Update rainfarm.py

doc/source/references.bib

@@ -51,6 +51,17 @@ @ARTICLE{CRS2004
   DOI = "10.1017/S1350482704001239"
 }
 
+@ARTICLE{DOnofrio2014,
+  TITLE     = "Stochastic rainfall downscaling of climate models",
+  AUTHOR    = "D'Onofrio, D and Palazzi, E and von Hardenberg, J and Provenzale, A and Calmanti, S",
+  JOURNAL   = "J. Hydrometeorol.",
+  PUVLISHER = "American Meteorological Society",
+  VOLUME    =  15,
+  NUMBER    =  2,
+  PAGES     = "830--843",
+  YEAR      =  2014,
+}
+
 @ARTICLE{EWWM2013,
   AUTHOR = "E. Ebert and L. Wilson and A. Weigel and M. Mittermaier and P. Nurmi and P. Gill and M. Göber and S. Joslyn and B. Brown and T. Fowler and A. Watkins",
   TITLE = "Progress and challenges in forecast verification",
@@ -306,6 +317,17 @@ @ARTICLE{SPN2013
   DOI = "10.1002/wrcr.20536"
 }
 
+@Article{Terzago2018,
+  AUTHOR = "Terzago, S. and Palazzi, E. and von Hardenberg, J.",
+  TITLE = "Stochastic downscaling of precipitation in complex orography: a simple method to reproduce a realistic fine-scale climatology",
+  JOURNAL = "Natural Hazards and Earth System Sciences",
+  VOLUME = 18,
+  YEAR = 2018,
+  NUMBER = 11,
+  PAGES = "2825--2840",
+  DOI = "10.5194/nhess-18-2825-2018"
+}
+
 @ARTICLE{TRT2004,
   AUTHOR = "A. M. Hering and C. Morel and G. Galli and P. Ambrosetti and M. Boscacci",
   TITLE = "Nowcasting thunderstorms in the Alpine Region using a radar based adaptive thresholding scheme",

pysteps/downscaling/rainfarm.py

@@ -4,7 +4,7 @@
 ============================
 
 Implementation of the RainFARM stochastic downscaling method as described in
-:cite:`Rebora2006`.
+:cite:`Rebora2006` and :cite:`DOnofrio2014`.
 
 RainFARM is a downscaling algorithm for rainfall fields developed by Rebora et
 al. (2006). The method can represent the realistic small-scale variability of the
@@ -20,38 +20,211 @@
 import warnings
 
 import numpy as np
-from scipy.ndimage import convolve
+from scipy.signal import convolve
+from pysteps.utils.spectral import rapsd
+from pysteps.utils.dimension import aggregate_fields
+
+
+def _gaussianize(precip):
+    """
+    Gaussianize field using rank ordering as in :cite:`DOnofrio2014`.
+    """
+    m, n = np.shape(precip)
+    nn = m * n
+    ii = np.argsort(precip.reshape(nn))
+    precip_gaussianize = np.zeros(nn)
+    precip_gaussianize[ii] = sorted(np.random.normal(0, 1, nn))
+    precip_gaussianize = precip_gaussianize.reshape(m, n)
+    sd = np.std(precip_gaussianize)
+    if sd == 0:
+        sd = 1
+    return precip_gaussianize / sd
+
+
+def _compute_freq_array(array, ds_factor=1):
+    """
+    Compute the frequency array following a given downscaling factor.
+    """
+    freq_i = np.fft.fftfreq(array.shape[0] * ds_factor, d=1 / ds_factor)
+    freq_j = np.fft.fftfreq(array.shape[1] * ds_factor, d=1 / ds_factor)
+    freq_sqr = freq_i[:, None] ** 2 + freq_j[None, :] ** 2
+    return np.sqrt(freq_sqr)
 
 
 def _log_slope(log_k, log_power_spectrum):
+    """
+    Calculate the log-slope of the power spectrum given an array of logarithmic wavenumbers
+    and an array of logarithmic power spectrum values.
+    """
     lk_min = log_k.min()
     lk_max = log_k.max()
     lk_range = lk_max - lk_min
     lk_min += (1 / 6) * lk_range
     lk_max -= (1 / 6) * lk_range
-
     selected = (lk_min <= log_k) & (log_k <= lk_max)
     lk_sel = log_k[selected]
     ps_sel = log_power_spectrum[selected]
     alpha = np.polyfit(lk_sel, ps_sel, 1)[0]
     alpha = -alpha
+    return alpha
 
+
+def _estimate_alpha(array, k):
+    """
+    Estimate the alpha parameter using the power spectrum of the input array.
+    """
+    fp = np.fft.fft2(array)
+    fp_abs = abs(fp)
+    log_power_spectrum = np.log(fp_abs**2)
+    valid = (k != 0) & np.isfinite(log_power_spectrum)
+    alpha = _log_slope(np.log(k[valid]), log_power_spectrum[valid])
     return alpha
 
 
-def _balanced_spatial_average(x, k):
-    ones = np.ones_like(x)
-    return convolve(x, k) / convolve(ones, k)
+def _compute_noise_field(freq_array_highres, alpha):
+    """
+    Compute a field of correlated noise field using the given frequency array and alpha
+    value.
+    """
+    white_noise_field = np.random.rand(*freq_array_highres.shape)
+    white_noise_field_complex = np.exp(complex(0, 1) * 2 * np.pi * white_noise_field)
+    with warnings.catch_warnings():
+        warnings.simplefilter("ignore")
+        noise_field_complex = white_noise_field_complex * np.sqrt(
+            freq_array_highres**-alpha
+        )
+    noise_field_complex[0, 0] = 0
+    return np.fft.ifft2(noise_field_complex).real
 
 
-def downscale(precip, ds_factor, alpha=None, threshold=None, return_alpha=False):
+def _apply_spectral_fusion(
+    array_low, array_high, freq_array_low, freq_array_high, ds_factor
+):
     """
-    Downscale a rainfall field by increasing its spatial resolution by
-    a positive integer factor.
+    Apply spectral fusion to merge two arrays in the frequency domain.
+    """
+
+    # Validate inputs
+    if array_low.shape != freq_array_low.shape:
+        raise ValueError("Shape of array_low must match shape of freq_array_low.")
+    if array_high.shape != freq_array_high.shape:
+        raise ValueError("Shape of array_high must match shape of freq_array_high.")
+
+    nax, _ = np.shape(array_low)
+    nx, _ = np.shape(array_high)
+    k0 = nax // 2
+
+    # Calculate power spectral density at specific frequency
+    def compute_psd(array, fft_size):
+        return rapsd(array, fft_method=np.fft)[k0 - 1] * fft_size**2
+
+    psd_low = compute_psd(array_low, nax)
+    psd_high = compute_psd(array_high, nx)
+
+    # Normalize high-resolution array
+    normalization_factor = np.sqrt(psd_low / psd_high)
+    array_high *= normalization_factor
+
+    # Perform FFT on both arrays
+    fft_low = np.fft.fft2(array_low)
+    fft_high = np.fft.fft2(array_high)
+
+    # Initialize the merged FFT array with low-resolution data
+    fft_merged = np.zeros_like(fft_high, dtype=np.complex128)
+    fft_merged[0:k0, 0:k0] = fft_low[0:k0, 0:k0]
+    fft_merged[nx - k0 : nx, 0:k0] = fft_low[k0 : 2 * k0, 0:k0]
+    fft_merged[0:k0, nx - k0 : nx] = fft_low[0:k0, k0 : 2 * k0]
+    fft_merged[nx - k0 : nx, nx - k0 : nx] = fft_low[k0 : 2 * k0, k0 : 2 * k0]
+
+    fft_merged[k0, 0] = np.conj(fft_merged[nx - k0, 0])
+    fft_merged[0, k0] = np.conj(fft_merged[0, nx - k0])
+
+    # Compute frequency arrays
+    freq_i = np.fft.fftfreq(nx, d=1 / ds_factor)
+    freq_i = np.tile(freq_i, (nx, 1))
+    freq_j = freq_i.T
+
+    # Compute frequency domain adjustment
+    ddx = np.pi * (1 / nax - 1 / nx) / np.abs(freq_i[0, 1] - freq_i[0, 0])
+    freq_squared_high = freq_array_high**2
+    freq_squared_low_center = freq_array_low[k0, k0] ** 2
+
+    # Fuse in the frequency domain
+    mask_high = freq_squared_high > freq_squared_low_center
+    mask_low = ~mask_high
+    fft_merged = fft_high * mask_high + fft_merged * mask_low * np.exp(
+        -1j * ddx * freq_i - 1j * ddx * freq_j
+    )
+
+    # Inverse FFT to obtain the merged array in the spatial domain
+    merged = np.real(np.fft.ifftn(fft_merged)) / len(fft_merged)
+
+    return merged
+
+
+def _compute_kernel_radius(ds_factor):
+    return int(round(ds_factor / np.sqrt(np.pi)))
+
+
+def _make_tophat_kernel(ds_factor):
+    """Compute 2d uniform (tophat) kernel"""
+    radius = _compute_kernel_radius(ds_factor)
+    (mx, my) = np.mgrid[-radius : radius + 0.01, -radius : radius + 0.01]
+    tophat = ((mx**2 + my**2) <= radius**2).astype(float)
+    return tophat / tophat.sum()
+
+
+def _make_gaussian_kernel(ds_factor):
+    """
+    Compute 2d gaussian kernel
+    ref: https://github.com/scipy/scipy/blob/de80faf9d3480b9dbb9b888568b64499e0e70c19/scipy/ndimage/_filters.py#L179
+    the smoothing sigma has width half a large pixel
+    """
+    radius = _compute_kernel_radius(ds_factor)
+    sigma = ds_factor / 2
+    sigma2 = sigma * sigma
+    x = np.arange(-radius, radius + 1)
+    kern1d = np.exp(-0.5 / sigma2 * x**2)
+    kern2d = np.outer(kern1d, kern1d)
+    return kern2d / kern2d.sum()
+
+
+def _balanced_spatial_average(array, kernel):
+    """
+    Compute the balanced spatial average of an array using a given kernel while handling
+    missing or invalid values.
+    """
+    array = array.copy()
+    mask_valid = np.isfinite(array)
+    array[~mask_valid] = 0.0
+    array_conv = convolve(array, kernel, mode="same")
+    array_conv /= convolve(mask_valid, kernel, mode="same")
+    array_conv[~mask_valid] = np.nan
+    return array_conv
+
+
+_make_kernel = dict()
+_make_kernel["gaussian"] = _make_gaussian_kernel
+_make_kernel["tophat"] = _make_tophat_kernel
+_make_kernel["uniform"] = _make_tophat_kernel
+
+
+def downscale(
+    precip,
+    ds_factor,
+    alpha=None,
+    threshold=None,
+    return_alpha=False,
+    kernel_type=None,
+    spectral_fusion=False,
+):
+    """
+    Downscale a rainfall field by increasing its spatial resolution by a positive
+    integer factor.
 
     Parameters
     ----------
-    precip: array-like
+    precip: array_like
         Array of shape (m, n) containing the input field.
         The input is expected to contain rain rate values.
         All values are required to be finite.
@@ -65,10 +238,14 @@
         Set all values lower than the threshold to zero.
     return_alpha: bool, optional
         Whether to return the estimated spectral slope ``alpha``.
+    kernel_type: {None, "gaussian", "uniform", "tophat"}
+        The name of the smoothing operator. If None no smoothing is applied.
+    spectral_fusion: bool, optional
+        Whether to apply spectral merging as in :cite:`DOnofrio2014`.
 
     Returns
     -------
-    r: array-like
+    precip_highres: ndarray
         Array of shape (m * ds_factor, n * ds_factor) containing
         the downscaled field.
     alpha: float
@@ -79,54 +256,74 @@
     Currently, the pysteps implementation of RainFARM only covers spatial downscaling.
     That is, it can improve the spatial resolution of a rainfall field. However, unlike
     the original algorithm from Rebora et al. (2006), it cannot downscale the temporal
-    dimension.
+    dimension. It implements spectral merging from D'Onofrio et al. (2014).
 
     References
     ----------
     :cite:`Rebora2006`
+    :cite:`DOnofrio2014`
 
     """
 
-    ki = np.fft.fftfreq(precip.shape[0])
-    kj = np.fft.fftfreq(precip.shape[1])
-    k_sqr = ki[:, None] ** 2 + kj[None, :] ** 2
-    k = np.sqrt(k_sqr)
+    # Validate inputs
+    if not np.isfinite(precip).all():
+        raise ValueError("All values in 'precip' must be finite.")
+    if not isinstance(ds_factor, int) or ds_factor <= 0:
+        raise ValueError("'ds_factor' must be a positive integer.")
+
+    # Preprocess the input field if spectral fusion is enabled
+    precip_transformed = _gaussianize(precip) if spectral_fusion else precip
 
-    ki_ds = np.fft.fftfreq(precip.shape[0] * ds_factor, d=1 / ds_factor)
-    kj_ds = np.fft.fftfreq(precip.shape[1] * ds_factor, d=1 / ds_factor)
-    k_ds_sqr = ki_ds[:, None] ** 2 + kj_ds[None, :] ** 2
-    k_ds = np.sqrt(k_ds_sqr)
+    # Compute frequency arrays for the original and high-resolution fields
+    freq_array = _compute_freq_array(precip_transformed)
+    freq_array_highres = _compute_freq_array(precip_transformed, ds_factor)
 
+    # Estimate spectral slope alpha if not provided
     if alpha is None:
-        fp = np.fft.fft2(precip)
-        fp_abs = abs(fp)
-        log_power_spectrum = np.log(fp_abs**2)
-        valid = (k != 0) & np.isfinite(log_power_spectrum)
-        alpha = _log_slope(np.log(k[valid]), log_power_spectrum[valid])
+        alpha = _estimate_alpha(precip_transformed, freq_array)
 
-    fg = np.exp(complex(0, 1) * 2 * np.pi * np.random.rand(*k_ds.shape))
-    with warnings.catch_warnings():
-        warnings.simplefilter("ignore")
-        fg *= np.sqrt(k_ds_sqr ** (-alpha / 2))
-    fg[0, 0] = 0
-    g = np.fft.ifft2(fg).real
-    g /= g.std()
-    r = np.exp(g)
-
-    P_u = np.repeat(np.repeat(precip, ds_factor, axis=0), ds_factor, axis=1)
-    rad = int(round(ds_factor / np.sqrt(np.pi)))
-    (mx, my) = np.mgrid[-rad : rad + 0.01, -rad : rad + 0.01]
-    tophat = ((mx**2 + my**2) <= rad**2).astype(float)
-    tophat /= tophat.sum()
-
-    P_agg = _balanced_spatial_average(P_u, tophat)
-    r_agg = _balanced_spatial_average(r, tophat)
-    r *= P_agg / r_agg
+    # Generate noise field
+    noise_field = _compute_noise_field(freq_array_highres, alpha)
+
+    # Apply spectral fusion if enabled
+    if spectral_fusion:
+        noise_field /= noise_field.shape[0] ** 2
+        noise_field = np.exp(noise_field)
+        noise_field = _apply_spectral_fusion(
+            precip_transformed, noise_field, freq_array, freq_array_highres, ds_factor
+        )
+
+    # Normalize and exponentiate the noise field
+    noise_field /= noise_field.std()
+    noise_field = np.exp(noise_field)
+
+    # Aggregate the noise field to low resolution
+    noise_lowres = aggregate_fields(noise_field, ds_factor, axis=(0, 1))
+
+    # Expand input and noise fields to high resolution
+    precip_expanded = np.kron(precip, np.ones((ds_factor, ds_factor)))
+    noise_lowres_expanded = np.kron(noise_lowres, np.ones((ds_factor, ds_factor)))
+
+    # Apply smoothing if a kernel type is provided
+    if kernel_type:
+        if kernel_type not in _make_kernel:
+            raise ValueError(
+                f"kernel type '{kernel_type}' is invalid, available kernels: {list(_make_kernel)}"
+            )
+        kernel = _make_kernel[kernel_type](ds_factor)
+        precip_expanded = _balanced_spatial_average(precip_expanded, kernel)
+        noise_lowres_expanded = _balanced_spatial_average(noise_lowres_expanded, kernel)
+
+    # Normalize the high-res precipitation field by the low-res noise field
+    norm_k0 = precip_expanded / noise_lowres_expanded
+    precip_highres = noise_field * norm_k0
 
+    # Apply thresholding if specified
     if threshold is not None:
-        r[r < threshold] = 0
+        precip_highres[precip_highres < threshold] = 0
 
+    # Return the downscaled field and optionally the spectral slope alpha
     if return_alpha:
-        return r, alpha
+        return precip_highres, alpha
 
-    return r
+    return precip_highres