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cusco_seismic_hazard_ttex.pdf
1. Seismic hazard for the Cusco city (Perú)
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through probabilistic analysis of historical
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and instrumental records of earthquakes
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Determining Cusco as a independent seismogenic region
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Manuel Abarca
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February 2, 2021
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Cusco city (and neighbourhoods, designate as region) is by itself a seismogenic
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zone, independent of Benioff zone. Historical seismic records shows earthquakes
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as great as 7.41 mw strike the city in the past. Being a UNESCO World Her-
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itage centre, the city needs clear rules for design, building and preservation of
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civil structures under seismic event conditions. This rules have to be based in a
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Seismic Hazard Analysis, specific for the city. The classical Probabilistic Seis-
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mic Hazard Analysis (PSHA) founded on a Poisson model it is not reliable for
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Cusco, because seismic data don’t fit the assumptions made by a Poisson pro-
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cess. We follow here another probabilistic way to determine some seismic hazard
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parameters, considering frequentist theory of probabilities. The Bayes theorem
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was applied to find the maximum magnitude earthquake for different periods of
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time. Peak Ground Accelerations (PGA) were determined through a well condi-
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1
2. tioned equation. We made also an analysis of tectonic stresses sources founding a
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new one. A quase-vertical contact between two different densities zones in upper
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mantle is origin of deviatoric stress. Additionally, macroseimic information is
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used to calculate an attenuation of intensities law.
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JUSTIFICATION
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Probabilistic Seismic Hazard Analysis (PSHA) were developed for the entire South America
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[19] [24] or for Peru as country [1] [25], but none city of Peru has been the object of a specific
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PSHA. Previous PSHA (based in a Poisson model) studies found a probable peak ground ac-
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celeration (PGA) of 0.59 gn (gravities) in 100 years for the Cusco region; but, macroseismic
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testimonials and in-situ studies after a big earthquake shows evidence of larger accelerations.
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In the 1986 earthquake some stones (more than 100 kg weight) collided between them over a
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flat surface in the epicentral area, this is possible just if the earth gravity force is surpassed
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[14] (personal communication of Dr. Huamán). Another point of disagreement with Pois-
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sonian PSHA is the zoning and distances to seismic sources, [1] and [25] uses earthquakes
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as far as 300 km from Cusco city to estimate PGA; but an intensities-magnitudes-distances
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relationship found in section An attenuation law for intensities says what an earthquake at
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300 km of epicentral distance needs to be 10.2 magnitude (mw) to reach the city with an
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intensity of VIII (MSK). This is unreliable. Still, the acceleration would be 0.154 gn, using
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the acceleration-intensities relationship of [7] (equation 5 in [7] ) , far below of predicted by
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PSHA studies. Then the zoning used in previous PSHA uses data of earthquakes too far to
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become a dangerous source of seismic waves for the city; by other side mixes earthquakes of
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intraplate type with seismicity related to Benioff zone, we will show that subduction earth-
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quakes are less relevant in hazard terms to Cusco city.
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However, prior to beginning a seismological research , one basic question about the ne-
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3. cessity to study a PSHA study in a city as Cusco has to be answered. The answer has two
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ways of analysis, one referring to monetary, historical and cultural values of goods resting
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on the city ; another way is respect to seismic sources of possible damage over those goods.
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Some of the architectural and artistic goods resting on the city (and neighbourhoods) are
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invaluable. From a historical viewpoint the city retains some buildings and streets of Inca
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empire epoch (and before also) in original form. The cultural importance of the city is great,
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because archaeological, anthropological, ethnological studies are in course. Every day new
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things are found underground the city; the complete image of old city is still to be discovered.
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The answer about possible sources of damage to the goods of the city is strictly seismolog-
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ical. From this viewpoint, Cusco and surrounded areas are very interesting because plentiful
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literature information respect to disastrous earthquakes felt in the city. The study of his-
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torical literature notice us of big earthquakes striking the city with notorious destructive
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effects on buildings and also with lost of lives. More detailed macroseismic and analytical
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information will be given in the next subsections, for now is sufficient to have a positive
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answer for the basic question.
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Cusco is by itself a seismogenic region. A preliminary process of instrumentally recorded
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data confirm to us the idea that Cusco has special characteristics as a seismically active zone,
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independent of Benioff zone. So, we will try to demonstrate in this section the following
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hypothesis: Cusco city is inmersed in a seismically active region with characteristics of
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intraplate seismicity, independent of earthquakes triggered by subduction mechanism and
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with its own seismotectonic signature.
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3
4. Earthquakes historically registered
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The most antique reference to an earthquake near Cusco is given by tradition, which talks
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about a big earthquake before the arrival of Spaniards; this could be occurred between 1438
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and 1491 [21] and affect buildings in Machu-Picchu.
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The first earthquake registered by Spanish chronicles is of 1590 [24] [26]. The most destruc-
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tive seismic event felt in Cusco was in 1650; really were two big shots (1650/03/31 16:10:00
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and 1650/03/31 19:00:00 and 260 aftershocks until April 3). It is assigned a intensity of X
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(MSK) for the two earthquakes; damages to buildings were severe (Fig. 1); produce damages
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to civil structures in distant cities as La Paz and Lima; near the epicentre were observed
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changes in the level of underground water and some streams deviate from original chan-
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nel. Magnitude of 7.41 is assigned based on this macroseismic information, being the most
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energetic earthquake in our catalogue (for the region of Cusco). The picture of Fig. 1 is
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interesting as document of epoch because it was commissioned by the bishop of the city to
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evaluate the damages due to the earthquakes.
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Another strong shaking of the earth was felt in Cusco in 1744 (1744/11/19 11:30:00) ;
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some churches had cracked walls and statues fell down in the cathedral. Intensity of VI and
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macroseismic magnitude 5.3 is assigned to this earthquake.
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There are instrumental recordings of seismic events after 1900 . However macroseismic
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information is important because this talks us about damages to civil structures, geological
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incidences and geographical scenery changes. Then the following earthquake which is still in
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memory of people is from 1950 (1950/5/21 18:37:40.00); it was felt with a intensity of VIII
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(MM) in Cusco , 120 lost lives, more than 50% of houses and buildings severe damaged (Figs.
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2 and 3, apparition of faults scarp [26], exchanges in underwater level and liquefaction of
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soils phenomena (personal communication of Vittorio Bonino). The news of the earthquake
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appearing in the journal of the capital of the country (Fig. 2) tell us about the importance
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5. of the city of Cusco. Fig. 3 is very illustrative about the severity of damages to houses and
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buildings, in the Santo Domingo church fell down some part of the main shrine, but the stone
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wall of Incas epoch at the basement of the church it remains intact; but this is another theme.
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The last big earthquake remembered by citizens is of 1986 (1986/4/5 20:14:29.20), magni-
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tude 5.2 mw, felt with intensity VIII (MM) in Cusco city. Were 9 killed persons; damages on
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houses and churches (statues of high part of cathedral fell down). A hundred of aftershocks
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registered and located near the Quenco location [14].
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A total of 88 earthquakes were felt and occurred in a radius of 60 km from Cusco between
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1590-1900 years. They are registered in our historical catalogue, but not all are able to apply
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for the PSHA study because some loss of seismic parameters, mainly magnitude.
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The zoning used by [1] and [25] takes both intraplate and subduction earthquakes, which
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means they are equally hazardous for the Cusco city. An analysis based in the attenuation
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law for seismic waves in Perú (section Intensity-magnitude-distance relationship) can clarify
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this point. Benioff zone is 125 km (Fig. 5 ) to 100 km under Cusco (Fig. 10), so, an earth-
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quake with magnitude 6.0 mw (there are not Benioff earthquakes with magnitude greater
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than this) directly below the city will produce an intensity of V. We are considering a tremor
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can be dangerous for lives and civil structures if have intensities of VIII or more. Then, the
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contribution of subduction earthquakes to the seismic hazard in the city is negligible.
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A last argument to contest our hypothesis could be: Any region of the Andes have high
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rate of seismicity, so, Cusco it is not a specially active region. We can find an answer to this
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in the macroseismic information. Making a search in the intensities catalogue [24] we find
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189 citations for the name Cusco (that is, declarations of people who felt an earth tremor
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in the region of Cusco), while neighbourhood regions as Puno or Apurimac receive 27 and
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6. 28 citations respectively. Of course regions located at coastal line appears with much more
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citations, but this is because they are directly above the Benioff zone. In conclusion, Cusco
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is the most seismically active region in the Peruvian Andes.
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Crustal thickness below Cusco
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Earthquakes with source mechanism of tectonic type can occur in the Earth crust. It is con-
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sidered in the mantle and in the asthenosphere constituent rocks have rigidity and viscosity
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can’t store deviatoric tectonic stresses in amount enough to fail. So, we need to know crustal
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thickness, in other words the limit until what rocks can suffer tectonic faulting. This is
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also known as Mohorovicic interface (Moho) and is determined by seismological methods as
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Receiver Function (RF) or surface waves tomography [3]. Lines of isopachs for crust below
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Cusco region shows Moho is between 60 to 65 km depth (Fig. 4).
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A more detailed study modelling the Moho and the flexure of oceanic lithosphere in the
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transect Cusco-Juliaca with RF inversion method [20] locates the Moho at 70-75 km depth.
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This is a relevant information to take in consideration at the moment to stablish the limits
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in depth for our seismic catalogue. The crust under Cusco region is more thicker than
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coastal areas or the continental interior (cratons); this have consequences in the isostatic
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equilibrium and in the emergence of deviatoric stresses, being the last directly related to
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tectonic earthquakes.
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Benioff zone under Cusco
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Benioff zone is defined as ”A dipping planar (flat) zone of earthquakes that is produced by the
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interaction of a downgoing oceanic crustal plate with a continental plate. These earthquakes
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can be produced by slip along the subduction thrust fault or by slip on faults within the
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downgoing plate as a result of bending and extension as the plate is pulled into the mantle.
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Also known as the Wadati-Benioff zone” (USGS, Earthquake Glossary). Below the oceanic
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6
7. lithosphere we find the asthenosphere which is the limit in depth to tectonic earthquakes. The
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upper limit for subduction type earthquakes is the contact between continental lithosphere
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(South American plate) and oceanic lithosphere (Nasca plate) [6], so, earthquakes to be
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considered in our study must be above this limit.
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The level curves in Fig. 5 shows a flexure of Nasca plate exactly below Cusco, passing
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from a flat subduction to a dipping slab in NW-SE direction [20]. In any case, focus more
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than 125 km depth occur into the oceanic lithosphere. This reason altogether with the Moho
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depth criteria are why we select earthquakes just until 90 km hypocenter depth; with this we
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are sure our earthquakes occur into de crust or in the upper mantle, but not into the oceanic
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lithosphere nor in the asthenosphere. None of the earthquakes in our seismic catalogue is of
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subduction type.
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Seismotectonic in the region
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We have seen in how the typical tectonic regime into a subduction zone is the thrust fault.
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This could be a criteria to determine if earthquakes in region under examination are triggered
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by stresses generated by the downgoing mechanism of oceanic plate, or they obey to another
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tectonic regime.
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The most of focal mechanisms in Fig. 6 are of normal fault type. This is corroborated
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by stress tensor (in the same figure) where a near vertical σ1 and near horizontal σ3 give us
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an seismotectonics scenery of normal faulting. Evidently the tectonic mechanics originating
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deviatoric stresses is not oceanic slab pull. We have to think in a local source of stresses as
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triggered mechanism of earthquakes occurred in Cusco region.
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Intensity-magnitude-distance relationship
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One of the first steps in a PSHA study is to choose seismic sources and determining the
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geometry of seismogenic zone. Viewing a seismicity map of Perú we can identify some
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possible seismic sources, but not all are useful for this study because they can be located too
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8. far as to strike the city with enough energy. We have to determine which is the maximum
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epicentral distance for an earthquake capable to produce damages to the city.
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We use an empirical Intensity-magnitude-distance relationship [4] as preliminary tool in
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evaluation of maximum distance, given a magnitude of reference and an expected intensity ,
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MMI = −3.29 + 1.68 × mw − 0.0206 × ∆ (1)
where,
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MMI : Mercalli Modified intensities scale;
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mw : Seismic moment magnitude,
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∆ : Epicentre distance in km.
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Solving (1) for the parameters MMI = V III, mw = 7.41 in a iterative way, we found
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the distance ∆ = 60km how the maximum limit at which an earthquake can produce serious
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damages to Cusco city.
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Then, we have delimited the area of interest for this PSHA study, a radius of 60 km
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around the city of Cusco and 90 km depth. This parameter for depth is very flexible because
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two limitations of data: the uncertainty in determination based on macroseismic informa-
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tion (historical chronicles) is too large, not even an error bar it is possible to know. For
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instrumentally recorded seismic events, hypocenter depth determination depends on veloc-
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ities model. Earth models as iaspei91 or ak135 locates Moho interface at 33-35 km depth,
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but the crust under Cusco is more thickest, with Moho 70-75 km depth, then hypocenter
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determination with programs using those earth models gives systematically bigger depth
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than real. So, although 90 km depth is a physical limit imposed by us for hypocenters, in
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catalogues can appear deeper focus but what really are into our limits.
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9. INTRODUCTION
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After to demonstrate the need of a PSHA study for Cusco region, we analyse the applicability
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of the Poisson model to seismicity of Cusco. The test methodology proposed by [11] is
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followed by us, over the seismic data. As the test shows a high discrepancy from the Poisson
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model, we choose the frequentist theory to find probabilities directly from relative frequencies
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of magnitudes. The return periods are taken also directly from averages of elapsed times
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between earthquakes of same magnitudes. These are the basic informations introduced to a
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formulation of Bayes theorem adopted from [27].
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DATA BASE
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We construct our data base for earthquakes, collecting information from different sources:
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a) The Centro Regional de Sismologia (CERESIS) [24];
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b) The International Seismological Centre (ISC) [12];
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c) The ANSS Comprehensive Earthquake catalogue (ComCat) [28];
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d) Incorporated Research Institutions for Seismology (IRIS);
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e) Instituto Geofísico del Perú (IGP).
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Most of the finally accepted earthquake data come from a) and b). IGP is mentioned just
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for completion of information, because all their data is included in the CERESIS catalogue.
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The constraints imposed to our catalogue are:
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1) epicenter located at 60 km of maximum distance from major Square of Cusco city;
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2) hypocenter at 90 km of maximum depth (some historical earthquakes located by macro-
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seismic information has hypocenter more than 70 km, but their precision is too poor, then
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we fixed to 33 km depth);
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3) earthquake should have magnitude parameter determined.
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In the elapse time 1590-1899 magnitudes were taken from a); for the period 1900-2018 the
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source of magnitudes is b). Magnitudes are normalized to seismic moment magnitude (mw)
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10. using the formulas of [23],
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mw = 0.67MS + 2.07; 3.0 ≥ MS ≤ 6.1 (2)
mw = 0.99MS + 0.08; 6.2 ≥ MS ≤ 8.2 (3)
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mw = 0.85mb + 1.03 (4)
Another types of magnitudes, as macroseismic magnitude (Mm) are considered equivalent to
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mw, and local magnitude (Ml) equivalent to mb, because absence of any kind of formulation
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for these. The complete catalogue of earthquakes for our study is presented in table 1.
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Due to magnitudes conversion to mw there are not earthquakes with magnitude less to
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4.0. This is a threshold level established by circumstances related to recording capabilities
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of epoch. Historical chronicles just register big earthquakes (or at least people feel like this),
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in the same way small tremors are neglected by historians. Looking to instrumental records,
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a world wide seismometers network begins their operation in 1960; the magnitude threshold
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for this network was 5.0 until 1990. In Perú a seismic network able to detect earthquakes
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below magnitude 5.0 (for all the territory) was installed in the new millennia. These are the
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reasons why the catalogue has few earthquakes in the low magnitudes range.
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Hypocentral distance, from focus to the main square of Cusco, is an interesting parameter
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because this is the real path travelled by seismic wave. This parameter tell us how much
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attenuation suffer the wave. The mean of hypocentral distances is 62.2 km, so considering
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the results obtained in we can say of all this earthquakes, they had potential to affect the
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city and produce serious damages.
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Some of the epicentres plotted in Fig. 7 match with traces of regional faults, this is in-
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teresting since they are confirming the theory of focal mechanism, one of the nodal planes
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11. would be related to geological fault. But, most of the epicentres no match with any fault; no
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nesting nor alignments of epicentres are seen. In a general view we can say epicentres have
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an spatial random behaviour. However, earthquakes in the crust must be result of a fault
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slip, so we can conclude that faults exists but have not superficial evidences.
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It is notorious the fact that the 3 biggest earthquakes are far from known geological faults.
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In the same sense the 2 earthquakes of 1650 appears plotted 55 km apart, but difference
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time between both is 3 hours, then any seismologist hope the second is an aftershock of the
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main and both must be located one near the other. Is very rare two big earthquakes occur
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close in time and in space but from independent geological phenomena. Also chronicles talks
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of 260 aftershocks felt in the city. Historical chronicles says nothing about location of those
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aftershocks. Most probably the location of those big earthquakes plus aftershocks happened
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in the same fault system.
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The 1986/04/05 earthquake had one hundred of aftershocks in an elapse time of 20 days.
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Twelve of the aftershocks were located in the same fault system of main shock; magnitudes
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diminishing to less values but focus depth upgoing in surface direction.
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Despite the numerous references to aftershocks in Cusco region they don’t appear in seismic
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catalogues. This is a declustering operation due to historical procedures. It is not a desirable
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effect over our data because we need the most complete relation of earthquakes along all the
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range of magnitudes.
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We collect also a catalogue of Centroid Moment Tensor (CMT) solutions for earthquakes
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inside the region of interest.
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Catalogues of seismic intensities are another source of information very useful in this
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research. CERESIS provides intensities in both formats, impress [24] and digital; the second
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12. catalogue we use is from ISC (via web). Seismic intensities deducted from historical chronicles
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are presented by Silgado [26] and Alva [24].
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GUTENBERG-RICHTER LAW
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The magnitude-frequencies relationship in essence says the less the magnitude the most the
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number of events; this is based in empirical observations and is valid for all the world [13].
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Looking to our bars graph of absolute frequencies (Fig. 8) it is clear what the magnitude-
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frequencies relationship for Cusco differs from the known shape for all the world. We can
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explain the low level of frequencies in magnitudes less to 6.0 with reasons exposed in above
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sections. The key factor could be the sensitivity of seismic networks; without stations close
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to the city until recent years, low magnitude earthquakes are under-registered. But never-
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theless the World-Wide Standardized Seismograph Network (WWSSN) since 1960 is able to
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detect any earthquake with magnitude 5.0 or greater; so we can expect the frequencies of
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events in the range 5.0-6.1 would be greater than frequency of 6.2 (frequency is used here in
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the statistical sense, number of occurrences). This is not the case, opening the possibility
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of an special distribution of seismic frequencies in Cusco founded in an own seismotectonic
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model.
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It is a established practice in PSHA studies, based in the Poisson model, to make a graphic
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of the logarithm of the exceedance cumulative relative frequencies versus magnitudes and fit
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to a curve of form,
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log(N) = a − b(M) (5)
which is known as Gutenberg-Richter law (G-R).
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But, in our case, due to dispersion of data, any interpretation over this curve (Fig. 9)
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12
13. have no sense; however if we look just to to the fitted curve, log(N) = 1.01 − 0.12 × mw,
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we can feel the temptation to follow an statistical analysis based in the Poisson model. It
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is necessary to see also the raw data and their frequencies distribution, finally, a test about
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compatibility with a Poisson distribution shows us this is not a good way of probabilistic
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analysis. This statement is substantiated in an analysis of discrepancy regarding Poisson
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distribution [11] ; the χ2
= 150.68 obtained from seismic data is too large (see table 2),
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enough to reject the zero hypothesis still with the lowest level of significance. Specifying this
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conclusion: our seismicity data don’t correspond with a Poisson process.
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By other side the Poisson probabilistic model requires events are random (in time and in
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space) and independent. Statistical independence implies future earthquake is not linked to
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occurrence of present earthquake, however Gutenberg and Richter says ”Further, the events
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are not strictly independent.” [13]. Another fact which breaks any probabilistic model ap-
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plied to earthquakes is the ”no repeatability” of events [8]. Clustering in time and space it
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seems to be the normal and universal behaviour of earthquakes in short period of time; but
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in the long period ”the waiting time until the next earthquake becomes longer the longer
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one has been waiting for it” [18]; none probability distribution is capable to take account of
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this phenomenology.
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Historical macroseismic information from Cusco indicates us a lot of aftershocks occur
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after a big shot. That facts it seems to be conformal to Bath’s law for the decay in the
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magnitude of aftershocks, and the Omori’s law for the temporal dropping in the number of
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aftershocks [22]. Apparently there is a magnitude threshold to trigger the swarm of after-
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shocks; in an empirical way we can say this limit is around 5.0 mw; we have not perceived
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sequences of tremors after an earthquake of 5.0 or less.
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The G-R law could be an expression of seismic activity at inter-plate zones; how 90% of
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13
14. earthquakes in the planet Earth occur at inter-plate regions [5] is logical to think it is of uni-
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versal validity. However we are studying seismicity into a continental plate, and the shape
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of their magnitudes-frequencies curve is the expression of a different kind of accumulation
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and release of elastic energy.
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Frequencies in Fig. 8 fails on fit the presumptions of G-R law, however maybe are ex-
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pressing a characteristic physical (geological) situation of the Cusco region. The mean of
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magnitudes is 5.4 mw, the median is 5.5, the mode is 6.3. The release of tectonic stresses by
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mean of earthquakes in the range 5.4-6.3 of magnitudes is the resultant of a defined strength
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of rocks in the region.
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We know how the contact between South-american plate (continental lithosphere) and
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Nasca plate (oceanic lithosphere) cumulates energy; Nasca plate is pushing the continental
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lithosphere, in a direction proximately W-E, and subducting in an amount of 61.9 mm/year
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[15] . This is a continuous process of accumulation of tectonic stress; if a quantity of energy
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is liberated in the form of seismic waves by cracking into the plate or by rupture of inter-plate
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asperities, immediately the driving forces restores the tectonic stress. While in the intraplate
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Andean region the role of ridge push is minor; being the main source of stresses the effect of
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topography and its compensating crustal root [16] , so the creation of seismic waves depends
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on internal strength of rocks. Little amounts of deviatoric stress can’t produce failure in
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rocks because their internal strength. This can explain why occur few small earthquakes in
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Cusco region. Most of the rocks in crust below Cusco fails and creates seismic waves in a
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certain interval of stresses, in the other extreme there are few rocks with high resistance to
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failure, but when cumulated stress is large enough those rocks trigger big earthquakes.
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14
15. PSHA BY BAYES THEOREM
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In spite of the Poisson model is not applicable to seismic data in Cusco region, still we
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have to deliver a result about seismic hazard in the Cusco region. The frequentist theory in
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statistics no made assumptions related to randomness of process; so this is the way to extract
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information strictly adjusted to objectivity of data. A probability based in frequencies of
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occurrence of events says nothing about time or periodicities; we can introduce this as prior
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data in estimation of probabilities applying the Bayes theorem.
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We can calculate the probability of an event m occurring t,
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Pr(m|t) =
Pr(m) × Pr(t|m)
∑n
i=1 Pr(mi) × Pr(t|mi)
(6)
In other words, we want to know the probability of occurrence of the earthquake with
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maximum magnitude m in the elapse time t. Pr(m) is also known as the priori probability,
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Pr(t|m) is the conditional probability of t given m and in the denominator of formula (6)
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we put the summation of probabilities for all the magnitudes until the maximum for that
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elapse time [27] .
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The prior probability is obtained from relative frequencies of earthquakes, in table 3. For
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example, 6.2 mw earthquakes have the higher probability of occurrence, with 17.6%. But
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this is a crude probability, without relation to periodicity nor considering the lower (or up-
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per) magnitude earthquakes. We need to introduce the parameter time in our formulation,
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because hazard is evaluated at different time lapses.
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The conditional probability is extracted from data of average periodicity, taken the inverse
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to obtain frequencies and normalized to relative frequency, table 4. The relevant data in this
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table is in ”T-cum” column; this shows the time return for a given magnitude earthquake, or
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lower. For example, an earthquake of magnitude 5.0 or lower can occur each 97.7 years. The
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15
16. conditional probability of 97.7 years against 5.0 mw is 0.26. The first column in the table
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reach the 6.2 magnitude as maximum, we can’t calculate the return period for an earthquake
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of magnitude 7.4, which appear in our catalogue, because with two occurrences in one day
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and related to the same faults system has to be considered one unique event.
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Another interesting data is in ”f-rel” column, this says us the probability of occurrence of
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an earthquake given their magnitude; for example a 4.1 mw earthquake has 35.7% probabil-
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ity of occurrence and a 6.2 mw earthquake has 0.8% probability of occurrence. We note this
377
results are best fitted to the G-R law assumptions. But here we lost the reference to time.
378
The conditional probabilities of magnitude as function of return periods is reached through
379
the Bayes theorem.
380
381
The Bayesian probability for the maximum magnitude earthquake, given an elapsed time,
382
is presented in the form of table 5. This is calculated for the expected maximum magnitude
383
in an elapsed time (time counted since the last earthquake of equal or bigger magnitude).
384
For example, in 50 years the expected maximum earthquake is of 4.7 mw and will occur with
385
a probability of 11.5% ; while in 350 years the expected maximum earthquake is of 6.2 mw
386
and will occur with a probability of 3.7%. This looks like a low probability, however we have
387
to interpret in statistical sense: for each 100 earthquakes occurred, 4 could be of magnitude
388
6.2; the others will have lower magnitudes. How many earthquakes can occur in Cusco in
389
350 years ? A lot, most than we think considering the seismic network is badly distributed
390
and undersensitive.
391
392
Acceleration of seismic waves for a given magnitude
393
The acceleration of seismic waves in Cusco is calculated from an empirical formula. Unfortu-
394
nately there is not a formula relating magnitudes and accelerations specifically for Perú. So,
395
16
17. we found a formula developed for interplate earthquakes of Chile and able to take account
396
of short epicentral distances [9], we use in the hope it is also applicable to intraplate zones
397
of Perú,
398
log(α) = c1 + c2mw + c3z + c4∆ − g × log(∆) + c5S
∆ =
√
R2
r + R2
m
Rm = c610c7mw
g = c8 + c9mw
S = 1
(7)
where α is acceleration; mw is moment magnitude; z is depth of focus; Rr is distance to
399
rupture fault; Rm is a near-source saturation term; g is the geometrical spreading coefficient
400
and S = 1 for soils. We choose the 1 second frequency constants,
401
c1 c2 c3 c4 c5 c6 c7 c8 c9
-3.3352 0.4013 0.0186 -0.0010 0.2839 0.0734 0.3522 1.5149 -0.1030
With these constants we calculate the PGA
402
can occur in Cusco for different magnitudes earthquakes and given a determined epicentral
403
distance in table 6. It is important to note the acceleration can surpass 1 gn with an
404
earthquake 7.4 mw (the maximum registered in Cusco); this will mean great destruction
405
in the city. We can see also with a 5.5 mw earthquake the acceleration is higher than
406
expected by previous PSHA estudies. So, in spite of low probabilities of big earthquakes,
407
the accelerations can reach high values.
408
An attenuation law for intensities
409
We use a preliminary attenuation law in Intensity-magnitude-distance relationship to de-
410
termine maximum distance of epicentres. But that was developed in a different geologic
411
environment; here we propose to find an attenuation law for intensities, fitted to the cur-
412
rent geotectonic conditions of Cusco. The data is extracted from intensities catalogue [24].
413
Intensities in table 7 were felt in Cusco city.
414
415
17
18. Solving the linear equations system we obtain the attenuation law:
416
I = −0.7482 + 1.2739 × mw − 0.0140 × ∆(km) (8)
If we choose the parameters mw = 7.41 and ∆ = 60.0km, the intensity gives us roughly
417
VIII. This result is consistent with the premise in Intensity-magnitude-distance relationship.
418
SOURCES OF TECTONIC STRESSES IN CUSCO
419
Different analysis of tectonic stresses in Andean region shown the main source of stresses are
420
the potential energy due to topographic inhomogeneities [16] [10] ; using the formula given
421
by Artyushkov [2],
422
∆σ = ρ1gh (9)
for an altitude of h = 3200m in Cusco; a density ρ = 2680kg/m3
and an acceleration of
423
gravity g = 9.8m/s2
we can estimate an amount of deviatoric stress of ≈ 84Mpa. A positive
424
stress means a tensional (horizontal) tectonic regime, which is corroborated by normal faults
425
found in the vicinities of Cusco city.
426
427
However the model of Benioff zone found by RF studio [20] under Cusco, shows a new
428
(unknown) source of tectonic stresses originated by the dipping contact between two regions
429
of upper mantle with different densities. This densities were derived from P wave seismic
430
velocities by the relationship,
431
ρ = 520 + 0.36 × Vp. (10)
Following the procedure for estimation of deviatoric stresses given by [17] and with the pa-
432
rameters values of Fig. 10, we find ∆σ = −58.8Mpa. The minus sign indicates this stresses
433
are compressive, but their direction is ≈ 90 rotated respect to the previously determined
434
tensional stresses. The resultant direction of both stresses is difficult to define; an evidence
435
18
19. of their interaction are some focal mechanisms solutions of normal fault type with strike slip
436
component (Fig. 6.
437
438
In the calculation of lithostatic pressures in both sides of dipping contact into upper mantle
439
(Fig. 10) we see the masses have different levels of compensation. Then, we can say this
440
region is not isostatically compensated. But, final demonstration of the phenomena is matter
441
of other study.
442
CONCLUSIONS
443
a) Cusco is a seismogenic zone by its own tectonics characteristics. Earthquakes capable to
444
hit the city with enough intensity to produce serious damages to lives and civil structures
445
have to occur in a maximum radius of 60 km around the Main Square of Cusco.
446
b) The main sources of tectonic stresses affecting Cusco are the topographic effect, which
447
produce ≈ 84Mpa of tensional stress in a direction normal to the alignment of Andes
448
Cordillera, and the semi-vertical contact between two regions of upper mantle with dif-
449
ferent densities which produce an compressional stress of ≈ −58.8Mpa; direction of this
450
last stress is the same of contour level lines of flat subduction zone. These are the reasons
451
for which all earthquakes in Cusco have focal mechanism solutions of normal fault type (or
452
normal fault with a small component of strike slip regime).
453
c) Data base of historical and instrumental records of earthquakes in Cusco region don’t
454
fit the requirements of a Poisson process; so, we can’t use the classical PSHA (based in a
455
Poisson model) to estimate seismic hazard parameters.
456
d) We use a frequentist approximation for prior probabilities of seismic events, a deter-
457
ministic return period as conditional probability and a Bayes model for the probability of
458
occurrence of the maximum magnitude earthquake. We hope a 6.2 mw earthquake with a
459
probability of 0.037, in an elapsed time of 400 years after the last earthquake with same
460
19
20. magnitude occurred. There are earthquakes with biggest magnitude occurred historically in
461
Cusco, but how they are unique it is not possible to extrapolate a return period for these
462
earthquakes.
463
e) The mean is 5.4 mw, the median is 5.5 and the mode of absolute frequencies of mag-
464
nitudes is 6.3 for Cusco region. This contradicts the assumptions of G-R law; but can say
465
something about mechanical strength of crustal rocks below Cusco. Those rocks has a frac-
466
ture limit which is reached by deviatoric stresses most than 84 Mpa (producing 5.4 mw or
467
higher earthquakes).
468
f) We found an attenuation law for seismic intensities in Cusco region, I = −0.7482 +
469
1.2739 × mw − 0.0140 × ∆(km).
470
g) The probable maximum acceleration in soil of Cusco, in case of a 7.4 magnitude earth-
471
quake is 1091.8 cm/s2
. This is more than one gravity, so, damages to civil structures and
472
buildings could be serious.
473
CONFLICT OF INTEREST
474
The authors declare that they have no conflict of interest.
475
References
476
[1] Z. Aguilar , M. Roncal , R. Piedra (2017), Probabilistic Seismic Hazard Assessment in
477
the Peruvian Territory, 16th World Conference on Earthquake, 16WCEE.
478
[2] E. V. Artyushkov (1973), Stresses in the Lithosphere Caused by Crustal Thickness In-
479
homogeneities, JOURNAL OF GEOPHYSICAL RESEARCH, 78, NO. 32, 7675-7708.
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[3] Marcelo Assumpção, Mei Feng, Andrés Tassara, Jordi Julià (2013), Models of crustal
481
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20
21. [4] W. H. Bakun and C. M. Wentworth (1997), Estimating Earthquake Location and Magni-
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tude from Seismic Intensity Data, Bulletin of the Seismological Society of America,87,No.
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[5] Bolt, Bruce (2005), Earthquakes: 2006 Centennial Update – The 1906 Big One (Fifth
487
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[6] Thomas Cahill, Bryan L. Isacks (1992), Seismicity and Shape of the Subducted Nazca
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[7] U. Chandra (1981), Different Magnitude-Epicentral Intensity Relations and Estima-
491
tion of Maximum Ground Acceleration, International Conferences on Recent Advances
492
in Geotechnical Earthquake Engineering and Soil Dynamics.
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[8] Heriberta Castaños and Cinna Lomnitz (2002) , PSHA: is it science?, Engineering Geol-
494
ogy, 66, 315–317.
495
[9] V. Contreras, R. Boroschek (2012), Strong Ground Motion Attenuation Relations for
496
Chilean Subduction Zone Interface Earthquakes, 15th World Conference on Earthquake
497
Engineering, Lisboa.
498
[10] Bernard Dalmayrac and Peter Molnar (1981), Parallel thrust and normal faulting in
499
Peru and constraints on the state of stress, Earth and Planetary Science Letters, 5, 473-
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[11] R. A. Fisher (1950), The Significance of Deviations from Expectation in a Poisson Series,
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Biometrics, 6, No. 1 , 17-24.
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[12] Di Giacomo, D., I. Bondár, D.A. Storchak, E.R. Engdahl, P. Bormann and J. Har-
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ris (2015), ISC-GEM: Global Instrumental Earthquake catalogue (1900-2009): III. Re-
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computed MS and mb, proxy MW, final magnitude composition and completeness assess-
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ment, Phys. Earth Planet. Int., 239, 33-47.
507
21
22. [13] B. Gutenberg and C. F. Richter (1944), Frequency of earthquakes in California, Bulletin
508
of the Seismological Society of America, 34, 4, 185-188.
509
[14] David Huamán Rodrigo (1987), Aspectos sismotectonicos del sismo del Cusco del 5 de
510
abril de 1986, Congress about Cusco april/5/1986 earthquake.
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[15] Eric Kendrick et al. (2003), The Nazca –South America Euler vector and its rate of
512
change, Journal of South American Earth Sciences, 16, 125-131.
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[16] P.Th. Meijer , R. Govers, M.J.R. Wortel ( 1997), Forces controlling the present-day
514
state of stress in the Andes, Earth and Planetary Science Letters 148, I57- 170.
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[17] Peter Molnar and Hélène Lyon-Caen (1988), Some simple physical aspects of the sup-
516
port, structure, and evolution of mountain belts, Geol. Soc. Am. Spec. Pap. 218, 179-207.
517
[18] Francesco Mulargia, Philip B. Stark, Robert J. Geller (2017), Why is Probabilistic
518
Seismic Hazard Analysis (PSHA) still used?, Physics of the Earth and Planetary Interiors,
519
264, 63–75.
520
[19] Mark D. Petersen, Stephen C. Harmsen, Kishor S. Jaiswal, Kenneth S. Rukstales, Nico-
521
las Luco, Kathleen M. Haller, Charles S. Mueller, and Allison M. Shumway (2018), Seis-
522
mic Hazard, Risk, and Design for South America, Bulletin of the Seismological Society of
523
America, 108 No. 2, 781–800.
524
[20] Kristin Phillips and Robert W. Clayton (2014), Structure of the subduction transition
525
region from seismic array data in southern Peru, Geophys. J. Int. , 196, 1889–1905.
526
[21] M. A. Rodríguez-Pascua, C. Benavente Escobar, L. Rosell Guevara, C. Grützner, L.
527
Audin, R. Walker, B. García, E. Aguirre (2020), Did earthquakes strike Machu Picchu?,
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J Seismol, 24, 883–895.
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[22] Robert Shcherbakov, Donald L. Turcotte, John B. Rundle (2004), A generalized Omori’s
530
law for earthquake aftershock decay, Geophysical Research Letters, 31, L11613.
531
22
23. [23] E.M. Scordilis (2006), Empirical global relations converting MS and mb to moment
532
magnitude, Journal of Seismology, 10, 225-236.
533
[24] Seismological Center for the region of South America (CERESIS) (1985), Earthquake
534
mitigation program in the Andean Region, 14 Vols. CERESIS, Lima.
535
[25] Sunil Sharma and Mario Candia-Gallegos (1992), Seismic hazard analysis of Peru, En-
536
gineering Geology, 32 , 73-79.
537
[26] Enrique Silgado (1978), Historia de los sismos mas notables ocurridos en el Perú (1513-
538
1974), pp. 111. Instituto de Geologia y Mineria, Lima.
539
[27] J. P. Wang, Su-Chin Chang and Yun Xu (2016), Best-Estimate Return Period of the
540
Sanchiao Earthquake in Taipei: Bayesian Approach, Nat. Hazards Rev., 06015001, 1-5.
541
[28] Young, J.B., Presgrave, B.W., Aichele, H., Wiens, D.A. and Flinn, E.A. ( 1996), The
542
Flinn-Engdahl Regionalisation Scheme: the 1995 revision, Physics of the Earth and Plan-
543
etary Interiors, 96, 223-297.
544
23
24. Figure 1: Picture of the epoch showing damages to Cusco city from 1650 earthquake.Source:
Life.
Figure 2: News about earthquake of 1950 in Cusco.Source: El Comercio.
24
25. Figure 3: Part of Santo Domingo church fell down in 1950.Source: Life
−80˚
−80˚
−79˚
−79˚
−78˚
−78˚
−77˚
−77˚
−76˚
−76˚
−75˚
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−74˚
−74˚
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−72˚
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−71˚
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−70˚
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−69˚
−69˚
−15˚ −15˚
−14˚ −14˚
−13˚ −13˚
−12˚ −12˚
−11˚ −11˚
−10˚ −10˚
Cusco
3
5
4
0
45
50
55
60
40
3
0
Figure 4: Moho depth under the region of Cusco. Dashed lines shows thickness of crust,
below Cusco it is more than 60 km. Source of data: [3]
25
26. −80˚
−80˚
−79˚
−79˚
−78˚
−78˚
−77˚
−77˚
−76˚
−76˚
−75˚
−75˚
−74˚
−74˚
−73˚
−73˚
−72˚
−72˚
−71˚
−71˚
−70˚
−70˚
−69˚
−69˚
−15˚ −15˚
−14˚ −14˚
−13˚ −13˚
−12˚ −12˚
−11˚ −11˚
−10˚ −10˚
Cusco
5
0
7
5
1
0
0
1
2
5
1
5
0
6
0
0
t
r
e
n
c
h
Figure 5: Nasca plate flexure below Cusco region.Source of Benioff contour lines: [6]
−72˚30'
−72˚30'
−72˚00'
−72˚00'
−71˚30'
−71˚30'
−14˚00' −14˚00'
−13˚30' −13˚30'
−13˚00' −13˚00'
Cusco
1
2
3
Figure 6: Focal mechanisms in region of Cusco and the stress tensor.Axis of main stress σ1
is indicated by number 1; the others main stresses are indicated in the same way.
Red dashed lines are fault systems recognized in the region.
−72˚30'
−72˚30'
−72˚15'
−72˚15'
−72˚00'
−72˚00'
−71˚45'
−71˚45'
−71˚30'
−71˚30'
−71˚15'
−71˚15'
−14˚15' −14˚15'
−14˚00' −14˚00'
−13˚45' −13˚45'
−13˚30' −13˚30'
−13˚15' −13˚15'
−13˚00' −13˚00'
669
789
909
1029
1149
1269
1389
1509
1629
1749
1869
1989
2109
2229
2349
2469
2589
2709
2829
2949
3069
3189
3309
3429
3549
3669
3789
3909
4029
4149
4269
4389
4509
4629
4749
4869
4989
5109
5229
5349
5469
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
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37 38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
Cusco
Urubamba
Urcos
Machu−picchu
Cotabambas
Figure 7: Epicentres of earthquakes accomplishing constraints for our catalogue are plotted
with red balls, numbers are from table ??; size of symbols are proportional to
magnitude. Black dashed lines are fault systems crossing the region.Scale color
bar is for altitude in meters.
26
27. 4.0 4.2 4.4 4.6 4.8 5.0 5.2 5.4 5.6 5.8 6.0 6.2 6.4 6.6 6.8 7.0 7.2 7.4 7.6 7.8 8.0
magnitude - mw
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
frequency
earthquakes frequency
Figure 8: Frequencies (number of events) of earthquakes in function of magnitude.
4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5
magnitude - mw
0.0
0.5
1.0
1.5
log(N)
observed earthquakes
log(N)=1.01-0.12*mw
Figure 9: G-R equation fitting seismicity data of Cusco.
27
![tioned equation. We made also an analysis of tectonic stresses sources founding a
19
new one. A quase-vertical contact between two different densities zones in upper
20
mantle is origin of deviatoric stress. Additionally, macroseimic information is
21
used to calculate an attenuation of intensities law.
22
JUSTIFICATION
23
Probabilistic Seismic Hazard Analysis (PSHA) were developed for the entire South America
24
[19] [24] or for Peru as country [1] [25], but none city of Peru has been the object of a specific
25
PSHA. Previous PSHA (based in a Poisson model) studies found a probable peak ground ac-
26
celeration (PGA) of 0.59 gn (gravities) in 100 years for the Cusco region; but, macroseismic
27
testimonials and in-situ studies after a big earthquake shows evidence of larger accelerations.
28
In the 1986 earthquake some stones (more than 100 kg weight) collided between them over a
29
flat surface in the epicentral area, this is possible just if the earth gravity force is surpassed
30
[14] (personal communication of Dr. Huamán). Another point of disagreement with Pois-
31
sonian PSHA is the zoning and distances to seismic sources, [1] and [25] uses earthquakes
32
as far as 300 km from Cusco city to estimate PGA; but an intensities-magnitudes-distances
33
relationship found in section An attenuation law for intensities says what an earthquake at
34
300 km of epicentral distance needs to be 10.2 magnitude (mw) to reach the city with an
35
intensity of VIII (MSK). This is unreliable. Still, the acceleration would be 0.154 gn, using
36
the acceleration-intensities relationship of [7] (equation 5 in [7] ) , far below of predicted by
37
PSHA studies. Then the zoning used in previous PSHA uses data of earthquakes too far to
38
become a dangerous source of seismic waves for the city; by other side mixes earthquakes of
39
intraplate type with seismicity related to Benioff zone, we will show that subduction earth-
40
quakes are less relevant in hazard terms to Cusco city.
41
42
However, prior to beginning a seismological research , one basic question about the ne-
43
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