A non-parametric, statistical downscaling algorithm applied to the Rohini River Basin, Nepal

Abstract

Climate change scenarios generated by general circulation models have too coarse a spatial resolution to be useful in planning disaster risk reduction and climate change adaptation strategies at regional to river basin scales. This study presents a new non-parametric statistical K-nearest neighbor algorithm for downscaling climate change scenarios for the Rohini River Basin in Nepal. The study is an introduction to the methodology and discusses its strengths and limitations within the context of hindcasting basin precipitation for the period of 1976–2006. The actual downscaled climate change projections are not presented here. In general, we find that this method is quite robust and well suited to the data-poor situations common in developing countries. The method is able to replicate historical rainfall values in most months, except for January, September, and October. As with any downscaling technique, whether numerical or statistical, data limitations significantly constrain model ability. The method was able to confirm that the dataset available for the Rohini Basin does not capture long-term climatology. Yet, we do find that the hindcasts generated with this methodology do have enough skill to warrant pursuit of downscaling climate change scenarios for this particularly poor and vulnerable region of the world.

Appendix

• Step 1. Let [X] represent the data matrix large-scale climate indices for n years (rows) and M grid boxes (columns). The NCEP reanalysis dataset contains variables from nine grid squares (nine columns). Matrix [X] represents data for the entire record period (1976–2006).

• Step 2. For feature year i (the year for which reconstruction is sought), select the corresponding large-scale climate predictors of year i. The x _i predictors of the feature year i represent the feature vector {F}.

• Step 3. Perform drop-one cross-validation. This involves dropping the year that is being hindcast from the matrix [X] to form a submatrix [S]. [S] contains predictor variables from all years of [X], except the feature year that is being hindcast.

• Step 5. Estimate correlation matrix [C] (order m _i × m _i ) from data matrix [S].

• Step 6. Perform Principal Component Analysis (see von Storch and Zwiers 2001; Wilks 2006) using matrix [C] to obtain the m _i eigenvalues λ ₍₁₎, ..., λ _(m),and the eigenmatrix (matrix of eigenvectors as columns) [E] (order, m _i × m _i ).

• Step 7. Project the feature vector {F} for feature year i onto the eigenvectors in matrix [E]. The projected feature vector {F’} is given by

  $$ {\left\{ {F\prime } \right\}_{1 \times {m_i}}} = {\left\{ F \right\}_{1 \times {m_i}}}{\left[ E \right]_{{m_i} \times {m_i}}} $$

  (3)

• Step 8. Calculate the m _i principal components. The principal component matrix [Z] is obtained from

  $$ {\left[ Z \right]_{n \times {m_i}}} = {\left[ S \right]_{t \times {m_i}}}{\left[ E \right]_{{m_i} \times {m_i}}} $$

  (4)

• Step 9. For each element m (m = 1, …, n), compute the weighted Euclidian distance (d _t ) between the projected feature vector {F’} (step 7) and the principal components contained in matrix [Z] (step 8).

  $$ {d_t} = {\left[ {\sum\limits_{j = 1}^{\rm{nret}} {\frac{{{\lambda_j}}}{{\sum\limits_{p = 1}^{{m_i}} {{\lambda_p}} }}{{\left( {f_j^\prime - {z_{tj}}} \right)}^2}\,} } \right]^{1/2}} $$

  (5)

  where, nret is the number of principal components retained such that $\sum\limits_{j = 1}^{\rm{nret}} {{\lambda_{(j)}} \approx 0.90}$; z _tj are the elements of [Z], and $f_j^\prime$ are the elements of the projected feature vector {F’}. This gives a set of n distances as possible neighbours from the overlap period to feature year i.

• Step 10. Sort the distances d _t in ascending order and retain only the first K-neighbours (Gangopadhyay et al. 2005). The prescribed choice for K is $\sqrt {n}$ ≈ 6 in this case. The K-nearest neighbours represent the K most similar years from the dataset to the feature year i.

• Step 11. Select the observed rainfall for each of the K neighbour years from the subset period; this represents the set of possible rainfall magnitudes for feature year i.

• Step 12. Assign weights to each of the K rainfall values. Several weighting schemes based on either K (Lall and Sharma 1996; Rajagopalan and Lall 1999) or distance such as the bi-square weight function (Gangopadhyay et al. 2005) and inverse distance weighting (Chow et al. 1988) are available. We tested our results using these different weighting schemes and found that they produce very similar results. We present results in this paper based on the bi-square weighting scheme. The bi-square weight, w _k , for neighbour k is given by

  $$ {w_k} = \frac{{{{\left[ {1 - {{\left( {\frac{{{d_{(k)}}}}{{{d_{(K)}}}}} \right)}^2}} \right]}^2}}}{{\sum\limits_{k = 1}^K {{{\left[ {1 - {{\left( {\frac{{{d_{(k)}}}}{{{d_{(K)}}}}} \right)}^2}} \right]}^2}} }} $$

  (6)

• Step 13. Bootstrap (Venables and Ripley 2002) the K rainfall values (step 11) using the weights w _k , k = 1, …, K (Step 12) to generate an ensemble of rainfalls for year i.

• Step 14. For each of the years 1976–2006, repeat steps 3 through 13 to obtain an ensemble rainfall reconstruction.