Meteorologically consistent bias correction of climate time series for agricultural models


Conventional bias correction of simulated climate time series for impact models is done separately for climate variables and hence leads to inconsistencies between them. However, agricultural models mostly use several variables, and meteorological consistency is essential. The present work points out meteorological inconsistency due to quantile mapping and describes a new method of consistent bias correction by an optimization approach. Time series of hourly precipitation and global radiation from the regional model REMO5.7 (Run UBA C20/A1B_1) were corrected with site observations from the German Meteorological Service. The results urge to check conventionally corrected series for consistency before using them for multidimensional models. Here, quantile mapping resulted in underestimation of diffuse radiation at hours with precipitation. This deficit was minimized by the developed procedure.

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The project was supported by the Ministry for Science and Culture of Lower Saxony within the network KLIFF—climate impact and adaptation research in Lower Saxony. We thank Dr. Daniela Jacob and Dr. Christopher Moseley (Max Planck Institute for Meteorology, Hamburg, Germany) for their support.

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Correspondence to Holger Hoffmann.



Basic functions

Arrays are denominated X, X′, X′′ ... X γ and their respective elements are x, x′, x′′, and x′′′, whereas the number of elements is designated with n, n′, n′′, and n′′′. s and z are arbitrary index variables; v and h are real values.

Unique function f α (X)

This equation can be used to reduce the domain X to unique values X′:

$$ \begin{array}{rll} f_{\alpha}(X)&=&X'\ \mbox{with} \\ \{X'\mid x'_{s}&=&x_{s},\ x_{s}' \neq{x_z'}\ \mbox{for all possible}\ z\ \mbox{and}\ z\neq{s}\}\\ \end{array} $$

Occurrence function f δ (X,X′)

This function sums the occurrences of values X in array X′:

$$ \begin{array}{rll} f_{\delta}(X,X')&=&X''\ \mbox{with}\\ \left. \Big \{X''\mid x''_{s}\right.&=& \left. \sum\limits_{z=1}^{n'}v_{s,z} \right.\Big\} \mbox{ with}\\ v_{s,z}&=& \left\{ \begin{array}{ccc} 1&\mbox{if}\ &x_{s}=x'_z \\ 0&\mbox{else}& \end{array} \right. \end{array} $$

Sort function f λ (X)

An array of sorted values X′ can be obtained from an input array X as follows:

$$ \begin{array}{rll} f_{\lambda}(X)&=&X'\ \mbox{with}\\ \{X'\mid x'_z&=&x_{s},x'_z\leq x'_{z+1}\} \end{array} $$

Percentile function \(f^{\beta}_\Delta(X)\)

The value of a given percentile β in an array X with n elements is calculated as:

$$ \begin{array}{rll} f^{\beta}_\Delta(X)&=&(1-\mbox{frac}(v))\cdot x_{\mbox{int}(v)}+\mbox{frac}(v)\cdot x_{\mbox{int}(v)+1}\\ v&=&n\cdot \frac{\beta}{100}+0.5 \\ \mbox{frac}(v)&:&\mbox{fractional part of}\ v \\ \mbox{int}(v)&:&\mbox{integer part of}\ v \end{array} $$

Rescale function \(f_\Omega^\beta(v,X)\)

This function rescales a value v by the corresponding percentile β of X.

$$ f_{\Omega}^\beta(v,X) = \frac{v}{f^{\beta}_\Delta(X)} $$

Lookup functions f π , f ω , and f ϕ

The f π function interpolates linearly in test array X′ for a given value v in the X domain:

$$ \begin{array}{rll} f_{\pi}(v,X,X')&=&\frac{v-x_h}{x_r-x_h}\cdot (x'_r-x'_h)+x'_h \\ h&=&\max(s)\ \mbox{with}\ x_{s}\leq v\\ r&=&\min(s)\ \mbox{with}\ x_{s}\geq v \end{array} $$

To interpolate a given value v of the X′ domain in test arrays X′′...X γ, the function f ω can be applied:

$$ \begin{array}{rll} f_{\omega}(v,X',X''...X^\gamma)&=&X\ \mbox{with}\\ \{X\mid x_{s}&=&f_{\pi}(v,X',X^s)\},\\ &&X^1=X',\ X^2=X'',... \end{array} $$

To interpolate two-dimensionally, f ω can be extended as follows:

$$ \begin{array}{rll} &&{\kern-8pt}f_{\phi}(v_1,v_2,X',X'',X'''\ ...\ X^\gamma)\\&&{\kern6pt}=f_{\pi}(v_2,X'',f_{\omega}(v_1,X',X'''\ ...\ X^\gamma)) \end{array} $$

If values v 1 and v 2 in Eq. 22 (first two parameters) are not single values but arrays, Eq. 22 should be applied for each element of the arrays separately. Therefore, the result is an array. For further explanation and higher dimensions, please see

Build function \(f_\theta^h(v_1,v_2)\)

To create an equally spaced array X′ between two values v 1 and v 2 with step size h, linear interpolation is conducted as follows:

$$ \begin{array}{rll} f_\theta^{h}(v_1,v_2)&=&(x_0\ ...\ x_{n})\ \ \ \mbox{with}\\ x_{s}&=&h\cdot s+v_1,\ x_{s}\leq v_2 \end{array} $$

Error identification

Standardized errors qst were calculated by:

$$ \begin{array}{rll} qst_i&=&\frac{q_i}{\sqrt{\frac{1}{n-1}\cdot \sum_{i=1}^{n}(q_i-\bar{q})^2}}\ \mbox{with }\notag\\ \bar{q}&=&\frac{\sum_{i=1}^n q_i}{n}\mbox{ with } \{q_1 ... q_n\}=Q \end{array} $$

where the mean deviation \(\bar q\) is the model bias. The model root mean squared error was determined as

$${\rm RMSE}= \sqrt{\frac{1}{n}\cdot \sum\limits_{i=1}^{n}q_i^2} $$

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Hoffmann, H., Rath, T. Meteorologically consistent bias correction of climate time series for agricultural models. Theor Appl Climatol 110, 129–141 (2012).

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  • Bias Correction
  • Global Radiation
  • Quantile Mapping
  • Diffuse Radiation
  • Simulated Time Series