Figure 1 shows diurnal and seasonal dependencies of the minimum slant range (1a), corresponding elevation angle (1b), and true reflection height (1c) for beam #0 of the Hokkaido East radar (azimuth of 5.7° clockwise from the north direction). The dependencies correspond to an operating frequency of 11 MHz and are shown for 2 years of low and high solar activities, which are 2010 and 2014, respectively. Echoes propagating in different HF channels (E, F1, and F2) were separated by checking the true reflection height. Figure 1d, e presents dependencies of the critical frequency and maximum height for the electron density profile corresponding to the coordinates of the ionospheric reflection point. The empty cells have no GB, as shown in Fig. 1. In the E and F1 channels, there is no GB in the nighttime for an entire year, in the daytime in winter, and partially during equinox. In the F2 channel, there is no GB during nighttime in winter. This phenomenon can be explained by decreases in the critical frequency below some level when the HF wave completely passes through the ionosphere without reflection (at any given elevation angle). With increases in solar activity, the local time interval of GB “absence” decreased in the F2 channel, increased in the F1 channel, and remained almost the same in the E channel.

### Minimum slant range and skip distance

The diurnal minimum of the slant range in the E channel was seen near noon (see left column of Fig. 1a). In the morning and evening, reflection from the layer occurred farther from the radar due to the lower critical frequency (see left column of Fig. 1d) and the corresponding slant range increased accordingly. The minimum slant range in the F2 channel showed a similar behavior in winter and equinox (see right column of Fig. 1a). However, for the F2 layer, decreases in the critical frequency were often accompanied by increases in the maximum height (and vice versa). Therefore, the slant range varied more rapidly in the F2 channel than in the E channel. There were two local maximums in the diurnal dependence of slant range in summer. In addition to the nighttime maximum, there was a maximum near noon. The latter became most prominent under low solar activity. During the minimum of solar activity, the critical frequency of summer daytime F2 layer decreased to the values corresponding to the E and F1 layers. This can lead to a partial or total screening of the F2 channel. In Fig. 1a, the F1 layer screens the F2 channel at 13 LT in June 2010, and the E layer screens both the F1 and F2 channels at 12 LT in May 2010.

A post-noon maximum is a feature of the minimum slant range in the F1 channel. The maximum became prominent under low solar activity. In Fig. 1e (middle column), the drastic increase in slant range is related with the significant increase in the F1 maximum height (up to the values of the F2 layer maximum height).

In general, the minimum slant range noticeably decreased with increases in solar activity. Under high solar activity, the critical frequencies became higher, the reflection from the layer occurred closer to the radar, and the corresponding slant range decreased. In case of the F2 channel, the decrease in slant range was partially compensated by a general lifting of the F2 layer (increases in the maximum height).

The behavior of the minimum slant range was repeated in detail in the seasonal and diurnal dependencies of the skip distance. The difference between the two characteristics varied in the following ranges: ~20–50 km in the E channel, ~80–150 km in the F1 channel, and ~100–200 km in the F2 channel.

### Elevation angle

In Fig. 1b, the diurnal and seasonal dependence of the elevation angle repeated the corresponding critical frequency behavior (Fig. 1d); the higher the critical frequency, the higher the elevation angle. This phenomenon directly follows Snell’s law (Davies 1990). In case of a flat ionosphere, a relation between the oblique and vertical sounding frequencies, *f*
_{oblique} and *f*
_{vert}, when the waves are reflected at the same true height, is given by *f*
_{oblique} = *f*
_{vert}/cos *θ*
_{0}. Here, *θ*
_{0} is the incidence angle measured from a normal to the ionosphere layer. The above relation can also be rewritten as

$$ {f}_{\mathrm{oblique}}={f}_{\mathrm{vert}}/ \sin \varDelta $$

(3)

Here, *Δ* is the elevation angle. Elevation measurements could be used for estimation of the plasma frequency in the reflection point (Hughes et al. 2002; Bland et al. 2014). Equation (3) is applicable for distances up to about 500 km. For longer distances, we should take into account the roundness of the Earth. A simple equation can be obtained for this case from the generalized Snell’s law, but it includes true reflection height as an additional parameter.

### True reflection height

Diurnal and seasonal dependence of the true reflection height in the F1 and F2 channels was determined mostly by the behavior of a corresponding maximum height (see middle and right columns of Fig. 1c, e). The true reflection height was lower than the corresponding maximum height by approximately 40 km. Since the IRI maximum height of the E layer was fixed at specific value of 110 km, variations in the true reflection height were caused only by the critical frequency variation (see left column of Fig. 1c–e). Deviations between the true reflection height and the maximum height of the E channel varied within a small range of ~6–10 km. The true reflection height in the F1 and F2 channels increased with increases in solar activity (lifting of the layers). On the contrary, the true reflection height in the E channel decreased with increases in solar activity.

### Application of the calculated characteristics

One of the important tasks, in monitoring the ionosphere using SuperDARN radars, is the mapping of the reflecting/scattering regions. In practical terms, while mapping, we should start by determining the effective reflection height. In a spherical ionosphere, assuming that the Breit and Tuve’s theorem holds (Davies, 1990), we determined the effective height, *h*
_{eff}:

$$ {h}_{\mathrm{eff}}={\left({a}^2+{r_{\mathrm{ref}}}^2+2a{r}_{\mathrm{ref}} \sin \varDelta \right)}^{1/2}-a $$

(4)

where *a* is the radius of the Earth and *r*
_{ref} is the slant range to the reflection point. In case of the absence of the elevation measurement, the effective height is usually assumed to be fixed at a specific value (Bristow et al. 1994; Grocott et al. 2013; Oinats et al. 2015), or some empirical height models could be used (Chisham et al. 2008; Yeoman et al. 2008). Figure 2a shows the dependence of the effective reflection height calculated using Eq. (4). It varied in a wide range from 200 to 500 km in the F1 and F2 channels and from 115 to 130 km in the E channel. The intervals did not overlap and therefore, if we could calculate the effective reflection height properly, we would be able to separate the E echoes from the F (F2 and F1) echoes at least.

Application of the fixed effective height leads to a systematic error in mapping. Here, we could estimate it using the following procedure. First, for some fixed *h*
_{eff} and certain *r*
_{ref} from the simulation, we determined an effective elevation angle from Eq. (4). Further, using the effective angle and *r*
_{ref}, we calculated the ground distance to the reflection point, *D*
_{ref}:

$$ {D}_{\mathrm{ref}}=a\cdot \mathrm{arctg}\left(\frac{r_{\mathrm{ref}} \cos \varDelta }{a+{r}_{\mathrm{ref}} \sin \varDelta}\right) $$

(5)

Figure 3a shows the distance offset, which is the difference between the two distances: the first one was calculated by Eq. (5) and the second one was the actual distance from the simulation. The effective height was equal to 115, 260, and 350 km in the E, F1, and F2 channels, respectively. The offset in the F2 channel varied in a wide range from −250 km in equinox under high solar activity (underestimation) to 100 km in summer under low solar activity (overestimation). The offset in the E and F1 channels varied from a few to tens of kilometers.

Using elevation angles, we could significantly minimize the error. However, there was a shortcoming in that HF radar that cannot measure the slant range to the reflection point directly (in case of GB). In practical terms, one often assumes that the reflection point is located in the path middle point, and the slant range to the reflection point is equal to half of the full GB slant range (Bristow et al. 1994; He et al. 2004). This is correct for the spherical ionosphere. However, in general cases, the reflection point can significantly move out from the middle point due to ionospheric gradient along the propagation path. Figure 2b shows a ratio between the full slant range and the slant range to the reflection point. The ratio is usually greater than two. Since beam #0 is oriented towards the North, it usually has negative electron density gradients. The greatest gradients would appear at local times when the solar terminator passes through the propagation path. However, as shown in Fig. 2b (right column), the ratio reaches values of ~2.3 even during daytime under low solar activity (the F1 layer was lifted to the heights of the F2 layer during this period). Figure 3b illustrates a systematic error due to regular gradients. The offset is shown for the distance calculated by Eq. (5) using the elevation angle and *r*
_{ref} = *r*
_{full}/2, where *r*
_{full} is the full GB slant range from the simulation (see Fig. 1a). The offset varied on average from 0 to 100 km in the F2 channel (overestimation). It reached values of ±30 km and of ±60 km in the E and F1 channels, respectively. To reduce the latter error while mapping, the use of the simulated slant range ratio might be effective (Fig. 2b).

Another important task is the estimation of the critical frequency of the ionosphere. Using the generalized Snell’s law in the spherical ionosphere (Davis 1990), the plasma frequency, *f*
_{
p
}, can be determined in the reflection point by

$$ {f}_p={f}_{\mathrm{oblique}}{\left(1-{\left(\frac{ \cos \varDelta }{1+{h}_{\mathrm{true}}/a}\right)}^2\right)}^{1/2} $$

(6)

Here, *h*
_{true} is the true reflection height. Figure 4a shows the ratio (red scatter plot) between the frequency calculated from Eq. (6) and the actual plasma frequency in the simulation (from IRI-2012) for the F2 channel and entire period from 2007 to 2014. The distance from the radar location to the reflection point is shown on the horizontal axis. The dispersion of the scatter plot increased with increases in distance. However, it did not exceed 10 % (from 0.9 to 1.1) for distances up to 2000 km. The blue dots showed a similar frequency ratio except for the first frequency calculated by Eq. (3). It is clear that Eq. (3) significantly underestimates the plasma frequency in the reflection point starting from about 500 km.

In practical terms, application of Eq. (6) is difficult because the true reflection height is unknown. To overcome this shortcoming, the effective height, Eq. (4), can be used instead of the true height. In this case, Eq. (6) can be rewritten as

$$ {f}_p\approx {f}_{\mathrm{oblique}} \sin \left(\varDelta +\frac{D_{\mathrm{ref}}}{a}\right) $$

(7)

A scatter plot for the ratio calculated using Eq. (7) is shown in Fig. 4a using green dots. Equation (7) on average gave better correspondence than those of Eq. (3) for distances longer than 500 km. The colored lines show the linear regression of the corresponding scatter plots. The green line did not exceed the accuracy of 10 % (~1.1 on figure) up to ~1000 km, but the blue line had only 30 % (~0.7 on figure) at that distance.

It is interesting to refer to the branches of the dots with gradients that are different from the average gradient shown by the lines in Fig. 4a (shallower gradient for the blue branch and steeper gradient for the green dots). These dots correspond the abovementioned period of the post-noon “lifting” of the F1 layer. During this period, the ionosphere gradients make some “compensation” for the Earth curvature. That is why the accuracy for these two branches is comparable in absolute values (or the accuracy of the blue branch appeared to be even better than of the green branch for distances longer than 1000 km). Such a lifting is a feature of low solar activity in summer post-noon. The number of dots in the branches did not exceed 4 % of the entire number of dots in the scatter plot; therefore, its influence was negligible.

The main contribution to a systematic error in Eq. (7) was the ionosphere gradients along the propagation path. Figure 4b shows the relationship between the frequency ratio (calculated using Eq. (7)) and the slant range ratio (Fig. 2b). The green dots correspond to distances up to 1200 km, and the black dots correspond to longer distances. The black line is a linear regression calculated for the fraction of green dots. It is clear that the regression coefficient is very close to the value of 2. Thus, we can clarify Eq. (7) by

$$ {f}_p\approx 2\frac{r_{\mathrm{ref}}}{r_{\mathrm{full}}}{f}_{\mathrm{oblique}}{\left(1-{\left(\frac{ \cos \varDelta }{1+{h}_{\mathrm{eff}}/a}\right)}^2\right)}^{1/2}=2\frac{r_{\mathrm{ref}}}{r_{\mathrm{full}}}{f}_{\mathrm{oblique}} \sin \left(\varDelta +\frac{D_{\mathrm{ref}}}{a}\right) $$

(8)

Equation (8) reduces the systematic error of Eq. (7) by a factor of 2 for distances lower than 1200 km (gray dots in Fig. 4c).

As mentioned earlier, the true reflection height was lower than the maximum height of the ionosphere layer; therefore, the frequency defined by Eq. (5) did not correspond to the critical frequency of the layer. Green dots on Fig. 4c show the ratio between the frequency calculated by Eq. (8) and the actual critical frequency of the F2 layer (from IRI-2012). It is clear that the frequency defined by Eq. (8) is lower than the actual critical frequency by 10–15 %.