Does increasing the spatial resolution of a regional climate model improve the simulated daily precipitation?

Abstract

Three different resolution (50, 12, and 1.5 km) regional climate model simulations are compared in terms of their ability to simulate moderate and high daily precipitation events over the southern United Kingdom. The convection-permitting 1.5-km simulation is carried out without convective parametrisation. As in previous studies, increasing resolution (especially from 50 to 12 km) is found to improve the representation of orographic precipitation. The 50-km simulation underestimates mean precipitation over the mountainous region of Wales, and event intensity tends to be too weak; this bias is reduced in both the 12- and 1.5-km simulations for both summer and winter. In south–east England lowlands where summer extremes are mostly convective, increasing resolution does not necessary lead to an improvement in the simulation. For the 12-km simulation, simulated daily extreme events are overly intense. Even though the average intensity of summer daily extremes is improved in the 1.5-km simulation, this simulation has a poorer mean bias with too many events exceeding high thresholds. Spatial density and clustering of summer extremes in south–east England are poorly simulated in both the 12- and 1.5-km simulations. In general, we have not found any clear evidence to show that the 1.5-km simulation is superior to the 12-km simulation, or vice versa at the daily level.

Appendix: The Ripley K function

The Ripley K-function (Ripley 1977) is a measure of spatial clustering which compares the number of near neighbours with the average spatial density of the whole region of interest (Fig. 8). Given any (time-varying) map (e.g. gridded precipitation), one marks all the events with ones (1) and non-events with zeroes (0). We denote that map (effectively a matrix/vector) with I:

$$ I(x,y,t) = \left\{\begin{array}{ll} 1, & \hbox{if}\,P\geq P_{\rm THRESHOLD}\\ 0, & \hbox{if}\,P < P_{\rm THRESHOLD} \end{array}\right. $$

(4)

In which, P _THRESHOLD is the threshold. The average spatial density of the events is simply defined as:

$$ \varrho(t)=\frac{N(t)}{A} $$

(5)

In which, N is the total number of events, and A is the area of the map (e.g. the number of grid points). By definition, ϱ(t) ≤ 1 (ϱ(t) = 1 implies events are occurring at every single grid point). The average number of near neighbours is a function of distance (or area which is proportional to the distance squared), and is given as:

$$ V(r,t)=\frac{1}{N} \sum_{i=1}^{N} \sum_{j=1}^{N} (-(\delta_{ij}-1)) I(d_{ij}\leq r,t) $$

(6)

Index i represents the summation over all existing events, and index j represents the other events. d _ij is the distance between them:

$$ d_{ij} = \sqrt{({\bf x}_i-{\bf x}_j)^{T} \dot ({\bf x}_i-{\bf x}_j)} $$

(7)

We have used the Kronecker delta function:

$$ \delta_{ij} = \left\{\begin{array}{ll} 1, & \hbox{if}\,i = j\\ 0, & \hbox{if}\,i \neq j \end{array}\right. $$

(8)

Therefore, −(δ _ij  − 1)) denotes the self-exclusion during near neighbour counting. If there is spatial clustering of events, the number of near neighbours (V(r, t)) to any existing event is higher than the value expected by computing the average background density:

$$ V(r,t) > \varrho(t) \pi r^{2} - 1 $$

(9)

The Ripley K-function is defined as the number of near neighbours divided by the average density:

$$ K(r,t)=\frac{V(r,t)}{\varrho(t)} $$

(10)

The Ripley K-function has the dimensions of area (radius squared), and is the non-clustered area (grid boxes) needed to match the number of events as observed in a localized clustered area. By definition, it is not defined if there is only one event (i.e. there are 0 near neighbours). If spatial density is perfectly uniform, then the Ripley K-function is exactly the geometric area of a circle with radius r. If events occur as Poisson processes in space (i.e. events occur at an average rate and independently with respect to each other), the Ripley K-function is approximately but not exactly the same circle geometric area. If events are clustered, then the Ripley K-function at a given radius exceeds the geometric area of the circle with the radius:

$$ K(r,t)\left\{ {\begin{array}{*{20}c} { \approx \pi r^{2} ,} \hfill & {{\text{if events are not clustered}}} \hfill \\ { > \pi r^{2} ,} \hfill & {{\text{if events are clustered}}} \hfill \\ \end{array} } \right.$$

(11)

A simulation study by Ripley (1979) showed that the spatial Poisson process null hypothesis can be rejected at the 0.05 and 0.01 level if the observed maximum Besag L-function exceeds a total area (A) dependent threshold:

$$ \hbox{sup}_{r}(L(r))\geq 1.42\frac{\sqrt{A}}{N},p=0.05 $$

(12)

$$ \hbox{sup}_{r}(L(r))\geq 1.68\frac{\sqrt{A}}{N},p=0.01 $$

(13)

In the present analysis, all individual K-functions for which the null hypothesis cannot be rejected at the 0.05 level are excluded from the time average.

The above assumes that event sampling is not limited by domain specifications. In our datasets, we have undefined points because of:

• No observations outside of model domain;

• Observations over water are undefined, and model non-land points are masked out.

This leads to under-sampling as there are unobserved events over the undefined area. Therefore, a correction factor (w) (Ripley 1977) should be used on V(r). We denote the corrected V(r) with a “hat”.

$$ \hat{V}(r,t)=\frac{1}{N}\sum_{i=1}^{N}\sum_{j=1}^{N}w|\delta _{ij}-1|I(d_{ij}\leq r,t) $$

(14)

$$ \hat{K}(r,t)=\frac{\hat{V}(r,t)}{\varrho(t)} $$

(15)

We have used an area based correction by Besag (1977) (discussed in the postscript of the original Ripley paper) due to its easy implementation with complex coastlines (problem degenerates to counting undefined grid boxes). Given a circle with radius r with only an area $\hat{A}$ within the defined domain (over land and within the SUK domain), the correction factor is defined as:

$$ w(r,x_{0},y_{0})=\frac{\pi r^{2}}{\hat{A}(r,x_{0},y_{0})} $$

(16)

Functions V and K are generally time dependent (maps of daily precipitation). We compute daily ϱ, V, and K values, and present only their time-averaged values.

$$ \overline{\hat{\varrho}} = \frac{1}{T}\sum_{t=1}^{T} \varrho(t) $$

(17)

$$ \overline{\hat{V}}(r) = \frac{1}{T}\sum_{t=1}^{T} \hat{V}(r,t) $$

(18)

$$ \overline{\hat{K}}(r) = \frac{1}{T}\sum_{t=1}^{T} \hat{K}(r,t) $$

(19)

If events are not clustered, K(A) is a linear function (K(A = π r ²) ≈ ϱ A). Since the area (A) is known for any given radius (r), K-function is often plotted as a square root (the Besag L-function) (Besag 1977).

$$ L(r)=\sqrt{\frac{K(r)}{\pi}}-r $$

(20)

Clustering is largest where L(r) is largest, and unclustered data will have L(r) ≈ 0. Note that both the K- and L-function are normalized in a way that they do not favour higher average spatial density. Both functions only measure the inflation of local density due to event clustering.

Since zonal (δx) and meridional (δy) grid point distances can only take on whole number values ($0, 1, 2, 3, \ldots$; “quantized grid space”), and can be diagonal. r is defined as:

$$ r(\delta x,\delta y)=\sqrt{\delta x^{2} + \delta y^{2}} $$

(21)

Note that L(r) and K(r) are functions of r. We have assumed the clustering and density to be isotropic (independent of direction), which is not true in general for precipitation (such as frontal and orographic precipitation). The sampled region are also assumed to be uniform. That is the same as saying the mechanisms behind rainfall within each region are assumed to be the same everywhere. We mimic that by sampling only the north-western orographic or south-eastern convective rain regions. That is, of course, only an approximation; non-uniformity clearly exists within each of the regions—such as non-uniform topography, changing land surface types, and irregular coastlines. A perfect uniform region is impossible to obtain, and we are only able to approximate that by slicing our domain with the Tees-Exe Line.