Abstract

The computation of the two-dimensional Fourier transform by the sampling points creates an ill-posed problem. In this chapter, we will cover this problem for the band-limited signals in the noisy case. We will present a regularized algorithm based on the two-dimensional Shannon Sampling Theorem, the two-dimensional Fourier series, and the regularization method. First, we prove the convergence property of the regularized solution according to the maximum norm. Then an error estimation is given according to the L2 -norm. The convergence property of the regularized Fourier series is given in theory, and some examples are given to compare the numerical results of the regularized Fourier series with the numerical results of the Fourier series.

Keywords: Fourier transform, band-limited signal, ill-posedness, regularization

AMS subject classifications: 65T40, 65R20, 65R30, 65R32


*2.3. VMD–HHT Model*

Since VMD could weaken the inherent end effect and mode aliasing of EMD, a new approach of VMD–HHT time–frequency analysis is suggested in this paper by replacing EMD with VMD. First, the acceleration data were decomposed using VMD. Secondly, the power spectrum density (PSD) was used to extract each mode frequency. As a result, the frequency range of dynamic responses could be defined, and the noise modes were eliminated. Then, the time–frequency–energy spectrum could be gained by employing HHT to extracted modes. Lastly, assisted by the spectrum to adjust and determine feature modes, characteristic information of dynamic responses with high accuracy could be obtained in all directions. Figure 1 shows the process of the VMD–HHT-based characteristic extraction model.

To verify the VMD–HHT-based characteristic extraction model for dynamic responses, a set of data was simulated to compare the performance between traditional HHT and VMD–HHT in extraction. The analog data consisted of five sinusoidal signals with different amplitudes and frequencies (the frequencies were 1 Hz, 5 Hz, 10 Hz, 15 Hz, and 35 Hz, respectively) at a sampling frequency of 100 Hz, and the effect of random noise was also added, as shown in Figure 2.

As shown in Figure 3, HHT cannot extract characteristic information of accelerometer signals. In contrast, the VMD–HHT model can accurately separate frequencies of analog signals into each band (separation frequency is consistent with setting frequency) and qualitatively analyze frequency distribution of energy based on color. Moreover, the analog data were made up of time-invariant stationary signals with constant statistics along the time axis. According to previous analyses and research, even for accessible composite signals, HHT cannot separate characteristic information effectively. Its algorithm EMD would cause mode aliasing, while the VMD–HHT would not. Therefore, this paper proposes an HHT method using VMD instead of EMD, from now on referred to as VMD– HHT time–frequency analysis model.

**Figure 1.** VMD–HHT-based characteristic extraction model.

**Figure 2.** Analog time series of acceleration.

**Figure 3.** Comparison between HHT and VMD–HHT: (**a**) characteristic information extraction of analog signals using HHT; (**b**) characteristic information extraction of analog signals using VMD–HHT.


**3. Hamoun lakes**

166 Atmospheric Aerosols – Regional Characteristics – Chemistry and Physics

beds that become the Hamoun wetlands during wet years.

the Hamoun lakes complex [10].

lakes in Sistan on June 15, 2004.

a halt.

capacity of 1,495,000 acre-feet (1,844 million cubic meters). About 300 miles (482.8 kilometers) of concrete-lined canals were built to distribute the reservoir waters [9]. The Hirmand river is the longest river in Afghanistan (ca. 1150 km; catchment > 160,000 km2) and the main watershed for the Sistan basin, finally draining into the natural swamp of

Severe dust plumes usually extend from Sistan into southern Afghanistan and southwestern Pakistan obscuring the surface over much of the region (Fig 2). Severe droughts during the past decades, especially after 1999, have caused desiccation of the Hamoun lakes leaving a fine layer of sediment that is easily lifted by the wind [55], thus modifying the basin to one of the most active sources of dust in southwest Asia [56, 50, 57]. Therefore, the Hamoun dry-lake beds exhibit large similarities with the other two major dust source regions of the world that comprise of dried lakes and topographic lows, i.e. Bodele depression in Chad [3] and lake Eyre in Australia [4]. The strong winds blow fine sand off the exposed Hamoun lake beds and deposit it to form huge dunes that may cover a hundred or more villages along the former lakeshore. As a consequence, the wildlife around the lake has been negatively impacted and fisheries have been brought to

Fig. 2 (left panel) shows a severe dust storm over the Sistan region as observed from the Terra-MODIS satellite's true color image on 15 June 2004. The intense dust plumes form a giant U-shape extending from Sistan into southern Afghanistan and southwestern Pakistan that obscures the surface over much of the region. The pale color of the dust plume is consistent with that of dried wetland soils. The dust is blowing off the dry lake

**Figure 2.** MODIS image of dust deflation over southern Afghanistan originating from the dry Hamoun

The Hamoun lakes are situated roughly at the termination of the Hirmand river's inland delta. The Hamoun lakes complex (Hamoun-e-Puzak, Hamoun-e-Sabori and Hamoun-e-Hirmand and Baringak) are located in the north of the Sistan region, which is also the largest fresh water ecosystem of the Iranian Plateau and one of the first wetlands in the Ramsar Convention [58]. Water in the Hamoun lakes is rarely more than 3 meters deep, while the size of the lakes varies both seasonally and from year to year. Maximum expansion takes place in late spring, following snowmelt and spring precipitation in the mountains. In years of exceptionally high runoff, the Hamoun lakes overflow their low divides and create one large lake that is approximately 160 kilometers long and 8–25 kilometers wide with nearly 4,500 square kilometres surface area. Overflow from this lake is carried southward into the normally dry Gaud-i Zirreh (Fig. 1), the lowest playa (463-meter altitude) in the Sistan depression. Furthermore, mountain runoff varies considerably from year to year; in fact, the Hamoun lakes have completely dried up at least three times in the 20th century [9]. The maximum extent of the Hamoun lakes following large floods is shown in Fig. 4, where a continuous large lake has been created covering an extended area of ~4,500 km2 with a volume of 13000 million m3 in Sistan and southwestern Afghanistan. This figure corresponds to spring of 1998 after snowmelt in the Afghanistan mountains that transferred large quantities of water into the Hamoun Basin. As a consequence, livelihoods in the Sistan region are strongly interlinked with and dependent on the wetland products and services, as well as on agricultural activities in the Sistan plain. Fishing and hunting represent an important source of income for many households and, therefore, the local and regional economy is strongly dependent on weather conditions, precipitation and land use – land cover changes. The political boundary between Iran and Afghanistan splits the Hamoun system, further complicating management possibilities in the area. Most (90%) of the watershed is located in Afghanistan and practically all of the wetlands' water sources originate there. The Iranian part is desert, and produces runoff only in rare cases of significant local rainfall [10].

In view of the Hirmand and the surrounding rivers that supply most of the sediments to the Hamoun lakes, a brief encapsulation of the relevant geology of the catchment area in Afghanistan is given. Afghanistan has a very complex geology, encompassing two major relatively young orogenies, Triassic and subsequent Himalayan, resulting in amalgamation of crustal blocks and formation of concomitant ophiolites and younger clastic and carbonate sedimentary rocks as well as basaltic lavas and, more recently, extensive alluvial and eolian detritus [59] . The Sistan region and Hamoun dry lake beds are mainly composed of Quaternary lacustrine silt and clay material as well as Holocene fluvial sand, silt and clay (Fig. 3). These materials have been carried to the basin by the rivers, while along their courses Neogene fluvial sand, eolian sand, silt and clay are the main constituents. Note also the difference in the soil-dust composition between two major desert areas, Registan and Dasht-i-Margo in Afghanistan. The former is composed of Neogene coarse gravels and the latter of Quaternary eolian sand. More details about the geology in the Sistan region can be found in [60].

Changes of Permanent Lake Surfaces, and Their Consequences for Dust Aerosols and Air Quality: The Hamoun Lakes of the Sistan Area, Iran 169

The Sistan Basin has recently experienced an unusually long 10-year drought starting in 2000 [10]. Combined with war and severe political disruption over the past 2 decades, the 10-year drought has created conditions of widespread famine that affected many people in eastern Iran and southwestern Afghanistan. A suggested, climatic forcing mechanism has been proposed for the recent drought by Barlow and others (2002). A prolonged ENSO (El Niño-Southern oscillation) cold phase (known as La Niña) from 1998 to 2001 and unusually warm ocean waters in the western Pacific appear to have contributed to the prolonged drought. The unusually warm waters (warm pool) resulted in positive precipitation anomalies in the Indian Ocean and negative anomalies over central Afghanistan [64], thus contributing to the drying of the Hamoun Basin. The contrast between a relatively wet year in 1976 and the nearly dry Hamoun lakes in 2001 is shown in Figs. 2 and 4. Millions of fish and untold numbers of wildlife and cattle died. Agricultural fields and approximately 100 villages were abandoned, and many

Most of the Sistan population lives near the Hamoun lakes and is employed in agricultural, fishery, handicrafts and other jobs. To counter the effects of droughts, the Iranian government prepares facilities such as food and flour supplies, medicine and health services and employment in the region to prevent the forced emigration of people, but the continuous and extreme droughts have forced some people to leave the Sistan region. Long droughts at the end of the 1960s, middle of the 1980s, and from 1999 to 2010 affected the Sistan region significantly and resulted in desiccation of the Hamoun lakes, making the surrounding lands saline and disturbing their soil fertility, while some places became barren (see Fig. 5). The most important findings in Fig. 5 are: (1) in 1976, the Hamoun lakes were still thriving. Dense reed beds appear as dark green, while tamarisk thickets fringing the margins of the upper lakes show up as pink shades in the satellite images (Fig. 5). Bright green patches represent irrigated agricultural lands, mainly wheat and barley. The lakes flood to an average depth of half a meter, denoted by lighter shades of blue, while dark blue

**Figure 5.** Satellite (Landsat) images of the Hamoun Basin in spring of different years. Hamoun lakes are fed primarily by water catchments in neighbouring Afghanistan. In 1976, when rivers in Afghanistan were flowing regularly, the lake's water level was relatively high. Between 1999 and 2011, however, drought conditions caused frequent dryness of the Hamoun lakes that almost disappeared in 2001 after

**4. Droughts in the Sistan**

a 3-year intense drought period [65].

succumbed to blowing sand and moving dunes [65].

**Figure 3.** Geological map of the Sistan Basin and southern Afghanistan *[61-63].*

**Figure 4.** Position of the Hamoun Lakes in Iran and Afghanistan, showing a maximum inundation period.


**Abstract**

Parameter estimation is an important step in the identification of systems. With the extension of systems, there needs the multi-parameter estimation of systems. The estimation of multi parameters of complex systems based on the extended PID controllers is considered in this chapter. As the related references proved that the integral item of the nonlinear PID controller could deal with the uncertain part of the complex system (which can also be called new stripping principle, simple notes as NSP). Based on this theory, new multi-parameter estimation method is given. Firstly, the unknown parameters are expanded to new states of the system. Two cases, parameters are constant or changing with time, are separately analyzed. In the time-variant case, the unknown parameters are extended to functions which actual forms are uncertain. Secondly the method NSP could be applied to cope with the uncertain part, and then reconstruction state observation to estimate the states. If the states are observed, the unknown parameters are obtained at the same time. Finally the convergence analysis of the error systems and some simulations will be given in this chapter to indicate the effectiveness of the proposed method.

**Keywords:** multi-parameter, parameter estimation, complex system, extended PID control, convergence analysis


*3.2. Convalescent Plasma*

Convalescent plasma therapy is the passive transfer of immunity from a convalescent donor to a patient with an acute infection [95]. The transfer of convalescent blood product to a patient is an old therapy which was already used to treat various infections in humans and animal models at the end of 1800's [96]. The theoretical basis for its use against EBOV infection is that EVD survivors are thought to be protected from re-infection through humoral immunity and so the transfer of this protective sera containing EBOV-specific antibodies could have therapeutic benefits for patients. EBOV outbreaks take place in low-income countries where expensive drugs are not always affordable and so convalescent plasma may provide a more cost-effective treatment solution. However, it comes with its complications as donor blood must be tested for a wide variety of contaminants including HIV, malaria and hepatitis A which are known to have high prevalence in many regions that suffer EBOV outbreaks [95]. Convalescent plasma therapies were tested to reduce the risks of transfusion-transmitted infections and their impacts on anti-EBOV antibodies were assessed. For example, Amotosalen/UVA pathogen reduction technology was tested to treat EVD convalescent plasma. Plasma were analysed by two types of ELISA and three neutralisation assays and it was found that anti-EBOV titres remained relatively unchanged following the treatment [97].

During the 2013–2016 West Africa epidemic, the Ebola Tx trial in collaboration with The Conakry Ebola Survivors Association organised a large plasma collection programme in Conakry in Guinea between November 2014 and July 2015 [98]. This consortium evaluated the efficacy of convalescent plasma in comparison with standardised supportive care in EVD patients in a phase 2/3 open-label non-randomised trial setting. Even though no serious adverse effects were observed with the use of the convalescent plasma, the therapy

was found to have no significant effect on mortality compared to the control group (31% of patients treated with convalescent plasma died 3–16 days post-diagnosis, compared to the 38% in the control group) [99]. These results may be due to the varying levels of EBOV-specific neutralising antibody in the plasma. At that time, they did not have any test to measure EBOV-specific antibody titres and neutralising antibodies in the field. Other clinical trials evaluated some methods to screen the antibody responses in donors' plasma in order to potentially select the plasma with high neutralising antibody titres. Brown et al. measured anti-EBOV antibodies in EVD survivors in Liberia in 2014–2015 using two ELISA and two neutralisation assays (microneutralisation, PRNT) [100]. They found that the four assays were concordant to measure donor antibody titres. However, 15 of 100 donors, including seven with a confirmed EBOV PCR positive result, did not have any detectable EBOV-specific antibodies. This trial found that viral load was reduced in EVD patients who received the convalescent plasma containing higher antibody levels, but not in patients who received the therapy with lower antibody levels [100]. Tedder et al. performed a similar evaluation in Sierra Leone using an IgG capture competitive doubleantigen bridging enzyme immunoassay and a pseudotyped virus assay [101]. Both studies demonstrate the benefit of screening donors' plasma for neutralising antibody titres even though neutralising antibodies are yet to be validated as correlates of protection.

#### *3.3. Vaccine-Induced Neutralising Responses*