A Systemic Analysis of Multiscale Deep Convective Variability over the Tropical Pacific

a. The multiplicative cascade model

The concepts of fractals and multifractals were first introduced by Mandelbrot (e.g., Mandelbrot 1974), and further developed to deepen the understanding of the intermittent turbulent phenomena (see Frisch 1995). The mathematical foundation of multifractal theories was provided by Hentschel and Procaccia (1983), Frisch and Parisi (1985), and Halsey et al. (1986), etc. Readers are referred to Gouyet (1996) for an excellent introduction.

In essence, multifractals are characterized by many or infinitely many power-law relations. To migrate the notion of a fractal set to one that manifests multifractal measures, one may partition the fractal set into many (typically interwoven) subsets, each characterized by a power law. This can either be characterized by a spectrum of fractal dimensions (i.e., the generalized dimensions) with each dimension for one subset, as we will introduce shortly; or it can be characterized by singular measures, which, in concept, gives the relative “weight” of each subset. A fuller picture of multifractals can be established by constructing a multifractal field. Multifractals can be constructed by a multiplicative cascade process model. Since the I_TBB is a one-dimensional time series, the definition for a 1D random multiplicative cascade process model following the spirit of Biferale et al. (1994) is given here, which refers to the following generating rules.

1. At stage 0, consider a unit length. Associate it with a unit weight.

2. Divide the unit length into two (say, left and right) segments of equal length.

3. Also, divide the weight into two fractions, r and 1 − r, and assign them to the left and right intervals, respectively. The parameter r, called the multiplier, is a random variable governed by a probability density function (pdf) P(r), 0 ≤ r ≤ 1.

4. Each new interval and its associated weight are further divided into two parts following the same rules.

A scheme is shown in Fig. 3. At stage i, the length of the interval is 2⁻ⁱ, and its associated multiplier is denoted as r_ij with j = 1, … , 2ⁱ. If the random multiplier distribution, P(r), is assumed to be symmetric about r = 0.5 and have successive moments μ₁, μ₂, … , then both r_ij and 1 − r_ij have marginal pdf P(r). At stage N, the weights (i.e., the measures) are denoted as {w_n(N), n = 1, … , 2^N}; each element is expressed as w_n(N) = u₁u₂ ⋯ u₁ ⋯ u_N, where u_l, l = 1, … , N, is either r_ij or 1 − r_ij, i = 1, … , N. Therefore, {u_l, l ≥ 1} are independent identically distributed (iid) random variables with pdf P(r).

Note that the above scheme is sometimes called microcanonical multiplicative cascades, where the multipliers add up exactly to 1. One may require that the weights are preserved only in terms of their expected value. Such a scheme is sometimes called canonical multiplicative cascades, and can easily be constructed by generating two random variables, for example, r_i1 and r_i2, where E(r_i1 + r_i2) = 1, and a weight w_i at stage i is split into two parts, wr_i1 and wr_i2. Multiplicative multifractals can even be nonconservative, as in Frisch (1995, p. 166). For our study, we shall employ the simplest conservative model, where weights simply add up to 1.

As an example, in Fig. 4 we have shown a portion (2880 points) of a realized field (4096 points) generated by a random multiplicative cascade process with stage number N = 12 and P(r) given by

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For random multiplicative cascade processes, at the stage N, a set of partition functions is computed as

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with E( ) the expected value real q. Here μ_q is the qth moment of P(r). A positive q value emphasizes large measures among w_n, while a negative q value emphasizes small measures. With large qs, M_q magnifies the intermittency and extreme events in the realization.

b. The definition of a multifractal field

The realization of the multiplicative cascade model has two properties that constitute our working definition of a multifractal field, which helps address the multiscale (hierarchical) structure of deep convective variability later in this paper. The first property is

M_qϵϵ^τ(q)ϵ(4)

Equation (4) indicates scaling laws between various orders of partition function and the scale ϵ. In the case of the multiplicative cascade model, ϵ = 2^−N and τ(q) = −ln(2μ_q)/ln2. If the measures w_n are uniform or only weakly nonuniform, the different emphases placed by q become trivial and the field is monofractal. In that case, w_i = w ∼ ϵ, and the number of terms in the summation in (3) is approximately ϵ⁻¹; therefore, M_q(ϵ) ∼ ϵ^q−1 and τ(q) = q − 1. Hence, the second property that determines a multifractal field is that the field has nonuniform measures, which can be estimated through the generalized dimensions:

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One may immediately notice that the computation of D₁ needs special attention because it is singular. Theoretically, one can employ l'Hôpital's rule, or use a Taylor series expansion for the numerator (e.g., Gouyet 1996). In practice, D₁ can sometimes be estimated via interpolation.

The first few dimensions of the dimension spectrum are of special interest. For instance, D₀ is the same as the fractal dimension α in (2), D₁ is called the information dimension, and D₂ is equivalent to the exponent of the power-law-decaying correlation function of the field (Hentschel and Procaccia 1983). When the measures are uniform or only weakly nonuniform (monofractal), D_q is independent of q and all D_q are reduced to the fractal dimension. This is equivalent to τ(q) being a linear function of q. When the measures are nonuniform (multifractal), D_q is a monotonically decaying function of q and τ(q) is a nonlinear function of q.

The above two properties or their similar forms are frequently used as the criteria to determine whether a dataset is multifractal (e.g., Davis et al. 1996; Ivanova and Ackerman 1999). However, the most strict definition of a multifractal field, as illustrated by the multiplicative cascade model, is that the multiplier distribution function P(r) is stage independent. Given a sufficiently long dataset, it is possible to estimate P(r) at different stages [e.g., Frederiksen et al. (1997), who used O(10⁵) samples]. This criterion is preferred since it is more free of artifacts. However, such datasets are rare in the atmospheric sciences.

If each measure, w_n(N), in Fig. 4 is interpreted as the total convective activity in a time interval of length 2^−NT, where T is the total time period of interest (2880 h), then the multiplicative cascade process becomes a primitive stochastic model for a time series of convective activity. Note the similarity between the intermittent features depicted in Figs. 1 and 4.

a. Multifractal characterizations

We now show that the multiscale I_TBB time series satisfies the two criteria that define it as a multifractal field. First, we compute the partition function, M_q(ϵ), and check if the following scaling relations hold

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for real values of q, where ϵ = 2^−N, N = 1, 2, … , 11, and 〈w^q_i〉 is the ensemble average of w^q_i. In order to compute (6), the total number of data points should have a base number of 2. We use 2048 points for computation each time. The computation is done twice, for the first and the last 2048 points in the I_TBB time series, respectively. For each computation, the total length as well as measure (weight) of the 2048 points are normalized to 1. The normalization of the weight yields Σ_i w_i = 1, which implies that τ(1) = 0 is the trivial scaling relation for M₁.

Figure 5a shows −log₂M_q(ϵ) versus −log₂ϵ for the 5° × 5° averaged I_TBB time series (Fig. 1). Only M_q functions with q ≥ 0 are shown since we are more interested in the large measures (intermittency) in the I_TBB. Computations that utilize different portions of the time series produce similar results. With linear regression, we clearly observe a series of power laws for different values of q. The power-law exponents, τ(q), are the slopes of the regressed lines. The regression lines fit less well for large values of q (e.g., q = 6) due to much rarer samples of extreme events. When ϵ = 2⁻¹, there are only two measures to evaluate (6) in each computation, so the resulting points are not used in the linear regressions. Nevertheless, in Fig. 5a, these points lie nicely on the regression lines for q = 0, 1, and 2. The scaling relations thus identified correspond to the time scales from 1 to at least 512 h (∼21.3 days, see the top of the figure, where the correspondence between the scale ϵ and the physical time is shown). The breakdown of scaling relations beyond 512 h could be due to insufficient sampling.

The second step toward characterizing the I_TBB dataset as multifractal is to check the degree of nonuniformity of the measures (w_i). Figure 5b shows the variation of τ(q) against q, in which τ(q) is the mean value of the power-law exponents of M_q(ϵ) derived from both portions of the data. The error bars defined by the standard deviations of the estimated power-law exponents of both data portions are plotted, indicating that the power laws are reasonably satisfied even for large q. In contrast to the reference straight line in the figure, τ(q) forms a concave curve. Hence, it is no surprise that the D_q shown in Fig. 5c (in which D₁ is interpolated from D_0.5 and D_1.5) is a monotonically decaying nonlinear function of q. The second criterion necessary for the I_TBB time series to be multifractal is thus satisfied.

In Figs. 5b,c, the broken curves depict the τ(q) and D_q functions obtained from the model time series (Fig. 4). In the model, the τ(q) behaves similarly to that of the observed I_TBB time series, with the exception that at larger values of q, the model τ(q) arches toward the abscissa. This tendency is reflected in D_q. In Fig. 5c, the model D_q is a more steeply declining curve, whereas the observed D_q shows a tendency to remain constant with larger q. This therefore indicates that the observed I_TBB time series contains fewer extreme events than the model time series. It is also noted that the fractal dimension (D₀) is 1.0 for the model and ∼0.97 for the observation. This is a fundamental difference between the model and the observed time series. By definition, the multiplicative cascade model can produce infinitesimally small positive measures but never zeros, which explains the fractal dimension of the model result. However, zeros are present when no significant deep convection is recorded in the 5° × 5° observational domain. The smaller than 1.0 fractal dimension indicates the sparseness of the observed I_TBB time series.

The I_TBB time series contains information on multiscale tropical convection-coupled waves and disturbances. The multiplicative cascade model captures the multifractal characteristics of the I_TBB time series reasonably well. As demonstrated by the model, multifractals can be created through the scale-invariant multiplicative process. In the tropical atmosphere, the multiplicative process could be partly responsible by continuous stretching and folding of dynamic and thermodynamic fields through the change of momentum and vorticity fields across various scales induced by convective updrafts and downdrafts. For example, over the same observation domain and period, convection has been found to generate perturbations of vorticity from the convection to large scales through modifying the momentum field (Tung and Yanai 2002a,b).

b. Consequence of spatial averaging

We now examine the impact of spatial averaging on the multifractal features of the I_TBB time series. In Fig. 6, seven domain-averaged I_TBB time series are plotted, with increasing domain size from top to bottom. The scale of the y axis is adjusted for domains 10° × 10° to better appreciate the variations of the time series. Although the spatial averaging appears to smooth the time series significantly, the irregularity within the time series is preserved. In fact, scaling relations held by the partition function of the 5° × 5° dataset are similarly found in other time series. This finding also confirms the conjecture by Yano et al. (2001) that the 1/f stochastic behavior of tropical convective variability should remain robust even after large-scale spatial averaging.

Figure 7 shows the generalized dimensions (D_q) of the seven domain-averaged time series, which are plotted as in Fig. 5c. The small error bars for D_q in the larger domains indicate that the scaling relations are more well defined there. Functions D_q of larger domain averages descend more slowly with q than those of smaller domain averages, particularly at smaller q values. This suggests that spatial averaging reduces the significance of the multifractal characteristics of the time series but does not eliminate it completely. The D_q for the single-point dataset appears to be horizontal with q ≥ 2, most likely the result of insufficient samples, particularly those of relatively extreme events, since the time series is most sparse. Such deficiency in sampling is also reflected by the large error bars for the D_q estimation. Furthermore, with the same q, larger-domain averages have bigger D_q. This may reflect that physical processes with better-defined long-range trends are dominant in the time series of larger-domain averaging. This aspect will be discussed along with the variance– time relations in section 5.

c. Consequence of temporal smoothing

A rather common picture of data analysts is: x(t) = y(t) + n(t), where y(t) is the true signal, n(t) is the measurement noise, often taken as white, and x(t) is the observed signal. One might expect x(t) to have a well-defined mean when averaged over a long enough period of time, while its standard deviation is insignificant when compared with the mean; or, by applying a moving average, seemingly “transient” noise could be removed. This picture is at least partly true, since measurement noise is always present. However, the averaging or smoothing tactic may not be as constructive as expected, especially when the true signal displays multifractal features. This is because the variations of y(t) over a broad range of time scales are the intrinsic features of such processes, as succinctly and precisely captured by the definition of multifractal in (6). More concretely, the variance and transient features of x(t) are driven from both y(t) and n(t). While the contribution from n(t) may be negligibly small when the average is taken over a long time interval, the contribution from y(t) can nevertheless be quite large. Often, the mean of the signal can be much smaller than its standard deviation. In some cases, the mean of x(t) may not even be well defined.

We can quantify how effective it is to temporally smooth the I_TBB time series by averaging the 5° × 5° time series over successively long time windows. For comparison, we also apply the same methods to the realization of the random multiplicative cascade model (Fig. 4), a white noise process (Fig. 8a), and a “shuffled” 5° × 5° I_TBB time series (Fig. 8b). White noise is defined as y_i = 1.0 + n_i. Time series n_i is obtained through a random number generator with mean and standard deviation equal to 0.0 and 0.25, respectively. The shuffling of the I_TBB time series is enabled by the following: (i) A sequence of uniformly distributed random variables, with length equal to that of the I_TBB time series, is generated. (ii) The random time series is sorted according to the magnitude of its elements. (iii) Suppose the ith element of the random sequence ranks the jth largest (or smallest), then the ith element of the I_TBB time series becomes the jth element of the shuffled time series.

Figures 9a–d show the results of daily and monthly nonoverlapping window averages for all four time series. Standard deviations are plotted along with the monthly averages. Evidently, the window averages become well-defined much sooner in the white noise process and shuffled I_TBB in Figs. 9c,d than the time series in Figs. 9a,b. In Figs. 9a,b, both monthly averages and standard deviations exhibit considerable variations from window to window. Figure 9a might be expected given that the slowest cycle resolved in the 120-day I_TBB time series is the 60-day signal of the MJO, so naturally the monthly mean values should vary. However, it is interesting but subtle that the I_TBB time series, as well as the random multiplicative cascade process, has a property called long-range dependence. This can be observed physically as a spectrum of smaller-scale convection-coupled waves and disturbances being enhanced at the convective phase of MJO, as implied by the large standard deviation in December in Fig. 9a.

a. Long-range dependence in the variance–time relation

In this section, we introduce an interesting property called long-range dependence in the I_TBB dataset. A random multiplicative cascade multifractal has long-range dependence. Such dependence is conventionally characterized by the strength of a so-called Hurst exponent (e.g., Mandelbrot and Van Ness 1968; Mandelbrot 1982). Moving averages are far less effective in removing irregularities from such time series or spatial fields than from white noise processes. Gao and Rubin (2001) found that the Hurst exponent is closely related to the second moment of the pdf for the multiplier [P(r)], as implied by the last line in Eq. (3).

The property of long-range dependence can be quantified independently from the partition function technique used in sections 3 and 4. Here, we start by further formalizing the phenomenon exhibited by Fig. 9. Given a time series, X = {X_i : i = 0, 1, 2, …}, for each m = 1, 2, 3, … , let X^(m) = {X^(m)_i : i = 1, 2, 3, …} denote the new time series obtained by averaging the original series X over nonoverlapping blocks of size m, that is,

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For each time series X^(m), we can compute its variance. For a white noise process, var[X^(m)] ∼ m⁻¹. For a random process with an exponentially decaying autocorrelation function, the relation var[X^(m)] ∼ m⁻¹ still holds when m is not too small: var[X^(m)] ∼ m⁻¹. This feature is readily verified with the presently quite short time series, as shown in the dashed line in Fig. 10. On the other hand, var[X^(m)] for all seven I_TBB time series decay with m much more slowly, as shown by the solid lines in Fig. 10. We find that for the I_TBB, var[X^(m)] ∼ m^2H−2, with H = ∼0.75–0.9. Following the definition in Mandelbrot and Van Ness (1968), the parameter H is the Hurst exponent. Mathematically, the autocorrelation function for these processes decays with a power law: r(k) ∼ k^2H−2, for large lag k [see derivation in Mallat (1999), section 6.4.3]. Note that when H > 1/2, the summation of r(k) goes to infinity, Σ^∞_i=1 r(k) → ∞. This quantifies the terms “long memory” or “long-range dependence.”

To better appreciate that var[X^(m)] ∼ m⁻¹ is alone true for short-range-dependent (in contrast to long-range dependent) red noise processes, we have simulated an autoregressive (AR) model defined by

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where A is the mean for the process A_n, constant a is an autoregressive coefficient, and η_n is a Gaussian random process with zero mean. One can readily show that the autocorrelation function for such a process decays exponentially, r(k) = a^k = e^klna. Autoregressive models like this have been used to model red noise process, for example, in convective heating by Lin and Neelin (2002).

Figure 11a plots a time series with length 2880 generated from such a process with a = 0.97, which is the lag-1 autocorrelation coefficient of the 5° × 5° I_TBB time series. Negative values in the time series can be truncated to further approximate the I_TBB time series. One may justify such truncation by picturing that the red noise represents vertical velocity, while only upward motion produces rainfall, which is related to the I_TBB. The time series in Fig. 11a is significantly different from the white noise and much more like the actual I_TBB time series, although it is still an inferior model to the multiplicative process in term of its ability to capture intermittency.

Figure 11b shows variance–time plots of the red noise processes from the AR model with a = 0 (white noise), 0.1, 0.5, 0.7, 0.9, 0.97, and 0.97 with only positive values, in addition to that of the 5° × 5° I_TBB time series. The truncation of negative values for the a = 0.97 process results in little difference in this figure. We observe from the figure that when a is much smaller than 1, the H value is very close to 1/2 for the entire range of m. When a is close to 1, however, H assumes a value of 1 when m is small. Thus, H = 1 implies a perfect correlation. This can be understood by noticing that when a is very close to 1, the value of A_n is somewhat directly passed to the value of A_n+1, resulting in a nice memory. Now it is clear that the region of m with H = 1 increases with the increasing of the coefficient a. However, even when a is as large as 0.97, the process still soon assumes an H value of 1/2 when m is not too small. These observations can be rigorously proven, for example, in Kaulakys and Meskauskas (1998). It is interesting to note that such AR processes with the coefficient a very close to 1 as well as other more general autoregressive and moving-average (ARMA) models (Box and Jenkins 1976) are sometimes used to model a random process with certain memory (Kaulakys and Meskauskas 1998). However, such processes are not very effective in describing processes with very long memory, since their autocorrelation functions decay exponentially.

The AR model is not entirely without merits. As indicated in Figs. 10 and 11b, the rate of decreasing variance versus increasing time interval m of the 5° × 5° I_TBB within around 1 day is better captured by the AR model with a = 0.9 than the multiplicative cascade process. It is possible that for modeling the second-order statistics of convective activities at time scales within ∼1 day, an AR model like the one shown in this paper (i.e., a = 0.9) may be sufficient. We note however, that a = 0.9 is smaller than the actual lag-1 correlation of the I_TBB time series, which is 0.97. Also note that its ability to produce any longer-range dependence as well as intermittency is mathematically limited. The multiplicative cascade process, however, has a much greater potential to be tuned into a desired fit to the real data in terms of not only the variance–time property but also the multifractal dimensions.

b. Properties associated with long-range dependence

Mandelbrot and Van Ness (1968) coined the term “fractional Brownian motions” (fBms) for an extended family of statistically self-similar Gaussian functions characterized by the Hurst exponent, H, 0 < H < 1. For ordinary Brownian motion (Bm, a nonstationary process), H = 1/2. White Gaussian noise is the derivative of the Bm; similarly, the fractional Gaussian noise (fGn) is the derivative of the fBms. The fBm family is often called the random-walk-type process, while the fGn is called the increment process. Both provide useful models for natural phenomena containing both macro- and microscales. The fGn is a much simplified prototype to address the power-law decaying autocorrelation function and power spectrum of the I_TBB as well as X time series in this study.

Furthermore, when the variance–time relation Var[X^(m)] ∼ m^2H−2 holds, through the Weiner–Khinchin theorem (Monin and Yaglom 1975), the power spectral density (PSD) for the time series X decays in a power-law manner: PSD_X(f) ∼ 1/(f^2H−1), with f the frequency. The PSD for the time series Y = {Y_n : n = 0, 1, 2, …}, with Y_n = Σⁿ_i=1 X_i an integration of X, can be proven to decay as PSD_Y(f) ∼ 1/(f^2H+1). These power-law-decaying PSDs justify the term of 1/f stochastic processes (e.g., Yano et al. 2001). It is noted that when H = 1/2, then the PSD for X (the white Gaussian noise) is flat, while that for Y (the Bm) is 1/f². The X time series is also often denoted as an increment process, while the Y time series as a random-walk-type process.

Suppose the mean value of the increment process X = {X_i : i = 0, 1, 2, …} is μ(x). If 0 ≤ H < 1/2, then when X_j > μ(x), it will be more likely to have X_j+1 < μ(x). If one takes X_j − μ(x) as the jump at time j, a jump-up is more likely followed by a jump-down when 0 ≤ H < 1/2. This is called the antipersistent correlation. The integration of X leads to a random-walk-type process (fBm) that is less nonstationary than the Bm. Conversely, if 1/2 < H < 1, then the times series X has the persistent correlation, or long-range dependence: a jump-up is more likely followed by another jump-up at the next time step, so is a jump-down followed by another jump-down. The integration X generates a random-walk-type process that is more nonstationary than the Bm.

Since 1/2 < H ≤ 1 for all seven I_TBB time series, we conclude that the I_TBB datasets possess the long-range dependence properties. As previously mentioned, during the IOP, smaller-scale convection-coupled waves and disturbances are enhanced during the convectively active phase of MJO. In observations, convective incidents associated with these waves/disturbances also gain intensity during this phase of the MJO until the full bloom of a westerly wind burst (WWB) that marks the end of this phase (e.g., Yanai et al. 2000). Interestingly, the long-range dependence in the 5° × 5° I_TBB data is well captured by the random multiplicative cascade model, as shown by the dash–dot–dot line in Fig. 10. Now it should be clear why temporal smoothing of the I_TBB time series cannot yield really smooth results. Moreover, the long-range dependence is more significant with a larger-domain average, according to the slopes of solid lines in Fig. 10, while the values of D₂ in Fig. 7 provide the same information. This may be due to the fact that larger domains encompass larger-scale variability that displays longer-range dependence.

In Fig. 10, the shuffled 5° × 5° I_TBB time series yields a similar slope to the white noise process. Evidently, the shuffling has destroyed the long-range dependence in the original time series. It is noted that the shuffling destroys all kinds of correlation in the original time series but preserves the original pdf, thus information of pdf is not sufficient to realize a fractal or multifractal process.

The pdf of I_TBB time series is expected to be non-Gaussian with a thick-tailed distribution, as indicated by the importance of higher-order partition functions. That means, extreme events of convective activity contribute significantly to the variability of the time series. However, given the currently short time series, it is difficult to quantify the pdf of I_TBB by computation, since those extreme events are considered too rare. It is important, however, to note that the non-Gaussian nature of I_TBB does not necessarily contribute to the multifractal variation we observe. A supporting example is that the pdf of Levy flight (Mandelbrot 1982) has a power-law tail, which is non-Gaussian; the process is nevertheless monofractal. As seen in Fig. 10, the shuffled 5° × 5° I_TBB time series yields a similar slope to the white Gaussian noise. Apparently, the shuffling has destroyed the long-range dependence in the original time series. In fact, the shuffling destroys all kinds of correlation in the original time series but preserves the original pdf, thus information of pdf is not sufficient to realize a fractal or multifractal process.