a. PDF and its parameters

We assume a vertically uniform, horizontally nonuniform rain model in an IFOV as shown in Fig. 1. This model is valid when we consider the SRT because it deals with only the path-integrated values. In this situation, we assume that we can measure a PIA with the SRT, a storm height, and a Z-factor profile that includes propagation loss due to rain.

If we assume that the probability density function (PDF) of rain rate in the IFOV is approximated by a lognormal model with the expectation of R, R, and the NSD of R, σ_R (i.e., we assume “uniform” statistics within the IFOV), then the PDF, f(R), is expressed as

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where μ_R and ξ²_R are the mean and the variance of lnR, which are related to R and σ_R as follows:

View Expanded

With this expression, the PIA measured with the SRT, A_SRT, in decibels, is given by

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with

ARLaR^b(4)

where L is the storm depth, and a and b are the coefficient and exponent of the k–R relation (attenuation coefficient versus rain-rate relation). We define the uniform PIA, A_u, as the PIA that corresponds to R with a usual k–R relation that can be obtained when rain is uniform within the IFOV:

A_uLaR^b(5)

Using R and σ_R as parameters, we can relate A_SRT to A_u. The relationship between those quantities assuming the lognormal PDF and a rain depth L of 5 km is shown in Fig. 2.

Since k–R and Z–R relations are generally expressed as power laws,

kaR^bZαR^β(6)

the distributions of k and Z are also lognormal and the means and variances of lnk (μ_k, ξ²_k) and those of lnZ (μ_Z, ξ²_Z) are given by

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Thus, the expectation and NSD of k (k, σ_k) and those of Z (Z, σ_Z) are expressed as

View Expanded

Equations (10) and (12) indicate that NSDs of k and Z are different from the NSD of rain rate but can easily be converted. Please note that the parameter σ_R in Fig. 2 is the NSD of rain rate (not of attenuation coefficient). In the calculation of A_u and A_SRT, the difference in PDF parameters is incorporated.

b. Relation between point and area-averaged quantities

Before describing the NUBF correction method, we consider the relationship between a “point” quantity Q and an “area-averaged” counterpart 〈Q〉 that is a basis of the current method. A similar problem has been discussed in the study of methods to predict PIA from a point measurement of rain rate with a rain gauge (Morita and Higuti 1978). They derived expressions for the mean and variance of PIA using a “normalized spatial autocorrelation function” (hereafter, ACF) of rain rate expressed by the form of exp(−ζr), where r is the distance between two points. In this paper, we derive an expression of Q (vertically averaged or integrated quantity with high spatial resolution) and 〈Q〉 (area-averaged value of Q over the IFOV). Note that the difference between the notation Q and 〈Q〉 is that the former is the expectation and the latter is still a statistical random variable.

First, we assume the following ACF model that depends on distance only:

ρrζr^η(13)

where r is the distance between two points on a two-dimensional rain field. As we see later in this paper, this model can well fit the “averaged” 1D ACF that is generated from the observed 2D ACF by making an “iso-range” circular averaging. Of course, there may be a wide range of directional properties in the 2D ACF, depending on storm type and orientation. Nevertheless, the approximation used here would be useful to obtain approximate properties of the relation between NSDs in connection to the ACF of rain field. Let Q(x, y) be the point value at (x, y) and 〈Q(x₀, y₀)〉 be the spatially averaged Q over a circular area having the diameter D centered at (x₀, y₀). In addition, let Q and σ^′_Q be the mean and NSD of Q. We add (′) to σ_Q because we implicitly assume here that rain-rate fluctuations having any spatial frequency can contribute to the variability of Q, which may be different from the variability of Q within an IFOV. Then we have the following relations:

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If we assume that ρ(r) is independent of the location (x, y), {〈Q〉² − Q²} can be written as

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where P(λ, z) is the incomplete gamma function, and the double integral over θ₁ and r₂ is represented by a function B(D, ζ, η). Note that the above 2D integration is made over the circular region having a diameter D. A point p(r₂, θ₂) is first specified in the circle. The variable θ₁ is a polar angle about p(r₂, θ₂), and r₁ is defined as a distance from p(r₂, θ₂) to the edge of the circle in the direction of θ₁. Thus the upper limit of the integration over r₁ (R_m) is a function of θ₁ and r₂. Thus, the relation between σ^′2_Q and σ²_〈Q〉 can be expressed as

View Expanded

which can be used to predict σ^′2_Q from σ²_〈Q〉 and the parameters of the spatial correlation function, although a 2D integration is needed to evaluate B(D, ζ, η). It should be noted that the PDF form has nothing to do with Eq. (16), but the PDF does affect the correction factor to estimate A_u (see Fig. 2).

The difference between σ_Q and σ^′_Q would be small when the diameter of IFOV is much wider than the correlation length of the ACF, which may be applicable for most spaceborne radiometers; however, this may not be the case for the 4-km IFOV we consider here. We will discuss this problem later in the discussion of the result of observation data analyses.

a. Data source

To study the rainfall characteristics statistically, we use 42 constant-altitude plan position indicator (CAPPI) maps at 2-km altitude obtained from the Massachusetts Institute of Technology (MIT) radar on board the R/V Vickers during the TOGA COARE campaign from November 1992 to February 1993. Those rain scenes are listed in Table 1 with a “subjective” rain-type classification. Though subjective, the classification may be useful to get approximate information about the rain-type dependence of the statistical characteristics. Examples of CAPPI data for types 1–4 are shown in Fig. 4. Please note that the classification is based on the characteristics of the area having a radius of 40 km centered at the radar location. The parameters to process the CAPPI Z-factor maps are listed in Table 2.

b. Spatial correlation properties of rain rate

As we discussed in section 3, the relation of variabilities between “point” rainfall quantity (Q) and area-averaged (〈Q〉) quantity can be estimated if we know the parameters of the ACF of Q, and the averaging area (represented by D). This is approximately true when we relate R and 〈Q〉, where Q is not necessarily equal to R. In other words, characterizing the ACF of R would be useful to estimate the σ^′_R–σ_〈Q〉 relationship. For this purpose, we calculate the 2D ACF of rain rate. This is then averaged over an isorange circle so that we can obtain an averaged 1D ACF. Examples of the averaged 1D ACF are shown in Fig. 5 as well as the fitting with the model function of Eq. (13). The calculation of the 2D ACF is made over two regions on a CAPPI scene;64 km by 64 km squares to the left and right of the radar site. This is because there is always a “blind” region near the radar site and sometimes to the north or south of the radar as well. It is found that Eq. (13) provides an excellent fit to the ACFs for almost all cases. Figure 6 shows the histograms of the averaged 1D ACF parameters ζ and η. On average, ζ ≈ 0.135 and η ≈ 1.13 for convective types (types 1, 2, and 4) and ζ ≈ 0.064 and η ≈ 1.10 for the stratiform type (type 3). Using Eq. (16) with D = 4 km, the relation between σ_〈R〉 and σ^′_R can be obtained.

c. Relation between finescale and coarse-scale rainfall variabilities

The relation Eq. (23a) can also be obtained directly from actual radar data by a regression analysis, although we have to notice that c_〈Q〉 represents the coefficient relating σ̂_R to σ̂_〈Q〉 when we use the sample data. In the following, we use A_SRT and 〈R〉; the former is used to obtain a statistical relation to estimate σ_R from the downlooking radar measurement, and the latter to compare with the theoretical calculation discussed above. The difference between 〈R〉 and A_SRT will be evaluated in section 4e. Figure 8 shows histograms of the coefficient c_ASRT, which corresponds to c_〈Q〉 in Eq. (23a), obtained with a linear regression of σ̂_R to σ̂_ASRT and the correlation coefficient r_ASRT for the convective (types 1, 2, and 4) and stratiform (type 3) cases. For the convective cases, histograms are given for two cases: (case 1) the minimum rain rate of 0.1 mm h⁻¹ and (case 2) the 5 mm h⁻¹ rain rate to investigate the dependence of the regression result on rain rate since the NUBF correction is more important for higher rain rates. Note that the correlation coefficient is about 0.5 and slightly higher at the stratiform case and that in the convective case the coefficient c_ASRT is generally greater than it is in the stratiform case. As for the rain-rate dependence, case 2 gives slightly lower and more stable c_ASRT values than case 1.

Let us compare the result of the regression analysis between σ̂_R and σ̂_〈R〉 with the result obtained from the ACF parameters (Fig. 7). Figure 9 shows the histogram of c_〈R〉 for 32 convective scenes. The mean value of 0.77 (case 2) compares fairly well with the calculation result (about 0.72) based on the ACF analysis shown in Fig. 7 with ζ = 0.135 and η = 1.13. On the other hand, the result of case 1 (mean = 0.96) is significantly higher than the calculation result. Although the sample NSDs should underestimate the NSD of the parent PDF by the factor of [(N − 1)/N)], where N is the number of data, the effect of this bias error, which causes a slight positive bias, would be rather small (about 2%) considering that both σ̂_R and σ̂_〈R〉 underestimate the parent PDF counterparts. Other possible causes of the discrepancy are (i) using the averaged 1D ACF may be a too simplified way to accurately predict the σ̂_R−σ̂_〈R〉 relationship, and (ii) more rigorous treatment is necessary of the window effect to the rain-rate variability within a 4-km IFOV. The scattergram between c_〈R〉 obtained from the σ̂_R–σ̂_〈R〉 regression and the correlation distance r_c derived from ACF analyses is shown in Fig. 10, where r_c is defined as the distance at which the ACF is reduced to e⁻¹. It is found that there is a clear correlation between them, suggesting the possibility to relate the the ACF properties to the σ_R–σ _〈R〉 relationship.

d. Lognormality of rain-rate PDF

To validate the assumption of lognormal distribution of rain rate within an FOV, the lognormality of rain rate for the spatial domain of several kilometers squares is examined by a χ² test. For this test, 49 rain-rate samples over a domain of 7 km by 7 km square are used, in which the χ² value is calculated only when at least 45 samples out of 49 are recognized as “rainy.” An example of the results is shown in Fig. 11, where χ² values are expressed as a cumulative distribution. In the distribution, χ² greater than about 10 is recognized as nonlognormal with the level of significance of 0.05 since we used an eight-category histogram to calculate the χ² value that should follow the χ² distribution for five degrees of freedom. Note that about 50% of the samples do not follow a lognormal distribution. We should note that a no-rain IFOV is not used to calculate a χ² value since there is an ambiguity in the treatment of “zero” rain rate. Figure 12 shows a summary of lognormality check of 42 rain scenes; the histogram of percentage of rain areas judged nonlognormal with 5% of significance from the χ² test. It is found that about 50% of the rain area may not be expressed as a lognormally distributed random field.

Including the zero rain rate in the χ² test should increases the χ² value, that is, nonlognormality. This suggests that the nonlognormality may be significant at storm edges and small isolated convective cells. In such cases, significant no-rain areas may exist within an IFOV, and we need to assume a delta-function PDF for the no-rain areas and another PDF for rainy areas. Such a mixed model was used in the NUBF study for microwave radiometer measurements (Graves 1993), but in the case of the spaceborne radar measurement, a model including a nonrainy region in an IFOV would be more difficult to apply because of the saturation of A_SRT due to the dominance of echo power from the nonrainy region (Kozu and Iguchi 1996). It is a next-step study subject to find alternative PDF that can better describe the rain-rate variability, to develop a method to detect a mixed rainy/nonrainy IFOV, and to apply a special procedure to such a mixed IFOV.

In spite of the above problems, we will assume the lognormal PDF in the simulation (described in the next section) to relate A_SRT to A_u because of its simplicity.

e. Radar measurable quantity other than A_SRT

In the above discussions and the investigations on rainfall characteristics, we have used A_SRT as the measurable quantity 〈Q〉. In cases where either rain attenuation is too small or normalized radar cross section (σ⁰) of the surface varies considerably, we may not be able to use A_SRT. There may be other quantities that can be used to estimate PIA, however. One possibility is the PIA obtained from the sum of attenuations at each range gate estimated from the profile of Z factor including attenuation (Z_m) with a k–Z relation (A_SU or A_ST). The quantity Z_m(r) is given by

View Expanded

where r₀ is the range at the storm top, and γ and δ are the coefficient and the exponent of the k–Z relation. PIA estimates (A_SU and A_ST) are expressed as follows:

View Expanded

The difference between A_SU and A_ST is that the former uses Z_m factors at high altitudes only, so that it may better be used for relatively intense rain cases than the latter, although it does not use rain information at rain bottom. Since A_SU and A_ST depend on the vertical structure of rain rate, the values obtained from the vertically uniform rain rate model assumed in this paper would not be the same as those from natural rain. Nevertheless, investigating the properties of these quantities under the vertically uniform rain would be useful for obtaining an approximate idea about the feasibility of using the Z_m profile alone. To test if we can use any quantity as 〈Q〉, depending on the rain and surface characteristics, correlations between c_〈Q〉’s obtained with the use of 〈R〉 and those obtained with the use of A_SRT, A_SU, or A_ST are plotted in Fig. 13, suggesting the validity of using A_SRT, A_SU, and A_ST in place of 〈R〉.

In the standard Tropical Rainfall Measuring Mission (TRMM) precipitation algorithm (Iguchi et al. 1997), the quantity

View Expanded

is used. The factor ε is a parameter to be adjusted by using A_SRT or by assuming the Z-factor profile is vertically constant just above the surface. Although we do not test this type of PIA estimate, we can expect that ξ is similar to the quantities we have tested here since ξ is the same as A_SRT when SRT is reliable, and it should be generally a better estimate of PIA than A_SU or A_ST.

a. Simulation procedure

For a test of the NUBF correction method, simulations are performed using the same radar dataset as those used in the above analyses. The procedure of the simulation is depicted in Fig. 14. For the preparation of the simulation, we generate a lookup table (actually a set of regression lines of the result shown in Fig. 2, see Table 3) to obtain A_u from A_SRT and σ_R.

We first convert the 1-km spatial resolution Z-factor map to the corresponding rain rate and PIA. Next, we calculate the coarse-resolution rain rate and PIA of 4-km resolution by using Eqs. (17), (5′), and (19). At the same time, we calculate σ̂_R using Eq. (20) and σ̂_ASRT using Eq. (21). The relationship between σ̂_R and σ̂_ASRT [Eq. (23a) is also used for the relation between “sample” NSDs] has been established from a linear regression analysis of these quantities as we discussed in section 4c. In the current simulation, the mean c_ASRT for convective rain, case 2 (=0.723, see Fig. 8) is used for all types of rainfall.

With the above preparation, the simulation starts with an examination of the coarse-resolution radar measurement (A_SRT) of each IFOV. For a specific IFOV, we obtain A_SRT and σ̂_ASRT. The σ̂_ASRT is then converted to σ̂_R using Eq. (23a). Here, A_SRT and the estimated σ̂_R, which is treated as σ_R, are then input to the lookup table to obtain an estimate of A_u (Ã_u).

Since we use the same dataset for the simulation as that used to derive the σ̂_R–σ̂_ASRT relation, this simulation may not be a fully independent test of the method. It would still be useful and necessary to confirm the validity of this method with this type of simulation, however, because the coefficient c_ASRT to estimate σ̂_R from σ̂_ASRT shows fairly wide variation when we look at the individual rain scene consisting of nine IFOVs, and the estimation of A_u from A_SRT and the estimated σ_R is a highly nonlinear, error-sensitive process. In addition, the validity of assuming the lognormal PDF can be tested with this simulation.

b. Simulation result

Following the procedure described above, simulations of the NUBF correction are performed; the results are shown in Figs. 15–18. In the figures, three scattergrams are compared for each rain type; one is the correlation between A_u and A_SRT (without correction), the second is that between A_u and Ã_u (estimated A_u with the NUBF correction) using the estimated σ_R, and the third is the same as the second except that the “true” σ_R (strictly speaking, true σ̂_R) is used. As shown in the figure, the NUBF-corrected results shown in panels b and c can improve the degree of underestimation of PIA, which is significant when no correction is applied. It is also clear, however, that over- and undercorrections are seen in some cases, particularly in types 1 and 2. In type 1, using the true σ_R significantly improves the tendency of overcorrection, suggesting the errors in estimating σ_R. On the other hand, in type 2, improvement by using the true σ_R is not significant, suggesting an improper PDF model causing an erroneous A_SRT–A_u relation. In the cases of types 3 and 4, the result of NUBF correction is better than types 1 and 2; in particular, the NUBF effect is very small in type 3, leading to stable NUBF corrections. From the above results it is clear that the “instantaneous” or “IFOV-based” NUBF correction can have considerable errors. Nevertheless, it can be concluded that the correction scheme tested here does work in general to reduce the significant negative bias errors due to the NUBF. Note that Durden et al. (1998) also obtained an encouraging result in testing an NUBF correction scheme using the data acquired by a KU-band airborne rain radar flown in the TOGA COARE field campaign.

The effectiveness of the NUBF correction can be seen more clearly from statistical comparisons. Figure 19 shows the ratio of cumulative probabilities of PIA for types 2 and 4. Each cumulative probability represents the percentage of occurrence of PIA that exceeds the given PIA value. In the figure, A_SRT/A_u and Ã_u/A_u are plotted. Note that the NUBF correction can significantly improve the estimation accuracy of the cumulative distribution of PIA.