Phase structure functions for ionospheric radio sounding: Dependence on irregularity scale



[1] Because the ionospheric plasma drifts, radio sounding signals encounter different irregularities of electron density, even at only slightly different times, causing temporal phase variations; these may be characterized conveniently by the structure function. We have proposed in several recent publications a new irregularity diagnostic method based on this effect, and it is now useful to consider the range of irregularity scales affecting the observed phase fluctuations. We show, in particular, that for the two-dimensional irregularity power spectrum index 2 < ν < 3 (quite typical for the ionosphere) the phase structure function value at the smallest available lag (0.01 s) is sensitive to irregularities in a broad range of scales, between several meters and several kilometers.

1. Introduction

[2] In modern HF pulsed sounding of the ionosphere using research ionosondes, the phase of the radio echo is measured precisely and frequently; the ‘same echo’ can be identified from one time to the next. Thus, a continuous series of phases can be obtained and analyzed by statistical methods. Since an ionospheric layer is generally in motion (primarily in a horizontal direction), the echo ray path passes through different electron density irregularities at each time, producing random temporal phase variations. Note that such short-period phase variations are distinct from the Doppler imposed by drift or wave motions. One of the most convenient statistical instruments for characterization of short-period variations is the structure function, SF, defined as the square of the difference of a random function, averaged over all pairs of points a certain distance apart. For temporal phase differences at lag τ, this is Dϕ(τ) = 〈[equation image (t + τ) − equation image (t)]2〉 and for electron density irregularities, δN (equation image) = equation image (equation image) at spatial lag equation image, it is DN (equation image) = 〈[δN (equation image + equation image) − δN (equation image)]2〉. Among the convenient features of the structure function, and unlike the correlation function or variance, large-scale variations (much larger than ∣equation image∣) only weakly influence the structure function at scale lengths less than ∣equation image∣. Thus, this investigation of small-scale irregularities and short-period phase variations can ignore the behavior of the mid-scale irregularity spectrum (scales exceeding the first Fresnel zone), where good statistical characteristics are difficult to obtain in practice.

[3] A theoretical relation between structure functions of the phase variations and the small-scale ionospheric irregularities was obtained in Zabotin and Wright [2001] for a model of frozen irregularities moving with a drift velocity equation image, on the assumption that the irregularities are infinitely stretched along geomagnetic field lines and their two-dimensional (transversal) spectrum is described by a power law Φ(κ) ∝ κ−ν. From the technical viewpoint, this work represented development of earlier results [Gailit et al., 1983; Denisov and Yerukhimov, 1966]. In practice, it is found that the SF is log-log-linear in the small-lag regime, and is therefore defined by simply an intercept and slope; we call them structure indices SIA and SIB. Using the dynasonde ‘B’ ionogram mode (blocks of a few repeated frequency ramps), one can define a ‘rudimentary structure function’ (RSF), containing only a few initial lags, but these are sufficient to determine SIA and SIB. Thus standard dynasonde ionograms, themselves available in long systematic time-series, become practical sources for irregularity diagnostics [Zabotin and Wright, 2001, 2002a, 2002b] in addition to their many other uses.

[4] The ray path traverses many irregularities with various scale lengths, thus integrating their effects. We must not expect a direct relation between specific spatial (l) and temporal (τ) scales, such as l = Vτ. In this paper, we conduct a special investigation, based on analysis of theoretical relations from Zabotin and Wright [2001], to reveal the range of scales to which the new diagnostic method is most sensitive. In the next section, we reduce the theoretical relations to a form appropriate for further numerical analysis. We find that the phase structure function may be represented as an integral over the irregularity spectrum, and the integrand shows directly the relative contribution of different spatial harmonics. A subsequent section is devoted to detailed numerical investigation of the integrand, and our conclusions are presented in the final section.

2. Analytical Relationships

[5] For convenience, we first repeat the basic assumptions of our model as stated in Zabotin and Wright [2001]. We consider a layer of plasma with isotropic refractive index and small-scale random irregularities infinitely stretched along the inclined magnetic field. The isotropic refractive index provision neglects additional refraction effects caused by the geomagnetic field. We do not take into account explicitly the effects of multibeam (“glint”) interference. Formally, the expressions used to describe phase variations of the sounding signal are based on considerations of geometrical optics. However, it is known that diffraction influences phase variations to a smaller extent than amplitude. Similar expressions characterizing statistics of phase variations can be obtained from the method of smooth perturbations of the parabolic equation approximation [Rytov et al., 1989, section 2].

[6] In our model, the “frozen” irregularity field drifts horizontally with velocity equation image through the reflection region, so that the sounding signal passes through different irregularity realizations at different times. This is considered the primary reason for the short-period phase variations observed at a fixed point equation image on the ground.

[7] Let z0 be the reflection altitude; z = 0 is the altitude of the ionosphere base, k0 = ω/c, equation image (z) is the unperturbed part of the layer profile (X = ωp22; ωp(z) is the plasma frequency profile). Then the unperturbed phase accumulation for vertical propagation in the plasma layer is

equation image

[8] The factor 2 allows for upward and downward propagation through the layer. The corresponding phase accumulation in the irregular medium is

equation image

where equation image(equation image) is the irregular part of X = equation image(z) + equation image(equation image), connected with the electron density irregularity field by the relation equation image(equation image) = equation image(zN (equation image). Random phase variations due to irregularities at an arbitrary time t are given by the well-known relation which is obtained by linearization of the difference of expressions (2) and (1):

equation image

[9] Here, the radius-vector equation image is represented in a form where its vertical component z (and the integration variable in expression (3)) is explicitly discriminated from its horizontal component equation image. In the spirit of perturbation theory, the integration over an undisturbed ray path is implied in (3). At the time t + τ, due to drift motion of the irregularities, the random phase attains another value

equation image

[10] Using the definition of the structure function cited in the Introduction and generalizing calculations for the case when reflection occurs along a straight oblique ray path with polar angle θ0 and azimuth angle ϕ0, one obtains the following relation:

equation image

where k0 = ω/c, and f(x) = {(1 − x2/8) ln [(2 − x + 2 equation image)/x] − (equation image + equation imagex) equation image}.

[11] Details of the derivation can be found in Zabotin and Wright [2001]. For a linear layer of isotropic plasma we have X(z) ≡ ωp2(z)/ω2 = z/z0.

[12] For a statistically homogeneous field of irregularities, infinitely stretched along magnetic field lines determined by the unit vector equation image = (0, sin γ, cos γ), the structure function DN (equation image) depends only on the absolute value of the component of vector equation image transverse to vector equation image (i.e., on R). Thus, the three arguments of function DN(equation image) in equation (5) are

equation image
equation image
equation image

[13] These expressions describe reflection along a straight oblique ray path with polar angle θ0 and azimuth angle ϕ0. The components of the horizontal drift velocity are Vx = V cos ϕ and Vy = V sin ϕ.

[14] For irregularities infinitely stretched along magnetic field lines, and isotropic in the transverse plane, the three-dimensional spatial spectrum has the general form

equation image

where δ(κ) is the delta-function, and C is a normalizing constant. The irregularity structure function may then be expressed as:

equation image

where J0(z) is the Bessel function.

[15] Substituting equation (7) into equation (5) and introducing a new integration variable x = ξ/z0, we obtain the following expression for the phase SF:

equation image

[16] The order of integration in (8) is easily changed, showing that the relative contribution into Dϕ(τ) of irregularities belonging to the spectral interval κ: κ + dκ depends on the shape of the function

equation image

[17] According to (9), two factors influence the function Δ(κ). The first, Φ(κ), is determined only by the statistical properties of ionospheric irregularities. It shows directly how irregularities of different scale length are populated in the propagation medium. As chosen for the phase SF method itself [Zabotin and Wright, 2001], we adopt the following two-parameter model for the function Φ(κ):

equation image

[18] This form specifies two typical features of ionospheric irregularity spectra: a power-law behavior in the intermediate-scale band lL1 = 2π/κ1, approaching flatness in the mid-scale band lL1. The parameter L1 separating the two bands is of order ten kilometers (roughly the density gradient scale in the ionosphere) and the spectral index in the lower band lies within the limits 2 < ν < 4.

[19] The second important factor entering function Δ(κ),

equation image

is determined exclusively by the nature of the phase SF method; its shape may be found by numerical methods. In spite of its apparent simplicity (containing just one integration) calculations for the whole range of κ (1 < κz0 < 106) are not trivial. The main problem is that the integrand represents a quickly oscillating function for large values of κz0. We overcome this difficulty, using a combination of the Lobatto integration formula with seven nodal points and the Gaussian integration formula with 96 nodal points [Abramovitz and Stegun, 1964, section 25.4].

3. Results

[20] The main question of interest is the lower limit of irregularity scales (or upper limit of the spatial harmonics κ) that contribute to the phase SF. Note that if we had only the statistical factor κΦ(κ) in the expression for Δ(κ), there would be no influence of small scales on the phase statistics: For ν > 2 the integral ∫ κΦ(κ)dκ does not depend on its upper limit if it significantly exceeds κ1. Therefore, the factor I) is of critical significance for small-scale irregularity diagnostics. Only this factor is able to give additional weight to the effect of small scales. Our calculations confirm that this is indeed the case.

[21] The function I), represented in dimensionless form as Iz0), then contains another dimensionless parameter, Vτ/z0, and four angular variables (θ0, ϕ0, ϕ, and γ). Our calculations show the dependence on the angular variables to be relatively weak, leading to only insignificant qualitative changes, except when directions of the ray and of the magnetic field line nearly coincide (a peculiar case to be considered later). Thus, we first examine the general properties of the function Iz0) and its significant dependence on the parameter Vτ/z0 for a more common situation.

[22] For Figure 1, the parameters θ0, ϕ0, ϕ, and γ were chosen quite arbitrarily (θ0 = 12°, ϕ0 = −60°, ϕ = 45°, and γ = 28°). The function Iz0) is shown for four values of Vτ/z0 (0.00002, 0.0002, 0.002, and 0.02). For clarity, the curves are marked with the corresponding value of the temporal lag τ for V = 0.1 km s−1 and z0 = 50 km; these are typical for the ionosphere. Note that τ = 0.01 s is the smallest temporal lag (just the inter-pulse interval) adopted in contemporary dynasonde data. We denote by (κz0)max the position of the main maximum in each case. The corresponding scale length is L2 = 2πz0/(κz0)max. In all of the curves of Figure 1, there are then three characteristic intervals of the parameter κz0: (a) for κz0 < 10 we have I ∝ (κz0)2; (b) for 10 < κz0 < (κz0)max, we have I ∝ (κz0)1.1±0.1 with exponent usually varying from ∼1.2 in the beginning of this interval to ∼1.0 at its end; and (c) for (κz0)max < κz0, the function Iz0) begins to decrease according to the law I ∝ (κz0)−1, with slight oscillations superimposed.

Figure 1.

Dependence of the “weighting factor” I) on the dimensionless spatial spectral harmonic κz0, for four values of the parameter Vτ/z0 (0.00002, 0.0002, 0.002, and 0.02). The curves are marked by the value of temporal lag τ (in seconds) corresponding to V = 0.1 km s−1 and z0 = 50 km.

[23] The approximate functional dependences were obtained by power-law fits to the calculation results. Numerical calculations show that the position of the first inflection (κz0 ≈ 10) depends weakly on ϕ0, ϕ, and γ. Its dependence on the ray path polar angle θ0 is a little stronger but still not very significant: when θ0 changes from 0° to 60° the inflection shifts by approximately one order of magnitude for smaller κz0, or larger scale lengths. For a typical z0 = 50 km, the equality κz0 ≈ 10 corresponds to the irregularity scale length ∼30 km. Since our main purpose is investigation of properties of the function Δ(κ) in the region of small scales, we shall neglect the change of this boundary, and the difference between the corresponding scale length and the scale L1 introduced earlier to characterize the inflection in the spectrum. The existence of a transitional interval caused by a mismatch of the two scales cannot influence our conclusions about the role of small-scale irregularities.

[24] Gathering all factors constituting the function Δ(κ), one may draw the following conclusions:

[25] 1. In the interval of small κz0 < 10), Δ(κ) varies as κ3. The lower limit of the integral ∫Δ(κ) dκ, if it were located within this interval, would not make a noticeable contribution. This means that, independently of the exact shape of the irregularity spectrum Φ (κ) in the interval κ ≪ κ1, irregularities with scales larger than L1 cannot influence the phase structure function at small τ.

[26] 2. In the interval of large κz0 > (κz0)max), Δ(κ) varies as κ−ν; the function always decreases quickly enough (ν > 2), that the upper limit of the integral ∫Δ(κ) dκ cannot be important. Thus, irregularities with scales less than L2 cannot contribute to the phase SF for even small lags.

[27] 3. In the interval of intermediate κ (10 < κz0 < (κz0)max), Δ(κ) varies as κ2−ν and the function decreases. However, if ν < 3, the integral ∫ Δ(κ) dκ is determined by its upper limit, provided that this lies in the interval of intermediate κ. Evidently, in this case, the phase SF values are sensitive to the whole range of irregularity scale lengths from L2 to L1. Fortunately, this case is the most important in practice, since the great majority of experimentally measured values of the irregularity spectrum index lie within the interval 2 < ν < 3 [Zabotin and Wright, 2001, 2002a, 2002b]. If ν > 3, the integral ∫Δ(κ) dκ does not depend on its upper limit, even if the latter belongs to the interval of intermediate κ; the phase SF then pertains primarily to scales close to L1. In fact, small-scale irregularity amplitudes may not correspond to the power law Φ(κ) ∝ κ−ν.

[28] It is seen from these considerations that the position of the main maximum (κz0)max of the function Iz0) is of great importance. It determines the lower boundary of the scale interval to which the phase SF method is most sensitive. In Figure 2 the dependence of (κz0)max on the parameter Vτ/z0 is shown, as it results from our numerical calculations. The two quantities are inversely proportional to a very good approximation ((κz0)max ≈ 3.25 (Vτ/z0)−1), which justifies the following deceitfully simple estimate for the irregularity scale corresponding to the main maximum position:

equation image
Figure 2.

Dependence of the main maximum position (κz0)max of the function Iz0) on the parameter Vτ/z0, which results from numerical calculations.

[29] That the parameter Vτ appears in this estimate is not surprising: in this problem, there are no other length quantities pertaining to the small-scale range. But the numerical coefficient (≈2) is not trivial and cannot be obtained from qualitative considerations.

[30] The expression (12) permits quantitative estimation of the minimum irregularity scale detectable by the phase SF method in various situations. If, for example, we take the dynasonde inter-pulse interval, usually 0.01 s, as the minimum temporal lag τ, and a quite typical value of the drift speed V = 100 m s−1, we obtain L2 ≈ 2 m. For τ = 1 s, L2 ≈ 200 m. Sometimes, use of the smallest available lag in the SF (or RSF) may not be desirable, if the structure value is comparable to the prevailing noise level, whereupon the next smallest lag (the B-mode ramp interval, about an order of magnitude larger), must be substituted into (12) as τ. In any case, these estimates show that very small ionospheric irregularities, comparable to dissipative scales, are accessible to diagnostics by the phase structure function method, provided of course that they are not too weakened within the spectrum itself (by the condition ν < 3). This conclusion suggests the value of comparisons with other methods, such as the CUPRI 50 MHz radar which is sensitive to 3 m irregularities.

[31] We need to consider separately one peculiar case when the direction of propagation is very close to the direction of the magnetic field (ϕ0 = 90° and θ0 = γ), with results illustrated in Figure 3. When these directions differ by less than several degrees (a relatively narrow angular area), the behavior of the function Iz0) changes for intermediate and small scales. For κz0 < (κz0)max, we see that I ∝ (κz0)2, while for (κz0)max < κz0, the function Iz0) is approximately constant with slight oscillations superimposed. The position of the inflection (κz0)max does not depend on the angles and, therefore, remains as before.

Figure 3.

Illustration for the special case when direction of propagation of the sounding signal is very close to the direction of the magnetic field. Parameters not shown in the figure are τ = 0.1 s, γ = 29°, ϕ0 = 90°, ϕ = 45°.

[32] What are the consequences for the relative contribution of different scales? Nothing is changed qualitatively for small scales, κz0 > (κz0)max: a general decrease of the function Δ(κ) ∝ κ1−ν, and the condition ν > 2, assure a negligible role for the irregularities with scales lL2. For intermediate scales L2 < l < L1 we have Δ(κ) ∝ κ3−ν, and the situation depends on the spectral index ν. When ν < 3, the function Δ(κ) is increasing, and the only significant contribution to the phase SF comes from irregularities of scales close to L2. When 3 < ν < 4, the function Δ(κ) is decreasing; however, the essential contribution to the integral ∫ Δ(κ) dκ still comes from its upper limit. In this case the whole scale interval L2 < l < L1 is effective in the resulting SF. To some extent, this situation is complementary to the more common case considered above. In principle, there is an experimental possibility to select field-aligned reflections (when available) to verify information about the smallest-scale irregularities observed from other directions in the case 3 < ν < 4, since angles of arrival are measured by the dynasonde.

4. Conclusions

[33] The general idea of using the phase structure function method for irregularity diagnostics was discussed in detail in a previous publication [Zabotin and Wright, 2001]. The present paper examines the actual scale length ranges accessible by this method. Our analyses carried confirm the sensitivity of the phase structure function to ionospheric irregularities in a broad range of scales, from several meters to several kilometers, with a more specific estimate of the minimal scale given by equation (12), provided that the small-scale irregularities are sufficiently represented in the ionosphere itself. This is the case if the index ν of their spatial power spectrum, Φ (κ) ∝ κ−ν, is less than three. In the opposite case, ν > 3, such small scales may still be observable by selection of echoes with ray paths close to the magnetic field, an experimental option made practical by the dynasonde's ability to measure echo angles of arrival.


[34] This work is supported by the National Science Foundation grant ATM-0125297. We are grateful to G. A. Zhbankov (Rostov State University) for assistance in carrying out numerical calculations.