Sensitivity to Mass Changes of Lakes, Subsurface Hydrology and Glaciers of the Quantum Technology Gravity Gradients and Time Observations of Satellite MOCAST+

Abstract

The quantum technology absolute gravimeters, gradiometers, and clocks are at the forefront of the instrumentation to be exploited in a future gravity mission (the QSG mission concept). Apart from the quantum payload, the mission design defines the choice of the number of satellites and the satellite orbit constellation, with the goal of optimizing the observation of the earth’s gravity field and reducing aliasing phenomena. Our goal is to define the realistic gravity field changes generated by glaciers and lakes and define the sensitivity of the quantum gravity mission for the detection of hydrologic and cryospheric mass changes. The analysis focuses on mass changes in the high mountains of Asia and the South American continent. The mass changes are based on terrestrial and satellite observations and are of a climatic origin. We show that compared to the existing GRACE-FO mission, a quantum gravity mission significantly improves the detection of the climatic mass gain of lakes and mass loss of glaciers, allowing for smaller mass features to be distinguished, and smaller mass losses to be detected. The greater signal is the seasonal signal with a yearly period, which would be detected at the 10 Gt level for areas > 8000 km². The yearly mass loss of the Patagonian glaciers can be detected at the 5 Gt/yr level, an improvement from the 10 Gt/yr detectable by GRACE-FO. Spatial resolution would also be improved, with an increase of about 50% in spatial frequency for the detection of the mass change rate of lakes and glaciers in Tibet. The improved spatial resolution enables an improved localization of the lakes and glaciers affected by climatic mass change. The results will contribute to defining the user requirements of the future QSG missions.

1. Introduction

The aim of this work is to define realistic user requirements for a future quantum gravity satellite mission concerning the observation of mass variations involving the cryosphere and hydrosphere. To reach this aim, first, a database of lakes, sub-surface storage variations, and glaciers is defined; then, the gravity fields are calculated; and lastly, the detectability is obtained by comparing the signals with the sensitivity of a quantum space mission carrying the payload of a gradiometer and clocks. In the frame of the MOCAST+ project supported by the Italian Space Agency (ASI), the spectral error curve of the MOCAST+ (MOnitoring mass variations by Cold Atom Sensors and Time measures) mission was defined by an interdisciplinary group of physicists, geodesists, and mission design specialists [1,2,3]. In the present work, the sensitivity of this satellite for observing the geophysical phenomena of interest is evaluated, and its improvement in relation to different payloads and existing mission architectures is made. This includes a quantum technology satellite mission, with the observation of the gravity gradient, such as the MOCASS (Mass Observation with Cold Atom Sensors in Space) mission [4,5,6] and the successful GRACE [7] and GRACE-FO missions. The definition of the user requirements must be made knowing the approximate spectral error properties of the past and future candidate gravity missions, as requirements that are much too small to be sensed from space would not contribute a credible accomplishment goal to allow for a satellite mission to be flown in the next two decades. In this work, the realistic observable geophysical signal concerns hydrology and cryosphere, considering a selection of focus regions of broad and general interest, such as the Tibetan plateau and Himalayas and the South American continent. These regions are particularly difficult to reach with terrestrial observations, and a complete and systematic coverage of on-site observations is lacking, making satellite observations particularly relevant and primed for yielding significant scientific contributions. The user requirements of a planned mission should be sufficient to gain a significant improvement in the observation of the given mass changes of a geophysical phenomenon, which can be measured in terms of the completeness of the phenomenon, recovering the signal down to small geographical extensions or masses and small mass changes in time.

The spectral error curves of the satellite missions are colored, each with different spectral characteristics, which influence the detectability of a phenomenon depending on its lateral extent and the amount of gravity signal it produces. The GRACE mission is characterized by high precision at long wavelengths, longer than 330 km, whereas GOCE (The Gravity Field and Steady-State Ocean Circulation Explorer) achieved a better sensitivity at smaller wavelengths down to 160 km, but with less sensitivity compared to GRACE at long wavelengths. Defining a database of geophysical signals, and the spectral characteristics of the static and time-variable gravity signal they generate, contributes to setting the observational requirements and guiding mission scenarios and technological choices concerning the instrumentation for the next planned gravity missions. The pathway for the next gravity missions has already been defined; specifically, the plan involves a double pair GRACE-type mission extending the observations of GRACE and GRACE-FO beyond the end of 2027, termed the MAGIC mission [8,9]. Each of the couples of the MAGIC mission shall have a laser-ranging instrument measuring the distance between the two satellites of the couple, combined with an ultra-sensitive three-component accelerometer (with a noise floor in a measurement bandwidth of 10⁻¹¹ m/s²$/\sqrt{\mathrm{Hz}}$ [8,9]) on board each satellite for correcting non-gravitational forces acting on the satellite. One couple shall be in polar orbit, the other in an inclined orbit in the order of 65 degrees (exact value to be defined), a configuration called “Bender” [10]. The Bender configuration has been shown to recover atmospheric and oceanic disturbing signals, reducing the value of the spectral gravity error curves by orders of magnitude compared to GRACE-FO, because aliasing is greatly reduced. Furthermore, the launch of a Quantum Pathfinder Mission by end of the decade is planned, and the quantum technological breakthrough shall be reached by 2035 with a Quantum Mission for Climate and the Green Deal transition (Olivier Carraz, ESA Quantum Satellite Gravity Consultation Platform, 12 November 2021).

This study is an improvement with respect to our previous geophysical performance evaluation of the MOCASS satellite mission [4,5], which concerned the extension of the glaciers and hydrology from the high Mountains of Asia to the South American continent, and the methodology in the evaluation of the satellite sensitivity to the geophysical gravity signals. Previously, the satellite error curves were compared to the amplitude of the dominant signal wavelength of the geophysical signal; here, we use a localized spectral analysis of the geophysical signals, which we describe below in general terms and in greater detail in a companion paper [11].

The considered mass sources vary in time, with a seasonal and yearly variation for hydrology, or with a trend-like change-rate for lakes and glaciers. The time-frame of the geophysical quantity is relevant for the observations, since the error on the gravity signal recovered from the satellite mission reduces with a longer observation window, due to the increased number of observations and the increased reliability of the average and the reduced root mean square error of the observable signal. We define a signal to be detectable when part of the spectral energy of the signal is above the spectral error curve of the satellite observations. This definition relays the separation of a given signal from the multitude of signals that the satellite observes to a different analysis, which aims at a complete knowledge of the observed gravity signal in space and time. We think this is justified, because the question of whether a signal is in principle observable is different from the question of whether this signal can be separated from the other signals sensed by the satellite. Furthermore, the recent advances in the geodetic approach to calculating the gravity field includes the definition of the oceanic and atmospheric tidal and non-tidal variations, which greatly reduces the error caused by these sources [12,13].

2. Method

In this section, we illustrate the method to define whether a geophysical phenomenon is observable by the satellite mission. Our approach operates in the spectral domain and requires successive steps until we can compare the signal’s spectral curves with the satellite error curves. The different steps are explained in the subsections of this section. We define the database of hydrologic and glacier mass changes, which is relevant for the evaluation of the sensitivity of the MOCAST+ satellite, and then calculate the gravity signals these mass changes generate. First, we introduce the method with which the sensitivity of the satellite to detecting the signals is defined.

2.1. Method to Calculate the Gravity Signal, the Gravity Spectrum on the Sphere, and the Sensitivity to the Satellite Noise Spectrum

The sensitivity of the satellite observations to the detection of lake-level variations or the deglaciation process requires the conversion of the level changes to mass changes and the calculation of the gravity field generated by these mass changes. This last step is performed by discretizing the 3D mass change into tesseroids, which are elementary building blocks on the sphere and for which an algorithm exists for calculating its gravity potential [14,15], and then by applying spatial gradients to the potential to obtain the gravity field at ground or satellite height. In the next paragraphs, the gravity field change rates are shown for a selection of hydrologic basins and glacierized areas. The gravity field is calculated for each individual mass distribution at different time steps, and the gravity change rate for each calculation point is obtained by fitting a time function, for instance, a linear or higher order time variation.

We investigated whether the gravity field solutions obtained by the mission profile could sense the modelled fields (“detectability assessment”) using the error-modeling results from [3].

The detectability criterion is defined in the spectral domain by comparing the spherical harmonic (SH) expansion of the signal of the simulated geophysical phenomena and the error estimates of the mission simulations. The signal is deemed detectable if the power of its SH transform in a given SH degree range is larger than the power of the degree error variance of the retrieved gravity model [16]. Since the mass changes have local and not global character, a spatial localization of the spectrum is required. Therefore, we resorted to the spatio-spectral localization method in the spherical harmonic domain proposed by [17] via its implementation in the SHTOOLS software [18]. We adopted the spherical cap as the localization domain, defined by its center coordinates and angular radius, and use the solutions to its concentration problem, defined as localization windows. The localization function parameters depend on the extent of the chosen phenomenon, on the calculation area used for its forward modeling, and on the degree range of interest. A pitfall of localization is loss of information on the signal spectrum in the lowest and highest SH degree range due to the convolution-like operators employed. The smaller the cap radius, the more SH degrees at the spectrum ends cannot be used. This imposes an additional constrain on the cap radius to be employed.

2.2. Definition of the Modelled Geophysical Gravity Signals-South America

The South American (SA) continent hosts several different climatic conditions due to its peculiar orography and wide latitudinal extension. The northern part of the continent is characterized by tropical climate, and hosts one of the largest hydrologic basins worldwide (the Amazon basin), which is recharged by the large precipitations occurring at the orographic barrier of the Andes. The precipitations feed the sub-surface aquifer system of the Amazon plain and a network of rivers and lakes, among these is the Amazon River, which then drains the waters to the Atlantic Ocean.

The southern sector of the SA continent is more arid and is characterized by a large glaciered area, presently subjected to systematic melting episodes associated with climate change [19]. The waters drain to a nearby drainage system constituted by lakes and rivers, which then contribute to the Atlantic Ocean and in particular to the sea water level variations.

These processes are associated with large amounts of water involving seasonal variations, inter-annual transients, and long-period trends. Tracking and understanding these water mass transports is fundamental for the scientific community, especially for constraining models for evaluating climate change impacts. Gravity missions such as GRACE have already proved to be fundamental to such scopes [19]; the modeling of the sensitivity of MOCAST+ allows for the quantitative assessment of the impact of the mission in observing these mass changes.

2.2.1. Glaciers

The mass change due to glaciers melting in South America is derived from integrating the information on areal extent of glaciers from the RGI (Randolph Glaciers Inventory) catalogue [20] and ice thickness variations from glaciological observations available for a selection of glaciers in the WGMS (World Glacier Monitoring Service) database [21]. RGI outlines are shown in Figure 1a, while the locations of the glaciological observations are reported in Figure 1b (yellow dots). The glaciological observations have been elaborated so that a yearly thickness variation for each glacier was obtained.

The WGMS observations cover only a subset of the glaciers of South America, so we extrapolated average deglaciation rates to the other unmonitored glaciers in order to obtain a more realistic mass change estimation. The average rates were estimated by analyzing the database WGMS and RGI and looking at relations between deglaciation rate and area, height, and geographic position. We found no significant statistical relation between all the parameters. We only noted that the large glaciers (area > 10 km²) are systematically melting with yearly average rates of −1.0 m/yr +/−0.91 m/yr. Smaller glaciers show larger dispersion of the rates, with an average variation of −0.40 m/yr +/−0.50 m/yr. These two rates were used for extrapolating the rates on the unmonitored glaciers, dividing them into the above-mentioned two area-classes.

The final model of the yearly mass change variation for each glacier in SA is shown in Figure 1d: the plot illustrates the various contributions as function of latitude. The brown curve plots a cumulative variation, where we can see that Patagonian glaciers (at latitudes −50°) lose over 20 Gt/yr of mass. Plot of the gravity effect at 250 km for the glaciers thickness variation is shown in Figure 1c.

2.2.2. Lakes

The other phenomenon of interest regards water level variations occurring in lakes. In South America, Patagonian lakes are sensitive to climate change since they intercept the waters melted from glaciered areas nearby [22]; these lakes are also quite extensive in terms of area and their seasonal water level oscillations involve several cubic kilometers of water (1 cubic kilometer of water corresponds to 1 Gt of mass). Other interesting lakes are located in the tropical area, and some of them are associated with hydroelectric power plants, where large and rapid variations of the storage water volumes are observed. Figure 2a reports the areas of major lakes (blue) and reservoirs (red).

The mass variations occurring in these superficial water bodies are calculated in a similar way as that performed for the glaciers. The areal extents are constrained by the GLWD [23] database, while the time variations of water level are derived from the DAHITI database [24]. Both long-period variations and seasonal signal amplitudes (period 1 year) are considered.

Figure 2b shows the long-term mass variations for about 16 lakes and 7 reservoirs, together with their locations in the map. The plots on the bottom and left side of the map report the yearly mass rates as functions of longitude and latitude. We can see that the largest mass variations are due to reservoirs (red lines), with peaks exceeding the 2 Gt/yr level. Natural lakes (blue) show lower variations, generally below 1 Gt/yr. We also considered the seasonal (annual and semi-annual variations) lakes variations, which are generally larger compared to long-period trends and for reservoirs can be quite large (>20 Gt, half peak-to-peak amplitude; Figure 2b).

2.2.3. Soil Moisture Variations over SA

We employed the GLDAS product [25] to predict the temporal variations in soil moisture and the effects on the gravity field over South America. GLDAS reports the total water mass in the first 2 m of soil; it is a global model, with several spatial and temporal resolutions. We employed monthly models with 0.25° of spatial resolution.

For the scopes of the sensitivity analysis, the GLDAS model was analyzed together with a database of hydrologic basins outlines in order to obtain soil moisture time variations for each hydrologic basin. This allowed us to express the sensitivity in terms of the smallest basin, which can be monitored. The outlines and names of hydrologic basins are derived from the watershed database of the USGS (HydroSHEDS [26]), which maps the principal drainage systems worldwide from the analysis of SRTM products at 500-m resolution. The watersheds for South America are shown in the following Figure 3b in grey, while three basins, which hereinafter are analyzed in detail, are reported with three different colors (green—Orinoco; yellow—Amazon; red—Paraná).

From these databases, we calculated the gravity variations at 250 km above ground level employing a tesseroid discretization.

Once we obtained the 4-D gravity and mass models for the basins, we extracted amplitude and long-period trends by fitting every gravity/moisture time-series at a given longitude and latitude with the following relation (Equation (1)):

$$y\left(t\right)=k+mt+{\displaystyle \sum }_{i=1}^2{a}_i\cos \left(i\frac{2\pi t}{365.25}\right)+b_i\sin \left(i\frac{2\pi t}{365.25}\right)$$

(1)

Figure 3a,b show the amplitude of the seasonal component and the long-period trend of the gravity effect of the GLDAS. The side-plots report the mass variations as functions of longitude and latitude. We see that seasonal effects are clearly larger compared to long-period variations (about a factor 100) with respect to both the mass and gravity variations. The largest variations are mostly related to the three basins considered; the Amazon in particular is associated with both strong seasonal signals and a long-term depletion signal. Paraná and Orinoco show important seasonal variations but generally less pronounced long-period trends. This is also more evident when observing the mass variations of the single basins (last three plots of Figure 3).

2.3. Lakes in the Tibetan Plateau

With the aim of defining the lakes’ mass variations in Tibet, two databases have been analyzed: one based on outlines of lakes [27], and the other is DAHITI [24]. The gravity field has been calculated for the lake outlines of 2020 and the given lake height variation rates, based on satellite altimetry. In general, it is found (already seen by [28]) that the lake levels are systematically rising in a range between 0.2 m/yr and 0.6 m/yr (Figure 4). The rate is independent of the lake area and is consistent over the entire plateau. For example, the rate of one of the biggest lakes, the Qinghai Lake, is 0.2 m/yr. The only two lakes with negative rates are the Rakshastal Lake in SW Tibet and the Taruo Co.

The database of the lakes’ outlines [27] comprises over 2000 lakes; the histogram of the areal distribution (Figure 5) shows the prevalence of small lakes, between 1 and 100 km² (Figure 5a,d); rare lakes have a considerable area up to >4000 km² (e.g., Qinghai); the smallest lakes have an area in the order of 10⁻⁵ km². Plots in Figure 5e,f show for each area bin the total area covered by the lakes; it is evident that lakes <10 km², although still numerous, will not contribute significantly to the total surface lake area.

The monitored lakes of the DAHITI database pertain to the largest ones (areas > 100 km²) and the corresponding mass variations of individual lakes vary between 0.1 and 1 Gt/yr, with very rare negative rates (Figure 6c).

The mass variations of the 14 lakes for which the increase rates are known have been used to calculate the total gravity change at 250 km altitude, which is shown in Figure 7a. The maximum signal amounts to 0.1 µGal/yr, over an area of about 5° radius. The dominant signal comes from the Siling Co, Nam Co, Tangra Yum Co, and Zhari Namco lakes.

As already shown, the database of lake outlines shows that in Tibet many more lakes than those monitored in DAHITI exist, but for which the time series are not available. Given the homogeneity of the level increases over the Plateau, and the lack of correlation of the rates with latitude/longitude (Figure 6d), height (Figure 6b), or lake area (Figure 6a), we extrapolated the rates to the entire lake database. Here, only the lakes with area above 4 km² are taken into account, neglecting the very small lakes, which we had already shown to scarcely contribute to the total surface area. The rate was calculated from averaging the rates of the known lakes and it amounts to +0.25 m/yr +/−0.15 m/yr. For the estimate, we excluded the two lakes that show negative rates, because they are not representative of the overall level changes. Figure 6d shows the mass change for every lake as a function of longitude and latitude. We also computed the yearly cumulative water mass change for the entire Tibetan plateau, which amounted to 8.8 Gt/yr +/−5.3 Gt/yr (dashed and solid red lines in Figure 6d bottom). The gravity effect of the remaining lakes, with interpolated level variation of 0.25 m/yr, is given in Figure 7b, and amounts to 0.2 µGal/yr. The total signal, considering the lakes in DAHITI, summed to the other lakes, gives a positive gravity change of 0.32 µGal/yr, and an estimated mass variation of about 11 Gt/yr +/−5.3 Gt/yr. This shows that the extrapolation of the rates to the smaller lakes is necessary to retrieve a realistic mass change, since the small lakes add up to a comparable mass change, or greater, compared to the big lakes.

2.4. Summary of the Simulated Hydro-Glacio Phenomena and Gravity Signals

In

Table 1. Summary of the simulated phenomena and gravity signals.

Phenomenon	Localization	Area/Equivalent Circular Radius (r) or Equivalent Ellipse Semi-Axes (a, b)	Mass Change Seasonal Amplitude (Half-Peak2peak)	Gravity Change Seasonal @250 km Altitude. Full Spatial Resolution (Half-Peak2peak)	Mass Change Long-Term (Absolute Value)	Gravity Change Long-Term @250 km Altitude. Full Spatial Resolution	Spatial Scale = Radius of a Spherical Cap, Which Includes the Gravity Anomalies
Glaciers	Patagonia-Andes	3 104 km2 a = 500 km b = 20 km	-	-	Single glacier: >1 Gt/yr Overall (clustering): 20–30 Gt/yt	0.001 mGal/yr for Patagonia (clustering)	8° including the whole Patagonia cluster
Natural Lakes	Whole South America	Titicaca 9 103 km2 R = 50 km	Single, 1–5 Gt	0.0001 mGal	Generally <1 Gt/yr	<0.00001 mGal/yr for single lake (Titicaca)	5–6° for single lake (i.e., Titicaca)
Reservoirs	Whole South America	Sobradinho 9 103 km2 r = 50 km	Single, can be >15 Gt	0.001 mGal	<3 Gt/yr	Max 0.0002 mGal/yr for single reservoir	5–6° For single reservoir (i.e., Sobradinho)
Natural Lakes	Tibet	Qinghai 4.5 103 km2 r = 35 km Whole Tibet 5 104 km2 r = 125 km	-	-	Single lake 1 Gt/yr (i.e., Qinghai); whole Tibet 11 Gt/yr	Single Lake 0.0001 mGal/yr; 0.0003 mGal/yr for clustering	3–5° Single Lake 12–15° for whole Tibetan plateau
Sub-surface hydrology (Soil Moisture GLDAS)	Amazon basin	6 106 km2 r = 1350 km	>600 Gt	>0.01 mGal	20 Gt/yr	0.0005 mGal/yr	17° for whole basin.
Sub-surface hydrology (Soil Moisture GLDAS)	Paraná basin	2.5 106 km2 r = 900 km	>100 Gt	0.005 mGal	2 Gt/yr	0.0001 mGal/yr	13° for whole basin
Sub-surface hydrology (Soil Moisture GLDAS)	Orinoco basin	0.9 106 km2 r = 530 km	>80 Gt	0.005 mGal	1 Gt/yr	0.0001 mGal/yr	10° for whole basin

, for each phenomenon considered we report the area and equivalent radius of the basins/lakes/glaciers, the mass changes involved for both long-period trends and seasonal variations (if available), and the gravity peak amplitude of the gravity anomaly calculated at 250 km altitude and an estimate of the spatial scale of the phenomenon. We define the spatial scale as the radius of a spherical cap that includes the gravity anomaly. Such caps are also the base for the localized spectral analysis for comparing the signals with the error curves, as explained below.

2.5. Definition of the Spectra of Geophysical Phenomena and the Error Curves of MOCAST+ and GRACE

Having calculated the space-time variations of the gravity signals of our geophysical database, we then proceed to the calculation of the localized spectra. Depending on the spatial extent of the phenomenon, the radius of the spherical cap was chosen accordingly to entirely contain the gravity signal.

The detectability is then defined by requesting the signal spectrum or at least part of it to be larger than the mission error spectral curves. We consider the error degree variances of GRACE [7,29] (GRACE Sci. team meeting), which is defined as the average of 150 monthly error curves for the interval 2002–2017. For MOCAST+, different configurations have been calculated, including Bender configurations with couples or triplets of satellites [10], and with payload of a gradiometer (Tzz or Tyy components; x,y,z Cartesian right-handed system centered on the satellite; x along orbit, z towards center of earth, and y orthogonal to orbit plane) and a supplemental clock, assuming “nominal” and “improved” scenarios for the clock’s performance [3]. In particular, the “nominal” and “improved” clocks differ in terms of sampling frequency, which in the “improved” clock’s case rises to 1 Hz from the 0.1 Hz of the nominal frequency. For both the clocks, the sensitivity is at the level of 0.2 m²/s²$/\sqrt{\mathrm{Hz}}$ for a broad frequency band. The clocks are used to measure the potential difference between the satellites and this is performed by employing three clocks in combination with an optical link [3]. This strategy has been demonstrated to better mitigate both the noises arising from baseline fluctuations as well as the phase noise of the local oscillator. Apart from this, no other laser or optical links are required. The gradiometers have similar performance to those already simulated in the frame of the MOCASS study, with a sensitivity better than 10 mE $/\sqrt{\mathrm{Hz}}$ in the frequency range of 10⁻⁶–10⁻² Hz. In the testing phase, different distances between couples or triplets of satellites were also considered (100 km or 1000 km). A further factor affecting the instrumental noise is the attitude control error, which affects the Tzz component, while the out of orbit plane component (Tyy) is insensitive to rotations around the y-axis, where the largest control error occurs. The attitude control in the simulations, with respect to the optimal noise scenario for the gradiometers, is set at the 0.1 nrad/s level, which is below that of the present gyroscope technology. For this reason, the error curve for the Tyy recovery was included, which due to its lower criticality to attitude control enables the full exploitation of the instrumental noise level. The following table,

Table 2. Mission configurations simulated in the frame of the MOCAST+ project. D = distance between couples or triplets of satellites.

Name Legend in Figure 8	Configuration
MOCAST+ Bender 2 couples 4 Tzz+ Cl nominal	Bender configuration (2 couples, polar+inclined, 4 Tzz gradiometers) D = 100 km; 0.1 Hz clocks; optimal noise PSD for the gradiometers
MOCAST+ Bender 2 couples 4 Tzz + Cl improved	Bender configuration (2 couples, polar+inclined, 4 Tzz gradiometers), D = 1000 km, 1 Hz clocks, optimal noise PSD for the gradiometers
MOCAST+ Bender 2 triplets 6 Tzz + Cl improved	Bender configuration (2 triplets, polar+inclined, 6 Tzz gradiometers), D = 1000/2000 km, 1 Hz clocks, optimal noise PSD for the gradiometers
MOCAST+ Bender 2 triplets Cl improved	Bender configuration (2 triplets, polar+inclined), D = 1000/2000 km, 1 Hz clocks, clock-only solution
MOCAST+ Bender 2 triplets 6 Tyy Cl improved	Bender configuration (2 triplets, polar+inclined, 6 Tyy gradiometers), D = 1000/2000 km, 1 Hz clocks

, summarizes the different configurations simulated, while the corresponding error curves are shown in Figure 8.

As is evident, the best configuration, which corresponds to the lowest error curve in Figure 8, is obtained with 2 triplets carrying improved clocks, the gradiometer, and an inter-satellite distance (D in Table 2) of 1000 km. For a more complete discussion of the mission scenarios and simulations we refer to [2,3].

All the error estimates have been simulated for a timespan of one month [2]. In the case of the “MOCAST+ Bender 2 couples 4 Tzz+ Cl nominal” configuration, we were also provided with the results of simulations covering two and four months. Apart from this specific case, timespan scaling was applied to account for the reduction in error due to longer observation intervals. This reduction, arising from the improved coverage and stacked measurements, was modelled with the following scaling relationship:

$${\sigma }^2{\left(l\right)}_{{\mathsf{\Delta }\mathrm{t}}_1}={\sigma }^2{\left(l\right)}_{\mathsf{\Delta }{t}_0}\cdot \left(\mathsf{\Delta }{t}_0/\mathsf{\Delta }{t}_1\right)$$

(2)

where ${\sigma }^2\left(l\right)$ is the error degree variance at SH degree $l$ formulated for the solution timespan $\mathsf{\Delta }{t}_0$ (original time interval) and $\mathsf{\Delta }{t}_1$ (time interval we are scaling to).

To simplify the figures, hereinafter, the sensitivity to the geophysical signal is determined by employing only the best configuration (blue curve in Figure 8).

3. Result

We now proceed to define the benefit of the quantum gravity mission for detecting the hydrologic and glacier mass changes of the lakes and glaciers defined in the previous section. The sensitivity has been calculated separately for South America and Tibet.

3.1. Sensitivity of MOCAST+ to Hydrology and Glaciers in the Andes

3.1.1. Sensitivity to South America Glaciers

As previously seen, most of the ice-loss mass in South America occurs over a circumscribed area in Patagonia, which causes a long-term gravity trend, already detectable by the GRACE and GRACE-FO missions [30].

Other areas in the Andes are covered by glaciers and could experience similar ice-thickness variations and sequent-mass variations; given the smaller extent and the probable minor deglaciation rates, in this case, the expected signals are smaller in terms of amplitude.

Figure 9b reports the error curves of the MOCAST+ mission after 1 year of data accumulation together with GRACE’s (red) error curve and glacier signals (black/grey lines). Regarding the signals, we considered two areas subject to deglaciation for which we calculated the localized spectra. The areas and caps employed for the analysis are shown in Figure 9a with dashed ellipses.

By comparing the signal and noise curves, we observe that the glacier signal originating in Patagonia (black solid line) is observable by both missions; however, MOCAST+ improves the detectability in terms of a higher achievable spatial resolution. This increase in the spatial resolution is relevant, as now the anomaly is seen up to almost degree and order of 53 in the SH expansion, while GRACE stops at degree 28. This corresponds to a leap ahead in spatial resolution from about 715 km to 380 km in terms of half-wavelength. A visual representation of the improvement in terms of resolution can be seen by comparing Figure 9c,d.

The other glaciers considered are located in the Andes and cover a smaller area compared to the Patagonian ones. For this reason, we localized the spectrum on a smaller cap with a radius of 5°. The corresponding spectrum is shown in the plot in Figure 9b with the solid dark grey line and, as clearly observable, such a phenomenon is only detectable with the MOCAST+ mission. Even when increasing the radius of the localized window—thus allowing for a consideration of larger wavelengths and thereby favouring a GRACE-like mission concept—this localized region remains undetectable by GRACE.

The last aspect we aim to remark upon is that the lower error curve of MOCAST+ compared to GRACE implies that smaller yearly mass change rates occurring in the Patagonian glaciers can be monitored. In fact, if we scale the Patagonia signal spectrum (black line in Figure 9a) by a factor of 10 we reach the noise level of MOCAST+ (black dotted curve), while with a factor 5 we approach the GRACE error curve (black dashed curve). This implies that with a MOCAST+ mission we can monitor yearly mass variations over three times smaller than the actual trends (about 20–25 Gt/yr), while with GRACE the sensitivity is increased roughly by a factor of less than two.

3.1.2. Sensitivity to Subsurface Hydrology in South America

We considered long-period trends and seasonal signals for three major basins in South America, namely, the Amazon, Paraná, and Orinoco.

In Figure 10a–c the yearly long-period gravity rates in these three basins and the outline of the caps considered for the spectral analysis are reported. For the Amazon, given its large dimensions and large signals, we considered two windows centred on the two local minima in the gravity anomalies. A comparison with the yearly error curves of MOCAST+ and GRACE is offered in Figure 10d.

The impact of MOCAST+ is again evident in terms of an increase in spatial resolution; in particular, for the Amazon basin, the two “blobs” are better depicted. The largest minimum (bounded by white dots in Figure 10a) is theoretically observable by both the GRACE and MOCAST+ missions; while the other local minimum, located towards the Andes orogeny, would be detectable after 1 year of observations only by a MOCAST+ mission concept.

In addition, a smaller basin, such as the Paraná basin, generates long-period signals just visible by the MOCAST+ mission and clearly not at all detectable by GRACE. The last basin considered, the Orinoco, is the smallest basin and also has the smallest long-term mass variations. It also shows a more complex pattern of anomalies in the spatial domain (Figure 10a). The signal degree variance for the whole basin, calculated within a cap of radius 10°, is lower than the MOCAST+ error by about one order of magnitude. Even considering a more localized window centred on the largest anomaly, the signal remains below the MOCAST+ sensitivity.

Regarding the subsurface hydrology, we further tested the improvement of the MOCAST+ mission for detecting the seasonal signals.

In order to verify a detectability criterion, the monthly error curves of various missions were compared to a 1-month signal during the seasonal peak. The three snapshots of the gravity field for the Amazon, Paraná, and Orinoco are shown in Figure 11a–c, and were taken at the months of February, January, and July, respectively. These months correspond to the maximum seasonal peak.

We see that the Amazon has the largest signal both in terms of amplitude and of areal extension; the Paraná has a seasonal signal mostly localized nearby the Amazon, which is also in phase with the Amazon. The Orinoco on the other side has an important gravity signal, as large as the Paraná, but more localized. It is also not in phase with the seasonal signals of the other two basins. Considering the different areal extensions of the anomalies, we opted for caps with different radii for the computation of the localized spectra, in particular, the radii of 15°, 10°, and 7°, for the Amazon, Paraná, and Orinoco basins, respectively.

The contribution of MOCAST+ is again its ability to recover the anomalies at a higher spatial resolution. The improvement is, as expected, more pronounced for the smaller basins, such as the Orinoco and Paraná, compared to the Amazon. This is because the Amazon seasonal signal has most of the energy at low degrees with a spectrum rapidly decaying at higher frequencies. For the other two basins, the spectra decay more slowly, resulting in a remarkable leap in resolution. The Orinoco could be seen up to d/o 53 with MOCAST+, while GRACE stopped at 35. A similar increase in resolution was obtained for the Paraná basin.

Apart from the increase in resolution, MOCAST+ was able to detect smaller amplitude seasonal signals with respect to GRACE. For instance, we can lower the Orinoco signal curve by two orders of magnitude in order to reach the noise level of MOCAST+. Since the seasonal mass variation of the Orinoco is about 80 Gt, MOCAST+ would be able to monitor a basin with a similar areal extension but with a lower seasonal mass variation, down to about 10 Gt. For GRACE, the threshold is higher, so that a basin with seasonal mass variations of about 25 Gt is observable.

3.1.3. Sensitivity to Lakes/Reservoirs in South America

The following Figure 12a,b reports the yearly long-period gravity rates and the seasonal amplitude signal for the considered lakes and reservoirs in South America. Dotted circular areas show the window outlines employed for the localized spectral analysis.

As described in the previous sections, the reservoirs generate the largest mass variations both concerning the long-period variations and the seasonal signals. The reservoirs’ long-term trends are in the order of a few Gt/yr signals, while the seasonal mass changes can be an order of magnitude larger (>10 Gt). The Patagonian lakes (Argentina, Buenos Aires, and Viedma) are the only ones experiencing a systematic mass increase, which is due to the deglaciation occurring in the Patagonian glacierized region. However, the amplitude is more than two orders of magnitude lower than other natural lakes (Titicaca) or reservoirs (Figure 12b).

The yearly long-period trends, expressed as the gravity variation in one year, have been again compared with the yearly error curves of the various missions (Figure 12c).

We see that both the reservoirs are close to the noise level of the MOCAST+ mission and in principle with two years of observations their effect would be detectable. Titicaca, given the smaller mass rates observed, would be monitored at this spatial resolution after 5 years of observations. We did not plot the spectra of the Patagonian lakes’ effects since the signal amplitude is several orders of magnitude below the sensitivity of all the curves. Even considering all the nearby smaller lakes in the Patagonian region to be subjected to the same average level rise would result in an undetectable signal. In any case, GRACE would be inadequate for monitoring the superficial water storage variations in all these reservoirs/lakes.

Regarding the seasonal variations of lakes and reservoirs, the spectra of Figure 12d report the sensitivity between the amplitude signal recorded at the month when the maximum seasonal signal is observed. It appears that only the hydrologic variations in the Sobradinho would be above the detection threshold of MOCAST+.

3.2. Sensitivity to Lake Hydrology in Tibet

We considered the long-period gravity trends caused by a systematic water mass increase in the lakes over the entire Tibetan plateau. In order to test the sensitivity of the MOCAST+ mission, we considered two models; one model includes only the 14 lakes for which we have a DAHITI time-series at our disposal. The other model includes the effect of all the lakes of the plateau; as detailed previously, the effect was estimated by extrapolating an average lake level rate derived from the DAHITI observations to all the remaining lakes. In the following Figure 13a,b, we report the two models considered. The colour scale is the same for both plots, so that the potential important contributions—in terms of the gravity signal—given by all the unmonitored lakes appear clearly.

For the known lakes, we report the localized spectra for Qinghai Lake and for a cluster of lakes in the Central Tibet. For this central area, two window caps were considered, one centred on the largest anomaly and the other, of greater size, which encloses the entire central Tibetan region.

Considering the yearly solutions, all the yearly gravity signals were below the noise level of both MOCAST+ mission and GRACE. Therefore, we rescaled the curves considering the expected errors of a two-year solution and the trend signal accumulated after two years (Figure 13c). The anomalies of the Qinghai and the central Tibet plateau (the one with smaller localization cap) after two years are clearly below the sensitivity of GRACE but would be detectable by the MOCAST+ configuration. We remark that in both cases the mass changes of the single lakes are very small and generally less than 1 Gt/yr. However, the spectral analysis proves that a long-period monitoring of these subsurface storage units is feasible only with mission concepts with noise characteristics at least as adept as MOCAST+. The spectral analysis of the cumulative effect of all lakes in the Tibetan plateau is given in the plot of Figure 13d. We see that when considering this window with a radius of 8°, only a MOCAST+ mission would be able to detect the yearly long-period trend. To capture a picture at this spatial resolution GRACE requires at least 2 years of observations. GRACE can see only a low-pass filtered anomaly, with a maximal d/o 15 as evident from Figure 13e, where the signal curve is higher than the GRACE noise only for a very wide cap window of an 18°radius.

3.3. Trade-Off between Spatial Resolution and Time-Span of Observations

For all the long-period trends considered, by augmenting the timespan of observations (i.e., from 1 year to 2 year solutions) we obtain a combined effect of lowering the error curves and contemporarily increasing the signal amplitude since the trend signal cumulates over the years. This results in an increase in the spatial resolution as the solution is integrated in time, and this increase in resolution helps to better localize and depict such long-period trends. In the context of the MOCAST+ simulations, the Paraná basin and Tibetan lakes offer interesting examples, demonstrating at best the impact of MOCAST+ with respect to GRACE in depicting the anomaly patterns at higher resolution.

Figure 14 shows the impact of longer observations in terms of the maximum degree/order resolvable for MOCAST+ (top plot) and GRACE (bottom plot). The Paraná basin’s sensitivities are included in the left column, while the sensitivities for the Tibetan lakes are shown in the right column. In these plots, the green-tinted curves are the error curves of the missions scaled down by a factor proportional to the increase in the timespan of observation (Equation (2)). The purple-tinted lines are the trend signal, which cumulates from 1 up to 5 years, so the spectrum of the gravity changes for up to 5 years has been calculated. The black circles show the intersection between the signal and noise curves, defining the highest resolution obtainable.

For the Paraná basin, we can see that with a 5-year solution of GRACE the anomaly is resolved up to a maximum degree and an order of 32, while after 5 years a MOCAST+ type mission would be able to recover the signal up to a degree and order of 50. For the Tibetan lakes, the situation is analogous, with a maximum d/o of 49 for MOCAST+, while GRACE stops at around d/o 32 (Figure 15 bottom row).

The improvement in the resolution results in a better definition of the maximum amplitude and smaller leakage effects (Figure 15).

3.4. Setting the South American and Tibetan Lakes and Basins in a Global Context

As a final consideration, we placed the lakes’ and hydrologic basins’ mass variations into a global context, which was possible by merging the different databases of the lakes/basins’ outlines with the time series of the lake level/moisture variations. The procedure entails calculating the time series of the mass variation for each water body/basin and then extracting the seasonal amplitude through the best fit of Equation (1). The Gigaton seasonal variation vs. the basin area is plotted in Figure 16, distinguishing between the basins of Africa, South America, and Asia. The analysis has considered basins with areas > 5 10⁵ km². Given the previous analysis, we have an estimate of the detectability limit of GRACE and MOCAST+ for basins of the size of the Orinoco and Paraná (Figure 11); these thresholds are shown as horizontal lines for the basin areas of 9 10⁵ km² and 2.5 10⁶ km². In order to be monitored by satellite gravimetry, a basin should be above the detectability threshold. In South America, we did not find basins with areal extensions similar to the Orinoco (about 10⁶ km²) with seasonal mass variations at the detectability limit (8–10 Gt). However, in other climatic contexts, as in South Africa and Asia, there appear to be some possible basins fitting these criteria. Monitoring such basins would definitely benefit from a high-sensitivity mission such as MOCAST+.

Similar to what we performed previously for the GLDAS seasonal component, we verified which other potential lakes/reservoirs worldwide could benefit from a MOCAST+ monitoring. We considered seasonal mass variations occurring in the lakes/reservoirs of the DAHITI database [24] for Asia and Africa. For each lake/reservoir time-series of the DAHITI database we extracted the annual seasonal lake level amplitude and multiplied it by the area of the basin, obtained from the watershed database [23]. This way we could easily obtain the seasonal mass variation.

Figure 16 reports the seasonal mass variation for the considered lakes as a function of the area for lakes/reservoirs in Africa (blue), Asia (red), and South America (black). There is clearly a scaling law that causes the large trend in the scatter plot; however, inside a small interval of areas, we also observe a remarkable variability in terms of the seasonal mass component.

According to our simulations and detectability criteria, the Sobradinho reservoir acts as a lower bound for a clear detection of the seasonal signals of lakes/reservoirs. Therefore, lakes and reservoirs with similar seasonal amplitudes and equal or larger areas would certainly benefit from a MOCAST+ mission. This area of interest is bounded in the graph by two vertical and horizontal dashed lines. Several lakes in the Rift valley are particularly interesting targets for such monitoring approach.

4. Discussion

The process of defining and evaluating the performance of the future innovative gravity mission uses the Scientific Readiness Levels (SLR) as a guideline. The SLRs extend from SRL1, the consolidation of the Initial Scientific Idea through SRL5, the End to End Performance Simulations, to reach SRL 9, the Scientific Impact Quantification [31]. We collocate the work we have performed in the frame of the MOCAST+ project in relation to SRL4 (Proof of concept and Measurement concept validation). Our work is based on the Initial Scientific Idea (SRL1) that the determination of the gravity field with respect to GRACE and GOCE can be improved in order to observe a number of geophysical phenomena that are presently below the sensitivity of the actual missions. The hypothesis is that a larger impact can be achieved by observing a larger percentage of the overall existing phenomena and a at higher completeness level. In the example of hydrologic basins, it has been shown that the drawdown of large-scale hydrologic basins is observed by GRACE/FO, but it presents only a limited number of applications and coverage, and a small percentage of the total hydrologic volume. Therefore, a more complete identification including smaller basins is needed and is an observational innovation, since deep hydrologic reservoirs may not be accessible by terrestrial observations.

Within the MOCAST+ study we assessed the value of this new mission concept by monitoring a wide range of geophysical phenomena that cause temporal variations of the mass on the Earth surface and underground. The whole study comprised the construction of a database of realistic mass models including the earthquake processes [11], the surface and subsurface hydrology, and mass changes related to glaciers. Herein, we discussed the lakes, hydrology, and glaciers in South America and the lakes in the Tibetan plateau. The database comprises phenomena at different spatial and temporal scales, including localized mass variations as deglaciation in Patagonia as well as widespread spatially continuous mass changes such as those due to the subsurface soil moisture over the entire Amazon basin. Seasonal and multi-year long-period trends were also considered.

The database also includes the forward modelled gravity field; thus, we were able to create a link between mass variations involved in the process, their spatial extension, the temporal duration of the phenomenon, and the simulated gravity anomalies.

With this database, we then quantitatively assessed the impact of the mission by firstly defining the detectability threshold of each phenomenon; this was performed by comparing the noise error curves of the satellite missions with the localized spectra of the geophysical phenomena.

On the basis of this comparison, we then defined the improvement of the new mission with respect to the present state of the art, represented by GRACE and GRACE-FO observations. In particular, the improvement was defined in terms of:

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The ability to observe a larger number of phenomena (i.e., smaller glaciers, smaller basins);

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An increase in the spatial resolution for the better characterization of a specific phenomenon;

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An increase in the sensitivity to lower mass changes (i.e., sensitivity to lower deglaciation rates).

We summarize our main findings in

Table 3. Overview of the detectability of geophysical signals by GRACE and MOCAST+. For MOCAST+, the comparison is relative to GRACE. d/o = degree and order; λ/2 = half-wavelength of highest resolvable d/o.

Phenomenon	Mass Variation/Gravity Signal	GRACE	MOCAST+ (Best Configuration)
Glaciers’ long-term variations in Patagonia	Patagonia cluster: 20–25 Gt/yr; Area 30,000 km2; Cap area: 8°; Signal spectra 1yr: d/o 25 @ 250 km → Tr2 = 10−8 mGal2 d/o 45 @ 250 km → Tr2 = 2 × 10−9 mGal2	-Detectable after 1yr. Max d/o 32 (λ/2 = 625 km). -Minimum rate observable (about 10 × 12 Gt/yr)	-Detectable after 1yr. Max d/o 53 (λ/2 = 380 km). -Minimum rate observable (about 5–8 Gt/yr)
Natural Lakes of South America: long-term trends	Lake Titicaca: 0.5–1 Gt/yr; Lake area 9000 km2; Cap area = 6° Signal spectra 1yr: d/o 32 @ 250 km → Tr2 = 7 × 10−11 mGal2 d/o 45 @ 250 km → Tr2 = 1 × 10−11 mGal2	Lake Titicaca: not detectable even after 5 years	Lake Titicaca trend: detectable after more than 5 yr
Reservoirs of South America: long-term trends	Sobradinho: 2–3 Gt/yr; Reservoir area 9000 km2; Cap area= 6° Signal spectra 1yr: d/o 32 @ 250 km → Tr2 = 9 × 10−9 mGal2 d/o 45 @ 250 km → Tr2 = 1 × 10−10 mGal2	Detectable after 5 years	Detectable after 2 years.
Natural Lakes in Tibet: long-term trends	Whole Tibet: 10–12 Gt/yr; Lakes area 50,000 km2; Cap area = 8° Signal spectra 1yr: d/o 25 @ 250 km → Tr2 = 0.5 × 10−9 mGal2 d/o 45 @ 250 km → Tr2 = 5 × 10−11 mGal2	Cumulative effect detectable after 2 years. Max d/o 26 (λ/2 = 770 km).	-Detectable after 1 year. Max d/o 28 (λ/2 = 715 km). -In 2 years increase in resolution with respect to GRACE. Max d/o: 34 (λ/2 = 590 km).
Amazon sub-surface hydrology (Soil Moisture GLDAS): Long-period trends	Amazon 1: 15 Gt/yr; Basin area 6 106 km2; Cap area = 8° Signal spectra 1yr: d/o 25 @ 250 km → Tr2 = 1 × 10−8 mGal2 d/o 45 @ 250 km → Tr2 = 5 × 10−11 mGal2	Detectable after 1 year. Max d/o 28.	Detectable after 1 year with higher resolution. Max d/o 32 (λ/2 = 625 km).
Paraná sub-surface hydrology (Soil Moisture GLDAS): Long-period trends	Paraná: 2–3 Gt/yr; Basin area 2.5 106 km2; cap area: 7° Signal spectra 1yr: d/o 28 @ 250 km → Tr2 = 1 × 10−9 mGal2 d/o 45 @ 250 km → Tr2 = 1 × 10−10 mGal2	Detectable after 2 years	Detectable after 1 year. On the noise level.
Orinoco sub-surface hydrology (Soil Moisture GLDAS): Long-period trends	Orinoco whole basin: <1 Gt/yr; Basin area 0.9 106 km2; cap area: 10° Signal spectra 1yr: d/o 20 @ 250 km → Tr2 = 7 × 10−11 mGal2 d/o 45 @ 250 km → Tr2 = 3 × 10−12 mGal2	Not detectable	Not detectable
Sub-surface hydrology (Amazon, Paraná, Orinoco): seasonal signals	Orinoco: 80 Gt; Basin area 0.9 106 km2; Cap area: 7° Signal spectra 1 month: d/o 28 @ 250 km → Tr2 = 1 × 10−6 mGal2 d/o 45 @ 250 km → Tr2 = 5 × 10−7 mGal2	Detectable. Max d/o 35 (λ/2 = 570 km).	-Improvement in spatial resolution of monthly solutions. Max d/o 50 (λ/2 = 400 km). -Higher Sensitivity to smaller seasonal mass changes in basins. For a basin such as Orinoco 10 times smaller mass variations can be detected.
Lakes/Reservoirs in South America: seasonal signals	Sobradinho: 20 Gt/yr; Reservoir area 9000 km2; Cap area: 6° Signal spectra 1 month: d/o 32 @ 250 km → Tr2 = 7 × 10−11 mGal2 d/o 45 @ 250 km → Tr2 = 1 × 10−11 mGal2	Hardly detectable	Sobradinho reservoir seasonal component detectable

.

The comparison of the signal spectra with the satellite error curves has shown that a higher sensitivity regarding the deglaciation rates and height variations in the lakes was gained, both in terms of the ability to resolve smaller glaciers and lakes and the sensitivity to capture smaller mass changes. The localized spectrum depends on the cap radius, with a greater radius lowering the average spectral amplitudes and extending the spectra to lower harmonic degrees. Thus, there is a trade-off between the spectral amplitudes and the spectral interval over which the amplitudes are distributed.

5. Conclusions

The spectral curves we have obtained for the different phenomena enable the provision of feedback to the choices of the payload and satellite mission concept, which both affect the steepness of the error curves in the spectrum. Our assessment of the MOCAST+ results with respect to the possibility of identifying geophysical phenomena is summarized via the following improvements:

• The monitoring of seasonal components of reservoirs, lakes, and glaciers with areas > 8000 km² and seasonal mass variations of 10 Gt: the estimate of the seasonal component allows for the retrieval of long-period trends with minor uncertainty;
• The sensitivity to deglaciation processes: in the case of Patagonian glaciers, the minimum rate observable with MOCAST+ amounts to 5 Gt/yr, an improvement compared to the GRACE observations, which detect a level of 10 Gt/yr;
• The spatial resolution for the long-term monitoring of hydrologic basins and lakes: considering the example of the Tibetan lakes, which gain about 10 Gt/yr, MOCAST+ after 1 year resolves the variation that GRACE resolves after two years; after 5 years, MOCAST+ can retrieve the signal up to degree 50 in the spherical harmonic expansion, whereas GRACE resolves this signal only up to degree 32.

This research could be brought to the level SRL5, which is focalized on the end-to-end simulations. Our database and the strategy to assess the impact of the mission could be developed into a flexible tool, which could be used to assess the performance of the next generation of gravity-mission-concept choices.