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  • View in gallery

    Schematic diagram of a liquid water content retrieval. Area B (bounded by the dashed line box) is the target area of the retrieval; areas A and C represent the lateral boundaries. Lines with arrows indicate the direction of signals received at a slanting antenna.

  • View in gallery

    Schematic diagrams of (a) MAF and (b) SAF. The dashed-line box represents the target area and the thick solid lines with arrows represent flight paths.

  • View in gallery

    Simulated liquid water fields of typical cloud types: (a) weak onion cloud, (b) broken cumulus cloud, (c) stratocumulus cloud, and (d) horizontally homogeneous cloud.

  • View in gallery

    Simulated effects of different detection schemes on (a) V, (b) condition number, and (c) retrieval error. For single-level detection schemes, the x axis represents cases in which the number of measurements is adjusted to be nearly identical to the number of measurements made by MAF with the specified number of levels (±1 measurement).

  • View in gallery

    Retrieval image of the broken cumulus cloud (Fig. 3b) using a multiaircraft flight with five detection levels.

  • View in gallery

    Simulated variations of V, condition number, and retrieval error as the number of scanning angles in a multiangle detection scheme is increased.

  • View in gallery

    Simulated variations of condition number (solid circles) and retrieval error (open circles) for the horizontally homogeneous cloud with (a) the number of detection levels and (b) the number of detection angles.

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    Retrieval images of horizontally homogeneous cloud when a multilevel detection scheme is used with six detection levels.

  • View in gallery

    Retrieval images based on (left) the single-level detection scheme and (right) the MAF scheme with two levels for (a),(b) weak onion cloud, (c),(d) broken cumulus cloud, (e),(f) stratocumulus cloud, and (g),(h) horizontally homogeneous cloud.

  • View in gallery

    Error distributions of the retrieval images in Fig. 9, where the error in each grid cell is calculated by subtracting the true value from the retrieved value, and the solid and dashed lines are for the positive and negative errors, respectively.

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Improvement of Liquid Water Content Retrieval Accuracy by Multilevel Detection in Cloud Tomography

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  • 1 Laboratory of Cloud-Precipitation Physics and Severe Storm, Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing, China
  • 2 Beijing Weather Modification Office, Beijing, China
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Abstract

A new multilevel detection scheme for cloud tomography is developed. This scheme solves problems intrinsic to conventional single-level detection, such as the lateral boundary problem and the low accuracy of liquid water content (LWC) retrieval for clouds without distinct liquid water cores. Sensitivity studies show that the new multilevel detection scheme can significantly enhance the well posedness of the inverse problem and increases the accuracy of the retrieval. These improvements are achieved not only for clouds with distinct liquid water cores but also for clouds with weak or no liquid water cores, which are difficult to accurately reconstruct using a single-level detection scheme. The settlement of the lateral boundary problem also leads to a natural and easy way of solving the detection time limit problem in cloud tomography. By using a multi-aircraft flight (MAF) scheme, segmental retrieval can be applied to make the applicable scope of cloud tomography much broader. Considering the detection time limit and the cost in practice, the feasible flight scheme at present is MAF with two detection levels. Although only one detection level is added to the conventional single-level scheme, the accuracy of LWC retrieval can be improved by 1.4%–13.1%.

Corresponding author address: Dr. Jun Zhou, Laboratory of Cloud-Precipitation Physics and Severe Storm, Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing 100029, China. E-mail: zhoujun@mail.iap.ac.cn

Abstract

A new multilevel detection scheme for cloud tomography is developed. This scheme solves problems intrinsic to conventional single-level detection, such as the lateral boundary problem and the low accuracy of liquid water content (LWC) retrieval for clouds without distinct liquid water cores. Sensitivity studies show that the new multilevel detection scheme can significantly enhance the well posedness of the inverse problem and increases the accuracy of the retrieval. These improvements are achieved not only for clouds with distinct liquid water cores but also for clouds with weak or no liquid water cores, which are difficult to accurately reconstruct using a single-level detection scheme. The settlement of the lateral boundary problem also leads to a natural and easy way of solving the detection time limit problem in cloud tomography. By using a multi-aircraft flight (MAF) scheme, segmental retrieval can be applied to make the applicable scope of cloud tomography much broader. Considering the detection time limit and the cost in practice, the feasible flight scheme at present is MAF with two detection levels. Although only one detection level is added to the conventional single-level scheme, the accuracy of LWC retrieval can be improved by 1.4%–13.1%.

Corresponding author address: Dr. Jun Zhou, Laboratory of Cloud-Precipitation Physics and Severe Storm, Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing 100029, China. E-mail: zhoujun@mail.iap.ac.cn

1. Introduction

The spatial distribution of liquid water content (LWC) in clouds is an important physical quantity. Existing methods for observing LWC, such as in situ measurements and active remote sensing, all have limitations. For example, the sampling volume of in situ measurements is too small to acquire a full view of a cloud, while radar reflectivity is complicated by its strong dependence on the particle size distribution. Microwave radiometers operating at a single frequency are mainly used to provide path-integrated liquid water content. (Westwater et al. 2004; Jiang 2004; Wang et al. 2012). The spatial resolution of retrieved LWC can be improved by applying the tomography technique in microwave remote sensing. This technique has been used in the United States since the 1980s to retrieve 2D and 3D images of the LWC distribution in clouds.

Warner and Drake (1985) proposed two possible configurations for cloud tomography based on the microwave emission from an observed cloud. In the first configuration, two ground-based radiometers scan the cloud in the same plane. In the second configuration, an airborne radiometer with two fixed antennas scans the cloud while the aircraft flies under the cloud base. In both configurations, the radiometer(s) are configured to scan along a horizontal line under the cloud; this approach is referred to as single-level detection in this paper. The single-level detection method has been widely used and has remained virtually unchanged over the past 25 years (Warner and Drake 1986; Drake and Warner 1988; Koldaev et al. 1990; Bobylev 1997; Huang 2010a), but it introduces two problems that seriously impact the accuracy of LWC retrievals.

The first problem is associated with the lateral boundary. Referring to Fig. 1, suppose that we need to retrieve the LWC in target area B. In this case, only those detections represented by the solid lines can be used in the inversion, as those represented by the dashed–dotted lines also observe unknown LWCs in the lateral boundary areas A and C. As a result, a portion of the target area cannot be “seen” by the slanting antenna; this portion of the target area is a blind zone that significantly complicates the inversion calculation. Several methods have been used to cope with this problem. One method involves assuming that LWC in the lateral boundary is zero, either because the target area is sufficiently large that it includes the whole cloud (Huang et al. 2008a) or because clouds outside the target area are sufficiently far away that their emission is not observed by the radiometer (Drake and Warner 1988). Another method involves acquiring the LWC in the lateral boundary with some other method, such as radar observations or a cloud model (Bobylev 1997). These assumptions and uncertainties limit the applicability of cloud tomography and lead to large errors in retrievals of LWC (Zhou et al. 2010). To address these limitations, Zhou et al. (2010) developed a nested retrieval method that provides information about the lateral boundary for use in retrievals of the LWC distribution within the target area. If the number of scanning angles is limited to two, however, as is the case in our current available airborne microwave radiometer, the retrieval images will still be seriously distorted. In addition, the nested retrieval method is time consuming and inconvenient for practical use. In this paper, we will focus on how to improve the retrieval accuracy when the distribution of LWC in the lateral boundary is unknown.

Fig. 1.
Fig. 1.

Schematic diagram of a liquid water content retrieval. Area B (bounded by the dashed line box) is the target area of the retrieval; areas A and C represent the lateral boundaries. Lines with arrows indicate the direction of signals received at a slanting antenna.

Citation: Journal of Atmospheric and Oceanic Technology 30, 2; 10.1175/JTECH-D-12-00054.1

There are two ways to reduce the area of the blind zone. One way is to increase the number of the scanning angles in single-level detection (multiangle detection). This flight scheme has been used in a field test and in numerical simulations (Huang et al. 2010a,b). The other way is to increase the number of detection levels (multilevel detection) by utilizing multiple passes of the cloud by one or more aircraft. In this paper, we compare the benefits of these two methods relative to each other and to the conventional single-level method. Furthermore, the settlement of the lateral boundary problem may lead to a natural and easy way to solve the detection time limit problem in cloud tomography. This will be specifically discussed in section 2b.

The second problem intrinsic to single-level detection is that it is difficult to accurately retrieve LWC for clouds that lack distinct liquid water cores. This problem is particularly pronounced for clouds that lack liquid water cores entirely, such as horizontally homogeneous clouds. The series of observed brightness temperatures for a cloud of this type traces a nearly horizontal line. In this case we can determine that the observed cloud does not vary much horizontally, but cannot describe its vertical structure; additional information about the vertical profile of LWC is needed to reconstruct such clouds. This paper will evaluate whether multilevel detection can increase the accuracy of LWC retrievals in such clouds.

Observations in northeastern China showed that the thickness of nonprecipitating stratiform clouds can usually extend 4–6 km (Jin et al. 2004; Liang et al. 2007). If the zenith angle of slanting antennas is fixed, the thicker the cloud is, the more serious the lateral boundary problem becomes. The thickness of these kinds of clouds and their nearly horizontally homogeneous LWC distribution will make the conventional cloud tomography very difficult. So how to solve these above two problems becomes very necessary and urgent.

Multilevel detection requires an aircraft flight through the cloud, which may introduce problems associated with adherent water on the antennas. In this study, we observe nonprecipitating stratiform clouds with low LWCs, the antenna radome is water resistant, and high-speed airflow from the aircraft motion acts to clear adherent water automatically. For these reasons, we believe that signal contamination by adherent cloud particles can be neglected in this study (Lei et al. 2003).

The remainder of this paper is organized as follows. Section 2 describes the numerical simulation scheme for generating the intended flight patterns and discusses the way to solve the detection time limit problem in cloud tomography by using a multilevel scheme. Section 3 presents sensitivity studies that quantify how multilevel detection improves the retrieval accuracy of LWC for clouds with and without liquid water cores. Section 4 presents numerical simulations of the currently feasible multilevel detection schemes based on the consideration of the detection time limit in cloud tomography and the cost in practice. Finally, we summarize the findings of this study and discuss the conditions of the proposed flight schemes in section 5.

2. Design of the numerical simulation

a. Parameters of the observing system

Since a certain degree of intersection between microwave beams is necessary for cloud tomography (Warner and Drake 1984; Huang et al. 2010a), two antennas working at different zenith angles would be the minimum requirement to produce such intersecting beams during the single-level detection. Therefore, double-antenna microwave radiometer is chosen as the basic configuration in the following numerical simulations. So strictly speaking, the multilevel scheme based on a double-antenna microwave radiometer is a combination of multiangle (two angles) and multialtitude observations.

The double-antenna airborne microwave radiometer has been developed by the Institute of Atmospheric Physics, Chinese Academy of Sciences (CAS), and the Northeast Institute of Geography and Agroecology, CAS. The central frequency of this radiometer is 31.65 GHz and the double-antenna elevation angles are 30° and 90°. This has proven to be the best combination of elevation angles when the number of the antennas is limited to two (Zhou et al. 2010). The beamwidth of the radiometer is 4.2°, the integration time is 0.3 s, the antenna switching time is 0.2 s, and the standard deviation of the measurement error is 0.2 K. The radiometer is fixed near the top of the cabin of a YUN-12 aircraft, and the two external antennas are covered in streamline radomes. The cruising speed of the aircraft is approximately 70 m s−1. Temperature and humidity profiles are obtained from a radiosonde that is released close to the target area at the approximate observation time. In this paper, we use a sounding profile that was obtained at Changchun station in China at 0000 UTC 8 July 2010. Atmospheric variability is simulated in the synthetic measurements by drawing temperature and humidity from a Gaussian distribution with means equal to the Changchun sounding profile values at the altitude of the center of the grid cell and standard deviations equal to those proposed by Drake and Warner (1988). For the inversion calculations, temperature and humidity are assumed to be horizontally homogeneous, and their values in each grid cell are equal to the Changchun sounding profile values at the altitude of the center of the grid cell. Sounding errors are added prior to the inversion. The water vapor mixing ratio is accurate to within ±5% and the temperature is accurate to within ±1.0 K (Huang et al. 2008b). The numerical simulations discussed in this paper use these default parameters except when otherwise specified.

b. Flight scheme

Multilevel detection can be implemented using either a multiaircraft flight (MAF) or a single-aircraft flight (SAF) with multiple passes. In the MAF approach, several aircrafts fly through the target area simultaneously at different altitudes (Fig. 2). To avoid the influence of the fuselage on the antennas, the highest aircraft should travel through the target area 1 min behind the next highest aircraft, and so on. In SAF, one aircraft flies through the target area multiple times at different altitudes. The first detection level in the numerical simulation is located at the bottom of the target area. The interval between two neighboring levels is equal to the depth of the target area divided by the number of levels.

Fig. 2.
Fig. 2.

Schematic diagrams of (a) MAF and (b) SAF. The dashed-line box represents the target area and the thick solid lines with arrows represent flight paths.

Citation: Journal of Atmospheric and Oceanic Technology 30, 2; 10.1175/JTECH-D-12-00054.1

Theoretically, the LWC distribution of the observed cloud should be roughly constant during the observation. This precondition leads to a strict limit in detection time, which largely restricts the application of cloud tomography. There are two ways of dealing with this problem. First, the width of the target area must be restricted to the time limit, which makes the application of cloud tomography confined to some small-scale clouds. In the field experiment performed over the Gulf of Mexico in 1985 (Warner and Drake 1988), the target area was set to be 6 km wide to include cumulus clouds and the detection was completed within 3–4 min. The statistical comparison between tomography retrievals and the in situ measurements showed excellent agreement. This time tolerance indicates that tomography does not require the target cloud to be absolutely static. The objective of the inversion is the average of the LWC in grid cells of the specified area; therefore, small changes in LWC at isolated points in space do not obviously impact the results of the retrieval. Second, if the target area is long and narrow, the correlation between scanning cycles becomes weak; the retrieval performed in the entire target area can be considered to be several individual retrievals in continuous subdomains. Therefore, the retrieval is not an instantaneous image of the LWC distribution but, rather, is actually a mixture of both spatial and temporal LWC distributions (Huang et al. 2010a; D. Huang et al. 2010, personal communication). But it should be noted that this insufficient overlap between the successive scan cycles also produces some false LWC distribution in the retrieval images and thus decreases the retrieval accuracy. In the Wakasa Bay, Japan, experiment in 2003 (Huang et al. 2010a), the target area was 176 km wide and 4 km high and the entire detection process was completed in 20 min by the research aircraft, which flew at 144 m s−1. The retrieval results showed that the tomographic image appeared to be physically plausible and consistent with the time series of the vertically scanning cloud radar image.

By using a multilevel scheme, this time limit problem can be solved in a more natural and easy way. As will be shown in section 3, the lateral boundary problem can be settled by this flight scheme, so it does not need the target area, including the entire cloud, or any prior information about the LWC distribution in the lateral boundary. Therefore, we can either intercept part of the cloud as the target area or use a segmental retrieval to avoid exceeding the time limit. In the former, a proper length of the target area is determined to include the part of the cloud that we are most interested in and that meets the time limit as well. In the latter, the whole target area is divided into several subdomains according to the requirement that the time limit in each subdomain must be satisfied. Then, the LWC distribution in subdomains will be retrieved separately. It must be pointed out that the segmental retrieval can only be realized through an MAF scheme in which the detection of each level is finished nearly at the same time.

MAF is strongly preferable if only the detection time limit is considered, as the detection can be completed in a much shorter time than SAF and its observations can be used in segmental retrievals to avoid the breakthrough in the detection time limit of the cloud tomography. This flight scheme is challenging to implement in practice, however, because of the cost and the difficulty in coordinating and commanding several aircrafts simultaneously. SAF costs much less, but the width of the target and the number of detection levels must be limited to keep from exceeding the detection time limit. The SAF flight path will provide more beam intersections than the MAF flight paths, which may help to improve the accuracy of the retrievals (Warner and Drake 1984). Everything taken into consideration, MAF with two detection levels is a practical option in field experiments at present. We will now present sensitivity studies that quantify how the number of detection levels affects the retrieval accuracy without consideration of the time limitation (section 3), and then we compare MAF with two levels (the feasible flight scheme given our limited budget and the detection time limit in cloud tomography) with conventional single-level detection (section 4). As in previous cloud tomography studies (Bobylev 1997; Drake and Warner 1988; Huang et al. 2008a,b; Koldaev et al. 1990; Warner and Drake 1984, 1985; Zhou et al. 2010), our numerical simulations do not include temporal changes of the LWC distribution during the detection. Therefore, segmental retrieval is not necessary in the numerical simulations in this paper. In real-world applications, the retrieval method for each subdomain in the segmental retrieval is the same as that shown in the following section.

c. Retrieval method

The retrieval method is identical to the rough scheme used in the nested retrieval method (Zhou et al. 2010). We give a brief introduction here. Considering the continuity and the possible maximum value of LWC in a nonprecipitating cloud, the double-sided constraint and Tikhonov regularization are used to build the objective function:
e1

The first term is the residual term, where x is the LWC vector, the radiative transfer operator, and b the observation data vector. The second term is the regular term, where α is the regularization parameter and L is the Laplace operator.

The choice of α is critical, as its value determines the relative weights of the residual and regular terms. Usually, two different strategies are used to determine α. The first is the priori strategy, in which the empirical value of α is determined in advance using a large set of numerical simulations; however, sensitivity analysis indicates that the optimum α for cloud tomography differs substantially by cloud type. It is therefore impossible to identify a universal empirical value of α. The second is the posteriori strategy, in which the optimum α is the one that most closely matches the error level of the raw data according to a set of guiding principles. The posteriori strategy is specifically divided into two sets of methods. One set of methods requires that the error level of the raw data be estimated before the determination of α (e.g., the Morozov deviation principle, the Engl criterion). Numerical simulations show that an inaccurate estimate of the error level will lead to large errors in the retrieval. In this paper, the error level includes the atmospheric variability, the sounding error, and the microwave radiometer noise. It is difficult to accurately estimate the overall error level. The other set of methods does not require a priori information about the error level [e.g., the generalized cross-validation (GCV) principle, the L-curve method]. The GCV principle can only be applied to linear problems, while the L-curve method can be applied to both linear and nonlinear problems (Wang 2007). The mathematical model that we use in this paper is nonlinear; we therefore use the L-curve method to determine α. This method has been widely used in a variety of inverse problems, and has previously been successfully applied in cloud tomography (Huang et al. 2008b).

The inversion in this study is a nonlinear optimization problem with simple bounds. Therefore, the limited-memory Broyden–Fletcher–Goldfarb–Shanno (L-BFGS-B) algorithm is applied to solve this problem. This algorithm, which is based on the gradient method, uses a limited-memory BFGS matrix to approximate the Hessian matrix of the objective function. It is known to be suitable for solving large nonlinear optimization problems with simple bounds on the variables (Zhu et al. 1997). In this paper, we always start the iteration of L-BFGS-B from an initial guess of zero. The retrieval error is defined as the RMS error divided by the maximum LWC in the simulated true field (Drake and Warner 1988).

d. Simulated liquid water fields

The simulated liquid water fields include weak onion cloud, broken cumulus cloud, stratocumulus cloud, and horizontally homogeneous cloud. These cases are similar to those studied by Zhou et al. (2010). The onion cloud as defined by Warner and Drake (1985) consists of a central core of relatively high LWC surrounded by uniform rings of decreasing LWC toward the cloud edge. The four cases are shown in Fig. 3. The target area is a two-dimensional 20-km-wide × 4-km-high vertical slice. This target area is then discretized into 200 (20 × 10) 1-km-wide × 0.4-km-high grid cells. If the altitudes of the cloud base and top can be roughly estimated from the ceilometer, cloud radar images, the thermodynamic profile from radiosonde, and/or direct observation by the aircrew, the target area can generally be constructed to contain both the cloud base and top. The LWC outside of the lower and upper boundaries of the target area is then known to be zero.

Fig. 3.
Fig. 3.

Simulated liquid water fields of typical cloud types: (a) weak onion cloud, (b) broken cumulus cloud, (c) stratocumulus cloud, and (d) horizontally homogeneous cloud.

Citation: Journal of Atmospheric and Oceanic Technology 30, 2; 10.1175/JTECH-D-12-00054.1

3. Sensitivity studies

We perform numerical simulations of multilevel, single-level, and multiangle detection to evaluate the ability of multilevel detection to improve the accuracy of LWC retrievals for clouds with and without distinct liquid water cores. We examine the ill-posed nature of the inverse problem by linearizing the forward model F(x) about some reference state x0. We show that
e2
defines the weighting function matrix (Hansen 1997). Here, x0 is taken to be the true liquid field to minimize the error ɛ introduced by the linearization. The condition number of is defined as the ratio of the maximum and minimum singular values. The value of the condition number characterizes the degree to which the underlying problem is ill-posed (Rodgers 2000). A small singular value means that there exists a certain combination of the columns (or rows) of that guarantees that the detection equations are linearly dependent. A situation with one or more small singular values implies that is nearly rank deficient (Hansen 1997). A larger condition number therefore means that the problem is more ill-posed and the solution is more sensitive to measurement noises or numerical errors (Huang et al. 2008b). For convenience, we use the base 10 logarithm of the condition number in place of its numeric value.

It should be noted that the retrieval accuracy depends not only on the structure of the equations that is reflected in the condition number of , but also on the observational signal strength and the agreement between the regular term and the true LWC distribution. The retrieval accuracy will be higher if the liquid water core in the cloud is stronger, if the noise level of detection [defined as the product of the variance of the radiometer noise and the number of detections (Wang 2007)] is lower, or if the regular term is more consistent with the true LWC distribution (Zhou et al. 2010). For now, the variance of the instrument noise has already been determined by the certain microwave radiometer. Due to the variability of cloud LWC distributions and the general lack of understanding regarding cloud physical characteristics, it is impossible to find a regular operator that is consistent with all cloud types. In this paper, we try to optimize the structure of the equations by improving the detection scheme.

The quantity V is defined as the percentage of grid cells detected by the beams from more than one visual position. In this case, beams from multiple visual positions include both beams from different scanning angles and beams from different detection levels. The value of (1 – V) can then be taken to approximate the area of the blind zone.

a. Clouds with distinct liquid water cores

The broken cumulus cloud (Fig. 3b) is one example of a cloud with distinct liquid water cores. Figure 4 shows that MAF and SAF have identical values of V and similar condition numbers and retrieval errors for this cloud. The addition of another detection angle from the shuttle flight path used in SAF does not strongly impact the inversion. We therefore take the MAF retrieval results as being representative of multilevel detection in the following discussion.

Fig. 4.
Fig. 4.

Simulated effects of different detection schemes on (a) V, (b) condition number, and (c) retrieval error. For single-level detection schemes, the x axis represents cases in which the number of measurements is adjusted to be nearly identical to the number of measurements made by MAF with the specified number of levels (±1 measurement).

Citation: Journal of Atmospheric and Oceanic Technology 30, 2; 10.1175/JTECH-D-12-00054.1

For MAF (Fig. 4), increasing the number of detection levels from 1 to 6 leads to a rapid growth in the value of V from 65% to 96.5%, meaning that the area of the blind zone is significantly reduced. The condition number decreases until the number of levels reaches 5, at which point it plateaus. This lack of further improvement is because additional detection levels no longer provide independent information. The retrieval error decreases from 12.1% to 3.2% as the number of levels increases from 1 to 5, then increases slightly as the number of levels reaches 6. This is because both the information and the noise level increase as the number of levels increases. The retrieval accuracy begins to decrease when the increase in the noise level exceeds that in the information. Figure 5 (and later Fig. 9c) shows that the retrieval image is much improved when a five-level detection is used in place of a conventional single-level detection. This numerical simulation shows that an appropriate increase in the number of detection levels can widen the instrument field of view and significantly improve the retrieval accuracy.

Fig. 5.
Fig. 5.

Retrieval image of the broken cumulus cloud (Fig. 3b) using a multiaircraft flight with five detection levels.

Citation: Journal of Atmospheric and Oceanic Technology 30, 2; 10.1175/JTECH-D-12-00054.1

The number of measurements increases with the number of detection levels. It has previously been shown that retrieval accuracy depends upon the number of measurements (Drake and Warner 1988). Is the improvement in the retrieval accuracy from increasing the number of detection levels simply the result of increasing the number of measurements? To address this question, we adjust the integral time and the antenna-switching time to make the number of measurements in single-level detection nearly identical to that in MAF multilevel detection with a specified number of levels (difference are within ±1). Figure 4 indicates that V, the condition number, and the retrieval accuracy are not improved by increasing the number of single-level detections. The improvement in the retrieval accuracy from multilevel detection does not result from the increase in the number of detections but from the increase in the number of detecting altitudes.

Figure 6 shows the results of simulated retrievals using a multiangle detection scheme. The combination of scanning angles is identical to that used by Zhou et al. (2010): the three-angle detection scheme uses angles of (45°, 90°, 135°), and the 5-, 7-, 9-, 11-, 13-, and 25-angle detection schemes use angles in the range 30°–150° spaced at intervals of 30°, 20°, 15°, 12°, 10°, and 5°, respectively. The value of V grows quickly to 100% as the number of scanning angles increases, indicating that multiangle detection is indeed an effective way to reduce the blind zone. However, the condition number does not decrease as sharply as when using MAF (Fig. 4b), quickly stabilizing at approximately 7.2 when the number of scanning angles reaches 5. The retrieval error reaches its minimum value of 7.1% when three angles are used, and it oscillates around 10% as the number of angles increases. This suggests that the amount of independent information provided by additional scanning angles quickly becomes saturated. The ability of multi-angle detection to improve retrieval accuracy is limited accordingly.

Fig. 6.
Fig. 6.

Simulated variations of V, condition number, and retrieval error as the number of scanning angles in a multiangle detection scheme is increased.

Citation: Journal of Atmospheric and Oceanic Technology 30, 2; 10.1175/JTECH-D-12-00054.1

Figure 6 also shows that the retrieval error and the condition number do not always change in the same way as the number of angles is increased. In particular, the retrieval error oscillates around a fixed value (~10%) as the number of angles is increased. This oscillation might arise because the sampling frequency is fixed. As the number of angles increases, the number of detections and the spatial extent of the scan at each angle both decrease. This means that some antennas may observe the target area incompletely. Thus, multiangle detection introduces some new uncertainties into the inversion calculation. To summarize, this comparison indicates that increasing the number of detection levels is a more effective and stable way to improve the accuracy of LWC retrievals using cloud tomography when the liquid water distribution in the lateral boundary is unknown than increasing the number of detection angles.

b. Clouds without liquid water cores

The information about the LWC distribution contained in observations of clouds without liquid water cores is much less than that contained in observations of clouds with strong liquid water cores. The matrix may be rank deficient if the number of detection levels is insufficient, meaning that more than one solution will satisfy the equations. This does not often happen when observing clouds with distinct liquid water cores. The setup of the grid cells and the detection schemes are unaltered in this section from those used in section 3a; the variations of V are therefore identical to those shown in section 3a.

Figure 7a shows that the condition number of a horizontally homogeneous cloud is effectively invariant as the number of detection levels increases from 1 to 4, but the rank of increases from 192 to 198 so the retrieval error decreases from 28% to 12.1%. The matrix becomes full rank when the number of levels reaches 5, so there is a sharp decrease in the condition number of and a decrease in the retrieval error to 3.3%. As shown later (Fig. 9g), single-level detection retrieves an incorrect horizontally homogeneous structure in which the maximum LWC appears in the middle of the target area, lower than that in the prescribed field (Fig. 3d). The reason is that the single-level detection observations contain very limited information about the cloud structure, so that the retrieved LWC distribution depends primarily on the regular term in Eq. (1). The Laplace operator in the regular term acts to drag the maximum LWC band into the middle of the target area, while the constraint that the LWC is zero outside of the lower and upper boundaries of the cloud induces a gradual decrease in retrieved LWC from the middle to both boundaries. The average of the retrieved LWC field is close to that of the true field, because the retrieval is constrained by the magnitude of the observed data. The retrieval is very close to the true LWC in the target area when the number of detection levels is increased to 6 (Fig. 8). Figure 7b shows that increasing the number of scanning angles is much less effective at improving the condition number and retrieval accuracy than increasing the number of detection levels. The rank of matrix remains deficient even when the number of angles reaches 25.

Fig. 7.
Fig. 7.

Simulated variations of condition number (solid circles) and retrieval error (open circles) for the horizontally homogeneous cloud with (a) the number of detection levels and (b) the number of detection angles.

Citation: Journal of Atmospheric and Oceanic Technology 30, 2; 10.1175/JTECH-D-12-00054.1

Fig. 8.
Fig. 8.

Retrieval images of horizontally homogeneous cloud when a multilevel detection scheme is used with six detection levels.

Citation: Journal of Atmospheric and Oceanic Technology 30, 2; 10.1175/JTECH-D-12-00054.1

To summarize, increasing the number of detection levels offers useful additional information about the LWC distribution of clouds without distinct liquid water cores. Multilevel detection improves the accuracy of LWC retrievals for these types of clouds significantly, while multiangle detection does not.

4. Current feasible flight scheme

Compared with SAF, the advantage of MAF is in the speedy completion of the whole detection process and the ability to solve the detection time limit problem in cloud tomography by segmental retrieval, while the disadvantage is in the great expense and difficulty in practice. Considering all these elements, MAF with two detection levels will be a feasible option in field experiments at present. We now compare the performance of the MAF with two levels and the conventional single-level scheme for the typical cloud types.

Table 1 and Fig. 9 show that adding one additional detection level reduces the retrieval error by 1.4%–13.1% and significantly improves the quality of the retrieval images. Error distributions of the retrieval results are showed in Fig. 10. The error in each grid cell is calculated by subtracting the true value from the retrieved one. As we can see, for the single-level scheme, the negative errors concentrate mainly in the area where the true LWC is relatively high while the positive errors concentrate mainly in the area where the true LWC is relatively low. This is because the information about the cloud structure offered by single-level detection is so insufficient that the retrieval result relies heavily on the regular term in Eq. (1). The smoothing effect of the Laplace operator in regular terms inclines us to make the LWC distribution uniform (i.e., to decrease the relatively high LWC and increase the relatively low LWC in the true field). This has resulted in the special characteristic of the error distributions shown in the left column of Fig. 10. This phenomenon is not as clear in the error distribution of the MAF scheme with two levels. This indicates that observations from one additional detection level greatly improve the well posedness of the inverse problem and weaken the dependence of inversion on the regular term.

Table 1.

Retrieval error of the one- and two-level detection schemes for typical cloud types.

Table 1.
Fig. 9.
Fig. 9.

Retrieval images based on (left) the single-level detection scheme and (right) the MAF scheme with two levels for (a),(b) weak onion cloud, (c),(d) broken cumulus cloud, (e),(f) stratocumulus cloud, and (g),(h) horizontally homogeneous cloud.

Citation: Journal of Atmospheric and Oceanic Technology 30, 2; 10.1175/JTECH-D-12-00054.1

Fig. 10.
Fig. 10.

Error distributions of the retrieval images in Fig. 9, where the error in each grid cell is calculated by subtracting the true value from the retrieved value, and the solid and dashed lines are for the positive and negative errors, respectively.

Citation: Journal of Atmospheric and Oceanic Technology 30, 2; 10.1175/JTECH-D-12-00054.1

In conclusion, the accuracy of LWC retrievals can be significantly improved even when only one additional detection level is added to the conventional single-level detection scheme. This improvement is obtained for simulated observations of clouds both with and without distinct liquid water cores.

5. Conclusions

We have developed a new multilevel detection scheme for cloud tomography that addresses problems intrinsic to the conventional single-level detection scheme. These problems include the lateral boundary problem and the low accuracy of liquid water content (LWC) retrievals for clouds without distinct liquid water cores. Sensitivity studies indicate that a new flight scheme that increases the number of detection levels at different altitudes can widen the instrument field of view and significantly improve the accuracy of LWC retrievals for clouds both with and without liquid water cores. We have examined the performance of this multilevel detection scheme relative to that of the multiangle detection scheme, which is also capable of reducing the area of the blind zone that arises from the lateral boundary problem. Our numerical simulations show that strong correlations among detection angles in the multiangle scheme limit improvements in the accuracy of retrievals for clouds with distinct liquid water cores. Furthermore, the multiangle scheme is wholly unable to retrieve clouds without distinct liquid water cores, as increasing the number of scanning angles offers no additional information regarding the structure of clouds of this type. The settlement of the lateral boundary problem also leads to a natural and easy way to solve the detection time limit problem in cloud tomography. Segmental retrieval or local retrieval can be applied to make the applicable scope of cloud tomography much broader.

By using MAF, the whole scanning process may be completed in much less time and segmental retrieval can be applied to avoid breaking the time limit in cloud tomography, but the cost will be very high if there are many detection levels. Consequently, MAF with two detection levels is a practical option in field experiments at present. Our numerical simulations show that observations from one additional detection level can offer important information about the cloud structure, which greatly improves the well posedness of the inverse problem and raises the accuracy of the LWC retrievals by 1.4%–13.1%.

There are some important limitations on the use of this remote sensing technique. First, the physical model assumes that all cloud droplets are small in comparison to the wavelength of the radiometer, so the scattering can be neglected. This assumption would be invalidated for precipitation amounts that exceed a threshold of roughly 1 mm h−1 (Drake and Warner 1988). Second, the LWC distribution in the target area must be nearly invariant during the observation time. From the perspective of hardware facilities, a faster-moving aircraft can reduce the detection time and make the retrieval more efficient; however, faster cruise speeds will also require that the microwave radiometer sampling frequency be higher to ensure that enough observations are obtained for the reconstruction of the cloud. These improvements in equipment would be very helpful in broadening the application of the multilevel detection scheme. In addition, as has already been shown in some previous studies, combining active, passive, and in situ measurements can yield more complete descriptions of clouds (Wang et al. 2012), while additional information from other detection modes may make the retrieval problem of cloud tomography easier to deal with.

Acknowledgments

This research was supported by the National Natural Science Foundation of China (Grant 41105016), the Knowledge Innovation Program of the Chinese Academy of Sciences (Grant KZCX2-EW-203), and the Elitist Training Project of Beijing, China (Grant 2012D002034000001). It is a pleasure to acknowledge insightful discussions with Wei Chong and Shen Zhilai at LACS IAP CAS. We thank two reviewers for constructive comments that improved the manuscript.

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