Long memory and trend in time series of precipitation in Mozambique

Many climate studies in Mozambique have clearly identified signals of climate change, especially changes in the extreme temperatures. Regarding precipitation, there is still a gap on the knowledge of how it is behaving due to both internal and external factors in the climate system. In this study, we have investigated the existence of long-term correlations and trend in time series of precipitation. Two databases were used for this purpose: in situ observations along the period of 1960–2020 and the Climate Hazards group InfraRed Precipitation with Stations (CHIRPS) dataset, along the period from 1981 to 2021. We have applied the rescaled-range analysis and the detrended fluctuation analysis for long memory investigation, and the linear regression and Mann-Kendall methods for trend analysis. Results have shown the existence of long memory in precipitation in most parts of Mozambique, being stronger in the southern and central regions and weakening toward the north of the country. On the other hand, significant trend signals of precipitation were detected in some isolated areas of Mozambique, presenting an increase in some regions such as the southern part of Manica and eastern of Inhambane provinces and a decrease in other regions such as the coastal areas of Zambezia and Nampula. These findings indicate that the probability of a random occurrence of precipitation is minimal, and the observed trends are likely to continue for a long period in future. Dry land agriculture should be prepared to adapt to new precipitation regime in the regions mentioned hereof.


Introduction
Climate variability usually exhibits long memory or a longrange dependence, meaning that the present climate observations may have long-term influence on future climate states (Beran 1994;Yuan et al. 2014;Qiu et al. 2020).Long memory in time series can be understood as the existence of autocorrelation in observations at long lags, that is, observations are not independent; each observation is affected by the precedent events.This long-range dependence in climate variables is associated with the slowly responding subsystems such as oceans and large forestry areas (Yuan et al. 2014).Mandelbrot (1967) related this scaling behavior to fractals.A fractal is an object whose geometry presents infinite self-similarities at different scales.In the context of climate variables, self-similarity is defined in terms of the distribution of the data series.The concept of fractal was introduced after Hurst (1951) had formulated the rescaled-range (R/S) analysis, while studying dams dimensioning over the Nile River.Hurst's work had culminated in an empirical relation, whose parameter, known as Hurst exponent (H ), carries information about the long-range dependence (long memory) of the time series.
Besides the R/S analysis, many other methods have been proposed for detecting the existence of long memory in time series, such as the detrended fluctuation analysis (DFA), periodogram method, aggregated variance method, and structure function method.With these methods, fractal and multifractal properties in different climatic variables have been investigated by many researchers (Caballero et al. 2002;Vyushin and Kushner 2008;Yuan et al. 2014;Qiu et al. 2020;Vera-Valdés 2021).
Each proposed method has its own potential.When using only one method, in some cases, a false detection of long-range dependence is possible, due to the existence of noise and strong trend signals in the data.While long memory is intrinsic in the data, noise and trend signals are caused by external effects, and in particular, trend is usually supposed to have a smooth and monotonous or slowly oscillating behavior (Kantelhardt et al. 2001).One example is the global warming that is being driven by the intensification of greenhouse gases concentration in the atmosphere, leading to an increasing heat-storage capacity.This particular climate driver is increasingly being associated to human activities.According to Arias et al. (2021), from the IPCC-AR5 to IPCC-AR6 (fifth to sixth assessment reports of the Intergovernmental Panel on Climate Change), new techniques and analyses have provided greater confidence in attributing changes in regional weather and climate extremes to human influence.
Climate observations and projections for Mozambique are reported by various researchers (Queiroz et al. 2007;Mcsweeney et al. 2010;Machaieie et al. 2020;Ussalu and Bassrei 2021;Mavume et al. 2021), and all have found climate change signals in Mozambique.For instance, significant changes in maximum and minimum temperatures have been observed throughout all regions of the country.On the other hand, precipitation behavior throughout the country is poorly known, principally at local scale.This is partly due to the fact that, unlike temperature, precipitation is by nature less representative, generally presenting great variability, both spatially and temporarily.It is significantly influenced by local factors, so that globally, it does not present clear trend (IPCC 2014(IPCC , 2021;;Arias et al. 2021), even at regional scale (Lim Kam Sian et al. 2021).
The occurrence of precipitation over the southeastern Africa region is influenced by the following climate drivers: (i) The Inter-tropical Convergence Zone (ITCZ), (ii) cold fronts from the south associated with the right arm of the South Atlantic anticyclone, (iii) continental depressions of thermal origin, (iv) Indian Ocean sea surface temperature (SST) patterns, (v) tropical cyclone invasions, (vi) El-Niño South-Oscillation (ENSO), and (vii) the Mozambique Channel trough (MCT) (Reason and Jagadheesha 2005;Pomposi et al. 2018;Ambrosino et al. 2011;Barimalala et al. 2020).While the ITCZ has influence only over the northern region of Mozambique, cold fronts affect the southern region only.The last two factors present a dipole-like impact on precipitation throughout Mozambique.For instance, with El-Niño, the southern and central regions experience a prolonged dry period, while excessive rainfall is observed in the northern.The reverse happens with La-Niña (Reason and Jagadheesha 2005).On the other hand, Barimalala et al. (2020) had observed that strong MCT summers cause an increase in rainfall over northern Mozambique, while a rainfall deficit is seen in southern Mozambique, and the opposite is observed with weak MCT summers.
The knowledge of local precipitation regime and dynamics is crucial, specially for water resources management and for the agricultural sector in Mozambique, where in most rural areas (90%), it is mainly practiced under dry land conditions, with limited usage of irrigation technologies, resulting in lower production levels (Salite and Poskitt 2019;Manuel et al. 2020).It is important to understand the current state and the future climate dynamics in the three major regions of Mozambique (northern, central, and southern) for capacity building in this sector.For the southeast Africa, which includes Mozambique, a general picture is provided through the IPCC regional projections, indicating that precipitation is generally decreasing (IPCC 2014;Seneviratne et al. 2021).This trend signal had been partially supported by Ussalu and Bassrei (2021) in their study about the climate dynamics of the southern region of Mozambique, and Machaieie et al. (2020) have also found a decreasing trend in inter-annual variation of precipitation and increase of drought frequency and severity over Quelimane city, located in the central region of Mozambique.
In this work, we investigate the existence of long memory and trend signals in the time series of precipitation over all regions of Mozambique using two databases.One consisting of in situ observations from a gauging network belonging to the National Institute of Meteorology (INAM) and the other is a high resolution (0.05 o ) gridded remote sensing data combined with in situ stations from the Climate Hazards group InfraRed Precipitation with Stations (CHIRPS) dataset.While the R/S analysis and DFA have been applied for long memory detection, the linear regression and Mann-Kendall techniques were applied for trend analysis.

Study area and data description
Mozambique is located between the parallels 10 • 27'S and 26 • 52'S and the meridians 30 • 12'E and 40 • 51'E, on the southeastern coast of Africa.Figure 1 shows the study area, Mozambique territory, and its ten provinces (major administrative division): three in the southern region (Maputo, Gaza, and Inhambane), four in the central region (Manica, Sofala, Tete, and Zambezia), and three in the northern region (Nampula, Niassa, and Cabo-Delgado).The general climate in Mozambique is tropical, with annual precipitation amount increasing toward the northern region of the country.From the plotting of annual precipitation in Fig. 1, it is observed that, while Gaza is the driest province with minimal annual precipitation close to 300 mm, Zambezia is the wettest province of the country, with maximum annual precipitation reaching 1900 mm.  of missing data along the entire period and having no less than 30 years of observations.Table 1 shows the names and geographical locations of gauging stations used in the study, including their respective percentage of missing data.

123
At each station, in situ and CHIRPS datasets from the same period are highly correlated, as can be seen in Table 2 and in the Taylor diagrams presented in Fig. 3.Although in situ dataset is given importance as ground data at specific point, yet, due to its poor spatial coverage, the use of CHIRPS dataset was necessary, for assessing the spatial variability of long memory and trend signals and for guarantying consistent results.

Rescaled-range analysis
The rescaled-range R/S analysis was formulated by Hurst (1951) while studying dams dimensioning over the Nile River.Hurst's initial idea was to determine the maximum and minimum volumes in reservoirs (ideal capacity) taking into account the annual flows associated with the river during a certain period of some decades.For this purpose, Hurst analyzed a statistical variable called adjusted range (R) from the cumulative river flows over time.Then, Hurst normalized this value of R by the standard deviation (S) of the sequence Fig. 2 An illustrative example of gap filling in monthly precipitation series from Chimoio using SARMAX predictions.Shaded areas are the existing gaps in the time series.CHIRPS dataset was used as exoge-nous variable.The SARMAX linear order was ( p = 3, q = 3), and the seasonal order was to obtain what he called rescaled adjusted range, which is the R/S statistics, a dimensionless quantity.
Considering a certain sequence X t (t = 1, 2, ..., N ) of random numbers (with N observations), not necessarily independent, and defining the k-th partial sum as ), τ being the number of elements of the partial sequence.Then, we define R τ as and the R/S (τ ) is given by In analyzing this statistics for observations involving different natural phenomena, Hurst found that there was a function relating the values of R/S to the number of observations that entered in the calculations (τ ), and it is given by the following expression: where H is a constant known as the Hurst exponent.It is estimated as the slope of a straight line fit in a log-log graph of R/S against τ .For stationary processes, H ranges from 0 to 1 (Mandelbrot and Wallis 1969), having the following interpretation: (a) If 0.5 < H < 1, the time series is said to be persistent; it indicates positive long-term correlations, which means that an increase in the past will tend to be followed by another increase in future; the opposite situation is also true, a decrease in the past will probably be followed by another decrease in future; (b) If 0 < H < 0.5, the time series is said to be anti-persistent, increasing and decreasing values occur alternately in adjacent pairs, meaning that an increase in the past is likely to be followed by a decrease in future and vice-versa; (c) If H = 0.5, the time series has a purely random behavior also known as white noise.
Here, the null hypothesis (H 0 ) is that the Hurst exponent H = 0.5 (white noise) and the alternative hypothesis (H 1 ) is H = 0.5.A standard normal distribution of H is considered, and a Z score is computed by the expression: Z = (H − 0.5)/σ H , where σ H is the standard deviation of H values obtained by repeating the process over different lags in the range of τ .Given that we expect both anti-persistence and persistence in the time series, a two-sided Z test is used so that at a selected level of significance (1 − α), the null hypothesis of white noise is rejected if the absolute value of Z is equal or greater than Z α/2 ( Z ≥ Z α/2 ).In this work, we consider a confidence level of 90% (α = 0.1), and in some cases, 95% (α = 0.05).

Detrended fluctuation analysis
The detrended fluctuation analysis (DFA) was introduced by Peng et al. (1994) when studying Mosaic organization of DNA nucleotide, and it is a method for determining the statistical self-affinity of a signal.It is useful for analyzing time series that appear to be long-memory processes.The obtained exponent is similar to the Hurst exponent (Kantelhardt et al. 2001), except that DFA may also be applied to signals whose underlying statistics such as mean and variance are non-stationary (changing over time).Given a time series x i (i = 1, 2, ..., N ), the DFA procedure consists of the following steps (Kantelhardt et al. 2001).In the first step, we determine the profile X t : where x denotes the mean value of the time series, and X t is called the cumulative sum or profile.
Fig. 3 Taylor diagrams, comparing in situ and CHIRPS datasets over the same period at each used station 123 Next, X t is divided into time windows of length n samples each, and a local least squares polynomial fit (the local trend) is calculated by minimizing the squared errors within each time window.If the local fit applied to the profile is linear, thus it is called DFA1, it removes constant trends in the original sequence (stationary process), and in this particular case, DFA is equivalent to R/S analysis.For polynomial fit of order j, we have DFA j, and this may remove trends of higher order (non-stationary process).Let Y t indicate the resulting piecewise sequence of polyfits.Then, the root-mean-square deviation from the trend, the fluctuation, is calculated by This process is repeated over a range of different window sizes n > j + 2, and a log-log graph of F n against n is constructed, which indicates statistical self-affinity expressed as F n ∝ n α .Finally, the scaling exponent α is calculated as the slope of a straight line fit, using least-squares.This exponent is a generalization of the Hurst exponent, and it gives the following information about the series self-correlations: α < 0.5: anti-correlated; α = 0.5: uncorrelated or white noise; α > 0.5: correlated; α = 1: 1/f-noise, pink noise; α > 1: non-stationary; α = 1.5:Brownian noise.In this work, we have used the DFA1 (which is comparable to the Hurst's R/S analysis, α = H ) and the DFA2.Statistical significance of estimated α is computed under the same considerations described above.

Linear regression
Linear regression (LR) is a parametric method for trend analysis in climate series, and its application is well and deeply explained in Hartmann et al. (2013) and Heumann et al. (2016).This method has been adopted for general use in the IPCC (2014) report for trend calculation.The trend parameter is determined as a slope of the best fitting straight line in the dispersion of the data series.
Let X n be the n-th value of a time series observed at a time t n (n = 1, 2, ..., N ).The trend slope is obtained by the ordinary least squares solution given by the following expression: where a is the trend slope (or the change rate) whose sign determines the trend direction, either an increase or a decrease of the values in the time series, X is the average value of the time series, and t is the average value of t.
The estimated a is a random variable and has statistical properties such as standard error or bias.Confidence intervals (CI) for a were computed assuming a Student's t-distribution in the residual's variability of observations with regard to the straight line (Hartmann et al. 2013;Heumann et al. 2016).At a given statistical significance level or confidence level (CL), the estimated trend slope is given as a ± .While a is the best guess or a point estimate, CI = [a − ; a + ] is an interval estimate that informs how good a guess is.At a given confidence level 0 where a L = a − is the lower and a U = a + is the upper endpoint of the CI.
In this work, we consider 90% of CL (α = 0.1) for the CIs, which is a reasonable choice for climatological problems (Hartmann et al. 2013).CI has the property coverage defined by the confidence level CL and the interval length 2 , which should ideally be small.The null hypothesis of no trend is rejected only if both endpoints of CI have the same signal as a under the considered CL.Here, this is diagnosed by computing the ration between the absolute value of a and the value of , obtaining a confidence index a / , which should be greater than unit (confidence index > 1), to accept the alternative hypothesis of significant trend in the time series.

Mann-Kendall test
The Mann-Kendall (MK) test is a non-parametric trend statistics that can be applied to time series not conforming to a normal distribution.It is robust to time series presenting missing values, tied values, and values below the limit of detection (Taylor and Loftis 1989;Yu et al. 1993;Meals et al. 2011).
Let X = (x 1 , x 2 , x 3 , ..., x N ) be a deseasonalized time series of N observations at a given station.The Mann-Kendall statistic (Mann 1945;Kendall 1975) is computed by the following expression: where A positive value of s indicates an upward trend, and a negative value of s indicates a downward trend.According to Mann, the null hypothesis (H 0 ) of this test states that X is an independent and identically distributed random variable, 123 and the alternative hypothesis (H 1 ) of a two-sided test is that the values of X are not distributed identically.
The variance of s is estimated by the expression (Hirsch et al. 1982): where t is the size of any existing tied data (a sequence of repeated values) in the time series.The standard normal score for statistical significance is computed by For a two-sided Z test at a selected level of significance 1−α, the null hypothesis of no trend is rejected if the absolute value of Z is equal or greater than Z α/2 ( Z ≥ Z α/2 ).Confidence levels of 90% (α = 0.1) and in some cases of 95% (α = 0.05) are considered.
Up to this point, we have only identified whether a trend exists; still, it is also very important to estimate the magnitude of a trend or the change rate.We have used the slope estimator proposed by Sen (1968) and used by many other researchers (Gan 1998;Xu et al. 2007;Fu et al. 2009), defined as slope = median where 1 < k < j < N .The slope estimator is the median over all possible combinations of pairs in the time series.For a particular time series of N observations, the number of combinations will be N (N − 1)/2.

Results and discussion
Both long memory and trend analyses were evaluated to deseasonalized time series consisting of (i) total annual precipitation, (ii) total precipitation during the summer period, from November to February (NDJF), and (iii) total precipitation during the winter period, from May to August (MJJA).Mozambique presents two main seasons, the winter, which is relatively dry, and the summer, which is relatively wet (Machaieie et al. 2020;Ussalu and Bassrei 2021).Results derived from in situ observations are displayed through tables, as they could not be interpolated and presented through colored maps, for being very sparse and not uniformly distributed over the country, while results based on CHIRPS database are mapped.

Long memory
For long memory detection, three different approaches were considered, the R/S, DFA1, and DFA2 methods.Overall, precipitation had presented strong memory over the southern region of Mozambique, weakening toward the northern.This result is consistent from both in situ and CHIRPS data.
Long memory exponents (H and α) and their respective confidence intervals derived from in situ observations are listed in Table 3, while those derived from CHIRPS dataset are mapped in Fig. 4.
In particular, from Table 3, we observe that at 95% confidence level ( Z > 1.96), the southern and some central stations (from bottom-most to middle stations in the tables) had presented the H exponent varying from 0.55 to 0.91, the DFA1 whose exponent is compared to the R/S method (α = H ) ranged from 0.54 to 0.96 and the DFA2 presented values ranging from 0.56 to 1.26.These intervals indicate that the time series of precipitation were overall persistent in the southern and in some central areas of Mozambique, which means that the correlation between past and future trends exists and is positive.On the other hand, we can say that the possibility of a random occurrence of the data series is minimal.The behavior observed in the precipitation time series is more likely to persist for a long period in the future.Some northern stations (uppermost stations in the tables) such as Mocimboa and Pemba had presented H and α values below 0.5 with at least 90% significance level, suggesting that time series on these particular stations are likely to be anti-persistent or even white noise.
From the spatial mapping of H and α based on CHIRPS dataset in Fig. 4, we can observe the full picture of how long memory strength varies throughout Mozambique territory.Quite similar results in terms of long memory strength are presented between the annual precipitation (top panels) and precipitation within the two main seasons, during summer months (middle panels) and during winter months (lower panels), although slight differences in the spatial distribution exist.While persistency is generally exhibited in the southern and central regions, the northeastern part of Mozambique had presented anti-persistency tending to white noise (with H and α values ranging from 0.1 to 0.5), especially in annual and summer precipitation over Cabo-Delgado province (refer to Fig. 4a, b).This result is true from the three approaches.A synthesis of overall long memory over each province of Mozambique based on CHIRPS data is presented in Table 6.
The results presented above collaborate partially with the study from Araneda-Cabrera et al. (2021), which evaluated the spatio-temporal characteristics of drought in Mozambique and their relationship with large-scale climate variability, using data from the Climate Research Unit (CRU).In their study, they have computed the persistence of the Stan-123 dardized Precipitation and Evapotranspiration Index (SPEI) through R/S method, having found H values greater than 0.5 in all regions, but higher values of H were seen in the southern and central regions, suggesting that the negative trends of SPEI they obtained in all regions will persist with greater strength over these regions, while trend persistence strength will be weak in the northern region.Although the variable evaluated by Araneda-Cabrera et al. ( 2021) was SPEI and not precipitation directly, which is the case of our work, it is clear that an agreement exists on the distribution of the persistence strength throughout Mozambique.
According to Mandelbrot and Wallis (1969), when 0 < H < 1, the occurrence of the phenomenon presents selfsimilarities as those of the fractal.A change of scale in the sampling of the series would lead to approximately the same result.The DFA2 had indicated α > 1 (mapped with blank areas) in annual precipitation along the coastal area of Inhambane, extending to the southern Sofala, southeast Mozambique (refer to Fig. 4c), and in winter precipitation in the central area of Tete province (refer to Fig. 4g-i), suggesting that time series on these areas are non-stationary.It is very likely that some underlying statistics such as mean, standard deviation, or variance are not constant over time.If it is true, it is possible that over these particular areas, some local or regional climate controllers are modifying or new ones emerging over time.At least, we are certain of the changes in frequency and intensity of tropical cyclones landing the coastal area of Mozambique in connection to the global warming (Mavume et al. 2009).It might also be the effect of changes in other factors, and this could be a subject of further investigations, just as the study from Barimalala et al. (2020), where the variability and impacts of the Mozambique Channel Trough on southeast African rainfall were investigated, or the study from Ambrosino et al. (2011), where large-scale climate factors controlling the southern African rainfall were investigated.

Trend
For trend analysis, we have employed the parametric LR and the non-parametric MK tests.Trend in annual precipitation throughout Mozambique territory has generally shown a considerable spatial variability.Both methods indicated a decreasing rainfall in some regions and an increase in others, although some trend signs are not statistically significant.
Summer precipitation is behaving similarly to the annual, just as it could be expected, given that this period coincides with the rainy season in Mozambique, thus containing the major part of annual precipitation.Significant decrease in annual and summer precipitation is observed over the coastal areas of Zambezia and Nampula, while the southern Manica and eastern Inhambane is experiencing a significant increase (refer to Figs. 5a, c and 6a, d).Besides, winter precipitation 123 is decreasing over almost all the country, except at some isolated points where a positive trend is true (refer to Figs. 5e  and 6g).
Based on in situ observations, trend results by LR and MK methods are presented in Tables 4 and 5, respectively.Most stations have exhibited negative trend signals, that is, a decreasing trend over time, with differences in significance levels.Although most trend signals are not statistically significant, some stations had presented a trend with at least 90% confidence.Examples of these trends include the decreasing rainfall detected over Massingir and Inharrime and the increasing trend over Sussundenga and Manhiça.
Using CHIRPS data, trend in precipitation by the LR method is shown in Fig. 5, while trend by the MK method is shown in Fig. 6.It can be observed that some isolated areas exhibited an exceptionally significant trend with 95% confidence.While an exceptionally significant decrease in precipitation has been observed during both seasons in the northeast Zambezia and southeast Nampula, an increase is also observed in the southern Manica, eastern Inhambane, and isolated parts of Tete province, being this last trend more pronounced in the summer period (90% confidence).
Some regions have presented different trend signals in between summer and winter seasons, just as it could be expected.This is certainly due to seasonal or dynamic climate factors, such as the cold fronts which present seasonal variations in position and strength, the tropical cyclones which affect during the summer season only, SSTs with seasonal patterns, among others (Mavume et al. 2009(Mavume et al. , 2021;;Barimalala et al. 2020).As an example, in the eastern part of Inhambane province, precipitation has presented an increase during the summer and a decrease during the winter (90% confidence).This result is consistent from the two methods (refer to Figs. 5c, e and 6d, g), which converged to quite similar trend signals in both databases, differing slightly in the trend magnitudes.This agreement, somehow, provides more confidence in the results obtained hereof.The non-parametric MK test provides higher statistical power in the case of nonnormality and is robust against outliers, and similar to the LR method, it is also robust against large data gaps.Therefore, even if data gaps had not been filled, we would still expect overall trustable trend results.
Considering CHIRPS data, the decreasing rainfall at the coastal areas of Zambezia and Nampula provinces has presented a change rate of at least −4.1 mm/yr in annual precipitation, meaning that, during the period from 1981 to 2021, the total annual precipitation has dropped on average about 168.1 mm, which represents about -12% relative to the average annual precipitation of this area.On the other hand, the southern Manica and the eastern Inhambane had presented a significant increase in summer precipitation with a change rate of at least 3.3 mm/yr, equivalent to a total increase of 135.3 mm (about 15%) during the period of 1981-2021.Overall trend in precipitation in each province of Mozambique based on CHIRPS data is also synthesized in Table 6.
Most in situ stations are in line with the CHIRPS based results in terms of trend signals.For instance, stations located in the vicinity of coastal areas of Zambezia (Quelimane city) and Nampula (Angoche district) have also indicated a decreasing rainfall, where the MK determined an average change rate of −4.51 and −3.49mm/yr, equivalent to an average total decrease of about 181.6 mm (-13%) and 143.1 mm (-10%), respectively, in annual precipitation during the period of 1981-2021.It should be noted that these trend magnitudes cannot be compared to those referred previously from CHIRPS dataset, due to the fact that calculations were based on different periods of data availability.On this particular area, similar trend results were obtained by Machaieie et al. (2020), while studying the variability and trend of precipitation in Quelimane, having pointed as a possible cause of the decreasing rainfall, the strengthening of ENSO after the 1970s climate shift.
These findings converge to the macro scale IPCC AR5 and AR6 projections over southeast Africa (IPCC 2014;Seneviratne et al. 2021).For instance, the CMIP53 multimodel ensemble mean of projected changes in precipitation for 2016-2035 relative to 1986-2005 under RCP4.54scenario, indicated, on average, negative anomalies (-10%) in the northern Mozambique in all seasons, while a positive anomaly (10%) was also expected for summer months (DJF) in the southern Mozambique (IPCC 2014).On evaluating these regional projections for Mozambique, Mavume et al. (2021) have found similar spatial variability in precipitation trend.
The Working Group I of the IPCC-AR6 have presented precipitation projections separated into the average precipitation trend and the trend on extreme precipitation at a daily scale, having found that, while average precipitation is overall decreasing over the southeast Africa, the extreme precipitation at a daily scale is lightly increasing on the region (Seneviratne et al. 2021).This last trend signal is certainly connected to the intensification of extreme events, such as the tropical cyclones and ENSO.

Conclusions
Overall, precipitation exhibited strong long memory in southern and central Mozambique, weakening toward the northern region.The southern and central regions are generally per-  The trend slope or the change rate a is given in millimeter per year (mm/yr), evaluated for the deseasonalized time series: (i) the total annual precipitation, the total summer (NDJF) precipitation, and the total winter (MJJA) precipitation.Confidence intervals were computed under 90% confidence level, assuming the t-distribution of the estimated slope.Evaluated data consists of in situ observations during the period of 1960-2020 sistent with the H exponent and DFA α ranging from 0.54 to 1.26, what implies that the probability of a random occurrence of precipitation is minimal.Besides, the northeastern part of Mozambique had presented some isolated anti-persistent areas tending to white noise, especially over Nampula and Cabo-Delgado provinces, and over these specific areas, we can say that precipitation occurs randomly; past precipitation states have no influence on the future events.
On the other hand, trend in precipitation had presented great spatial variability.It is decreasing in some regions, with more significance over Zambezia and Nampula provinces, where annual precipitation is estimated to have dropped about -12% from 1981 to 2021, and increasing in other regions, with more significance over Inhambane and southern Manica, where a total increase of about 15% is estimated in annual precipitation, during the same period.
These findings are in line with the IPCC-AR5 and AR6 projections over the southeast Africa region.In locals where precipitation exhibits persistency, it is very likely that the observed trends prevail for a long period in future.This insight is useful for planning and decision-making in the agricultural sector, as well as for setting climate change adaptation plans of contingency in Mozambique and for supporting seasonal weather forecasting services.
The present work did not investigate contributing local factors to the observed memory and trend at different areas; thus, further studies aiming to explore the physical processes or possible local drivers that may have conditioned such results are necessary.The parameter s, the Z score for statistical significance, and the trend slope or change rate given in mm/yr were evaluated for time series consisting of (i) total annual precipitation, (ii) total precipitation during summer months (NDJF), and (iii) total precipitation during winter months (MJJA).Evaluated data consists of in situ observations during the period of 1960-2020

Fig. 1
Fig. 1 Map of the study area, Mozambique with province boundaries drawn, and the annual precipitation plotted (based on CHIRPS dataset from 1981 to 2021).Red stars are the gauging stations used in the study 123

Fig. 4
Fig. 4 Spatial distribution of long memory in precipitation from CHIRPS database.The H exponent (left column), DFA1 α (central column), and DFA2 α (right column) were evaluated for the deseasonalized time series consisting of (i) the total annual precipitation (top

Fig. 5
Fig. 5 Trend in precipitation by linear regression.The trend slope or the change rate is given in mm/yr (left column) and is evaluated for the deseasonalized time series consisting of (i) the total annual precipitation (top panel), (ii) the total summer (NDJF) precipitation (middle panel), and (iii) the total winter (MJJA) precipitation (lower panel).Statistical significance is shown in the right column.Confidence intervals were computed assuming a t-distribution of the estimated slope.Red plus signs in the right column indicate locals where the trend in precipitation is significant with at least 90% confidence level.Evaluation based on CHIRPS dataset during the period of 1981-2021

Fig. 6
Fig. 6 Trend in precipitation by the Mann-Kendall test.The parameter s (left column), the trend slope or change rate given in mm/yr (central column), and the Z score for statistical significance (right column) were evaluated for deseasonalized time series consisting of (i) a set of annual precipitation (top panel), (ii) a set of total summer precipitation (NDJF)

Table 3
Long exponents: Hurst exponent H , DFA1 α, and DFA2 α, with their respective Z score for statistical significance

Table 5
Trend in precipitation by the Mann-Kendall test