SCO adjusts terrain elevation on a structured grid using working bounds on eight-neighbour elevation differences derived from a prescribed slope angle θ. Water bodies and lateral boundaries remain fixed throughout. The adjustment proceeds through weighted pairwise projection. Whenever the elevation difference across a neighbour pair exceeds a local bound, the solver redistributes the excess between the two endpoints in proportion to their mobility weights. The bounds include a conservative geometric scaling, explained in Sect. 2.2. Inner stopping conditions and a separate outer acceptance test determine when iteration ends (Sect. ). The implementation assumes uniform grid spacing (Δx=Δy).

Because SCO operates on the standard geo_em files produced by geogrid, it integrates easily into the existing WPS pipeline between geogrid and metgrid with no modifications to the WRF source code. Here, geogrid interpolates static geographical data, including terrain, to the model grids, while metgrid interpolates meteorological fields horizontally to the same grids. The Python implementation reads and writes geo_em.d0X.nc directly and is used here for both offline diagnostics and terrain construction. Source code, documentation, and a worked example are archived on Zenodo (see Code and data availability).

The target angle θ is the user-prescribed slope parameter. Let hi,j denote terrain height on a grid with spacing Δx. The local eight-neighbour discrete slope angle is 1Si,j=max⁡(m,n)∈N8(i,j)arctan⁡|hm,n-hi,j|d(i,j),(m,n), where d=Δx for cardinal neighbours and d=2Δx for diagonal neighbours. The eight neighbours comprise the four cardinal points (i±1,j), (i,j±1) and the four diagonal points (i±1,j±1). Taking the maximum detects an exceedance in any sampled direction; averaging could conceal a single steep pair. The maximum over the active interior is denoted Smax⁡,int. This interior consists of land points with mobility weight w>0.5, excluding the frozen rim and the outer part of the boundary taper.

Cardinal and diagonal bounds of Δxtan⁡θ and 2Δxtan⁡θ each correspond to θ in their sampled direction. SCO multiplies both by cos⁡(π/8), giving the initial cardinal bound 2Δhmax⁡(0)=Δxtan⁡θcos⁡(π/8). The diagonal bound is 2Δhmax⁡(0). This common factor is a conservative geometric scaling for a locally planar surface: the gradient direction is within π/8 of a sampled direction, so satisfaction of all scaled bounds implies |∇h|≤tan⁡θ for that plane (Appendix ). The scaling does not remove the directional anisotropy of the eight-neighbour stencil.

The angle corresponding to the initial working bounds is 3θeff=arctan⁡[cos⁡(π/8)tan⁡θ], and the linear approximation θeff≈0.93θ holds to within 0.1° over 18–26°. Local tightening can reduce the working bounds further. A field that fully satisfies the initial bounds has Smax⁡,int≤θeff, apart from numerical tolerance, but acceptance at the nominal angle θ does not establish convergence to those bounds. The idealised example in Fig.  illustrates the basic contrast: SCO modifies only the cones whose slopes violate the working bounds, whereas WPS smoothing alters all four. RAW denotes the unsmoothed reference terrain.

Each grid point carries a mobility weight wi,j∈[0,1] that controls how freely the solver may adjust its elevation. Water points receive w=0 and are never modified. The outermost r rows and columns are likewise frozen at w=0; across the next t rows and columns inward, a cosine taper raises w smoothly from 0 to 1. All remaining interior land points carry w=1. The fixed settings used here are r=3 and t=5.

Grid points with w>0.5 define the active interior. This threshold coincides with the midpoint of the cosine taper and marks the region over which Smax⁡,int is evaluated and the outer acceptance test is applied. Boundary points (w≤0.5) may retain residual slope violations, but do not enter the outer acceptance test or local tightening. The inner monitor includes all neighbour pairs with wp+wq>10-6, including pairs adjoining frozen points.

The solver sweeps four classes of neighbour pairs in sequence (horizontal, vertical, main diagonal, anti-diagonal) using a red–black decomposition for vectorised execution. For a neighbour pair (p,q), the working bound is Lpq=(dpq/Δx)min⁡(fp,fq)Δhmax⁡(0), where dpq=Δx for cardinal pairs and 2Δx for diagonal pairs. The local tightening factor f is initially 1. It changes the bound, whereas the mobility weight w determines how the correction is shared. When the height difference δ=hq-hp exceeds Lpq in magnitude and the pair is movable, the excess 4e=δ-sgn(δ)Lpq is redistributed according to the endpoint mobilities: 5hp←hp+ewpwp+wq,hq←hq-ewqwp+wq. When both endpoints share the same mobility the pair mean is conserved. When they differ, as in the transition zone, the more mobile endpoint absorbs the larger share and the pair mean shifts toward the less mobile point. Frozen points (w=0) never move.

An outer loop progressively tightens the local bound at each active interior point through the factor fi,j: 6Δhmax⁡,i,j=fi,jΔhmax⁡(0).

For sweep s, the monitored quantity Es is the largest positive pairwise exceedance max⁡(|hq-hp|-Lpq,0) encountered over movable pairs during the sweep. The inner stopping conditions are Es<0.1 m, stagnation, or the limit of 500 sweeps. Stagnation means Es≥0.999Es-1 for three consecutive comparisons. Persistent exceedances beside the frozen boundary can trigger this exit before the active interior has fully converged.

After each inner solve, the outer acceptance test checks Smax⁡,int≤θ+0.01°. If it is not met, the factors at active interior points with Si,j>θ are tightened as 7fi,j←max⁡fi,jβθSi,j,fmin⁡, where β=0.97 and fmin⁡=0.30 are fixed. The next inner solve uses the updated bounds. The outer loop ends on acceptance or after 10 rounds. The full pseudocode and parameter values are given in Appendix . All offline diagnostics and figures in this study use max_outer = 10, max_sweeps = 500, inner_tol = 0.1 m, and outer_tol_deg = 0.01°. These are also the module defaults in the released code, which users may adjust for other domains.

The offline diagnostics quantify achieved slopes, the magnitude and extent of terrain modification, and retained small-scale terrain structure. The near-maximum count is the number of active interior points whose Si,j falls within ε=0.2° below the member's own Smax⁡,int. It describes concentration in the slope distribution, not geographic clustering.

Terrain fidelity is characterised by the mean absolute elevation change, MAEh=〈|hnew-hraw|〉, over interior land points. The modified-area fraction is the proportion of those points where |hnew-hraw|>0.1 m. MAEh measures the typical magnitude of elevation change. The modified-area fraction measures how widely those changes spread. Retained small-scale terrain structure is quantified by the root-mean-square (RMS) over interior land points of the unnormalised five-point second difference 8Li,j=hi+1,j+hi-1,j+hi,j+1+hi,j-1-4hi,j (in metres). On the uniform grid, this difference equals Δx2 times the discrete Laplacian. Because it responds to second-order terrain structure, it distinguishes terrain fields with similar Smax⁡,int but different small-scale content. Ridge-to-valley relief retention along cross-sections is defined as the ratio of the peak-to-valley elevation difference in the processed field to that in RAW.

The test region is the southeastern Tibetan Plateau (Fig. ), where raw discrete slope angles on a 1 km grid exceed 40° (Table ). Two one-way-nested WRF-ARW v4.6.1 domains are used. The outer domain (D01) covers the eastern Tibetan Plateau margin and the Sichuan Basin on a 360×360 grid at 3 km spacing; the inner domain (D02) spans the steep plateau–basin transition on a 420×420 grid at 1 km spacing. Both domains use 54 vertical levels with a model top at 50 hPa and the hybrid terrain-following coordinate. D01 supplies the lateral boundary conditions for D02, with no feedback from D02 to D01. D01 terrain is fixed after one WPS smth-desmth_special pass; all terrain-treatment differences are confined to D02. Time steps were reduced for the WPS configurations as needed for stability (Fig. ). With the parent-to-nest time-step ratio fixed at 3, these reductions also change the parent-domain integration.

All simulations are initialised at 00:00 UTC (Coordinated Universal Time) 18 July 2025 from the fifth-generation reanalysis (ERA5) of the European Centre for Medium-Range Weather Forecasts and integrated for 48 h. Static terrain for the 1 km inner domain is built from the 30 m Advanced Spaceborne Thermal Emission and Reflection Radiometer (ASTER) Global Digital Elevation Model GDEM; by area averaging onto the model grid. The physics configuration is identical across all experiments: Thompson microphysics , Rapid Radiative Transfer Model for general circulation models (RRTMG) radiation , the Yonsei University (YSU) planetary boundary layer scheme , the revised Fifth-Generation Penn State/National Center for Atmospheric Research mesoscale model (MM5) Monin–Obukhov surface layer, Noah land surface model with multiparameterization options (Noah-MP) , and gravity-wave drag on D01 only. No cumulus parameterisation is used on either domain. A reference WRF namelist is provided in the code repository.

For the WPS pathway, smth-desmth_special is applied for N passes to the aggregated D02 terrain, and the member label gives the pass count. Five members are tested: WPS01, WPS60, WPS100, WPS120, and WPS150. SCO members are generated by applying the slope-constrained adjustment (Sect. ) directly to the unsmoothed D02 terrain. Five members span θ=18 to 26° at 2° intervals, with the member label giving the target angle. All SCO members share the same solver settings (r=3, t=5, β=0.97, fmin⁡=0.30); only θ varies. RAW is retained as an offline reference. A control experiment, W1+SCO24, applies one WPS pass before SCO at θ=24° to test whether light pre-smoothing changes the stability outcome at fixed Smax⁡,int.

The cross-method comparison pairs the least-modified stable member from each method among the discrete candidates tested under their reported configurations and the same 48 h stability criterion. These selections are neither exact stability limits nor globally optimal terrain solutions. Members are not matched by slope or terrain cost, since neither method offers a natural counterpart at fixed values of these quantities. One additional stable member from each method enters the near-surface evaluation to gauge within-method sensitivity.

Near-surface verification uses hourly 10 m wind speed and 2 m temperature from 927 China Meteorological Administration (CMA) surface stations within the D02 interior (Fig. b). The denser local network improves spatial sampling in interior valleys, where terrain effects are expected to be strongest.

Stations within 30 grid points of the lateral boundaries are excluded to avoid relaxation-zone contamination, and each retained time series must contain at least 20 % valid observations during the 48 h window. At any given station, the terrain-height difference between members is much smaller than the model-minus-observation elevation offset, so systematic elevation bias largely cancels in the paired comparison. Model fields are bilinearly interpolated to station locations without elevation correction.

The final sample contains 44 200 wind pairs and 44 115 temperature pairs. Inter-member root-mean-square error (RMSE) differences are tested with a two-sided paired t-test across stations, each contributing one time-averaged RMSE value. The test does not account for spatial correlation among nearby stations, so the reported p values should be interpreted as approximate.

Under the common 48 h stability criterion, SCO22 and WPS120 are the least-modified stable members among those tested for the two methods (Table ). Their terrain costs differ by an order of magnitude. SCO22 modifies 14.0 % of interior land points with MAEh=8.2 m; WPS120 modifies the entire domain and has MAEh=108.1 m (Fig. ).

SCO22 also retains far more small-scale terrain structure. Its RMS of the five-point second difference reaches 279.8 m, nearly six times the 47.9 m of WPS120. Cross-sections confirm the same pattern: SCO22 retains 97 %–99 % of the RAW ridge-to-valley relief along two representative transects, compared with 79 %–86 % for WPS120 (Fig. ).

All SCO members attain the nominal target θ over the active interior (Fig. ). The maxima for SCO24 and SCO26 lie close to θeff, whereas those for SCO18–SCO22 remain above it (Table ; Fig. c). The latter fields pass the outer acceptance test without fully satisfying the tighter internal working bounds, consistent with the stagnation exit described in Sect. .

The WRF integrations with the unstable members terminated prematurely. Along the WPS pathway, WPS100 (Smax⁡,int=24.44°) fails while WPS120 (24.13°) completes the full 48 h integration under the reported configurations (Table ; Fig. ). Along the SCO pathway, SCO24 (22.36°) fails and SCO22 (21.12°) passes, a sampled passing–failing gap of 1.24°. The selected stable members, SCO22 and WPS120, differ by 3.0° in peak slope.

At the domain scale, the slope fields of SCO22, SCO24, WPS100 and WPS120 differ sharply. Of the active interior points in SCO24, 15 359 fall within ε=0.2° below its own Smax⁡,int; WPS120 has only 4 (Table ). The exceedance counts (Fig. b) illustrate the same contrast: SCO members drop steeply near their diagnosed maxima, whereas the WPS tail decays gradually over several degrees. Because repeated smoothing progressively lowers slopes, few WPS grid points remain near the residual peak. In the slope probability density functions (PDFs; Fig. a), the contrast takes a different form: the WPS distribution falls off gradually in the upper tail, with most grid points well below the peak, while SCO24 has a concentrated population near its working bound.

Figure  compares the local terrain around the reported failure site. Within this 41×41 neighbourhood, SCO24 retains small-scale ridge–valley structure and has 604 grid points above 22°. By contrast, WPS100 presents a narrow band of steep slopes against a smooth background, with only 23 grid points above 22°.

W1+SCO24 tests whether removing small-scale terrain structure at fixed Smax⁡,int can produce a stable integration. One WPS pass applied before SCO at θ=24° lowers the second-difference RMS from 289.6 to 186.7 m and approximately halves the near-maximum count (from 15 359 to 7681), while holding Smax⁡,int at 22.36° (Table ). Row-mean spectra confirm that about 70 % of the power near 2 km wavelength is removed, while the 10 km band remains largely intact (Fig. ). SCO24 and W1+SCO24 both fail. The tested pre-smoothing was therefore insufficient to restore stability under these settings.

Figure  shows SCO22 and SCO24 because they bracket the sampled SCO passing–failing interval; it also compares WPS120 and W1+SCO24. SCO22 and SCO24 retain most of the RAW power near 2 and 10 km, and SCO24 retains more than SCO22; WPS120 removes most of the power in those bands (Table ). SCO is a nonlinear local adjustment, so its spectral response depends on terrain amplitude and structure, grid spacing and θ rather than on a fixed transfer function. In the tested sequence, lower θ increases terrain modification (Table ). The slope distributions and second-order diagnostics show that the methods produce different terrain structures beyond the maximum slope. Those properties may influence numerical behaviour, but the present experiments do not isolate their effects.

Four stable members (WPS120, WPS150, SCO20, and SCO22) are evaluated against the 927-station network over the full 48 h period (Table ). Comparing the two least-modified stable members, SCO22 produces lower errors for both variables: wind RMSE is lower by about 0.15 m s⁻¹ with a weaker positive bias, and 2 m temperature RMSE is lower by about 0.16 K (Table ). The error differences between these successful configurations cannot be attributed to D02 terrain alone. The 2 m temperature bias reverses sign between the two members (±0.17 K) but at this magnitude it is too small to diagnose a systematic difference.

Within-method differences are much smaller. SCO20 and SCO22 show no detectable difference in wind and temperature RMSE under the approximate stationwise paired test (Sect. 3.3). The step from WPS120 to WPS150 changes scores only modestly. Because the comparison rests on a single 48 h case, these numbers are specific to the present domain and period.