Fourier analysis of low-resolution satellite images of cloud

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1 New Zealand Journal of Geology and Geophysics, 1991, Vol. 34: /91/ $2.50/0 Crown copyright Note Fourier analysis of low-resolution satellite images of cloud S. G. BRADLEY Department of Physics University of Auckland Private Bag Auckland, New Zealand S. R. GREEN DSIR Plant Physiology Palmerston North New Zealand C. D. STOW Department of Physics University of Auckland Abstract Case studies of Fourier transformed APT images of cloud scenes show that VIS and IR images exhibit similar features. In transform space it is possible to infer characteristic cloud shape, orientation, and size. Advantages arise from the ability to suppress high-frequency noise and from the transform providing an overview of spatial features of an entire image. It should prove possible to use IR data alone to obtain information regarding wind alignment as well as cloud type classification. Keywords cloud shape; cloud orientation; cloud images; remote sensing of cloud THE TWO-DIMENSIONAL FOURIER TRANSFORM A two-dimensional function f(x,y) has a two-dimensional transform F(u,v) and between the two the following relations exist: F(u,v) = J j f(x,y) exp[-2m(xu+yv)]dx dy, (1) DO f(x,y)= j J F(u,v)exp[+2m(ux+vy)]dudv. (2) ~oo oo These equations describe an analysis of the twodimensional function f(x,y) into components of the form exp[-2my^«+3^}]- Any such component can be split up into cosine and sine parts. In the transform domain, the components are characterised in wavelength and orientation by the point (u,v) in the M-V plane, and its amplitude by F(u,v). The interpretation of u and v as spatial frequencies is emphasised by dimensioning llu and IN, the wavelengths of sections taken in the x and y directions, respectively, as in Fig. 1. For a general treatment of two-dimensional Fourier transforms see Lipson (1972) or Papoulis (1968). The scale of the Fourier transform in M-V space is reciprocally related to the scale of objects in x-y space, thus f(ax,by) transforms to F (ulaylb). (3) ab Furthermore, the rotation of objects in x-y space leads to an identical rotation of their transform in M-V space. It may also be shown that a regular array of identical objects generate a transform that may be regarded as the transform of one object INTRODUCTION Bradley et al. (1987a, b, c) have described a low-cost facility for the reception and analysis of low-resolution (APT) images from the polar-orbiting NOAA satellites. In addition to the routine analysis of cloud using distributions of visible and infrared pixel values, spatial structures were studied, as described below, using two-dimensional Fourier transform techniques. These techniques provided both an objective and a quantitative evaluation of such structures as cellular formations in stratocumulus cloud, striations parallel to the wind in cirrus cloud, or wave clouds aligned orthogonally to the air flow. Whereas spectral (Fourier) analysis of ocean waves is commonly performed using remote-sensing data, this has not been the case for cloud fields, possibly because strong periodicities are less frequently observed. There follows a description of the Fourier spatial analysis together with preliminary results obtained from cloud in the New Zealand region. G90039 Received 16 July 1990; accepted 15 April 1991 Fig. 1 Interpretation of Fourier transform image in terms of spatial frequencies.

2 550 New Zealand Journal of Geology and Geophysics, 1991, Vol. 34 modulated by an "interference" function whose wavelength is inversely related to the array spacing (see Fig. 2). The above properties of the Fourier transform allow recognition of dominant scales, alignment, and geometrical order in satellite pictures of clouds. Fine structures in the cloud field are represented in the outer regions of the transform but because of irregularities in shape and separation in real clouds these regions of the transform cannot be expected to yield useful data. The value of the transform at its origin is a measure of the average scene brightness. The region of the transform surrounding the origin contains information concerning the size, orientation, and general shape of clouds that characterise the satellite image. In particular, the transform enables ordered structures to be identified within a field which also contains disorder. The function expt-nfjc 2 /^2 + y^ltp-)] which transforms to ab expt-nfa 2!* 2 + tpv 2 )], shows that ellipses (which often approximate real cloud shapes) are conserved during transformation but with axes inversely related. IMAGE PROCESSING Numerical arrays corresponding to visible (VIS) and infrared (IR) images were obtained by digitising the analog transmissions from the NOAA-7 and NOAA-8 satellites (see Bradley et al. 1987a, b). This process typically generates arrays of 600 X 800 pixels of which we have analysed subsets of size 64 X 64, where each pixel corresponds to an area of about 4 km by 4 km on the Earth's surface at a resolution of about 1% for reflectivity and 2 K for temperature. Transforms of the above subsets form 64 X 64 arrays of resolution 1/256 km" 1 with their origin at the centre of the array. Images are presented below using 8-level grey scales which are distributed linearly over the observed ranges of reflectance and temperature; transforms are displayed similarly. Low-pass filtered images are also provided which emphasise the large-scale features; these images are derived by reconstruction from the 32 X 32 array about the origin of each transform (spatial scales larger than 16 km are retained). RESULTS Figure 3 shows the visible and infrared images received from NOAA-7 at 0430Z on 5 November The 0300Z (local time) synoptic report showed an anticyclone northeast of New Zealand, a low-pressure centre south of New Zealand, and an occluded front extending over the North Island. The visible satellite image shows a large area of stratocumulus cloud west of the South Island, the occluded front over the North Island, and some frontal cloud to the southeast. The stratocumulus clouds have an open cellular structure to the north and a more closed structure to the south. Figure 4A shows a digitised visible image of a portion of the more open stratocumulus cloud referred to above. A site was chosen clear of land: the sea surface is shown in black and the cloud tops as varying shades of grey through to white. The clouds are scattered and have variable sizes; there is an overall trend in cloud size from a closed cellular structure on the left of the image (cloud cells typically 10 x 15 pixels) to a more open cellular structure on the right (cloud cells of the order of 5 x 10 pixels). Clouds of this size are more clearly seen in the low-pass filtered image (Fig. 4B) Most of the spectral content (Fig. 4Q is located about the origin, with u < 0.04 km" 1 and v< 0.02 km" 1. A Fig. 2 Typical spatial transforms. B, Fourier domain. B A, Spatial domain.

3 Note Fourier analysis of cloud 551 Fig. 3 VIS (left) and IR (right) images from NOAA-7 at 0430Z on 5 November Fig. 4 VIS images from open stratocumulus cloud. A, Digitised image. B, Low-pass filtered image. C, Fourier transform image. Fig. 5 IR images from open stratocumulus cloud. A, Digitised image. B, Low-pass filtered image. C, Fourier transform image.

4 552 New Zealand Journal of Geology and Geophysics, 1991, Vol. 34 Fig. 6 VIS images for an area of striated cirrus cloud. A, Digitised image. B, Low-pass filtered image. C, Fourier transform image. Fig. 7 IR images for an area of striated cirrus cloud. A, Digitised image. B, Low-pass filtered image. C, Fourier transform image. Fig. 8 IR images for an area of broken stratocumulus cloud. A, Digitised image. B, Low-pass filtered image. C, Fourier transform image. Considering the transform to be an ellipse with a semi-axes of km" 1 and km" 1, the spatial domain can be considered to contain clouds which appear as ellipses with corresponding semi-axes of 25 km and 50 km. It appears from the transform that clouds are oriented with these axes at about 15 to the co-ordinate axes; this is consistent with a subjective evaluation of the visible images. Figure 5 shows spatial and Fourier images of the same cloud field as that in Fig. 4 but obtained from the IR channel. The spatial scales and orientation derived from these images is in agreement with those found for VIS data. A similar analysis has been performed on a region of striated cirrus cloud situated near the top centre of Fig. 3. In Fig. 6 (VIS image) the variations in cloud-top brightness are

5 Note Fourier analysis of cloud 553 clearly visible with regions of similar brightness oriented at approx. 45 to the positive axis. The filtered image (Fig. 6B) has reduced some of the variability amongst adjacent pixels and displays the cloud orientation a little more clearly. Striations in the image, orientated at approx. 45 with an axial ratio of about 5:1, have transformed to an elliptical shape in the transform domain. The semi-axes of the ellipse have length km" 1 and km" 1 ; this implies elliptical shapes in the original image with semi-axes of 18 km and 85 km. Areas of brightness typical of this size and orientation can be seen in the filtered image. Compared with Fig. 6, the infrared image (Fig. 7) shows more variability amongst adjacent pixels. The striated cloud top is oriented at approx. 45, as before. Although the greyscales in the visible and infrared images relate to different optical properties of the clouds, there is close correspondence between the positions of the striations in each image. The similarity in spatial structure and orientation in the two images is demonstrated in the transform domain, where the main spectral components are identical. Roughly elliptical clusters of pixels of similar temperature, measuring about 18 X 85 km and oriented at 45, can be seen in the filtered image. The four previous examples, visible and infrared images of stratocumulus and striated cirrus cloud forms, show that it is possible to infer a realistic size and orientation for clouds from the Fourier transforms. Both the visible and infrared images have similar spatial structures and orientations. However, not all clouds are observed to display a Fourier transform leading to an unambiguous parameterisation of cloud size and orientation. Figure 8 shows an area of broken stratocumulus cloud taken from the NOAA-8 IR image on 11 November The filtered image indicates a range of cloud sizes with some orientations in the y direction and some at an angle of about 45. The cloud sizes are smaller than those of the previous examples and therefore the Fourier transform extends to higher spatial frequencies. The majority of the spectral energy is concentrated within an ellipse possessing semi-axes of 0.04 km" 1 and 0.02 km" 1. In the spatial domain this corresponds to ellipses similar to that found in Fig. 4. The transform also indicates an orientation of 30-40, consistent with orientation in the lower right of the image. In the above examples, orientation is much better assessed in the Fourier space than in the image space because random features in the latter are unobtrusive in the former. SUMMARY AND CONCLUSION Five cases of Fourier transformation of cloud scenes obtained from APT indicate that VIS and IR images have similar transforms. From these transforms it is possible to infer characteristic orientations, sizes, and shapes of clouds. Therefore it should be possible to use IR data alone, giving 24 h coverage, for these tasks. It is envisaged that the shape and orientation information could be useful in determining wind alignment. Furthermore, these data could be associated with known altitude levels by judicious selection of the temperature range to be transformed. REFERENCES Bradley, S. G.; Green, S. R.; Stow, C. D. 1987a: A low-cost satellite image reception and analysis facility. Part I: Objectives, image reception and data management. Weather 42: b: A low-cost satellite image reception and analysis facility. Part II: Image preprocessing. Weather 42: c: A low-cost satellite image reception and analysis facility. Part III: Display and analysis a case study. Weather 42: Lipson, H. S. ed. 1972: Optical transforms. New York, Academic Press. 436 p. Papoulis, A. 1968: Systems and transforms with applications in optics. New York, McGraw Hill.