DQM: Specificities in Xclim vs. (Cannon,2016)'s description

[coxipi]

Generic Issue

• xclim version: 0.51.0
• Python version:
• Operating System:

Description

I want to underline differences between our implementation in Xclim and the method by Cannon as described in the 2016 paper. In the end, I don't think that this has important consequences for the results, but I think it's worth explaining what I've done.

Let

• $X$ be a simulation, $Y$ a reference dataset;
• $X^h,Y^h$ be the datasets in a historical training period.
• $\overline{X}^T$ is the trend of a field $X_t$ centered on time $t$ (a rolling mean 15 years before/after $t$). Previous fields should also have a subscript $t$ but I only write here for clarity. The $T$ stands for trend, I really want to underline there is still a time dependance (smoothening over 30 years with a rolling window). If I write simply $\overline{X}$, then it's a mean, all points in the time series are taken, time dependance is lost.

For instance, in the Quantile Mapping method, we use $X^h, Y^h$, find a transfer function that computes adjustment factors $A_{QM(X^h, Y^h)}$ and apply this to $X$

$$X_{QM}^{\prime} = X + A_{QM(X^h, Y^h)}(X)$$

For the scaling transformation, we simply have a constant factor added
$$X_{Scaling}^{\prime} = X + A_{Scaling(X^h, Y^h)}$$

DQM Xclim

$$\begin{aligned} X_{1}&=X+A_{\text{Scaling}(X^{h},Y^{h})} \\\ X_{2}&=X_{1}- \overline{X_1}^T \\\ X_{3}&=X_{2}+A_{\text{QM}(X^{h}-\bar{X^{h}} ,Y^{h}-\bar{Y^{h}})}(X_{2}) \\\ X_{\text{BC}}&=X_{3} + \overline{X_1}^T \end{aligned}$$

DQM Cannon

$$\begin{aligned} X_{1}&=X- \overline{X}^T + \overline{X^{h}} \\\ X_{2}&=X_{1}+A_{\text{QM}(X^{h} ,Y^{h})}(X_{1}) \\\ X_{\text{BC}}&=X_{2} + \overline{X}^T - \overline{X^{h}} \end{aligned}$$

What I Did

I programmed Cannon's method to see the differences. In the training, I simply reused eqm_train, and instead of computing scale, I compute mu_hist = hist.mean(dim=dim). Then in the adjustment, I corrected the detrend sim with mu_hist as I would have done for scale in our DQM implementation:

    ds["sim"] = detrending.detrend(sim) 
    ds["sim"] = u.apply_correction(
        ds.sim,
        u.broadcast(
            mu_hist,
            ds.sim,
            group=group,
            interp=interp if group.prop != "dayofyear" else "nearest",
        ),
        kind,
    )

Here is a test:

ref = open_dataset("sdba/CanESM2_1950-2100.nc").tasmax.isel(location=0)
sim = open_dataset("sdba/ahccd_1950-2013.nc").tasmax.isel(location=0)
ref, hist = [da.sel(time=slice("1981","2010")) for da in [ref,sim]]
group = sdba.Grouper("time.dayofyear", 31)
ref = xclim.core.units.convert_units_to(ref,sim) # I want K
scenC = sdba.DetrendedQuantileMappingC.train(ref=ref,hist=hist, group=group).adjust(sim=sim)
scenX = sdba.DetrendedQuantileMapping.train(ref=ref,hist=hist, group=group).adjust(sim=sim)

The climatological means over each days of year (in the historical period) are very similar

Looking a time series, the differences are smaller that scen - sim but still not negligible:

Other thoughts

A possible alternative implementation could be to work with detrended fields in the training period:

DQM Xclim - alternate

$$\begin{aligned} X_{1}&=X+A_{\text{Scaling}(X^{h},Y^{h})} \\\ X_{2}&=X_{1}- \overline{X_1}^T \\\ X_{3}&=X_{2}+A_{\text{QM}(X^{h}-\bar{X^{h}}^T ,Y^{h}-\bar{Y^{h}}^T)}(X_{2}) \\\ X_{\text{BC}}&=X_{3} + \overline{X_1}^T \end{aligned}$$

Code of Conduct

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