# Reach Scale Hydrology

CDF Matching against discrete reference CDF quantiles for bias correction

## Sparse CDF Matching

Sparse CDF Matching performs bias correction against a known reference climatology (in the form of CDF) in a same way as a traditional CDF Matching procedure except that only a number of discrete points on the reference CDF are known instead of a complete CDF. The two methods are identical over the known points of reference CDF and the bias correction ratios in the gaps between the known points where interpolated with a log-normal function, i.e., linear interpolation of the logarithm of correction ratio.

### How it works

To deal with the model biases that may still be present after LSM calibration (e.g., biases related to errors from precipitation forcing, LSM structures/parameterizations, and underrepresented routing processes), we propose a new BC approach that corrects the VIC runoff biases referenced against nine *Q*_{c} maps also delivered by Beck et al. (2015). The problem is very similar to the CDF matching used in a traditional BC (Reichle & Koster, 2004), except here no full CDF is available except for some sparse percentile values (*Q*_{c}). Our assumption is that these *Q*_{c} maps trained from ML can potentially offer useful information on runoff signatures beyond our limited knowledge of model processes and parameters, which is in line with the increasing recognition that considers ML as a powerful approach to understand hydrology in ungauged basins (e.g., Zhang et al., 2018).

At each VIC grid cell (Figure below), the model‐simulated daily runoff (35‐year, 12,784 samples) is used to construct the empirical CDF (blue line), with runoff values at different exceedance probabilities computed as *R*_{99,m}, *R*_{95,m}, *R*_{90,m}, *R*_{80,m}, *R*_{50,m}, *R*_{20,m}, *R*_{10,m}, *R*_{5,m}, and *R*_{1,m} (blue dots). The corresponding nine runoff characteristics, denoted as *R*_{99,o}, *R*_{95,o}, *R*_{90,o}, *R*_{80,o}, *R*_{50,o}, *R*_{20,o}, *R*_{10,o}, *R*_{5,o}, *R*_{1,o}, respectively, are used as reference points for adjustment (red dots). To use the sparse reference information, the ratio correction factor *C*_{i} is calculated (Equations below) at all available reference points. Assuming the intermediate ratio correction factors (*C*_{ij}) between *C*_{i} and *C*_{i}_{+1} follow a loglinear relationship (where *j* and *N* stand for the *j*th point and the total number of points between *i* and *i*+1, respectively; *i*+1stands for the next available runoff characteristics), *C*_{ij} can be written as equation 2. For modeled runoff values (*R*_{m}) greater than *R*_{1,}_{o} and those less than *R*_{99,}_{o}, a simple extrapolation technique is applied by taking the correction factor as *C*_{99} and *C*_{1}. The bias‐corrected values are eventually computed by multiplying the original runoff time series by *C*_{ij}.

*C*_{i} = *R*_{i}_{,o} / *R*_{i}_{,}_{m}

*C*_{ij} = (*C*_{i}) ^{1-}^{j}^{/}^{N} (*C*_{i})^{j}^{/}^{N} , if *R*_{99}_{,}_{o} < *R*_{i}_{,}_{m} < *R*_{1}_{,}_{o}

*C*_{ij} = *C*_{99} , if *R*_{i}_{,}_{m} < *R*_{99}_{,}_{o}

*C*_{ij} = *C*_{1} , if *R*_{i}_{,}_{m} > *R*_{1}_{,}_{o}

### Sample code

### Reference

The method is described in this paper:

Lin, P., M. Pan, H. E., Beck, Y. Yang, D. Yamazaki, R. Frasson, C. H. David, M. Durand, T. M. Pavelsky, G. H. Allen, C. J. Gleason, and E. F. Wood, 2019: Global reconstruction of naturalized river flows at 2.94 million reaches. Water Resources Research, https://doi.org/10.1029/2019WR025287.

Contact Peirong Lin peirongl@princeton.edu or Ming Pan mpan@princeton.edu for questions.