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Analysis of the Spatial Interpolation Error associated with Maps of Median Annual Climate Variables

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Analysis of the Spatial Interpolation Error associated
with Maps of Median Annual Climate Variables
Andrew Tait and Xiaogu Zheng
National Institute of Water & Atmospheric Research Ltd.
May 2007
Contents
1. Description of the spatial interpolation methodology...................... 2
2. Description of the interpolation model prediction error .................. 3
3. Maps and error statistics .................................................................. 4
4. References...................................................................................... 21
Disclaimer
NIWA have prepared the maps presented in this document exercising
all reasonable skill and care. Nevertheless, NIWA can give no
warranty that the maps are free from errors, omissions or other
inaccuracies. Users of the maps will release NIWA from all liability
whether direct, indirect, or consequential, arising out of the provision of
the maps.
2
1. Description of the spatial interpolation methodology
Mid-latitude Southern Hemisphere westerlies and associated weather systems passing
over New Zealand interact with the country’s complex terrain to produce substantial
spatial variations in the climate (Coulter, 1975; Salinger et al., 2004). To produce
maps showing the full spatial variation of New Zealand’s climate, information is
needed with a spatial resolution of 500m or better. This requires spatial interpolation
of meteorological data from irregularly spaced climate stations onto high resolution
regular grids, through a procedure which can reflect the main causes of inter-station
variability (Wratt et al., 2006). Hutchinson (1991, 1995) describes the method of thin
plate (or Laplacian) smoothing spline interpolation, and its application to the
interpolation of climate data. This method is summarised in the following paragraphs.
Many standard interpolation packages use bivariate interpolation, or interpolation
based on two variables (usually x and y location parameters, e.g. latitude and
longitude). For example, inverse distance weighting (IDW) is an interpolation
method that estimates values between locations using a weighted average of the data
from those locations, where the weights are the inverse of the distance from the data
sources to the interpolation point. Bivariate interpolation therefore requires detailed
information about the location of the data (e.g. climate stations). However, these
methods do not incorporate any dependencies on additional geographic variables,
such as elevation and distance from the sea, which are often highly significant. Thin
plate smoothing splines provide a robust way of incorporating any number of
additional variables that may significantly improve the accuracy of the interpolation,
in addition to location.
Splines work by fitting a surface to the data with some error allowed at each data
point, so the surface can be smoother than if the data were fitted exactly. A single
parameter controls the smoothing and is normally chosen to minimise the mean
square error between the actual value at the stations and their values predicted by all
the other stations. Each station is omitted in turn from the estimation of the fitted
surface and the mean error is found. This is repeated for a range of values of the
smoothing parameter, and the value that minimises the mean error is taken to give the
optimum smoothing. This is called the method of generalised cross validation (GCV).
Most meteorological variables, such as air temperature, rainfall and evaporation, are
affected by altitude. Thus it makes sense to interpolate these parameters using a
spline model with two position variables (latitude and longitude or easting and
northing) and a sub-model which accounts for the dependence on elevation. The
broad spatial pattern is determined by the two position variables, while the inclusion
of elevation modifies the broad pattern to give more precise representations of the
higher resolution variability. An advantage of the spline formulation is that the
coefficient of the linear sub-model (e.g. the temperature rate of change with elevation,
or the lapse rate) is determined automatically from the data (Hutchinson, 1991), and
therefore does not need to be specified a priori. Additional geographic variables such
as distance away from the coast or proximity to the main divide, or meteorological
variables such as percent cloud cover (for mapping solar radiation) can also be
included in the spline model.
3
Maps of long-term statistics (e.g. the median and the 20
th
and 80
th
percentiles), based
on the climatological period 1971–2000, have been produced for New Zealand for
several climate parameters using the spline model ANUSPLIN (Version 4.3:
Hutchinson, 2007). A trivariate (three independent variables) thin plate smoothing
spline was used, as this model consistently resulted in lower interpolation errors over
the whole country compared with bivariate and bivariate plus a dependent variable
models. This result is similar to an analysis of spline model options for interpolating
climate variables for Australia (Hutchinson, 1989).
For the New Zealand maps, the third variable in the interpolation is dependent upon
the climate parameter being mapped (Table 1). Elevation above sea level was used
for temperature-related parameters such as the minimum, mean, and maximum air
temperature; soil temperature at 10cm depth, growing degree days base 5 and 10°C,
cold season chilling total, potential evapotranspiration, and relative humidity.
Rainfall total, number of days of rain, and days of soil moisture deficit were
interpolated using an expert-guided map of the 1951–80 mean annual rainfall as the
third independent variable (Tait et al., 2006). The average percent of days with cloud
cover (derived from satellite observations) was used to interpolate total sunshine
hours and daily solar irradiance, and the average annual daily temperature range
(derived from spatial interpolation using elevation as the third variable in the spline)
was used to interpolate mean daily wind speed.
Hutchinson (1989) used a trivariate thin plate smoothing spline to interpolate several
meteorological variables across Australia. The number of data points varied with
meteorological parameter; 150 points (i.e. climate stations) for solar radiation, 300 for
pan evaporation, 900 for maximum and minimum air temperature, and 10,000 points
for precipitation. The approximate standard errors calculated within the spline
program for this Australian study were: Solar radiation 3%; pan evaporation 5%;
minimum and maximum temperature 0.2–0.5°C; rainfall 10–15%, respectively. As is
shown in section 3 below, the errors for mapping climate variables for New Zealand
are in a similar range to those for Australia.
2. Description of the interpolation model prediction error
The ANUSPLIN program used for this analysis produces as part of the suite of
possible outputs an estimate of the model prediction standard error. This error
statistic can be calculated at all of the data sites used for the interpolation. The model
prediction standard error is a combination of the variance of the data values and the
model standard error estimate of the interpolated values.
Confidence intervals of the calculated spline values are estimated by multiplying the
prediction standard error by 1.96, the 95 percent two-sided confidence interval of the
standard normal distribution.
4
3. Maps and error statistics
Table 1 shows some standard statistical outputs from ANUSPLIN for each climate
parameter. The table lists the third independent variable used in the trivariate spline
model (along with the two positional variables, easting and northing), the signal to
noise ratio resulting from the GCV analysis, the root GCV, and the surface root mean
square error.
The climate maps presented in this section are the median annual statistic (e.g. the
median annual rainfall total), based on the period 1971–2000, for 15 climate
parameters. The parameters are:
1. Median annual daily minimum air temperature
2. Median annual daily mean air temperature
3. Median annual daily maximum air temperature
4. Median annual daily soil temperature at 10cm
5. Median annual growing degree days base 5°C
6. Median annual growing degree days base 10°C
7. Median annual cold season chilling total
8. Median annual potential evapotranspiration total
9. Median annual 9am relative humidity
10. Median annual rainfall total
11. Median annual number of days of rain ≥ 1mm
12. Median annual number of days of soil moisture deficit
13. Median annual sunshine hours total
14. Median annual daily mean solar irradiance
15. Median annual daily mean wind speed
Each map includes the location of the climate stations from which data were used in
the interpolation of the parameter. The error analysis box on each map describes the
number of data sites, the minimum, median, maximum, mean, standard deviation,
coefficient of variation, and mean relative prediction standard error at the data sites.
5
Table 1: Summary statistics from trivariate spline models for the interpolation of several climate
parameters for New Zealand.
Climate parameter Third independent
variable
Signal to
Noise ratio
Root GCV Root Mean
Square Error
Median annual daily
minimum air temperature
Elevation above sea
level (m)
2.77 : 1 0.73 °C 0.32 °C
Median annual daily mean air
temperature
Elevation above sea
level (m)
1.46 : 1 0.50 °C 0.25 °C
Median annual daily
maximum air temperature
Elevation above sea
level (m)
1 : 1.05 0.61 °C 0.30 °C
Median annual daily soil
temperature at 10cm
Elevation above sea
level (m)
1 : 1.67 0.70 °C 0.34 °C
Median annual growing
degree days base 5°C
Elevation above sea
level (m)
1.45 : 1 178 gdds 87.4 gdds
Median annual growing
degree days base 10°C
Elevation above sea
level (m)
1.34 : 1 138 gdds 68.3 gdds
Median annual cold season
chilling total
Elevation above sea
level (m)
1 : 1.16 194 hours 96.5 hours
Median annual potential
evapotranspiration total
Elevation above sea
level (m)
1 : 1.01 36.8 mm 18.4 mm
Median annual 9am relative
humidity
Elevation above sea
level (m)
1.20 : 1 3.29 % 1.64 %
Median annual rainfall total 1951-80 mean annual
rainfall (mm)
1.85 : 1 215 mm 103 mm
Median annual number of
days of rain ≥ 1mm
1951-80 mean annual
rainfall (mm)
1 : 1.01 13.9 days 6.9 days
Median annual number of
days of soil moisture deficit
1951-80 mean annual
rainfall (mm)
1 : 1.33 15.8 days 7.82 days
Median annual sunshine hours
total
Mean percent of cloud
cover (%)
2.72 : 1 161 hours 71.5 hours
Median annual daily mean
solar irradiance
Mean percent of cloud
cover (%)
1.85 : 1 0.79
MJ/m
2
/day
0.38
MJ/m
2
/day
Median annual daily mean
wind speed
Mean annual daily
temperature range (°C)
2.98 : 1 1.16 m/s 0.51 m/s
6
7
8
9
10
11
12
13
14
15
16
17
18
19
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4. References
Coulter, J.D. (1975) The Climate. Biogeography and Ecology of New Zealand. G.
Kuschel, Ed., Junk B.V. Publishers, 87–138.
Hutchinson, M.F. (1989) A new objective method for spatial interpolation of
meteorological variables from irregular networks applied to the estimation of
monthly mean solar radiation, temperature, precipitation and windrun. CSIRO
Division Water Resources Tech. Memo, 89/5, pp 95–104.
Hutchinson, M.F. (1991) The application of thin plate smoothing splines to continent-
wide data assimilation. BMRC Research Report 27, Bureau of Meteorology,
Melbourne, pp104–113.
Hutchinson, M.F. (1995). Interpolating mean rainfall using thin plate smoothing
splines. Int. J. Geographical Information Systems, 9(4): 385–403.
Hutchinson, M.F. 2007. ANUSPLIN, http://cres.anu.edu.au/outputs/anusplin.php.
Last accessed May 2007.
Salinger, M.J., Gray, W., Mullan, A.B. and Wratt, D.S. (2004) Atmospheric
circulation and precipitation. In Mosley, M.P. (Editor), Freshwaters of New
Zealand. NZ Hydrological Society and Limnological Society of New Zealand,
pp 2.1–2.18.
Tait, A.; R. Henderson; R. Turner and X. Zheng (2006) Thin-plate smoothing spline
interpolation of daily rainfall for New Zealand using a climatological rainfall
surface. International Journal of Climatology, 26, pp 2097–2115.
Wratt, D.S., Tait, A., Griffiths, G., Espie, P., Jessen, M., Keys, J., Ladd, M., Lew, D.,
Lowther, W., Mitchell, N., Morton, J., Reid, J., Reid, S., Richardson, A.,
Sansom, J., Shankar, U. (2006) Climate for Crops: Integrating climate data
with information about soils and crop requirements to reduce risks in
agricultural decision-making. Meteorological Applications, 13, 305–315.
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This study presents a method for estimating daily rainfall on a 0.05° latitude/longitude grid covering all of New Zealand for the period 1960–2004 using a second order derivative trivariate thin plate smoothing spline spatial interpolation model. Use of a hand-drawn (and subsequently digitised) mean annual rainfall surface as an independent variable in the interpolation is shown to reduce the interpolation error compared with using an elevation surface. This result is confirmed when long-term average annual rainfall data, derived from the daily interpolations, are validated using long-term river flow data. Copyright © 2006 Royal Meteorological Society
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Locally applicable information about climate and soil properties can help farmers identify opportunities and reduce risks associated with changing to new land uses. This article describes techniques for preparing high-resolution regional maps and GIS surfaces of agriculturally relevant climate parameters. Ways of combining these climate surfaces with soil data and information about the physical requirements of crops to identify areas likely to be the most suitable for new high-value crops are then outlined. Innovative features include methods for merging observations from temporary climate stations installed for one to two years in conjunction with longer-term climate station observations to improve input data for the maps, and techniques for mapping quantiles of climatic factors that may constrain agricultural operations. Examples are the expected ‘one-in-five year’ first and last frost dates, and the ‘one-in-five year’ lowest and highest seasonal rainfalls. The use of night-time satellite infrared observations to improve spatial resolution of frost hazard maps is also described. Typical standard errors of these climate mapping techniques are summarised. The benefits of ongoing consultation with local farmers and local government staff during the design and implementation of climate/soil/crop potential studies are described. These include optimising products to meet local needs, quality control of the resulting maps and GIS surfaces through local knowledge, and improved uptake of information by users. Further applications of techniques described in this paper include products useful to the energy sector, preparation of daily gridded climate data estimates for use in water quality and plant growth modelling, and development of regional climate change scenarios. Copyright © 2006 John Wiley & Sons, Ltd.
Atmospheric circulation and precipitation
  • M J Salinger
  • W Gray
  • A B Mullan
  • D S Wratt
Salinger, M.J., Gray, W., Mullan, A.B. and Wratt, D.S. (2004) Atmospheric circulation and precipitation. In Mosley, M.P. (Editor), Freshwaters of New Zealand. NZ Hydrological Society and Limnological Society of New Zealand, pp 2.1-2.18.
The Climate. Biogeography and Ecology of New Zealand
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Coulter, J.D. (1975) The Climate. Biogeography and Ecology of New Zealand. G. Kuschel, Ed., Junk B.V. Publishers, 87-138.