Calibration of the Cloud Particle Imager Probes Using Calibration Beads and Ice Crystal Analogs: The Depth of Field

Paul J. Connolly School of Earth, Atmospheric and Environmental Science, The University of Manchester, Manchester, United Kingdom

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Michael J. Flynn School of Earth, Atmospheric and Environmental Science, The University of Manchester, Manchester, United Kingdom

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Z. Ulanowski Department of Physics, University of Hertfordshire, Hatfield, United Kingdom

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T. W. Choularton School of Earth, Atmospheric and Environmental Science, The University of Manchester, Manchester, United Kingdom

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M. W. Gallagher School of Earth, Atmospheric and Environmental Science, The University of Manchester, Manchester, United Kingdom

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K. N. Bower School of Earth, Atmospheric and Environmental Science, The University of Manchester, Manchester, United Kingdom

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Abstract

This paper explains and develops a correction algorithm for measurement of cloud particle size distributions with the Stratton Park Engineering Company, Inc., Cloud Particle Imager (CPI). Cloud particle sizes, when inferred from images taken with the CPI, will be oversized relative to their “true” size. Furthermore, particles will cease to be “accepted” in the image frame if they lie a distance greater than the depth of field from the object plane.

By considering elements of the scalar theory for diffraction of light by an opaque circular disc, a calibration method is devised to overcome these two problems. The method reduces the error in inferring particle size from the CPI data and also enables the determination of the particles distance from the object plane and hence their depth of field. These two quantities are vital to enable quantitative measurements of cloud particle size distributions (histograms of particle size that are scaled to the total number concentration of particles) in the atmosphere with the CPI. By using both glass calibration beads and novel ice crystal analogs, these two problems for liquid drops and ice particles can be quantified.

Analysis of the calibration method shows that 1) it reduces the oversizing of 15-μm beads (from 24.3 to 14.9 μm for the sample mean), 40-μm beads (from 50.0 to 41.4 μm for the sample mean), and 99.4-μm beads (from 103.7 to 99.8 μm for the sample mean); and 2) it accurately predicts the particles distance from the object plane (the relationship between measured and predicted distance shows strong positive correlation and gives an almost one-to-one relationship). Realistic ice crystal analogs were also used to assess the errors in sampling ice clouds and found that size and distance from the object plane could be accurately predicted for ice crystals by use of the particle roundness parameter (defined as the ratio of the projected area of the particle to the area of a circle with the same maximum length). While the results here are not directly applicable to every CPI, the methods are, as data taken from three separate CPIs fit the calibration model well (not shown).

Corresponding author address: Paul J. Connolly, School of Earth, Atmospheric and Environmental Science, The University of Manchester, Simon Building, Oxford Road, Manchester, M13 9PL, United Kingdom. Email: p.connolly@manchester.ac.uk

Abstract

This paper explains and develops a correction algorithm for measurement of cloud particle size distributions with the Stratton Park Engineering Company, Inc., Cloud Particle Imager (CPI). Cloud particle sizes, when inferred from images taken with the CPI, will be oversized relative to their “true” size. Furthermore, particles will cease to be “accepted” in the image frame if they lie a distance greater than the depth of field from the object plane.

By considering elements of the scalar theory for diffraction of light by an opaque circular disc, a calibration method is devised to overcome these two problems. The method reduces the error in inferring particle size from the CPI data and also enables the determination of the particles distance from the object plane and hence their depth of field. These two quantities are vital to enable quantitative measurements of cloud particle size distributions (histograms of particle size that are scaled to the total number concentration of particles) in the atmosphere with the CPI. By using both glass calibration beads and novel ice crystal analogs, these two problems for liquid drops and ice particles can be quantified.

Analysis of the calibration method shows that 1) it reduces the oversizing of 15-μm beads (from 24.3 to 14.9 μm for the sample mean), 40-μm beads (from 50.0 to 41.4 μm for the sample mean), and 99.4-μm beads (from 103.7 to 99.8 μm for the sample mean); and 2) it accurately predicts the particles distance from the object plane (the relationship between measured and predicted distance shows strong positive correlation and gives an almost one-to-one relationship). Realistic ice crystal analogs were also used to assess the errors in sampling ice clouds and found that size and distance from the object plane could be accurately predicted for ice crystals by use of the particle roundness parameter (defined as the ratio of the projected area of the particle to the area of a circle with the same maximum length). While the results here are not directly applicable to every CPI, the methods are, as data taken from three separate CPIs fit the calibration model well (not shown).

Corresponding author address: Paul J. Connolly, School of Earth, Atmospheric and Environmental Science, The University of Manchester, Simon Building, Oxford Road, Manchester, M13 9PL, United Kingdom. Email: p.connolly@manchester.ac.uk

1. Introduction

The cloud particle size distribution (PSD) has been widely measured by counting and sizing particle shadows that are produced by illuminating particles with a HeNe laser, such that dark images on a bright background result. This shadow is magnified with some optical arrangement and is usually recorded by an array of photodiodes. In general, the particles are some distance away from the object plane (the plane that is focused onto the image plane) of the optical system. Instruments that use this method are called optical array probes (OAPs) and have been used by cloud physicists for over 30 yr. The “1D” versions record the 1D size of cloud or precipitation particles by recording the shadow of a particle with a linear array of photodiodes as it passes through the laser beam, while the “2D” versions are able to build up 2D images with the linear array by repeatedly sampling a slice of the shadow of the particle in a similar way.

The Cloud Particle Imager (CPI),1 manufactured by Stratton Park Engineering Company (SPEC), Inc., is somewhat different from an OAP, although there are some similarities. It is designed to take discrete images of cloud particles—liquid or ice—in the size range of 10–2000 μm. However, instead of measuring only the shadow of the particle, it measures aesthetically pleasing, 8-bit diffraction patterns of the particles at a size resolution of 2.3 μm. While this is an improvement for image quality, it has not been quantified for measuring PSDs. A major reason for this is that the probe is uncharacterized with regards to the errors associated with particle sizing and the depth of field (DOF).

For the 1D OAPs, Knollenberg (1970) described a DOF for the 1D OAPs as being the distance from the object plane above which determination of particle size would show a considerable loss in accuracy. For spherical particles, measurements using coherent illumination and a linear array of photodiodes found that Z = ±3.0(R2/λ), where R is the radius of the particle and λ is the wavelength of the illumination. An important aspect of the DOF is that it can cause particles to be missed if they are some distance from the object plane, and so if not corrected for this will tend to negatively skew the PSD.

Joe and List (1987), Korolev et al. (1991), Baumgardner and Korolev (1997), Reuter and Bakan (1998), Korolev et al. (1998), and Strapp et al. (2001) performed detailed evaluations of OAPs with regards to the PSD of cloud particles. Joe and List (1987) experimentally found the oversizing errors of two OAPs with three levels of grayscale and also improved concentration estimates by improving the way in which the DOF was defined (a polynomial regression fitted to the data). Later, by recognizing that particle images could be adequately described by the Fresnel–Kirchhoff (F–K) approximation, Korolev et al. (1991), Baumgardner and Korolev (1997), and Korolev et al. (1998) modeled the oversizing and sample volume for 1D and 2D probes. The modeling results of Korolev et al. (1998) have also been shown to agree with experiments by Strapp et al. (2001). These later papers also recognized the importance of the electronic response time of the probe’s photodiode elements and the effect that image discretization can have on detecting particles.

Some of the considerations in correcting the OAP data can be applied to CPI images. The most important findings for OAPs that still affect CPI data are that the particles are oversized with distance from the object plane and that reduction in image quality with distance from the object plane defines the DOF. Particles larger than 10 μm tend not to be rejected because of the discreteness of the CCD, as the CPI images have a 2.3-μm resolution. Also, the airspeed correction is not needed for CPI data taken from most aircraft (200 m s−1) as the electronic response time of the CCD camera is very fast (of the order of tens of nanoseconds) and the pulse width of the imaging laser is 25 ns. Consequently, the particle will only move 5 μm during exposure—this has negligible affect on oversizing (see section 4a). A preliminary study to assess the oversizing of spherical particles and the dependence of the DOF on particle size has been initiated by SPEC, Inc. (2007); this study was preliminary in that only one size of bead was used. In this paper we have conducted a more rigorous study into the DOF.

Other than problems in determining the DOF, there is the problem that the PSD may suffer from artifacts due to particle shattering on the inlet (e.g., Field et al. 2006). To quantify these problems we need to first characterize the normal operation of the probe. We do not doubt particle shattering could happen for the CPI, but in this first study we will not consider this. Hence we chose to evaluate the CPI at oversizing particles and attempt to define the DOF. Section 2 covers the Kirchhoff diffraction theory for circular opaque discs, section 3 describes the experiments used to calibrate the CPI, section 4 shows the results of the study, and section 5 describes the solution algorithm. The conclusions are in section 6.

2. Theory

In this section we will explain the principles of operation of the CPI, define some terms that will be used throughout the paper and cover some relevant theory.

a. CPI principles of operation

The CPI takes 8-bit images of particles that fall within the DOF and field of view (FOV) of its imaging optics. The probe can be operated in two main modes, the first most common mode is to first detect particles that are within the DOF with the particle detection system (PDS); the PDS consists of two intersecting elliptical cross section continuous laser beams (30 mW, 788 nm) that fall onto a dump spot of two separate photomultiplers (see Fig. 1). For detection, a particle must scatter light into the photomultiplers from both beams. The PDS detection then triggers the imaging laser (pulsed 80 W, 850 nm), which fires onto a CCD camera. Image processing subtracts a predetermined background image from the current image and, providing the darkness is less than 40 bits and consists of three or more neighboring pixels, regions of interest are cut from the full image and stored to disk. During in-cloud periods, the probe is typically in a busy state for 90% of the time. The maximum number of frames that can be taken is 40 s−1. The basic principles of operation are illustrated in Fig. 1.

The second, less common mode was designed primarily as a fault finder, although it has now been realized that it may provide the user with a simpler, more reliable way of calculating particle number concentrations. In this mode, the imaging laser is fired at a constant rate of 20 s−1. Images are therefore taken at more regular intervals—if we assume that the probe takes images at regular intervals, we can calculate the PSD without having to correct for the size and airspeed response of the PDS. PSDs are determined by dividing the total number of particles in a size interval imaged in a time period by the volume of air sampled (i.e., the products of the DOF, FOV, and the number of images taken).

In general imaged particles are some distance from the object plane of the imaging system. When the particles are on the side farthest from the objective lens, a diffraction pattern will result at the object plane, which will be directly transferred—although magnified—to the image plane (i.e., onto the CCD). When particles are on the near side of the object plane, a diffraction pattern still results because of the arrangement of the optical system (see the experimental findings of Korolev et al. 1991).

b. Diffraction theory for a circular opaque disc

In the regime where the wavelength λ of illumination of coherent light is significantly less then the size (radius R) of the object near-field diffraction of coherent light by a spherical droplet has been likened to diffraction by a circular opaque disc Korolev et al. (1991).

A circular opaque disc, a distance Z from a screen, when illuminated with coherent, monochromatic light will result in a diffraction pattern on the screen (see Fig. 2). Providing that λ is significantly less than R, we can use the Kirchhoff approximations. This reduces to separation of the disturbance into one predicted by geometrical optics U(g) and a disturbance from diffraction U(d) (see Born and Wolf 1970, p. 449). The disturbance from diffraction can be expressed as a boundary integral around the diffracting object,2 Γ (see Born and Wolf 1970, p. 452). Born and Wolf derived the integral for the circular discs complementary aperture (i.e., the inverse of a circular opaque disc: a circular aperture). Because of the Babinet principle, the disturbance from diffraction by a circular disc is the same as a circular aperture but has a negative sign.

This is described by Eq. (1):
i1520-0426-24-11-1860-e1a
i1520-0426-24-11-1860-e1b
i1520-0426-24-11-1860-e1c
i1520-0426-24-11-1860-e1d
where k = (2π/λ), Z is the displacement between the disc and its image normal to the direction of propagation of coherent light, and S = [Z2 + R2 + r2 − 2Rr cos(α)](1/2) and is measured from point P to the boundary. Here n denotes a unit vector normal to the direction of propagation of the primary wave, and dl is a line element on the boundary. The two arguments to the cosine and sine functions denote the angle between the two respective vectors (see Fig. 2).

Equation (1) is similar to the findings of Korolev et al. (1991); however, we have made the distinction that positive Z refers to the particle being on the side of the object plane farthest from the objective lens and vice versa.

If there were no optical system, clearly there would be no diffraction pattern for Z on the side closer to the laser—a consequence of the Huygens–Fresnel principle. However, when enlarged by an optical system, a diffraction pattern will still result for objects with negative Z because of the difference in optical path produced by the optical system when particles are away from the object plane. Korolev et al. (1991) showed from laboratory measurements that the two images on either side but for the same magnitude in Z are similar.

The F–K approximation is a useful tool to compare with CPI measurements of spherical droplets. It can also aid in the development of a correction algorithm for retrieving true droplet sizes from the CPI images by helping to choose the variables that are important to correct the oversizing. Furthermore, the theory also provides useful background for developing a correction for nonspherical particles, as we shall see when we develop the correction for ice crystals. In section 4, it will be shown that the F–K approximation for circular opaque discs cannot be applied directly to the CPI for droplets, but some modifications are needed because of the optical system.

c. Justification for the Fresnel–Kirchhoff approximation

The F–K approximation can be applied providing the following conditions are met.

  • There must be monochromatic illumination. The CPI imaging laser has a wavelength of 850 nm so that this condition is met.

  • There must be coherent illumination. This condition is also met (see above).

  • There must be parallel-plane wave illumination. The CPI imaging laser is elliptical and has higher divergence than a typical HeNe laser (1 mrad). However, the beam is collimated, reducing the divergence to less than a degree. This renders the slight curvature in the wave front unimportant as the particles that need to be corrected (i.e., the small particles) are much smaller than the laser beam cross section. Consequently, around the particle boundary, light rays have a phase difference of less than π/100 (1%).

  • The inequality Z > 400λ must be satisfied (see Papoulis 1968, p. 320). This condition is met by the majority of imaged particles for the CPI. Only particles that are very close to the object plane will not meet this inequality, and in any case, these particles are not oversized.

  • The imaging optics must be high quality. This condition may not be met with the CPI. This can be noted by the slight asymmetry of the rings of constructive and destructive interference in Fig. 3 and the suppressed Poisson spot.3 The reason for this may be that the imaging laser is not normal to the object plane; however, we can still assume the object to be a circular opaque disc, but the diffraction image in the image plane will be asymmetrical because the image plane is not normal to the direction of the propagated light. The overall effect this has on the apparent size of the particle is small. Another aspect of the imaging optics is that when particles are closer to the objective lens than the object plane, their images are not true representations of the arrangement in Fig. 2. On the other hand, particles on the other side of the object plane are. These particles are generally oversized less for a given |Zd| than those on the near side.

Figure 3 shows a comparison of the theoretical diffraction pattern from a circular opaque disc to that obtained from glass beads (99.4-μm diameter) by the CPI. It is shown that the images are similar in that as |Zd| increases they become increasingly oversized, and also that the spacing of concentric rings of maxima and minima are similar between theory and measurements. However, the magnitude and diameter of the central maxima in intensity—the Poisson spot—in the CPI images is not accurately predicted by the theory. The main reason for this may be because the imaging optics of the CPI are not ideal. In this paper, we are not concerned with the Poisson spot; rather we wish to be able to determine the particle edges to measure the size, Lapp. The important factors for determining this quantity from the images are the width of the minima in the diffraction pattern and the gradient at this point—these are in agreement between theory and measurement (see rightmost column of Fig. 3). It is also observed that diffraction patterns of particles having both negative and positive Z show qualitatively similar dependence on |Z|. Hence, the features of the F–K approximation can be used in a qualitative sense to correct the particle size.

This analysis shows some similarities and differences between the F–K approximation and the CPI data. The important aspect is that for a given particle true size, and sign of Z, the apparent size does show a strong dependence on Zd. So while the F–K approximation and the analysis of Korolev et al. (1991) does not strictly hold true for CPI images, we can use the result in an approximate sense. The following section is intended to provide some background for interpreting raw data from the CPI.

d. Oversizing of spherical objects and their sample volume

In this section we describe how CPI images are processed to obtain a measured particle size and other image properties. For simplicity in this discussion we will assume that particles are not affected by the discreteness of the image pixels. This is in fact an oversimplification but it facilitates the discussion here—in the final correction algorithm, this effect is included.

The data obtained by the CPI require additional processing to obtain statistical information on the imaged particle. First, a median filter (3 × 3, applied 12 times consecutively) is used to filter out any salt and pepper noise on the image. Second, CPIview—the CPI image-processing software provided by SPEC, Inc.—processes the image by searching for points on the image that fall below a darkness shadow depth (−25 is the default darkness threshold for the 8-bit image) below the predetermined background. A feature extraction routine is used to flag all distinct features conforming to this threshold criterion in the masked image. Second, a Sobel filter (which approximates the magnitude of the 2D gradient vector) is used to calculate the norm of the 2D gradient in intensity for the image. The previously flagged feature regions are tested to see if they conform to a gradient threshold (40% of the points in a feature must exceed the gradient threshold for the particle to be accepted). The apparent length Lapp is then given by the maximum length of the particle image determined from the features edge; the maximum length of the particle image is the longest distance that can be measured in a straight line between two points on the particle’s image edge. This is different to the true length L, which is defined as the true maximum length of the particle [note L = 2R so that Zd = 4(λ|Z|/L2)]. A schematic of a typical cross section of the intensity of a particles image is shown in Fig. 4. This is a simplification of the edge of a particles image but is useful to consider. From the image intensity a value called particle focus “foc” is determined by approximating the norm of the 2D gradient [i.e., Eq. (2a)] of the image with a Sobel filter and dividing the maximum value of this gradient along a line of length 15 pixels that is normal to the particles edge and crosses its boundary, by the full width at half maximum (FWHM) of the gradient; this is then scaled by an arbitrary factor of 0.9 (see Fig. 4b). This is repeated for 14 times around the edge of the particle and the mean is taken [see Eq. (2b)]:
i1520-0426-24-11-1860-e2a
i1520-0426-24-11-1860-e2b
where I is the intensity on the image, G(X, Y) is the norm of the 2D gradient of intensity, Gi,l is the value of G(X, Y) along a line normal to the particle edge, and FWHM is the full width at half maximum of Gi,l.

Particles with low |Z| have values of foc close to 100 or higher, while those with high |Z| have foc decreasing to values of 20. The particle foc value is useful for accepting or rejecting particles that are positioned significantly away from best focus (i.e., low foc of <20) and consequently show large oversizing. This also defines a DOF for the CPI that is the distance from the object plane for a particle whose foc value is equal to 20, ZDOF = Zfoc=20. ZDOF has two values, one on either side of the object plane.

Since the diffraction images are approximately a function of normalized position Zd, the ratio Lapp/L should be a function of normalized position Zd, as it is itself a property of the diffraction image. Therefore, a calibration curve that depends on Zd can be calculated. The problem with this is that the equation describing Lapp/L cannot be solved unless the distance from the object plane Z is known; hence, at least one further equation is needed to provide closure to find the position from the object plane. We chose to use the particle focus foc; the rationale is as follows. The change in relative intensity, I/I0 (where I0 is the illumination intensity and I is the intensity at the screen/camera) with respect to a change in r/R (see Fig. 4) is approximately a function of normalized distance from the object plane. The quantity in question (the invariant for constant Zd) is not the value of foc measured by the CPI processing software [foc≅(dI/dr)|r=Lapp/2] but can be determined as
i1520-0426-24-11-1860-e3a
i1520-0426-24-11-1860-e3b
where Iloc is the local image intensity, and L is the actual maximum dimension/length of the particle. By considering Eq. (3) it should be noted that for a given value of normalized position, Zd, the smaller the particle, the higher the foc value. In reality, the smaller particles do not display higher values of foc. This is because 1) the FWHM is evaluated over a larger percent of the particle edge (averaging out the sharp gradient) and 2) because of nonideal optics.

To choose a more suitable equation to provide a relationship to determine the particle oversize, the possibility that the foc value depends on distance from the object plane, Z (and not Zd) was explored; at least for the coarse gradients that are taken by the processing software. Diffraction patterns were calculated for several different sizes of circular opaque disc for constant values of Z. The results from the calculations are shown in Fig. 5. It is shown that, at least for coarse gradients, the foc value (i.e., the gradient at the particle edge) is indeed constant for a given distance from the object plane Z. This result is repeatable for all distances from the object plane within the DOF and therefore provides an equation to fit the data to, providing closure for determining Z when it is an unknown.

The DOF acceptance of particles due to diffraction arises both because of the specification that the foc value exceed some finite value when processing and also because of the efficiency of detecting particles that appear on dark regions of the image, where the available shadow depth can be below the threshold required for image detection. Since the available shadow depth changes with position on the image (because of optical effects, see Fig. 6), this effect is worse for smaller particles than larger particles as larger particles have a higher likelihood of shadowing points that have enough shadow depth. The corrections devised here will take both of these effects into account (see section 4).

3. Experiments

a. Method

The experiments used to evaluate the CPI sizing and sample volume issues consist of two datasets. One is from measurements using National Institute of Standards and Technology (NIST)-certified glass calibration beads, and the other dataset used ice crystal analogs described in Ulanowski et al. (2003, 2004). The analogs are grown from a sodium fluorosilicate solution and have a very similar crystallographic form to hexagonal ice. This enabled various surrogates for ice crystal habits observed in atmospheric clouds to be reproduced. Example scanning electron microscope (SEM) images of some of the ice crystal analogs that were used to evaluate the CPI in this study are shown in Fig. 7.

In two separate experiments, glass calibration beads and ice crystal analogs were placed upon a glass slide and were held in place on the slide quite well by Van der Waals forces. The glass beads were placed onto the glass slide using a spatula and dispersed with a stainless steel needle. The ice analogs, however, had to be placed onto the glass slides using a micromanipulator, so that individual crystals previously characterized by optical and/or electron microscopy could be used.

The outer aluminum case was removed from the CPI and a calibration rig that firmly housed the glass slides was attached to the outer inlet of the CPI. The calibration rig consisted of three micrometer gauges that enabled accurate positioning of the glass slide relative to the CPI object plane in three dimensions (see Fig. 8). It can be seen from Fig. 8 that the glass slide is positioned at 45° to the axis of the CPI sample tube; this was intentional so that every position on the glass slide was equidistant from the object plane of the CPI, which is also at 45° to the axis of the sample tube. This was not completely necessary but it aided in studying the dependence of the diffraction pattern with distance of the particle from the object plane.

During a typical experiment the images produced by the CPI were recorded by operating the probe in continuous triggering mode (see section 2)—hence, images were taken at approximately 20 s−1. The glass slide was manipulated from one side of the DOF, through the position of best focus and to the other side in 0.05-mm steps, so that the images produced were in very poor focus on one side of the object plane going into good focus and then out of focus on the other side of the object plane—occasionally the step size was reduced to 0.02 mm for better resolution. This method was then repeated in reverse so that errors in deducing the position from the object plane, such as any inherent “play” in the calibration rig, could be assessed; this was found to be small.

In addition to recording the images, the relative position of the glass slide from the micrometer starting points was recorded with the logged probe time. This was then used to determine the absolute position of the population of beads and analogs from the object plane and since the size of the beads and analogs were known, the normalized position, Zd was also deduced by calculating Zλ/R2.

b. Deduction of the absolute position of the object plane

Particles were moved though the object plane by adjusting the x and z micrometer gauges on the calibration rig (see Fig. 8). Since there is no easy way to determine the absolute position of the glass slide from the object plane (denoted by x = 0, z = 0), the central position has to be inferred by inspection and processing of the images. The data were processed by standard processing software supplied with the probe by SPEC, Inc., CPIview. This software calculates the particle foc value from each imaged particle, which is a measure of the sharpness of focus of the particle (see Fig. 4). The position of the object plane on the relative position scale (i.e., from the starting point of the micrometer gauges) was then deduced by noting the value of the relative position at which a maximum in the foc value was found.4

There were two issues with processing the particle data. First, finding the position from the object plane is crucial as it is established that the diffraction pattern produced by the objects is dependent on the normalized distance from the object plane, Zd = Zλ/R2—note it was convenient to neglect the λ in the expression for Zd, as this was the same in every image and so is effectively a constant for the CPI; in section 4, Zd is defined as Z/R2. So to calculate Zd, the position where particles were found to have the maximum foc value was taken to be the position of the object plane, as the theory suggests that this should be the case. The distance of each particle away from the object plane, measured normal to the object plane was then calculated by performing a coordinate rotation of 45° in the xz plane of the relative coordinates taken from the calibration rig. Second, although care was taken, dust and debris were always present on the glass slide and had to be flagged as bad data. Similarly, the particles had to be flagged for the actual particle that they corresponded to, so that the true size of the particle, L, and the particle type were known. A software approach was developed so that the images could be batch classified by hand into either glass beads of a known size, a particular ice analog, or as bad data. While the batch classification quickened the analysis considerably, this second step is still one of the most time-consuming steps of the whole calibration process.

4. Results

a. Particle oversize and sample volume

Examples of typical images obtained from the CPI for three different sizes of glass beads are shown in Fig. 9. The degradation of the images with increasing distance from the object plane is shown to be worse the smaller the particles. In addition, this is worse for particles with positive Z. To develop our calibration we have considered the approach of Korolev et al. of using Zd to describe the oversize, but we had to have a slight dependence on the sign of Zd. Also, because the edge-finding algorithm searches for an absolute difference from the background image instead of a relative difference, the oversize changes with relative illumination.

To scale for this, first the actual background mean used to process the images must be known; the SPEC processing software uses a nearest-neighbor interpolation in time to locate the nearest background image to the saved particle image in the logged file, while the background mean saved in the data file is from the latest background taken by the instrument (the two definitions of background mean are not the same so the correct background must be used). Second, the local illumination intensity Iloc (i.e., the illumination intensity in the vicinity of the particle) needs to be determined, as this is not calculated by the SPEC processing software but is required to correct particle sizes. Custom-written software was developed to locate the correct background and to calculate the local illumination intensity. From the cropped particle image coordinates, we approximate the local illumination by
i1520-0426-24-11-1860-e4
where Iloc is the mean of the local illumination intensity if there was no object present, BGloc is the mean of the local background image intensity, I is the whole image mean, and is the whole background mean.

It was found in accordance with the F–K approximation that the ratio Lapp/L depends on the normalized position from the object plane; however, several other factors influence the degree of particle oversizing.

  • The effects of the optical system not being ideal means that when imaged, the diffraction patterns are not the same on both sides of the object plane (see Fig. 3 and section 2). This effect is slight and would probably not be observed in the two-dimensional cloud (2D–C) probes because of lower image quality.

  • Since the processing software provided by SPEC, Inc., searches for image values with a darkness threshold less than 25 to detect particles (an absolute threshold as opposed to a relative threshold), the sizing of the particles depends on the ratio of the local image mean to the actual image mean (approximated by BGloc/; when there is no particle present in the image, this is synonymous with the relative illumination, Iloc/I). For particles that are in dark regions of the image, the threshold shadow depth occurs closer to the centroid of the particle; hence, particles are not oversized as much.

  • The discretization of CPI images, and to some extent the resolving power of the optical system, means that the oversize factor can be worse for small particles as the pixel resolution (2.3 μm) becomes a sizable proportion of the total particle, hence the degree of oversize of the particle depends on the actual size, L.

Similarly, it was found in accordance with the theory that foc decreases with normalized distance from the object plane, however, because of nonideal optics, not in the precise way that the theory suggests. The foc value decreases as a function of increasing actual distance from the object plane for all particle sizes (see Fig. 5).5 Notably, the value of foc also increases as the relative illumination BGloc/ [see Eq. (4)] increases in a linear fashion. This is to be expected, as when the gradient, foc ≅ (ΔIr)|r=Lapp/2 is taken, the change in image intensity, ΔI determines the value of the particle focus in a linear way. The relative illumination also affects the dispersion or spread of the foc relationship with Z, which arises because the edge detection of the particle changes with relative illumination.

The results for ice analogs showed a similar pattern to those for beads. We were able to empirically separate the results for beads and ice analogs by using one extra variable, particle roundness [Ar = 4(Area)/πL2app, defined as the ratio of the projected surface area to the area of a circle with equivalent maximum dimension]. Heymsfield and Miloshevich (2003) have shown that different ice crystal habits (e.g., plates, columns, bullet-rosettes) can be discriminated reliably by the use of the particle roundness parameter (Ar).

Figure 10 and rows 1, 2, and 3 of Fig. 11 show example CPI images of columnar ice analogs as the distance from the object plane is altered. The main features of the images are that as the beads demonstrated, the foc value decreases and oversizing increases with increasing distance from the object plane. Moreover, the small particles are oversized by a higher factor than the large particles. Similar images are shown for a rosette ice analog in row 4 of Fig. 11.

By examining the data, it was found that as the particle roundness decreased, the particle foc value tended to decrease. This is because for small particles with low roundness, the side with the shortest dimension tends to determine the foc value more than the longest dimension L of the particle. Hence, we modulated the whole foc function by L/Ar. The functional form is given by Eq. (5), and the parameters from the fit along with the confidence intervals (90%) are given in Table 1.
i1520-0426-24-11-1860-e5

The relationship for particle focus is a surface that depends on three independent variables, Z, BGloc/, and L/Ar; its form is depicted in Fig. 12. The surface is an upward sloping ridge (with increasing BGloc/) with a maximum at Z = 0 for a given relative intensity. The ridge’s spread changes as the relative illumination changes, which is because the edge-detection algorithm is affected by the relative illumination: the primary criterion for detection of particle edges is that a threshold of 25 bits below the background in image intensity must be exceeded. With decreasing local illumination, this threshold occurs at a high percentage of the total shadow of the particle. Also, as particle roundness increases for a given L, the foc value decreases in accordance with the data. This is purely empirical.

By examining the data it was found the oversize Lapp/L increased as particle roundness decreased. This is because we defined the normalized position as Z/L2, where L is the maximum dimension. In the case of the ice analogs, the use of L in the Zd expression tends to underestimate the amount of oversizing because the minimum dimension of the particle also acquires some importance in the diffraction pattern as the roundness decreases. We therefore defined the quantity “effective position” Zeff, which is defined as
i1520-0426-24-11-1860-e6
where Ar is the particle roundness. Note this is purely empirical. The data show that, to a first approximation, the particle oversize from a general object can be captured by Zeff where the length L is multiplied by the roundness Ar—this determines, in an empirical sense, the deviation from ideal, circular diffracting objects.
The functional form of the oversize relationship for ice crystals that was finally arrived at is shown by Eq. (7)—note the asymmetry of the curve with sign of Z—and the parameters for the fit along with confidence intervals are given in Table 2. This relationship is a function that depends on four independent variables—its form is depicted in Fig. 13. The second dimension, particle size, explains the effect of the discretization of the image; particles are oversized as they become smaller because of the uncertainty in the number of shadowed pixels. Furthermore, the third dimension of the function shows the dependence on relative illumination: as the relative illumination is decreased, the particles are oversized by less, which is because, as explained above, the relative illumination affects the edge-detection algorithm, causing the particle edge to be determined for a higher percentage of the total particle shadow and therefore decreasing the apparent particle size Lapp. The fourth dimension shows the dependence on particle roundness, which shows an increase in oversizing as roundness decreases. Please note, this is for a given Zd = Z/L2, where L is defined as the maximum length. Also, particles that have low roundness tend to produce images with darker edges. Consequently, the threshold in the image-processing software is met closer to the particle centroid for particles with low roundness than spherical particles of the same maximum dimension. Both the focus relationship and the oversize relationship [Eqs. (5) and (7), respectively] have very good fit parameters (typically less than ±1%), which gives confidence to our results,
i1520-0426-24-11-1860-e7

Determining either the particle true size, L, or the distance from the object plane, Z, from both Eqs. (5) and (7) presents some slight difficulty, since Z is unknown and clearly L is also unknown. The equations therefore have to be solved by numerical optimization (minimization). This will be explained in section 5.

5. Solution algorithm

Here we describe how to correct the apparent particle size and DOF by using Eqs. (5) or (7) (droplets and ice crystals). The two equations are optimized using a nonlinear least squares minimization routine. This routine minimizes the sum of the squares of the residuals of the equations to arrive at a best guess using the Levenberg–Marquardt method.

First, in order to correct the droplet size, the foc value and roundness Ar are known from image processing—in CPIview; the relative illumination BGloc/ is found by calculating the local background mean in the vicinity of the particle and dividing this by the whole background mean [see Eq. (4)] and Lapp is known from the image processing. The unknowns are the true length L and the position from the object plane Z. These are found by using the nonlinear least squares minimization routine for Eqs. (5) and (7).

Figure 14 shows histograms of three uncorrected bead sizes and corresponding corrected bead sizes using the correction algorithm. It can be seen that the algorithm vastly improves the sizing of particles. The mean of the measurements are improved and the spread in the data (standard deviation) is reduced in all three cases.

Figure 15 shows how well the algorithm predicts the distance Z of each particle from the object plane. The “bootstrap” is an analysis technique that investigates the robustness of a statistic by randomly resampling data points (in Fig. 15a, e.g.). This can be performed on any desired statistic. A bootstrap-type analysis on the correlation coefficient (Fig. 15b) and also on the gradient of the linear fit (Fig. 15c) of the data in Fig. 15a shows that the relationship between the measured distance Zmeas and the predicted distance Zpred has high correlation, a good one-to-one relationship, and is robust.

Figure 16 shows histograms of uncorrected ice analog sizes and corrected ice analog sizes using the correction algorithm. While the correction improves on the size estimate (by looking at the means and standard deviations), it is not as good as for the beads. One reason for this was that when the ice crystals were imaged far away from the object plane, the edge-detection algorithm did not work as well as it did for the beads. Occasionally the edges were cropped by the edge-finding algorithm, meaning that both the apparent length and the corrected length could be smaller than the true length. This is seen in the histograms of uncorrected and corrected ice analogs (Fig. 16).

Figure 17 shows how well the algorithm predicts the distance of each ice analog particle from the object plane. The correlation is again good and robust (as seen by the bootstrap analysis in Fig. 17b), but the gradient of the linear fit is slightly too low (0.90). By looking at the data in Fig. 17a, it appears that the algorithm tends to slightly underpredict the maximum and minimum distance from the object plane. This is for the same reason as above (i.e., that the particle image edges were cropped at large |Z|). Consequently, we underpredict the distance from the object plane because our estimate of oversize is too low.

To find the DOF, the true length L found in the first optimization is used and the foc value is set to 20. The value of 20 is the default value used to reject particles6 that are of poor focus—hence this defines the DOF. Both foc and Lapp are solved for twice, once with the constraint that the position from the object plane is positive and the other that the position from the object plane is negative. Subtraction of these two positions defines the total DOF. The value of Lapp|foc = 20 that is solved for represents the apparent length that the particles would have if it was imaged right at the edge of poor focus, foc = 20; however, this value is not used for anything. This procedure is repeated for a rectangular grid of 10 × 10 positions (in x and y) on the image plane to define DOF(x, y)—the DOF as a function of position on the image (note that the DOF changes with lateral position on the object plane because the relative illumination changes).

The DOF corresponding to the background image in Fig. 6 is shown for both round (Ar = 1.0) and nonround (Ar = 0.1) particles in Fig. 18. It is shown that the DOF decreases at dark points in the image and increases at brighter points in the image (X = 0, Y = 0 is a dark region and X = 1000, Y = 1000 is a bright region; see Fig. 6). It is also shown that the small least round particles have the lowest DOF (Fig. 18 top-right columns 3 and 4).

The double integral of the DOF7 over x and y in the object plane defines the sampling volume of the probe, VDOF,
i1520-0426-24-11-1860-e8
where the constant factor in front of the integral is the area of one pixel (m−3) and x and y are pixel coordinates. This is shown for a range of particle sizes and roundness in Fig. 19. It is shown that there is little variation in VDOF for round particles. However, nonround particles show a decrease in VDOF for sizes below 40 μm. It can also be seen that VDOF decreases slightly in the region of the graph where L ≃ 40 μm, Ar ≥ 0.6. The reason for this is that elongated particles have greater probability of overcoming the required darkness threshold as they extend over the dark and bright regions of the image.

6. Conclusions and discussion

Nonlinear regression was applied to a large calibration dataset on particle size and DOF for the CPI probe. Fitting the model to a large dataset of calibration data allowed a high degree of confidence in the model. The heuristic model was developed by considering the Kirchhoff theory for diffraction by circular discs, and also modifying the result from the theory by 1) approximating the response of the optical system and 2) introducing an empirical factor, dependent on the roundness of particles, to describe ice crystals.

There are two important factors in correcting the particle data, which are correcting the particle size and also defining the DOF. Both were achieved by optimizing two equations describing the oversize and distance from the object plane of spherical and complex particles.

The DOF of the CPI does not change drastically with particle size or habit until particle roundness becomes lower than 0.5 and particle size smaller than 40 μm. This result does not agree with the 2D OAP instruments, whose DOF depends approximately on the square of the particle size. The discrepancy is explained by the stringent rejection criteria used in the postprocessing software of the CPI (i.e., foc > 20 for accepted particles).

The image foc value was found to depend primarily on 1) the position from the object plane Z and 2) the relative illumination. Theory suggested that the change of relative intensity with r/R should depend on the normalized position Zd; however, this is limited by the optical system and the CCD resolution of the CPI. Numerical calculations were able to show an empirical relationship of foc on Z was adequate (see Fig. 5).

To put these corrections in to perspective, recent measurements of mixed phase clouds from the Facility for Airborne Atmospheric Measurements (FAAM) aircraft during the Ice and Precipitation Initiation in Cumulus (ICEPIC)8 campaign made airborne comparisons of size distribution data taken with several different microphysics probes. Those were a Fast Forward Scattering Spectrometer Probe (FFSSP; e.g., Field et al. 2003), the 2D-C, and the two-dimensional precipitation probe (2D-P). Figure 20 shows the probe PSD data averaged for one of the flights, B200—also shown are the uncorrected CPI data and 5% and 95% confidence intervals for the corrected CPI data (calculated from the Poisson distribution and knowledge of the DOF; see Fig. 19). It shows that there is reasonable agreement between the probes in the size interval 20–120 μm, which is improved over the old method. However, when corrected, the CPI seems to overestimate number concentrations slightly at the larger sizes. Preliminary calculations of the trajectories of ice particles within the airflow of the CPI inlet show that this may be due to inertial clustering of particles in the inlet, but this needs further work for publication.

The agreement between different probes in Fig. 20 is encouraging, but perhaps unexpected as such probes are thought to suffer from artifacts such as shattering of particles on the inlet (e.g., Field et al. 2006). It is not the aim of this present work to conclude why this might be, although one reason might be that this cloud was not strongly glaciated. Figure 21 shows the size distributions again, but with the CPI data classified into round and nonround particles, which may be a basic distinction between liquid and ice particles. It can be seen that the vast majority of particles, by number, are round (liquid) particles, which may be the reason for the good agreement. Another possibility is that the level of poor agreement associated with artifacts does not show clearly on the logarithmic scale. The study by Field et al. showed that the total number concentration may be affected by crystal shattering by up to a factor of 4 when large ice particles are present. This would be difficult to spot on the logarithmic scale, although this is another important area that needs to be considered.

Further work is needed to address the importance and validity of the DOF correction in calculating concentrations and ice water content (IWC) from the CPI data by comparing with other probes in stronger glaciated clouds. We also need to better understand and quantify the effects that particle shattering by the inlet might have on the PSD. One avenue of work that we may explore is that because we are able to look at the spacing of cloud particles in the CPI DOF—we can retrieve the distance Z from the object plane quite accurately (see Fig. 15)—we may be able to investigate shattering artifacts.

The results presented in this paper cannot be directly applied to every CPI as they depend on the optical alignment of the probe and other characteristics. However, we have performed the bead calibrations with two other CPI probes and found that the nonlinear regression model describes the data equally well, having slightly different fit parameters to those in Tables 2 and 1.

Acknowledgments

PJC would like to thank Andrew Heymsfield for helpful comments made to the manuscript and also for helpful discussions during a visit he made to the National Center for Atmospheric Research, Boulder, Colorado. We would also like to acknowledge the help of Paul Lawson, Brad Baker, and Pat Zmarsly of SPEC, Inc., for their e-mail discussions and helpful advice with the development of the CPI correction. The CPI was provided by the University of Manchester component of the NERC Centre for Atmospheric Research (NCAS) University Facility for Atmospheric Measurements (UFAM). Funds for Dr. Ulanowski to visit and work at the Manchester laboratory were also provided by NCAS UFAM. We would also like to thank Alan Blyth for allowing the use of ICEPIC data for comparisons of PSDs.

REFERENCES

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Fig. 1.
Fig. 1.

A schematic showing the CPI principles of operation during the normal operation/triggering mode. Cloudy air flows into the sample tube and intercepts an area defined by two intersecting elliptical cross section continuous laser beams: the PDS 90 and PDS 45 lasers. A scattering signal is monitored by detectors at the end of the optical path of both lasers. If the scattering signal is detected from both beams simultaneously, the imaging laser is armed and fires a 20-ns pulsed laser that casts a diffraction pattern of the object on to the CCD camera. The continuous triggering mode is similar, but it does not rely on the PDS lasers to arm the imaging laser.

Citation: Journal of Atmospheric and Oceanic Technology 24, 11; 10.1175/JTECH2096.1

Fig. 2.
Fig. 2.

A schematic of diffraction by a circular opaque disc. The figure shows how monochromatic, coherent light of wavelength λ is diffracted by a circular opaque disc of radius R, resulting in a diffraction pattern on a screen a distance Z measured normal from the object. Here k is the wave vector of the light (note the wavenumber k=2π/λ), S is the vector from a point P on the screen to a point P0 on the boundary, dl is the line vector element on the boundary, and Z is the vector from the object plane to the image plane. For other notation in this diagram, see Eq. (1).

Citation: Journal of Atmospheric and Oceanic Technology 24, 11; 10.1175/JTECH2096.1

Fig. 3.
Fig. 3.

Examples showing some agreement between diffraction theory and measured CPI images. Four different values of normalized position are shown (rows). The CPI-measured images are shown for objects on the far side of the object plane from the objective lens (second column) and on the near side of the object plane (third column). Radial diffraction patterns for both theory and measurements are shown in the fourth column. While the detailed features are not well produced (such as the Poisson spot and other slight maxima and minima) the gradient in intensity near the apparent edge of the particle and the width itself are reproduced well. Note that in the CPI images, the rings of maxima and minima are not always concentric (due to nonideal optics); however, this affects particle sizing only minimally.

Citation: Journal of Atmospheric and Oceanic Technology 24, 11; 10.1175/JTECH2096.1

Fig. 4.
Fig. 4.

A schematic showing how the CPI software processes image data. (a) A radial diffraction pattern from a circular disc with normalized position Zd = 1.0. (b) At i positions around the particle edge, the norm of the 2D gradient of intensity is sampled for 15 pixels normal to and crossing the boundary of the particle edge, Gi,l. The foc value is determined by dividing the maximum value of Gi,l by the FWHM of Gi,l and multiplying by 0.9. The foc value is approximately proportional to the gradient of intensity at the edge (dI/dr)|r=Lapp/2. An invariant property of the images is the quantity (dI/I0)/(dr/R), providing that the edge detection is at the same relative position for each particle with constant Zd. However, a better invariant property for the foc parameter is described in Fig. 5.

Citation: Journal of Atmospheric and Oceanic Technology 24, 11; 10.1175/JTECH2096.1

Fig. 5.
Fig. 5.

Derivation of the particle focus relation is an empirical relation. For ideal images with no discretization and an ideal optical system, the gradient of relative intensity [(dI/I0)/(dr/R)](r/R) is a function of normalized position Zd. However since the interval taken to calculate the derivative is large compared to the change in derivative, large gradients cannot be calculated accurately. A better relation for determining the particle focus relation was found by numerical calculation and is illustrated in this diagram. It was found that the gradient [(dI/I0)]/(dr)|r=Lapp/2 at the particle edge is approximately constant for a given value of the position from the object plane, Z, and can also be determined more accurately.

Citation: Journal of Atmospheric and Oceanic Technology 24, 11; 10.1175/JTECH2096.1

Fig. 6.
Fig. 6.

A typical background image taken with the Manchester CPI probe Ser 007. The image has 8 bits of grayscale. Note the dark region in the top-left-hand corner and the lighter regions near the center and to the right. CPI images have 1024 × 1024 pixels in the X and Y orientations and the resolution is approximately 2.3 μm.

Citation: Journal of Atmospheric and Oceanic Technology 24, 11; 10.1175/JTECH2096.1

Fig. 7.
Fig. 7.

SEM images of ice crystal analogs grown from sodium fluorosilicate. These are just some SEM images of the types of ice analog crystals used to develop the CPI corrections. Particle descriptions are given in the top left of each image. Abbreviations are as follows: bul = bullet, col = column, gros = germ rosette, pla = plate, plag = plate aggregate, ros = rosette, scol = short column. Numbers after the abbreviation are approximate particle lengths in micrometers.

Citation: Journal of Atmospheric and Oceanic Technology 24, 11; 10.1175/JTECH2096.1

Fig. 8.
Fig. 8.

The calibration rig used to move calibration beads and ice crystal analogs through the probe sample volume. The calibration rig was attached to the end of the probe sample tube and inserted so that the glass slide was within the image DOF. Note that the glass slide, positioned at the tip of the rig, can move in three dimensions and was orientated at 45° to the direction of airflow so that the slide was normal to the imaging laser beam.

Citation: Journal of Atmospheric and Oceanic Technology 24, 11; 10.1175/JTECH2096.1

Fig. 9.
Fig. 9.

Depiction of how image quality changes with distance from the object plane. Here the X scale is the distance of the beads from the object plane in millimeters while each row shows a different size of calibration bead: (top) 100-μm beads, (middle) 15-μm beads, and (bottom) 40-μm beads.

Citation: Journal of Atmospheric and Oceanic Technology 24, 11; 10.1175/JTECH2096.1

Fig. 10.
Fig. 10.

Depiction of how image quality changes with distance from the object plane for columns. Here the X scale is the distance of the ice analogs from the object plane in millimeters while each row shows a different ice analog: (first row) 135-μm column (col135), (second row) 45-μm column (col45), (third row) 60-μm column (col60), and (fourth row) 75-μm column (col75).

Citation: Journal of Atmospheric and Oceanic Technology 24, 11; 10.1175/JTECH2096.1

Fig. 11.
Fig. 11.

Depiction of how image quality changes with distance from the object plane for columns and rosettes. Here the X scale is the distance of the ice analogs from the object plane in millimeters while each row shows a different ice analog: (first row) 25-μm column (col25), (second row) 55-μm column (col55), (third row) 80.5-μm column (col80.5), and (fourth row) 80-μm rosette (ros80).

Citation: Journal of Atmospheric and Oceanic Technology 24, 11; 10.1175/JTECH2096.1

Fig. 12.
Fig. 12.

The CPI particle focus calibration curve has three independent variables; the relationship for each independent variable is depicted. For the first independent variable Z, foc changes as a ridge that is steeper on the positive side of the object plane. For the second independent variable, BGloc/, foc increases in a linear fashion as expected. Not shown is the fact that the full width at half maximum of the ridge decreases with decreasing BGloc/ because this variable affects the edge detection. For the third independent variable, particle focus increases for the least round particles because of the sharpness of the image features.

Citation: Journal of Atmospheric and Oceanic Technology 24, 11; 10.1175/JTECH2096.1

Fig. 13.
Fig. 13.

The CPI length calibration curve has four independent variables; the relationship for each independent variable is depicted. For the first independent variable, Z/L2, Lapp/L (the degree of oversizing) changes as a valley that is steeper on the negative side of the object plane. For the second independent variable, L, Lapp/L increases as L decreases, which is due to the discreteness of the image pixels. For the third independent variable, BGloc/, Lapp/L increases in a linear fashion because the edge detection occurs at a higher percent of the image intensity. For the fourth independent variable, Ar, Lapp/L decreases as the particle roundness increases. This is because the length L of the particle does not describe the normalized position of the particle for ice particles as it does for spherical particles.

Citation: Journal of Atmospheric and Oceanic Technology 24, 11; 10.1175/JTECH2096.1

Fig. 14.
Fig. 14.

Histograms of (a)–(c) uncorrected bead sizes and (d)–(f) corrected bead sizes. Also shown are respective means, standard deviations, and sample size N. (a), d (d) The uncorrected and corrected 15-μm beads; the actual NIST quoted sizes were 15 ± 1.1 μm. (b), (e) The uncorrected and corrected 40-μm beads; the actual NIST quoted sizes were 40 ± 2.8 μm. (c), (f) The uncorrected and corrected 100-μm beads; the actual NIST quoted sizes were 99.4 ± 1.1 μm.

Citation: Journal of Atmospheric and Oceanic Technology 24, 11; 10.1175/JTECH2096.1

Fig. 15.
Fig. 15.

(a) The predicted vs measured distance from the object plane. This shows strong positive correlation (Pearson’s r2 = 0.91) and a good one-to-one relationship (gradient of linear fit is 0.99). (b), (c) Both the correlation and gradient of the relationship are robust as a bootstrap analysis produces minimal spread.

Citation: Journal of Atmospheric and Oceanic Technology 24, 11; 10.1175/JTECH2096.1

Fig. 16.
Fig. 16.

Histograms of (a)–(c) uncorrected ice analog sizes and (d)–(f) corrected ice analog sizes. Also shown are respective means, standard deviations, and sample size N. (a), (d) The uncorrected and corrected 25-μm column. (b), (e)The uncorrected and corrected 80-μm column. (c), (f) The uncorrected and corrected 80.5-μm rosette.

Citation: Journal of Atmospheric and Oceanic Technology 24, 11; 10.1175/JTECH2096.1

Fig. 17.
Fig. 17.

(a) The predicted vs measured distance from the object plane for ice analogs. This shows strong positive correlation (r2 = 0.92) and reasonable one-to-one relationship (gradient of linear fit is 0.90). (b), (c) Both the correlation and gradient of the relationship are robust as a bootstrap analysis produces minimal spread.

Citation: Journal of Atmospheric and Oceanic Technology 24, 11; 10.1175/JTECH2096.1

Fig. 18.
Fig. 18.

The predicted DOF on both sides of the object plane for different sizes (10, 50, 100, and 150 μm) of round (Ar = 1.0) particles (columns 1 and 2); and for different sizes (10, 50, 100, and 150 μm) of nonround (Ar = 0.1) particles (columns 3 and 4).

Citation: Journal of Atmospheric and Oceanic Technology 24, 11; 10.1175/JTECH2096.1

Fig. 19.
Fig. 19.

A contour plot of the sampling volume (m3) of one image for different particle types (length versus roundness). Note that when imaged with the CPI, the roundness parameter is sometimes greater than 1.0 for very round particles because square pixels that are counted as being completely inside the surrounding circle by the feature extraction algorithm also actually have area outside of the encompassing circle.

Citation: Journal of Atmospheric and Oceanic Technology 24, 11; 10.1175/JTECH2096.1

Fig. 20.
Fig. 20.

A comparison of FFSSP, 2D-C, 2D-P, and CPI size distributions for ICEPIC flight B200. The line black lines show 5% and 95% confidence intervals on the corrected CPI size distribution data (calculated from the Poisson distribution and knowledge of the DOF; see Fig. 19). Confidence intervals become wider with decreasing time interval.

Citation: Journal of Atmospheric and Oceanic Technology 24, 11; 10.1175/JTECH2096.1

Fig. 21.
Fig. 21.

A comparison of FFSSP, 2D-C, 2D-P, and CPI size distributions for ICEPIC flight B200. The same as in Fig. 21, except that the particle phase (liquid or ice) is also shown from the CPI data. The criteria for classifying as liquid is that the particle roundness must be greater than 75% and the maximum deviation of the radius measured at several angles from the particle center must be less than 10% of the particle mean radius.

Citation: Journal of Atmospheric and Oceanic Technology 24, 11; 10.1175/JTECH2096.1

Table 1.

Calibration parameters for the CPI depth-of-field/focus correction for liquid and ice particles [see Eq. (5)].

Table 1.
Table 2.

Calibration parameters for the CPI depth-of-field/length correction for liquid and ice particles [see Eq. (7)].

Table 2.

1

The CPI is an instrument that is designed to be mounted on an aircraft and take images of individual cloud particles—in the size range 10–2000 μm. The images are taken with a high power, 25-ns pulsed laser that falls onto a charge-coupled device (CCD) camera.

2

This is the Rubinowicz representation of the Kirchhoff diffraction integral.

3

At larger Zd or for larger R the Poisson spot is in better agreement with theory.

4

This was well defined. Note, in this setup the relative position of the object plane changes from experimental run to run.

5

This is because of several combined factors, including the pixel discreteness, and the interval over which the gradient is taken.

6

This has been determined through trial and error in CPIview.

7

Note that DOF(x, y) is set to zero for values on the image (x, y) coordinates that are below the threshold for accepting images—40 bits.

8

More details of these data will be published elsewhere.

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  • Ulanowski, Z., Connolly P. J. , Flynn M. J. , Gallagher M. W. , Clarke A. J. M. , and Hesse E. , 2004: Using ice crystal analogues to validate cloud ice parameter retrievals from the CPI ice spectrometer data. Proc. 14th Int. Conf. on Clouds and Precipitation, Bologna, Italy, International Commission on Clouds and Precipitation, 1175–1178.

  • Fig. 1.

    A schematic showing the CPI principles of operation during the normal operation/triggering mode. Cloudy air flows into the sample tube and intercepts an area defined by two intersecting elliptical cross section continuous laser beams: the PDS 90 and PDS 45 lasers. A scattering signal is monitored by detectors at the end of the optical path of both lasers. If the scattering signal is detected from both beams simultaneously, the imaging laser is armed and fires a 20-ns pulsed laser that casts a diffraction pattern of the object on to the CCD camera. The continuous triggering mode is similar, but it does not rely on the PDS lasers to arm the imaging laser.

  • Fig. 2.

    A schematic of diffraction by a circular opaque disc. The figure shows how monochromatic, coherent light of wavelength λ is diffracted by a circular opaque disc of radius R, resulting in a diffraction pattern on a screen a distance Z measured normal from the object. Here k is the wave vector of the light (note the wavenumber k=2π/λ), S is the vector from a point P on the screen to a point P0 on the boundary, dl is the line vector element on the boundary, and Z is the vector from the object plane to the image plane. For other notation in this diagram, see Eq. (1).

  • Fig. 3.

    Examples showing some agreement between diffraction theory and measured CPI images. Four different values of normalized position are shown (rows). The CPI-measured images are shown for objects on the far side of the object plane from the objective lens (second column) and on the near side of the object plane (third column). Radial diffraction patterns for both theory and measurements are shown in the fourth column. While the detailed features are not well produced (such as the Poisson spot and other slight maxima and minima) the gradient in intensity near the apparent edge of the particle and the width itself are reproduced well. Note that in the CPI images, the rings of maxima and minima are not always concentric (due to nonideal optics); however, this affects particle sizing only minimally.

  • Fig. 4.

    A schematic showing how the CPI software processes image data. (a) A radial diffraction pattern from a circular disc with normalized position Zd = 1.0. (b) At i positions around the particle edge, the norm of the 2D gradient of intensity is sampled for 15 pixels normal to and crossing the boundary of the particle edge, Gi,l. The foc value is determined by dividing the maximum value of Gi,l by the FWHM of Gi,l and multiplying by 0.9. The foc value is approximately proportional to the gradient of intensity at the edge (dI/dr)|r=Lapp/2. An invariant property of the images is the quantity (dI/I0)/(dr/R), providing that the edge detection is at the same relative position for each particle with constant Zd. However, a better invariant property for the foc parameter is described in Fig. 5.

  • Fig. 5.

    Derivation of the particle focus relation is an empirical relation. For ideal images with no discretization and an ideal optical system, the gradient of relative intensity [(dI/I0)/(dr/R)](r/R) is a function of normalized position Zd. However since the interval taken to calculate the derivative is large compared to the change in derivative, large gradients cannot be calculated accurately. A better relation for determining the particle focus relation was found by numerical calculation and is illustrated in this diagram. It was found that the gradient [(dI/I0)]/(dr)|r=Lapp/2 at the particle edge is approximately constant for a given value of the position from the object plane, Z, and can also be determined more accurately.

  • Fig. 6.

    A typical background image taken with the Manchester CPI probe Ser 007. The image has 8 bits of grayscale. Note the dark region in the top-left-hand corner and the lighter regions near the center and to the right. CPI images have 1024 × 1024 pixels in the X and Y orientations and the resolution is approximately 2.3 μm.

  • Fig. 7.

    SEM images of ice crystal analogs grown from sodium fluorosilicate. These are just some SEM images of the types of ice analog crystals used to develop the CPI corrections. Particle descriptions are given in the top left of each image. Abbreviations are as follows: bul = bullet, col = column, gros = germ rosette, pla = plate, plag = plate aggregate, ros = rosette, scol = short column. Numbers after the abbreviation are approximate particle lengths in micrometers.

  • Fig. 8.

    The calibration rig used to move calibration beads and ice crystal analogs through the probe sample volume. The calibration rig was attached to the end of the probe sample tube and inserted so that the glass slide was within the image DOF. Note that the glass slide, positioned at the tip of the rig, can move in three dimensions and was orientated at 45° to the direction of airflow so that the slide was normal to the imaging laser beam.

  • Fig. 9.

    Depiction of how image quality changes with distance from the object plane. Here the X scale is the distance of the beads from the object plane in millimeters while each row shows a different size of calibration bead: (top) 100-μm beads, (middle) 15-μm beads, and (bottom) 40-μm beads.

  • Fig. 10.

    Depiction of how image quality changes with distance from the object plane for columns. Here the X scale is the distance of the ice analogs from the object plane in millimeters while each row shows a different ice analog: (first row) 135-μm column (col135), (second row) 45-μm column (col45), (third row) 60-μm column (col60), and (fourth row) 75-μm column (col75).

  • Fig. 11.

    Depiction of how image quality changes with distance from the object plane for columns and rosettes. Here the X scale is the distance of the ice analogs from the object plane in millimeters while each row shows a different ice analog: (first row) 25-μm column (col25), (second row) 55-μm column (col55), (third row) 80.5-μm column (col80.5), and (fourth row) 80-μm rosette (ros80).

  • Fig. 12.

    The CPI particle focus calibration curve has three independent variables; the relationship for each independent variable is depicted. For the first independent variable Z, foc changes as a ridge that is steeper on the positive side of the object plane. For the second independent variable, BGloc/, foc increases in a linear fashion as expected. Not shown is the fact that the full width at half maximum of the ridge decreases with decreasing BGloc/ because this variable affects the edge detection. For the third independent variable, particle focus increases for the least round particles because of the sharpness of the image features.

  • Fig. 13.

    The CPI length calibration curve has four independent variables; the relationship for each independent variable is depicted. For the first independent variable, Z/L2, Lapp/L (the degree of oversizing) changes as a valley that is steeper on the negative side of the object plane. For the second independent variable, L, Lapp/L increases as L decreases, which is due to the discreteness of the image pixels. For the third independent variable, BGloc/, Lapp/L increases in a linear fashion because the edge detection occurs at a higher percent of the image intensity. For the fourth independent variable, Ar, Lapp/L decreases as the particle roundness increases. This is because the length L of the particle does not describe the normalized position of the particle for ice particles as it does for spherical particles.

  • Fig. 14.

    Histograms of (a)–(c) uncorrected bead sizes and (d)–(f) corrected bead sizes. Also shown are respective means, standard deviations, and sample size N. (a), d (d) The uncorrected and corrected 15-μm beads; the actual NIST quoted sizes were 15 ± 1.1 μm. (b), (e) The uncorrected and corrected 40-μm beads; the actual NIST quoted sizes were 40 ± 2.8 μm. (c), (f) The uncorrected and corrected 100-μm beads; the actual NIST quoted sizes were 99.4 ± 1.1 μm.

  • Fig. 15.

    (a) The predicted vs measured distance from the object plane. This shows strong positive correlation (Pearson’s r2 = 0.91) and a good one-to-one relationship (gradient of linear fit is 0.99). (b), (c) Both the correlation and gradient of the relationship are robust as a bootstrap analysis produces minimal spread.

  • Fig. 16.

    Histograms of (a)–(c) uncorrected ice analog sizes and (d)–(f) corrected ice analog sizes. Also shown are respective means, standard deviations, and sample size N. (a), (d) The uncorrected and corrected 25-μm column. (b), (e)The uncorrected and corrected 80-μm column. (c), (f) The uncorrected and corrected 80.5-μm rosette.

  • Fig. 17.

    (a) The predicted vs measured distance from the object plane for ice analogs. This shows strong positive correlation (r2 = 0.92) and reasonable one-to-one relationship (gradient of linear fit is 0.90). (b), (c) Both the correlation and gradient of the relationship are robust as a bootstrap analysis produces minimal spread.

  • Fig. 18.

    The predicted DOF on both sides of the object plane for different sizes (10, 50, 100, and 150 μm) of round (Ar = 1.0) particles (columns 1 and 2); and for different sizes (10, 50, 100, and 150 μm) of nonround (Ar = 0.1) particles (columns 3 and 4).

  • Fig. 19.

    A contour plot of the sampling volume (m3) of one image for different particle types (length versus roundness). Note that when imaged with the CPI, the roundness parameter is sometimes greater than 1.0 for very round particles because square pixels that are counted as being completely inside the surrounding circle by the feature extraction algorithm also actually have area outside of the encompassing circle.

  • Fig. 20.

    A comparison of FFSSP, 2D-C, 2D-P, and CPI size distributions for ICEPIC flight B200. The line black lines show 5% and 95% confidence intervals on the corrected CPI size distribution data (calculated from the Poisson distribution and knowledge of the DOF; see Fig. 19). Confidence intervals become wider with decreasing time interval.

  • Fig. 21.

    A comparison of FFSSP, 2D-C, 2D-P, and CPI size distributions for ICEPIC flight B200. The same as in Fig. 21, except that the particle phase (liquid or ice) is also shown from the CPI data. The criteria for classifying as liquid is that the particle roundness must be greater than 75% and the maximum deviation of the radius measured at several angles from the particle center must be less than 10% of the particle mean radius.

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