1 Introduction

Seismic reality evidences many contradictions to the model assumption of a stationary Poisson point process. Eventually, a response to these leads to complications of the existing hypotheses by introducing sequences of main events and their associates (fore- and after-shocks) superimposed with hypothetical distributions of the associate size, time, and location. Therefore, among other fundamental shortcomings related with inadequate characterization of ground shaking, estimation of the annual rate of magnitude M earthquakes, N(M), at a given site of interest remains the basic source of erroneous seismic hazard assessment (Kossobokov and Nekrasova 2012; Panza et al. 2014; Nekrasova et al. 2014; Kossobokov et al. 2015), contributing to disastrous seismic engineering decisions (Davis et al. 2012).

The results of the global and regional analyses (Kossobokov and Mazhkenov 1994; Nekrasova and Kossobokov 2002, 2005, 2006, 2016; Kossobokov 2005; Nekrasova 2008; Nekrasova et al. 2011, 2015; Parvez et al. 2014) imply that the occurrence of earthquakes at a seismically prone site x, for a wide range of magnitudes M ∈ (M−, M+) and sizes L ∈ (L−, L+), can be described by the following formula:

$$ N\left(M,L\right)={10}^A\times {10}^{B\times \left(5-M\right)}\times {L}^C $$
(1)

where L × L is a square embedding seismic locus and A, B, and C are constants. An alternative formulation using the inter-event time instead of rate of occurrence has been suggested by Bak et al. (2002) as Unified Scaling Law for Earthquakes (USLE). The name introduced in (Bak et al. 2002; Christensen et al. 2002) applies to the generalization of Gutenberg-Richter relationship by Kossobokov and Mazhkenov (1994) given in the following form:

$$ {\log}_{10}N\left(M,L\right)=A+B\times \left(5-M\right)+C\times {\log}_{10}L $$
(2)

where N(M, L) is the expected annual number of earthquakes of a certain magnitude M within a seismically prone area of diameter L, A, and B characterize the annual rate of magnitude 5 events and the balance between magnitude ranges, correspondingly (analogous to a- and b-values of Gutenberg-Richter relationship), and C estimates the fractal dimension of the epicenter support.

Note that (1) the traditional estimations of seismic hazard for cities and urban agglomerations are usually underestimated when rescaling earthquake rate of occurrence from a larger surrounding territory (Kossobokov and Mazhkenov 1994; Nekrasova et al. 2015) with non-uniform distribution of earthquake sources (Beauval et al. 2006) and (2) a discrepancy in evaluation of seismic hazard propagates non-linearly into erroneous estimations of risks (Wyss et al. 2012). In a pilot application to the territory of Greater Caucasus and Crimea, we follow the methodology described below.

2 Methodology

In assessment of seismic hazard, we use an alternative to traditional methodologies. It is based on USLE and requires reliable estimation of the scaling properties of the earthquake sequences at the territory of concern. We keep applying the same methodology of seismic risks estimation as the one thoroughly described in (Parvez et al. 2014; Panza et al. 2014; Nekrasova et al. 2015; Parvez et al., 2018).

A space-time-magnitude volume, S × T × M is considered, where S is the territory, T is time interval from T0 to T1, and M is the magnitude range above M0; the events with magnitude m ≥ M0 are reasonably complete in the catalogue since T0. The algorithm for Scaling Coefficients Estimation, named SCE (Nekrasova et al. 2015), applies to a catalogue of earthquakes as the initial input data source.

In assessment of seismic hazard based on USLE, the coefficients A, B, and C defined in application of the SCE algorithm at grid points are used to determine the expected magnitude of maximum credible earthquake (MCE) events. Specifically, at the scale of tens of kilometers, the 0.25° × 0.25° cells are set at the grid points predetermined by the nodes of morphostructural zoning (Gorshkov et al. 2003) and/or the empirical distribution of epicenters, or even some regular mash. For each 0.25° × 0.25° cell centered at a grid point, we calculate the expected numbers of events from magnitude ranges Mj in 50 years, i.e., N50(Mj, 0.25°) = 50 × N(Mj, 0.25°), then find the maximum magnitude (Mmax), with the expected number N50(Mmax, 0.25°) ≥ 10%. Such a determination of Mmax implies expectation of one MCE event in 500 years and complies with a traditional probabilistic seismic hazard analysis (PSHA) “10% chance of exceedance in 50 years” in a cell. At this step, we assess seismic hazard potential in the distributed sources of earthquake occurrence. Note that by applying an empirical estimate of mathematical expectation, we avoid the model assumption of a stationary Poisson point process in favor of a stochastic repeatability of seismic regime.

In the next step, we expand hazard assessment from the distributed earthquake sources to the entire territory of a study and convert the expected maximum magnitude Mmax to a characteristic of macro-seismic effect by applying the empirical formulae. For example, when considering the peak ground acceleration, PGA, we apply, at each point of a regular grid, the empirical formula for acceleration produced by a source of Mmax at the irregular grid as inspired from the scenario-based neo-deterministic seismic hazard assessment study by Parvez et al. (2003), i.e.,

$$ \mathrm{Acc}\left(M\max, D\right)= const\times g\times D-1.5\times \exp .\left(M\max -5\right) $$
(3)

where D is the source-receiver distance on a 0.25° × 0.25° grid, const = 6 × 4.8, g = 9.81 m/s2 is the gravity constant, and exp(X) is the natural exponent of X. The maximum of acceleration values computed at a grid point is assigned to it. Usually, we opt the minimum and maximum distances of 10 and 500 km, respectively.

Thus, one can use the long-term estimates of the USLE coefficients to characterize seismic hazard in a rather traditional terms of maximum expected magnitude and/or macro-seismic intensity, peak ground acceleration (PGA), etc. It should be noted that at a finer scale of seismic hazard determination, the USLE approach may help improving the neo-deterministic procedure (Panza et al. 2001, 2012) that supplies realistic time histories of strong ground motions, which allows to retrieve peak values for ground displacement, velocity, and design acceleration based on earthquake scenarios (e.g., Parvez et al. 2011; Paskaleva et al. 2011) and to address some issues largely neglected in traditional hazard analysis (i.e., local crustal properties, soils) by making use of all the available empirical data on the Earth’s structure.

In assessment of seismic risks, we may face many different risk estimates even if the same object of risk is considered. Specifically, it may result from the different laws of convolution, as well as from different kinds of vulnerability of an object of risk under specific environments and conditions. Both conceptual issues must be resolved in a multidisciplinary problem-oriented research performed by specialists in the fields of hazard, objects of risk, and object vulnerability. Following (Parvez et al. 2014), to illustrate this general concept, we perform the oversimplified convolution of seismic hazard assessment map H(x) with the population and its vulnerability R(x) = H(x) · ʃxP · f(P), where ʃxP is the integral of the population density over the cell x, i.e., the number of individuals within the area of the cell x, and f(P) is individual vulnerability as a function of density distribution P. In application, we differentiate individual vulnerability in proportion to some power of the population density at a given site, which appears to be admissible due to specifics of man-made environment inflicted in the areas of high concentration of individuals, e.g., the number of floors in typical residential building changes with population density. Risk estimates of this kind are given in Parvez et al. (2014).

3 Data

As the input of our pilot application to the territory of Greater Caucasus and Crimea, we used the following seismic and population data.

3.1 Seismic data

Seismicity of Greater Caucasus region is considered within 40°–46°N and 36°–51.5°E. The regional catalogue is compiled using the annual periodicals Earthquakes in Northeastern Asia (1998–2008) and Earthquakes in Russia (2004–2014). The regional catalogue is sufficiently complete in reporting earthquakes of energy class K = 10.5 or above (officially adopted in the Soviet Union), which range corresponds to magnitude M = 3.6 or larger (Fig. 1) and provides the sample of 4941 earthquakes (with depth above 70 km).

Fig. 1
figure 1

Cumulative frequency-magnitude plot: Greater Caucasus, 1998–2014. The range of incomplete data is shaded in gray

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Seismicity of Crimea Peninsular region is considered within 44°–46.5°N and 32°–37°E. Regretfully, the data from the recent abovementioned sources appear incomplete and, perhaps, irregular. Therefore, for the purposes of this study, we have used the Composite regional catalogue of Crimea compiled by Tatyana Rautian and Vitaly Ivanovich Khalturin using a series of yearbooks Earthquakes in USSR in 1962–1991 published by Nauka Publishers in 1964–1997 (https://earthquake.usgs.gov/data/russia_seismicity/download/Crimea.ZIP). The earthquake hypocenters’ data in 1962–1990 appears reasonably complete at the energy class K > 7, i.e., above magnitude M = 1.7 (Fig. 2), which sample size is 388 events.

Fig. 2
figure 2

Cumulative frequency-magnitude plot: Crimea, 1962–1990. The range of incomplete data is shaded in gray

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3.2 Population data

The population data are taken from Gridded Population of the World (2005) that is a raster data product compiled at the Center for International Earth Science Information Network (CIESIN), Columbia University and Centro Internacional de Agricultura Tropical (CIAT). This population density distribution maps of the two regions based on the GPWv3 estimate for 2015 are given in Fig. 3. In addition to the distributed model data for the territory, we use the available census data from Federal State Statistics Service (2016) to characterize seismic risks for the cities with population above 100,000 inhabitants in the regions of Greater Caucasus and Crimea. As demonstrated by Parvez et al. (2018), using actual census data on population of municipalities could be critical for realistic estimations of the expected risks.

Fig. 3
figure 3

The population density P (in ind/km2): The GPWv3 estimate for the year 2015

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3.3 Morphostructural nodes

The question where catastrophic earthquakes can happen is highly important for knowledgeable seismic hazard and risk assessment. A simple answer that large earthquakes can happen only in places where smaller magnitude quakes were registered in the past is not always correct as confirmed by seismological practice, which has many case histories of “surprises.” In our study, we expand the answer to the sites, which did not yet explicitly show up as earthquake-prone and consider the intersections of morphostructural lineaments of Crimea and Greater Caucasus (denoted as “nodes” below) as the suitable network of grid points for an USLE application. The principles of Morphostructural Zoning (MSZ), its formalization, and performance in recognition of the areas prone to large earthquakes in advance of their occurrence are described in Gorshkov et al. (2003) and Soloviev et al. (2014). The MSZ map of Greater Caucasus (Fig. 4) resulted from field investigations by E.Ia. Rantsman, A.I. Gorshkov, and M.P. Zhidkov in the period from 1979 to 1984 (Gorshkov et al. 1986) and had been tested by seismic evidence after the first publication on several occasions (Soloviev et al. 2014). Specifically, all 13 earthquakes of magnitude 5 or larger in Greater Caucasus, 1988–2014, occurred in less than 25 km from the M5.0-prone nodes; nine of them from the areas where such earthquakes were not registered before 1988. The MSZ map of Crimea (Fig. 5) is compiled recently by A.I. Gorshkov (Gorshkov and Soloviev 2016) based on the same principles and, presumably, with the same accuracy. Thus, in the following application of the USLE-based approach, we consider 107 and 38 intersections of the MSZ lineaments in Greater Caucasus and Crimea, respectively.

Fig. 4
figure 4

The MSZ map of Greater Caucasus (after Gorshkov et al., 2003)

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Fig. 5
figure 5

The MSZ map of the Crimea Peninsula (after Gorshkov and Soloviev 2016)

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4 Application of the SCE algorithm

In each of the two regions, the SCE algorithm was applied to the sample of earthquakes defined in Section 3.1 considered within the hierarchy of square boxes with linear size of 2°, 1°, 1/2°, and 1/4° centered at each of the 145 nodes of the MSZ maps.

4.1 Greater Caucasus

Reliable estimates of coefficients A, B, and C were obtained for each of the 102 out of 107 nodes of Greater Caucasus, for which the number of earthquakes recorded in 1998–2014 is sufficient for application of the SCE algorithm. Figure 6 shows the maps of the estimates of coefficients A, B, and C together with the squared sum of their standard errors σA, σB, and σC. The densities of their distributions are given in Fig. 7. The error of determination of the USLE coefficients at all 102 nodes does not exceed 0.05, which confirms rather high quality of the mapped values for the entire territory of Greater Caucasus.

Fig. 6
figure 6

The maps of the A, B, C estimates and of the squared sum of their errors at the MSZ nodes of Greater Caucasus

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Fig. 7
figure 7

The density distributions of the A, B, C estimates at the MSZ nodes of Greater Caucasus

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The distribution of the level of seismic activity (coefficient А) has four characteristic maxima, of which the two largest at A about 0 and 0.3 correspond to the expectation of about one and two earthquakes with М = 5.0 in a year. Most of the positive As are located in Dagestan. The mean values of А, in which the earthquake rate of occurrence is three times smaller, are spread in the central and eastern regions of Greater Caucasus. The other seismically active regions here are characterized by even lower level of recurrence of moderate earthquakes.

Most of the B values, which characterize the slope of the frequency-magnitude graph, vary from 0.7 to 1.4 with two pronounced peaks at 0.8 and 1.2 (a 2% peak at about 0.5 is related to the nodes in the south-eastern part of Greater Caucasus where the catalogue is apparently incomplete). The estimates of B below 0.6 to the west of 40°E appear to be outlier-related local incompleteness of the catalogue due to deficiency of seismographic stations, which network is gradually improving from two stations before 2003 to eight in 2013 (Table 1.5 in Gabsatarova et al. 2015).

The fractal dimension of distributed epicenters C varies from below 0.6 to 1.5 with the two clear peaks about 1.1 and 1.4. The first generally corresponds to the linear extension of dominant morphostructures of Greater Caucasus, and the second—to the most fractured areas in the south-east of it, where the values of C > 1.2, i.e., the seismically active regions of Dagestan, Chechnya, and Karachay-Cherkessia.

4.2 Crimea Peninsula

Reliable estimates of coefficients A, B, and C were obtained for each of the 38 nodes of Crimea, where earthquake catalogue, 1962–1990 appears sufficient for application of the SCE algorithm. Figures 8 and 9 for Crimea are analogous to Figs. 6 and 7. The error of determination of the A, B, and C coefficients at all 38 nodes does not exceed 0.02, which is indicative of a higher quality of the mapped values in Crimea than in Greater Caucasus and, apparently, related to much longer period covered by the catalogue, i.e., 28 and 17 years, respectively.

Fig. 8
figure 8

The maps of the A, B, C estimates and of the squared sum of their errors at the MSZ nodes of Crimea

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Fig. 9
figure 9

The density distributions of the A, B, C estimates at the MSZ nodes of Crimea

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The distribution of coefficient А ranges from − 2 to − 1.4, which corresponds to the expectation of one earthquake with М = 5.0 in 100 and 25 years, respectively, i.e., much lower rate of moderate earthquakes than in Greater Caucasus. The values of coefficient B vary in a narrow range from below 0.5 to 0.9 with a peak at 0.7. The values of coefficient C are mainly in range from 0.8 to 1.0 with a high peak at about 0.9, which reminds the classical 1D Cantor “dust” with P = 0.9 and df = log(P)/log(2) ≈ 0.85, a possible indication of a linear fault zone in decline of tectonic activity.

5 Hazard and risk determination

The USLE coefficients determined at the nodes of morphostructures were used to determine the distribution of the magnitude of the MCE events in the two regions (as described in step 6 of Section 2). As an exception, we used the pattern recognition results from Gvishiani et al. (1988); Gorshkov et al. (2003); and Soloviev et al. (2013) for the eight nodes with B below 0.6 to the west of 40°E. The resulting maps of Mmax given in Fig. 10 disclose significant difference of seismic hazard expectations in Greater Caucasus, where Mmax above 6.5 is widespread along the entire mountain range, and Crimea, where the exceedance of such a level appears to be rather unlikely. The seismic hazard, in terms of PGA with 10% of exceedance in 50 years, for the entire territories of the two regions is given in Fig. 11; according to computations based on the distribution of Mmax, the ground shaking in Greater Caucasus may exceed the level of 1g, while in Crimea, it is hardly possible, except for the south-eastern part of the peninsula.

Fig. 10
figure 10

The maps of the USLE-based maximal magnitude, Mmax, at the MSZ nodes of Greater Caucasus and Crimea

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Fig. 11
figure 11

The seismic hazard maps of Greater Caucasus and Crimea based on the USLE approach in terms of expected peak ground acceleration, PGA, with 10% chance of exceedance in 50 years

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As an illustration of the complexity of seismic risk assessment, we calculated (step 7, Section 2) and plotted the maps of Rii(x) = H(x) · ʃxP · P and Riv(x) = H(x) · ʃxP · P3 where individual vulnerability is proportional to the 1st and the 3rd power of population density, correspondingly (Fig. 12). For each of the two maps, the risk is given in arbitrary units proportional to the maximum value defined as 1000, so that the risk estimates provided in terms of a non-dimensional index, ranging from 0 to 1000. The color of a cell corresponds to the decimal logarithmic scale in range from 1000 to 0.01. Not a surprise that the maximum risk for population is concentrated at the urban agglomerations and drops dramatically when moving to the rural districts of the two regions.

Fig. 12
figure 12

The seismic risk maps of Greater Caucasus and Crimea based on the USLE approach in terms of Rii(x) = H(x) · ʃxP · P (top row) and Riv(x) = H(x) · ʃxP · P3 (bottom row)

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Table 1 summarizes the results of seismic hazard and risk estimates for the 18 cities with population above 100,000 inhabitants affected by earthquakes of Greater Caucasus and the five largest cities of Crimea. The top five values of Population, city area S, PGA, Rii, and Riv are highlighted disclosing the complex combinations of possibilities. For example, the city of Sevastopol and its agglomeration from the top five in population is among the lowest of the 23 cities in respect to the risk estimates due to the largest area and moderate seismic hazard; the capital of Chechen Republic, Grozny, although of the highest expected ground shaking, is on the 8th and 15th place in regard to Rii and Riv; the city of Novorossiysk gets the highest risk estimates due to the 3rd largest PGA and its area just below the median of S. The matrix of correlation coefficients (Table 2) between the city parameters with or without the Sevastopol agglomeration from Table 1 shows up quite expected dependence of the risk estimates on the number of citizens and seismic hazard, while their anti-correlation with the city area.

Table 1 Seismic hazard and risk at major cities of the two regions
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Table 2 Correlations of city parameters
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Let us emphasize that the hazard and risk estimates for the two regions under study are presented here for academic methodological purposes highlighting the general problem-oriented approach based on USLE. Evidently, our estimates do not use more adequate though complicated procedures of convolutions of seismic hazard, objects of risks, and their vulnerability. The estimations addressing more realistic and practical kinds of seismic risks should involve regional experts in earthquake engineering, social sciences, and economics.

6 Discussion and conclusions

The results of our pilot investigation of seismic hazard and risks of Greater Caucasus and Crimea disclose their natural mid-term and spatial variability (Nekrasova and Kossobokov 2016). The maps of seismic hazard in terms of PGA with 10% rate of exceedance in 50 years for the two adjacent regions are dramatically different (Figs. 10 and 11). Moreover, the accuracy of this parameter determination differs from place to place. We should emphasize once again that, despite an apparent similarity with the PSHA estimation, the proposed maps based on USLE do not involve assumptions about temporal probability of earthquake occurrence. This is one of the basic differences between PSHA and neo-deterministic USLE-based approach, in general, as well as essentially deterministic USLE-based realization presented in this study. The basic seismic hazard estimates determined by means of the neo-deterministic seismic hazard assessment (NDSHA) is substantially different from popular deterministic seismic hazard assessment (DSHA): the in-detail discussion of the NDSHA enhancement and other alternatives to probabilistic seismic hazard analysis (PSHA) can be found in Panza et al. (2011, 2014). In particular, the existing seismic history is short; getting, experimentally, reasonable confidence limits on an objective estimate of recurrence rate of an earthquake on a local, or even on a regional scale, requires a geologic span of time above 10,000 years, which is unreachable for instrumental, or even historical, seismology (see, e.g., Beauval et al. 2008). Therefore, seismic hazard estimates may not rely exclusively on earthquake instrumental data and require the neo-deterministic assessment based on a solid geological evidence and pattern recognition of earthquake-prone areas (Panza et al. 2001, 2011, 2012, 2014; Peresan and Panza 2012; Peresan et al. 2011). Nevertheless, just the 5 out of 344 moderate (M ≥ 5.0, 550 BC–present) earthquakes in Greater Caucasus and 7 out of 22 of such earthquakes in Crimea might indicate the sites where the seismic hazard is underestimated in our analysis presented in terms of Mmax at MSZ nodes (Fig. 10). All these “surprises” are listed in Table 3, and all of them are from the seismic history of the regions when determination of earthquake parameters is very uncertain (Kondorskaya et al. 1982); enough mentioning of the deadliest 1668 Shemakhi earthquake, which magnitude could be as low as 7.0 with a possibility of at least two alternative dates either on February 17, 1667 or January 04, 1668 (Utsu 2002). The reported magnitude determination error (above 0.5 attributed to 9 out of these 12 earthquakes in Kondorskaya et al. (1982)) permits limiting the scores of “surprises” to the two cases of 1668 Shemakhi and September 11, 1927 Yalta earthquakes and conclude quite acceptable the level of underestimation of the obtained seismic hazard maps of Greater Caucasus and Crimea.

Table 3 The earthquakes, 550 BC–2016, which magnitude M is larger than Mmax at the distance of ½° or less
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On the other hand, the hazard maps in Fig. 11 are much more cautious and site-specific than the one of the Global Seismic Hazard Assessment Program (Giardini et al. 1999; Giardini et al., 2003). In particular, the GSHAP PGA values are in contradiction to the historical evidence of 21 and 6 “surprises” of the GSHAP hazard maps for Greater Caucasus and Crimea, respectively. The scores to be compared with the only 6 “surprises” out of 366 historical trials of the hazard maps of the regions in Fig. 11. All this favors the neo-deterministic USLE-based approach to the problem of assessing seismic hazard and risks showing that its hazard maps perform more reliably than those of PSHA.

Our study attempts to contribute an urgent revision of the probabilistic seismic hazard maps by an improvement of background methodologies and implementation in assessment of seismic hazard and risks. Evidently, it does not take into consideration economic and social factors of risk assessment, neither account for the role of site effect due to topography and soils, nor make use of an earthquake rupture size for the large and/or complex seismic events. These essential considerations should be addressed in the future practical estimations and mappings.