## Introduction

The thermal structure and evolution of the oceanic lithosphere govern different geodynamic processes as a function of age1,2. The oceanic lithosphere is considered the upper thermal boundary layer of the mantle convection3, defined by the 1200 °C isotherm4,5, which is formed at the mid-ocean spreading centers. The evolution of this geotherm is associated with the square root of age6,7. Hence, the systematic variation of some physical parameters with age became the primary factor to consider as model for the thermal evolution of the lithosphere. The first models were proposed by realizing that the ocean floor heat flow is highest at mid-ocean ridges (MOR) and decreases with distance7,8. In this way, these models predict the variation in bathymetry and heat flow with age to describe how the lithosphere cools as it moves away from the spreading center. We recently proposed a model that describes the variation of seismic energy release sum with age in the East Pacific Rise (EPR)9. It is based on the main idea that on MOR systems, there is high seismic activity in the ridge axis10,11, and due to more stable and colder conditions of the older lithosphere, it decreases as an asymptotic function9,12. It proves that the main factor controlling the seismicity of MOR systems is the thermal structure of the lithosphere13, and describes the general shape of the depth curve of 1200 °C isotherm, including the flattening for ages > 2 million years old (Myr)9. Its main limitation is the unpredictability of seismic energy sum for ages > 4 Ma, which is thought to be due to local instabilities in remote regions from ridge centers or near to boundaries of other tectonic plates.

The thermal structure at a spreading center is predominantly influenced by two factors: (1) the rate of magma supply, and (2) the efficiency of hydrothermal circulation in removing heat14. Thus, the spreading rate dependence of seismicity on MOR’s systems becomes the main constraint to relate seismicity on MOR’s with the seafloor age15,16. The EPR is considered a fast-spreading ridge (> 6 cm/year)17, where the relatively thin brittle lithosphere produces scarce earthquakes that are extremely difficult to locate with the present OBS networks18, then, seismically its dynamic spreading process is a little less known in comparison to slow-spreading centers (< 4 cm/year) such as the Mid-Atlantic Ridge (MAR) or the Central Indian Ridge (CIR), where spreading dynamics enhance seismic activity. In this work, our study area includes environments with different spreading rates. We focus on the northern EPR, whose spreading rate ranges from 4 to 7 cm/year17, the MAR, in which case the spreading rates vary from 0.6 to 3 cm/year11,19, and the CIR where half-spreading rate range from 0.9 to 3.7 cm/year20. Specifically, we focus our main observations in areas of the ridge where a relatively high density of earthquakes has been recorded.

The sum of seismic energy released by earthquakes constitutes an important tool for characterizing geodynamic processes. The sum of seismic energy is proportional to the seismic activity in terms of the moment magnitude and the number of earthquakes recorded in a certain area17,21. According to Bird et al. (2013), the corner magnitude of spreading earthquakes is about 5.8, independent of the spreading rate. Nevertheless, the number of earthquake events decreases gradually, where rates are close to 40 mm/year. Then decrease considerably at spreading rates larger than 80 mm/year22. In this paper, we demonstrate the variations and contrasts in the sum of seismic energy as a function of lithospheric age between different spreading centers and the main implications on the thermal structure beneath each ridge system. Based on these observations, we propose a general model, with a broader scope for the variation of the sum of seismic energy released with age and cooling of young oceanic lithosphere.

## Methods

### Seismic catalog, age, heat flow, and bathymetry

We examined the seismic catalog from the United States Geological Service (USGS) and the Advanced National Seismic System (ANSS), which can be downloaded via GeoMapApp 3.6.1023. The database collects earthquakes of magnitude $$4.5 \le {m}_{b}\le 7.0$$, located between the years 1960 and 2020 for the EPR, MAR, and CIR. The bathymetry data of MOR’s was requested and provided by the National Oceanic and Atmospheric Administration (NOAA). Lastly, the data for ocean crust age were obtained from the digital model of age, asymmetry, and propagation rates of the oceanic crust carried out by Müller et al.24 (see Data availability).

### Transect considerations and energy release sum

We derived twelve perpendicular transects in different MOR areas, three transects located on the EPR, six on the MAR, and three on the CIR (Fig. 1). We selected areas with a special high density of earthquakes recorded in the ridge area located towards the flanks and ridge areas whose transforming boundaries were relatively spaced. Hence, areas with very few events and/or with significantly affected by transform boundaries were avoided. The transect longitudes, and the corridors, depended on the distribution and scatter of events, and longitudes vary between 3° to 5° (approx. 350–550 km), and the transect corridor widths range from 0.3º to 1°. Additionally, due to the fact that some study areas were located close to transforming zones, it was necessary to check the focal mechanisms to exclude the strike-slip mechanism and only to consider earthquakes associated with spreading centers. Also, the corridor is broad enough to prevent the relocation process. We divide the ocean basins into age intervals of 1 Ma, and then we computed the sum of seismic energy released by earthquakes located within each age interval along the transects.

### Seismic energy calculations and global model

The seismic catalog used in this work has limited magnitude resolutions. It is due to the location of seismic networks and limitations for deployments of ocean-bottom seismometers. Hence, the computation of seismic energy released by earthquakes, for each transect, would be restricted to the completeness magnitude (Mc), which varies from 4.8 to 4.5. Furthermore, the transect with the most number of events involved (Fig. 1D,L) has almost 70 earthquakes. Thus the low number of events try to estimate makes it challenging to use the b-value parameter for evaluating the number of earthquakes below the Mc for each age interval. Then, it is important to note that we are not considering the energy released by microseismicity. However, we consider that the contribution of microseismicity to empirical relationships is not significant. Since the sum of seismic energy is calculated using the moment magnitude (Mo) formula, in this work, we use a series of empirical relationships to obtain the seismic energy released. First, knowing the body-wave magnitude (mb) from the seismic catalog, we used the empirical approach given by Shapira and Hofstetter25 to estimate Mo from mb (Eq. 1).

$$Log\left( {M_{O} } \right) = 1.59m_{b} + 15.63$$
(1)

Then we used the model proposed by Hofstetter and Shapira26 that relates the seismic energy with Mo (Eq. 2), where Mo is expressed in dyncm, and Log (EO) is in ergs.

$$Log\left({E}_{O}\right)=1.19{log(M}_{o})-8.81$$
(2)

Once we computed the sum of seismic energy released for the age intervals along the twelve transects, we performed an average energy release sum for EPR, MAR, and CIR (Fig. 5). Finally, following the methodology proposed by Sclater et al.8 and Stein & Stein1, we estimated a general relationship between the sum of seismic energy released and the oceanic lithosphere age based on a linear regression by least-squares fit between the binary combination of these parameters.

## Results

### MOR systems and seismic energy released

It is noticeable in the transects that, independent of the spreading rate, there are more seismic events along the ridge axis. The event count decreases towards the axial flanks. Also, scarce earthquakes of high and intermediate magnitude occur preferentially in the young oceanic lithosphere (Fig. 2). Interestingly, we observe that as the spreading rates increase, the number of spreading earthquakes reduces. In contrast, in slow-spreading centers like MAR and CIR (Fig. 2D–L), we observed up to twenty events for the lithosphere younger than 1 Ma. In the EPR, for the same age interval, there occurred only five to ten events (Fig. 2A–C). The EPR transect generally shows a higher frequency of seismic events of low and moderate magnitudes (mb < 5.5), mostly concentrated on the ridge axes. At the ridge axes, the 1200 °C isotherm emerges, and as the oceanic lithosphere gets colder, the seismicity production decreases.

Our results show that changes in spreading rates lead to changes in the seismic regime on MOR’s and, therefore, calculating the sum of energy released for individual transects, we find that it is a little higher in fast-spreading MOR’s systems than for slow-spreading rates regimes (Fig. 3). Also, as the spreading rate decreases, the seismicity has a greater range with respect to the age of the lithosphere. In the EPR (Fig. 3A–C), the sum of energy released decays rapidly with age so that it is only possible to make an approximation at ages no older than 4 to 5 Myr. Instead, for fast-spreading centers like MAR (Fig. 3D–I), the seismic occurrence at older ages allows us to determinate energy values to a maximum limit of 10 Myr. Finally, in the CIR (Fig. 3J–L), our observations show that sum of seismic energy decreases and can be calculated for the lithosphere no older than 6 Myr. On the EPR, we found that the sum of seismic energy released decreases exponentially. However, these observations are restricted to 4 Myr since there are no recorded seismic events at older ages (Fig. 4a). The computed sum of seismic energy released on the youngest lithosphere reaches a value of Log(Eo) = 20.528 ± 0.253 ergs. It decreases gradually with age to a value of around Log(Eo) = 18.60 ± 0.432 ergs for the lithosphere not older than 4 Myr. The MAR includes regions with the lowest occurrences of seismic events of moderate to high magnitude (Fig. 2D,E). Therefore, for the first two transects in the north part of MAR, we obtained the lowest values of the sum of energy released (Fig. 3D,E), whereas the energy release values increase in the central and southern part of the ridge (Fig. 3F–I). On the ridge axis, in the lithosphere younger than 1 Myr, we estimated a mean of energy released sum value around Log(Eo) = 20.399 ± 0.408 ergs, and it decreases steeply until ages around 3 Myr with an energy release value of Log(Eo) = 18.692 ± 0.583 ergs. In the older lithosphere, between 3 to 10 Myr, the sum of seismic energy released decreases gradually to a value of Log(Eo) = 18.020 ± 0.765 erg (Fig. 4b). On the CIR, we found that the sum of seismic energy released decrease rapidly for younger ages until 3 Myr approximately.

Moreover, for the lithosphere older than 3 Myr, it decreases more gently until 6 Myr. For the lithosphere younger than 1 Myr, the sum of seismic energy released is Log(Eo) = 19.984 ± 0.665 erg, and it shows a rapid decay until 3 Myr. For lithosphere ages ranging between 3 and 6 Myr, the sum of energy released decreases gradually since Log(Eo) = 18.995 ± 0.583 erg to a value of Log(Eo) = 16.840 ± 0.339 ergs respectively (Fig. 4c).

### Global model

The systematic decrease of the seismic energy released, in the young oceanic lithosphere, from the ridge axis towards the older flanks, is notable in each of the cross-sections made on EPR, MAR, and CIR (Figs. 3, 4). The same behavior according to global bathymetry and heat flow were related to the lithosphere age1,8,27. Similarly, we calculated the average of the energy released sum for all MOR systems and proposed a comprehensive model as a function of age and independently of the spreading rate (Fig. 5a). According to our results, the average of the sum of the seismic energy released for a lithosphere no older than 1 Myr is Log(Eo) = 20.344 ± 0.282 erg. We also found that this energy released decreases rapidly with the square root of age, and it occurs over a lithosphere age of about 3 Myr where the energy released sum is Log(Eo) = 18.899 ± 0.583 erg. In crusts older than 3 Myr, this relation breaks down, and the energy decreases exponentially to a constant value of Log(Eo) = 18.02 ± 0.765 erg for lithosphere ages of 10 Myr (Fig. 5b). The statistical adjustment for $$t\le 3 Myr$$ (see Equations on Fig. 5a) shows a correlation coefficient of $$\left|{R}^{2}\right|=0.970$$, with a constant value $${K}_{1}=20.344$$ and with confidence boundaries of 95% ($${K}_{1}[17.795 21.392$$]). In a lithosphere with $$t>3 Myr$$ (see Equations on Fig. 5a), our adjustment shows a correlation coefficient of $$\left|{R}^{2}\right|=0.730$$, with a constant value $${K}_{2}=19.887$$, and confidence boundaries of 95% ($${K}_{2}[18.183 18.70$$]).

The expressions in Fig. 5a show that the seismic energy released over the axis (t = 0) is infinite. For this reason, we carry out the integration of the formulas to remove the singularity in the proposed system. The limits of the integration extending from $${t}_{i-1}$$ to $${t}_{i}$$, where $${t}_{0}$$ is zero (see Eqs. 3 and 4), and the integral is given in ergMyr. Thus, a finite value for the average energy released in a young lithosphere above the ridge’s axis is found easily.

$$t_{i} \le 3{ }Myr{ }\mathop \smallint \limits_{{t_{i - 1} }}^{{t_{i} }} {\text{log}}\left( {E_{o} \left( t \right)} \right) = 40.688{ }\sqrt {\left( {t_{i}^{{c_{6} }} } \right)} { }{-}\sqrt {\left( {t_{i - 1}^{{c_{6} }} } \right)} { }$$
(3)
$$t_{i - 1} > 3{ }Myr{ }\mathop \smallint \limits_{{t_{i - 1} }}^{{t_{i} }} log\left( {E_{o} \left( t \right)} \right){ } = - 1807.9\left[ {{ }e^{{ - c_{7} \cdot t_{i} }} { }{-}{ }e^{{ - c_{7} \cdot t_{i - 1} }} } \right]$$
(4)

where $${c}_{6}=0.126 {Myr}^{-1}$$ and $${c}_{7}=-0.011 {Myr}^{-1}$$.

It is necessary to bear in mind that these expressions are restricted to the seismicity around the ridge. Thus, the main limitation to generate these empirical relations are aseismic zones in MOR’s; hence, we considered only the most seismically active zones. According to the study areas, it was possible to compute the sum of seismic energy for the lithosphere no older to 10 Myr (Fig. 5a). Furthermore, we consider that to generate a more representative model, observations in more ridge systems must be rendered. In that case, the outcome of new scenarios may gradually modify the expressions proposed. furthermore, to the extent that digital signals are used to calculate energy, it is expected to have less uncertainty with respect to the models presented in this work.