6139145 VINELAND STATION 43.18333 -79.40000 79.2
4733 6140818 BLUE SPRINGS CREEK 43.63333 -80.11667 373.4
4756 6142285 ELORA RESEARCH STN 43.65000 -80.41667 376.4
4763 6142627 FULLARTON 43.38333 -81.20000 335.3
4774 6143083 GUELPH OAC 43.51667 -80.23333 333.8
4832 6149387 WATERLOO WELLINGTON A 43.45000 -80.38333 317.0
4866 6150830 BOWMANVILLE MOSTERT 43.91667 -78.66667 99.1
4881 6151042 BURKETON MCLAUGHLIN 44.03333 -78.80000 312.4
4900 61515DE CLAREMONT FIELD CENTRE 43.95000 -79.08333 182.9
4937 6153300 HAMILTON RBG 43.28333 -79.88333 102.1
4945 6153545 HORNBY 43.56667 -79.85000 198.1
4959 6154611 LONG SAULT IHD 44.05000 -78.71667 342.9
5100 6158740 TORONTO MET RES STN 43.80000 -79.55000 193.5
5105 6158749 TORONTO NEW INT’L A 43.95000 -79.13333 262.7
5178 6164433 LINDSAY FROST 44.33835 -78.74028 262.1
5192 6166455 PETERBOROUGH TRENT U 44.36667 -78.30000 198.1

Hourly scalar data from the MSC was collected from 105 stations over the same period. These data contain the necessary input \((T_a, p_a, r, u)\) for the model. The file 199611010000_exportMSChourlyNetcdf.nc was exported from FEWS.

Automatic calibration

Parameters \(a\) and \(b\) were determined automatically using a shuffled complex evolution (SCE) global optimization routine. The Nash-Sutcliffe (1970) efficiency factor (NSE) is computed for both daily and monthly timeseries, and the average of these 2 objective functions is automatically maximized.

The goal here is not to create a predictive model, rather an empirical function that allows me to distribute \(E_a\) estimates regionally.

Model testing

MSC station 4832: WATERLOO WELLINGTON A maintained a 10+ years pan evaporation station with concurrent hourly meteorological data; however, reported only at daytime hours. This station was used as a benchmark in validating the above model formulation. Using the SCE optimization, \(a=0.0107\) and \(b=7.8\times 10^{-4}\) yielded the following results.

Monthly evaporation amounts followed seasonal trends well. In the image above, simulated and observed monthly evaporation flux are in good agreement (NSE=0.90).

From visual inspection, the model shows good match at the daily timescale (r²=0.55).

Or, as an alternate view, a scatter plot of daily data:

Regional calibration

Due to the poor coordination among pan evaporation stations and hourly meteorological stations, the 17 locations where pan evaporation was measured had hourly climate data interpolated to it’s location. The SCE optimization routine was then applied independently to all stations using the interpolated input data.

Using pyFEWS\prep\interpolateMSCpanET.py to create 199611010000_exportPanEvap-InterpolatedMSChourly.nc ~9hrs
Using src\ORMGP Modelling\PanEvap\opt\main.go to optimize ~10min

The resulting weighted (by data length) model parameters are \(a=9.3\times 10^{-3}\) and \(b=7.8\times 10^{-4}\) shown below as the red diamond:

Discussion

It is worthy of note that \(b\) is an order of magnitude less than \(a\). With wind speeds at times exceeding 10 m/s, any effect wind has on evaporation occurs on infrequent windy days. As a long term model, further simplification of the above equation may be allowed to something like \(E_a=Cd_a\), a function of \(T_a\) and \(r\) — maybe next time.

There is also an apparent linear trend to the optimized set of parameter pairs, which may offer leads to further simplifying the equation.

Limitations

A problem with calibrating to evaporation data is the seasonal constraints to southern Ontario. The parameters determined above are entirely conditioned on the summer season. Projecting the results into the winter season using the above formulation and parameterization appears to look consistent where winter rates of evaporation potential fall to ~15% that of summer rates.

The model also has a lower limit of zero meaning that condensation/dew collection is not represented. It is then assumed that this is a negotiable source to the land surface water balance.

Source data

Python code snippet

to read *.nc files

import np
import netCDF4 as nc

with nc.Dataset('199611010000_exportMSChourlyNetcdf.nc') as ds:
  ds.set_auto_mask(False) # https://github.com/Unidata/netcdf4-python/issues/785
  for v in ds.variables: print(v) # print available variables
  
  tim = ds.variables['time']
  tims = nc.num2date(tim[:],tim.units).astype('datetime64[ns]')
  nt = len(tims)        
  
  lats = ds.variables['lat'][:]
  lngs = ds.variables['lon'][:]
  elev = ds.variables['z'][:]
  
  def getVar(vnam):
      print("  - "+vnam)
      v = ds.variables[vnam][:,:]
      v[v == -9999] = np.nan
      return v
  
  sids = getVar('station_id')
  smns = getVar('station_names')
  pa = getVar('air_pressure')
  ta = getVar('air_temperature')
  rh = getVar('relative_humidity')
  wd = getVar('wind_direction')
  ws = getVar('wind_speed')

References

Nash, J.E. and J.V. Sutcliffe, 1970. River flow forecasting through conceptual models, Part I - A discussion of principles. Journal of Hydrology, 10. pp. 282-290.

Novák, V., 2012. Evapotranspiration in the Soil-Plant-Atmosphere System. Springer Science+Business Media Dordrecht. 253pp.

Penman, H.L. (1948) Natural evaporation from open water, bare soil and grass. Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences 193(1032): 120-145.