Let Xts represent the occurrence of daily precipitation for a location s (s=1,…,S) on day t (t=1,…,n; n is the number observed days) and let Xts be either 0 for dry days or 1 for wet days. The first-order Markov chain model for Xts is defined with the assumption that the occurrence probability of a wet day is fully defined by the previous day as 1PrXts=1|Xt-1s=0=p01s2PrXts=1|Xt-1s=1=p11s. Also, p00s=1-p01s and p10s=1-p11s, since the summation of 0 and 1 should be unity with the same previous condition. This consists of a transition probability matrix (TPM) as 3TPMs=p00sp01sp10sp11s=1-p01sp01s1-p11sp11s. The marginal distributions of TPM (i.e., p0 and p1) can be expressed with TPM and its condition of p0+p1=1 as 4p0s=p01s1+p01s-p11s5p1s=1-p11s1+p01s-p11s. Note that p1 represents the probability of precipitation occurrence for a day, while p0 represents non-occurrence. The lag-1 autocorrelation of precipitation occurrence is the combination of transition probabilities as 6ρ1(s,s)=p11s-p01s. The simulation can be done by comparing TPM with a uniform random number (uts) as 7Xts=1ifuts≤pi1s0otherwise, wherepi1s is the selected probability from TPM regarding the previous condition i (i.e., either 0 or 1). Wilks (1998) suggested a different method using a standard normal random number wts∼N[0,1] as 8Xts=1ifwts≤Φ-1(pi1s)0otherwise, where Φ-1indicates the inverse of the standard normal cumulative function Φ.

Wilks (1998) suggested a multisite occurrence model using a standard normal random number (here denoted as MONR) that is spatially dependent but serially independent. The correlation of the standard normal variate for a site pair of q and s can be expressed as 9τ(q,s)=corr[wtq,wts]. Also, the correlation of the original occurrence variate is 10ρ(q,s)=corr[Xtq,Xts]. Once the correlation of the standard normal variate is known, the simulation of multisite precipitation occurrence is straightforward. Multivariate standard normal distribution is used with a parameter set of [0, T], where 0 is the zero vector (S×1) and T is the correlation matrix with the elements of τ(q,s) for q∈{1,…,S} and s∈{1,…,S}.

Since direct estimation of τ(q,s) is not feasible, a simulation technique is used to estimate τ(q,s) from ρ(q,s). A long sequence of the occurrences is simulated with different values of τ(q,s) and its corresponding correlation of the original domain ρ(q,s) is estimated with the simulated long sequence by the inverse standard normal cumulative function (i.e., Φ-1). A curve between τ(q,s) and ρ(q,s) is derived from this long simulation with the MONR model and is employed for parameter estimation for a real application.

In the current study, a novel multisite simulation model for discrete occurrence of precipitation variable with the KNNR technique (Lall and Sharma, 1996; Lee and Ouarda, 2011; Lee et al., 2017) for a discrete case (denoted as discrete KNNR; DKNNR) is proposed by combining a mixture mechanism with GA. Provided the number of nearest neighbors, k, is known, the discrete k-nearest neighbor resampling with genetic algorithm is done as follows:

Estimate the distance between the current (i.e., time index: c) multisite occurrence Xcs and the observed multisite occurrence xis. Here, the distance is measured for i=1,…,n-1 as11Di=∑s=1SXcs-xis.

The selection of the number of nearest neighbors (k) has been investigated by Lall and Sharma (1996) and Lee and Ouarda (2011). A simple selection method was applied in the current study as k=n. For hydrometeorological stochastic simulation, this heuristic approach of the k selection has been employed (Lall and Sharma, 1996; Lee and Ouarda, 2012; Lee et al., 2010; Prairie et al., 2006; Rajagopalan and Lall, 1999). One can use generalized cross-validation (GCV) as shown in Lall and Sharma (1996) and Lee and Ouarda (2011) by treating this simulation as a prediction problem. However, the current multisite occurrence simulation does not necessarily require an accurate value prediction and not much difference in simulation using the simple heuristic approach has been reported. Also, this heuristic approach of the k selection has been popularly employed for hydrometeorological stochastic simulations (Lall and Sharma, 1996; Lee and Ouarda, 2012; Lee et al., 2010; Prairie et al., 2006; Rajagopalan and Lall, 1999).

In Appendix A, an example of the DKNNR simulation procedure is explained in detail.

The capability of model to take climate change into account is critical. For example, the marginal distributions and transition probabilities in Eqs. (5) and (3) can change in future climate scenarios. It is known that nonparametric simulation models have difficulty adapting to climate change, since the models employ in general the current observation sequences. However, the proposed model in the current study possesses the capability to adapt to the variations of probabilities by tuning the crossover and mutation probabilities in Pcr (Eq. 13) and Pm (Eq. 14), adding the condition when applied.

For example, the probability of P11 can be increased with the crossover probability Pcr by adding the condition to increase the probability of P11 as Xc+1s=15xp∗+1sifε<Pcr&xp∗+1s=1&Xcs=1xp+1sotherwise. It is obviously possible to increase the probability of P1 by removing the condition of Xcs=1.

In addition, further adjustment can be made with the mutation process in Eq. (14) as 16Xc+1s=xξ+1sifε<Pmandxξ+1s=1xp+1sotherwise. This adjustment of adding the condition xξ+1s=1 can increase the marginal distribution as much as Pm×P1. This has been tested in a case study.

The roles of crossover probability Pcr (Eq. 13) and mutation probability Pm (Eq. 14) were studied by Lee et al. (2010). In the current study, we further tested these by selecting an appropriate parameter set of these two parameters with the simulated data from the DKNNR model and the record length of 100 000. RMSE (Eq. 18) of the three transition and limiting probabilities (P11, P01, and P1) between the simulated data and the observed was used, since those probabilities are key statistics that the simulated data must match the observed data, and no parameterization of these probabilities was made for the current DKNNR model. Results are shown in Figs. 2 and 3 for Pcr and Pm, respectively. For Pcr in Fig. 2, the probability of 0.02 shows the smallest RMSE in all transition and limiting probabilities. The RMSE of Pm in Fig. 3 shows a slight fluctuation along with Pm. However, all three probabilities (P11, P01, and P1) have relatively small RMSEs in Pm=0.003. Therefore, the parameter set 0.02 and 0.003 was chosen for Pcr and Pm, respectively, and employed in the current study. We also tested the simulation without the GA mixing procedure (results not shown). The results showed that no better result could be found from the simulation without GA mixing. The necessity of the GA mixing is further discussed in the following.

We further tested and discuss why the GA mixing is necessary in the proposed DKNNR model as follows. For example, assume that three weather stations are considered and observed data only have the occurrence cases of 000, 001, 011, 010, 011, 100, and 111, among 23=8 possible cases. In other words, no patterns for 110 and 101 are found in the observed data. Note that 0 indicates dry days and 1 indicates rainy (or wet) days. The KNNR is a resampling process in which the simulation data are resampled from observations. Therefore, no new patterns such as 110 and 101 can be found in the simulated data.

This can be problematic for the simulation purpose in that one of the major simulation purposes is to simulate sequences that might possibly happen in the future. The wet (1) or dry (0) for multisite precipitation occurrence is decided by the spatial distribution of a precipitation weather system. A humid air mass can be distributed randomly, relying on wind velocity and direction, as well as the surrounding air pressure. In general, any combinations of wet and dry stations can be possible, especially when the simulation continues infinitely. Therefore, the patterns of simulated data must be allowed to have any possible combinations (here 4096), even if they have not been observed from the historical records. Also, the probability to have this new pattern must not be high, since it has not been observed in the historical records, and this can be taken into account by low probability of the crossover and mutation.

This drawback of the KNNR model frequently happens in multisite occurrence as the number of stations increases. Note that the number of patterns increases as 2n, where n is the number of stations. If n=12, then 4096 cases must be observed. However, among 4096 cases, observed cases are limited, since the amount of data is limited. The GA process can mix two candidate patterns to produce new patterns. For example, in the three-station case, a new pattern of 101 can be produced from two observed occurrence candidates of 001 and 100 by the crossover of the first value of 001 with the first value of 100 (i.e., 001→101), which is not in the observed data.

Note that the data employed in the case study are 40 years and 122 days (summer months) in each year. The total number of the observed data is 4880 and the number of possible cases is 4096. We checked the number of possible cases that were not found in the observed data. The result shows that 3379 cases were not observed at all for the entire cases as shown in Fig. 4.

We further investigated the number of new patterns that were generated with the probabilities Pcr=0.02, Pm=0.001 by the proposed GA mixing. The generated data for 100 sequences from DKNNR with the GA mixing show that the number 3379 was reduced to 1200, which is not in the dataset among the 4096 possible patterns. Therefore, more than 2000 new patterns were simulated with the GA mixing process. The KNNR model without the GA mixing did not produce any new patterns in the 100 sequences with the same length of the historical data.

The data simulated from the proposed DKNNR model and the existing MONR model were analyzed. The estimated transition probabilities (P11 and P01 in Eq. 3) as well as the occurrence probability (P1 in Eq. 5) are shown in Table 2 and Figs. 5–7 for the observed data and the data generated from the DKNNR and MONR models. In Table 2, the observed statistic shows that P11 is always higher than P01, and P1 is between P11 and P01. Site 6 shows the lowest P11 and P1, and site 12 shows the highest P11.

As shown in Fig. 5, the probability P11 of the observed data shows that sites 6, 7, 8, and 9 located in the northern part of the region exhibited lower consistency (i.e., consecutive rainy days) than did the other sites, while sites 5 and 12 had higher probability of P11 than did other sites. Both models preserved well the observed P11 statistic. It seems that the MONR model had a slightly better performance, since this statistic is parameterized in the model, as shown in Sect. 2.2, and that is the same for P01 and P1, as shown in Figs. 6 and 7. Note that the MONR model employed the transition probabilities in simulating rainfall occurrence, while the DKNNR model did not. The occurrence probability P1 can be described with the combination of transition probabilities as in Eq. (5). Even though the transition probabilities were not employed in simulating rainfall occurrence, the DKNNR model preserved this statistic fairly well.

In the DKNNR modeling procedure, the simple distance measurement in Eq. (11) allows to preserve transition probabilities in that the following multisite occurrence is resampled from the historical data whose previous states of multisite occurrence (xis) are similar to the current simulation multisite occurrence (Xcs). This summarized distance (Di) is an essential tool in the proposed DKNNR modeling. The condition of the current weather system is memorized and the system is conditioned on simulating the following multisite occurrence with the distance measurement like a precipitation weather system dynamically changes, but often it impacts the system of the following day.

As shown in Fig. 6, the P01 probability showed a slightly different behavior such that sites 1, 2, and 3 located in the middle part of the Yeongnam province showed a higher probability than did other sites. A slight underestimation was seen for sites 2 and 11 but it was not critical, since its observed value with a cross mark was close to the upper IQR representing the 75th percentile.

The behavior of P1 was found to be the same as that of the P11 probability. It can be seen in Fig. 7a that no significant underestimation is seen for the DKNNR model. The P1 statistic was fairly preserved by both DKNNR and MONR models. Note that the MONR model parameterized the P1 statistic through the transition probabilities as in Eq. (5), while the DKNNR model did not. Although the DKNNR model used almost no parameters for simulation, the P1 statistic was preserved fairly well.

Cross-correlation is a measure of the relationship between sites. The preservation of cross-correlation is important for the simulation of precipitation occurrence and is required in the regional analysis for water resources management or agricultural applications. Furthermore, lagged cross-correlation is also as essential as cross-correlation (i.e., contemporaneous correlation). For example, the amount of streamflow for a watershed from a certain precipitation event is highly related to lagged cross-correlation.

Daily precipitation occurrence, in general, shows the strongest serial correlation at lag-1 and its correlation decays as the lag gets longer. This is because a precipitation weather system moves according to the surrounding pressure and wind direction that dynamically change within a day or week. Therefore, we analyzed the lag-1 cross-correlation in the current study as the representative lagged correlation structure.

The cross-correlation of observed data is shown in Table 3. High cross-correlation among grouped sites, such as sites 6, 7, and 8 (northern part) and sites 3, 4, and 5, as well as 12 (southeast coastal area, 0.68–0.87), was found. As expected, sites 5 and 12 had the highest cross-correlation (0.87) due to proximity. The northern sites and coastal sites showed low cross-correlation. This observed cross-correlation was well preserved in the data generated from both DKNNR and MONR models, as shown in Fig. 8 as well as Tables 4 and 5. However, consistently slight but significant underestimation of cross-correlation was seen for the data generated by the MONR model (see Fig. 8b). Note that the error bars are extended to upper and lower lines of the circles to 1.95 times the standard deviation. The difference of RMSE in Table 6 showed this characteristic, as most of the values were positive, indicating that the proposed DKNNR model performed better for cross-correlation.

The lag-1 cross-correlation of observed data, as shown in Table 7, ranged from 0.22 to 0.35. The lag-1 cross-correlation for the same site (i.e., ρ1(q,s), q=s) was autocorrelation and was highly related to P01 and P11 as in Eq. (6). All the lag-1 cross-correlations exhibited similar magnitudes even for autocorrelation. This implies that the lag-1 cross-correlation among the selected sites was as strong as the autocorrelation and as much as the transition probabilities P01 and P11 thereof.

The observed lag-1 cross-correlations were well preserved in the data generated by the DKNNR model, as shown in Fig. 9a, while the MONR model showed significant underestimation, as seen in Fig. 9b. The difference of RMSE shown in Table 8 reflects this behavior. In Fig. 9b, some of the lag-1 cross-correlations were well preserved, that were aligned with the baseline. From Table 8, the MONR model reproduced the autocorrelations well with the shaded values. It is because the lag-1 autocorrelation was indirectly parameterized with the transition probabilities of P11 and P01 as in Eq. (6). Other than this autocorrelation, the lag-1 cross-correlation was not reproduced well with the MONR model. This shortcoming was mentioned by Wilks (1998). Meanwhile, the proposed DKNNR model preserved this statistic without any parameterization.

We further tested the performance measurements of mean absolute error (MAE) and bias whose estimates showed that MAE had no difference from RMSE. In addition, bias of lag-1 correlation presented significant negative values, implying its underestimation for the simulated data of the MONR model as shown in Table 9, while Table 10 of the DKNNR model showed a much smaller bias.

Also, the whole-year data instead of the summer season data were tested for model fitting. Note that all the results presented above were for the summer season data (June–September), as mentioned in Sect. 4 in the data description. The lag-1 cross-correlation is shown in Fig. 10, which indicates that the same characteristic was observed as for the summer season, such that the proposed DKNNR model preserved better the lagged cross-correlation than did the existing MONR model. Other statistics, such as correlation matrix and transition probabilities, exhibited the same results (not shown). Also, other seasons were tried but the estimated correlation matrix was not a positive semi-definite matrix and its inverse cannot be made for multivariate normal distribution in the MONR model. It was because the selected stations were close to each other (around 50–100 km) and produced high cross-correlation, especially in the occurrence during dry seasons. Special remedy for the existing MONR model should be applied, such as decreasing cross-correlation by force, but further remedy was not applied in the current study since it was not within the current scope and focus.

Model adaptability to climate change in hydro-meteorological simulation models is a critical factor, since one of the major applications of the models is to assess the impact of climate change. Therefore, we tested the capability of the proposed model in the current study by adjusting the probabilities of crossover and mutation as in Eqs. (15) and (16). A number of variations can be made with different conditions.

In Fig. 11, the changes of transition and marginal probabilities are shown, along with the increase of crossover probability Pcr from 0.01 to 0.2 with the condition that the candidate value is 1 and the previous value is also 1 as in Eq. (15) for the selected 5 stations among the 12 stations (from station 1 to station 5; see Table 1 for details). The stations were limited in this analysis due to computational time. In each case, 100 series were simulated. The average value of the simulated statistics is presented in the figure. It is obvious that the transition probability P11 increased as intended along with the increase of Pcr. As expected from Eq. (5), P1 presents that the change of P1 is highly related to P11. However, the probability P01 fluctuated along with the increase of Pcr. Elaborate work to adjust all the probabilities is however required.

The changes in transition and marginal probabilities are presented in Fig. 12 with increasing mutation probability Pm from 0.01 to 0.2 under the condition that the candidate value is 1, so that the marginal probability P1 increased. P01 also increased along with increasing P1. The change of P11 was not related to other probabilities. The combination of the adjustment of Pcr and Pm with a certain condition to the previous state will allow the specific adaptation for simulating future climatic scenarios.

As an example, assume that the occurrence probability (P1) of the control period is 0.26 (see the dotted line with a cross in the bottom panels of Figs. 11 and 12) and global circulation model (GCM) output indicates that the occurrence probability (P1) increases up to 0.27. This can be achieved with increasing either the crossover probability to 0.1 or the mutation probability to 0.05. Note that the crossover probabilities might cause the stations to affect each other, while the mutation probabilities do not.

Climate change, however, may refer to a larger phenomenon, which cannot be addressed directly through modifying only the marginal and transition probabilities as in the current study. Further model development on systematically varying temporal and spatial cross-correlations is required to properly address climate change of the regional precipitation system.